Research Article An Optimal Portfolio and Capital...
12
Research Article An Optimal Portfolio and Capital Management Strategy for Basel III Compliant Commercial Banks Grant E. Muller and Peter J. Witbooi University of the Western Cape, Private Bag X17, Bellville 7535, South Africa Correspondence should be addressed to Peter J. Witbooi; [email protected] Received 3 October 2013; Accepted 5 January 2014; Published 19 February 2014 Academic Editor: Francesco Pellicano Copyright © 2014 G. E. Muller and P. J. Witbooi. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We model a Basel III compliant commercial bank that operates in a financial market consisting of a treasury security, a marketable security, and a loan and we regard the interest rate in the market as being stochastic. We find the investment strategy that maximizes an expected utility of the bank’s asset portfolio at a future date. is entails obtaining formulas for the optimal amounts of bank capital invested in different assets. Based on the optimal investment strategy, we derive a model for the Capital Adequacy Ratio (CAR), which the Basel Committee on Banking Supervision (BCBS) introduced as a measure against banks’ susceptibility to failure. Furthermore, we consider the optimal investment strategy subject to a constant CAR at the minimum prescribed level. We derive a formula for the bank’s asset portfolio at constant (minimum) CAR value and present numerical simulations on different scenarios. Under the optimal investment strategy, the CAR is above the minimum prescribed level. e value of the asset portfolio is improved if the CAR is at its (constant) minimum value. 1. Introduction Successful bank management can be achieved by addressing four operational concerns. Firstly, the bank should be able to finance its obligations to depositors. is aspect of bank management is called liquidity management. It involves the bank acquiring sufficient liquid assets to meet the demands from deposit withdrawals and depositor payments. Secondly, banks must engage in liability management. is aspect of bank management entails the sourcing of funds at an acceptable cost. irdly, banks are required to invest in assets that have a reasonably low level of risk associated with them. is process is referred to as asset management. It aims to encourage the bank to invest in assets that are not likely to be defaulted on and to adopt investment strategies that are sufficiently diverse. e fourth and final operational concern is capital adequacy management. Capital adequacy manage- ment involves the decision about the amount of capital the bank should hold and how it should be accessed. From a shareholder’s perspective, utilizing more capital will increase asset earnings and will lead to higher returns on equity. From the regulator’s perspective, banks should increase their buffer capital to ensure the safety and soundness in the case where earnings may end up below an expected level. In this paper, we address problems associated with asset and capital adequacy management. e Basel Committee on Banking Supervision (BCBS) regulates and supervises the international banking industry by imposing minimal capital requirements and other mea- sures. e 1998 Basel Capital Accord, also known as the Basel I Accord, aimed to assess the bank’s capital in relation to its credit risk, or the risk of a loss occurring if a party does not fulfill its obligations. e Basel I Accord launched the trend toward increasing risk modeling research. How- ever, its oversimplified calculations and classifications have simultaneously called for its disappearance. is paved the way for the Basel II Capital Accord and further agreements as the symbol of the continuous refinement of risk and capital. e 2004 (revised) framework of the Basel II Capital Accord (see [1]) laid down regulations seeking to provide incentives for greater awareness of differences in risk through more risk-sensitive minimum capital requirements based on numerical formulas. e Total Capital Ratio or Capital Adequacy Ratio (CAR) (see for instance [2–6]) measures the amount of the bank’s capital relative to its amount of credit exposures. Internationally, a standard has been adopted Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 723873, 11 pages http://dx.doi.org/10.1155/2014/723873
Transcript of Research Article An Optimal Portfolio and Capital...
Research ArticleAn Optimal Portfolio and Capital Management Strategyfor Basel III Compliant Commercial Banks
Grant E Muller and Peter J Witbooi
University of the Western Cape Private Bag X17 Bellville 7535 South Africa
Correspondence should be addressed to Peter J Witbooi pwitbooiuwcacza
Received 3 October 2013 Accepted 5 January 2014 Published 19 February 2014
Academic Editor Francesco Pellicano
Copyright copy 2014 G E Muller and P J WitbooiThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
We model a Basel III compliant commercial bank that operates in a financial market consisting of a treasury security a marketablesecurity and a loan and we regard the interest rate in themarket as being stochasticWe find the investment strategy that maximizesan expected utility of the bankrsquos asset portfolio at a future date This entails obtaining formulas for the optimal amounts of bankcapital invested in different assets Based on the optimal investment strategy we derive a model for the Capital Adequacy Ratio(CAR) which the Basel Committee on Banking Supervision (BCBS) introduced as ameasure against banksrsquo susceptibility to failureFurthermore we consider the optimal investment strategy subject to a constant CAR at the minimum prescribed level We derive aformula for the bankrsquos asset portfolio at constant (minimum) CAR value and present numerical simulations on different scenariosUnder the optimal investment strategy the CAR is above theminimumprescribed levelThe value of the asset portfolio is improvedif the CAR is at its (constant) minimum value
1 Introduction
Successful bank management can be achieved by addressingfour operational concerns Firstly the bank should be ableto finance its obligations to depositors This aspect of bankmanagement is called liquidity management It involves thebank acquiring sufficient liquid assets to meet the demandsfrom deposit withdrawals and depositor payments Secondlybanks must engage in liability management This aspectof bank management entails the sourcing of funds at anacceptable costThirdly banks are required to invest in assetsthat have a reasonably low level of risk associated with themThis process is referred to as asset management It aims toencourage the bank to invest in assets that are not likely tobe defaulted on and to adopt investment strategies that aresufficiently diverse The fourth and final operational concernis capital adequacy management Capital adequacy manage-ment involves the decision about the amount of capital thebank should hold and how it should be accessed From ashareholderrsquos perspective utilizing more capital will increaseasset earnings and will lead to higher returns on equityFrom the regulatorrsquos perspective banks should increase theirbuffer capital to ensure the safety and soundness in the case
where earnings may end up below an expected level In thispaper we address problems associated with asset and capitaladequacy management
The Basel Committee on Banking Supervision (BCBS)regulates and supervises the international banking industryby imposing minimal capital requirements and other mea-sures The 1998 Basel Capital Accord also known as theBasel I Accord aimed to assess the bankrsquos capital in relationto its credit risk or the risk of a loss occurring if a partydoes not fulfill its obligations The Basel I Accord launchedthe trend toward increasing risk modeling research How-ever its oversimplified calculations and classifications havesimultaneously called for its disappearance This paved theway for the Basel II Capital Accord and further agreementsas the symbol of the continuous refinement of risk andcapital The 2004 (revised) framework of the Basel II CapitalAccord (see [1]) laid down regulations seeking to provideincentives for greater awareness of differences in risk throughmore risk-sensitive minimum capital requirements basedon numerical formulas The Total Capital Ratio or CapitalAdequacy Ratio (CAR) (see for instance [2ndash6]) measuresthe amount of the bankrsquos capital relative to its amount ofcredit exposures Internationally a standard has been adopted
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 723873 11 pageshttpdxdoiorg1011552014723873
2 Journal of Applied Mathematics
that requires banks to adhere to minimum levels of capitalrequirements By complying with minimum capital levelsbanks are guaranteed the ability to absorb reasonable levelsof losses before becoming insolvent Thus CARs ensure thesafety and stability of the banking system Under the BaselII Accord banks were required to maintain a CAR thatis a ratio of total bank capital to total risk-weighted assets(TRWAs) with a minimum value of 8
In response to the 2007-2008 financial crisis the BaselCommittee on Banking Supervision (BCBS) introduced acomprehensive set of reform measures known as the BaselIII Accord The Basel III Accord is aimed at improving theregulation supervision and risk management within thebanking sector and is part of the continuous effort madeby the BCBS to enhance the banking regulatory frameworkThe Basel III Accord builds on the Basel I and II documentsand seeks to improve the banking sectorrsquos ability to deal withfinancial and economic stress improve riskmanagement andstrengthen the banksrsquo transparencyThe focus of the Basel IIIAccord is to foster greater resilience at the individual banklevel in order to reduce the risk of system wide shocks In thisregard the Basel III Accord contains changes in the followingareas (i) augmentation in the level and quality of capital(ii) introduction of liquidity standards (iii) modifications inprovisioning norms (iv) introduction of a leverage ratio Fora detailed discussion on the aforementioned enhancementsin the Basel III Accord over the Basel II Accord see the paper[7] of Jayadev or the book [8] of Petersen and Mukuddem-Petersen
In our paper we are concerned with the level of bankcapital Thus we study the Total Capital Ratio or CAR whichif denoted by Λ is defined as
Λ =119862
119886rw (1)
In the expression for the CAR above 119862 denotes the totalbank capital and 119886rw the TRWAs of the bank Under BaselIII rules the minimum prescribed value of the bankrsquos CARremains unchanged at 8 However banks are now requiredto hold a Capital Conservation Buffer (CCB) of 25 andCountercyclical Buffer Capital (CBC) in the range of 0ndash25 The CCB ensures that banks can absorb losses withoutbreaching the minimum capital requirement and are able tocarry on business even in a downturn without deleveragingThe CBC is a preemptive measure that requires banks tobuild up capital gradually as imbalances in the credit marketdevelop It may be in the range of 0ndash25 of risk-weightedassets which could be imposed on banks during periods ofexcess credit growth There is also a provision for a highercapital surcharge on systematically important banks (see [78])
In finance the stochastic optimal control method is apopular optimization technique and was used for the firsttime by Merton [9 10] This technique involves solvingthe Hamilton-Jacobi-Bellman (HJB) equation arising fromdynamic programming under the real world probabilitymeasure Examples of the application of this technique inthe banking literature can be the papers by Mukuddem-Petersen and Petersen [11] Mukuddem-Petersen et al [12]
Mukuddem-Petersen et al [13] and Gideon et al [14] forinstance In their paper Mukuddem-Petersen and Petersen[11] studied a problem related to the optimal risk manage-ment of banks in a stochastic setting The aforementionedauthors minimized market and capital adequacy risk thatrespectively involves the safety of the securities held andthe stability of sources of funds In this regard the authorssuggested an optimal portfolio choice and rate of bank capitalinflow that will keep the loan level as close as possible toan actuarially determined reference process In the paper[12] by Mukuddem-Petersen et al the authors solved astochastic maximization problem that is related to depositoryconsumption and banking profit on a random time intervalIn particular they showed that it is possible for a bankto maximize its expected utility of discounted depositoryconsumption on a random time interval [119905 120591] and its finalprofit at time 120591 In the paper [13] of Mukuddem-Petersenet al (see also the book of Petersen et al [15]) the authorsanalyzed the mortgage loan securitization process whichwas the root cause of the subprime mortgage crisis (SMC)of 2008 The authors considered a bank that has the cashoutflow rate for financing a portfolio of mortgage-backedsecurities (MBSs) and studied an optimal securitizationproblemwith the investment in theMBSs as controls Gideonet al [14] quantitatively validated the new Basel III liquiditystandards as encapsulated by the net stable funding ratio Byconsidering the inverse net stable funding ratio as a measureto quantify the bankrsquos prospects for a stable funding overthe period of a year they solved an optimal control problemfor a continuous-time inverse net stable funding ratio Morespecifically the authors made optimal choices for the inversenet stable funding targets in order to formulate its cost Thelatter was achieved by finding an analytical solution for thevalue function
In theoretical physics the Legendre transform is com-monly used in areas such as classical mechanics statisticalmechanics and thermodynamics The Legendre transformalso has its uses in mathematics of finance For instance inpension fund optimization problems the Legendre transformis generally applied to transform nonlinear HJB PDEs arisingfrom the optimal control technique to linear PDEs for whichit is easier to find a closed form solution Authors who haveapplied the Legendre transform in pension funds are Xiao etal [16] on the constant elasticity of variance (CEV) modelfor a defined contribution pension plan and the Legendretransform-dual solutions for annuity contracts Gao [17] onthe stochastic optimal control of pension funds Gao [18] onthe optimal investment strategy for annuity contracts undertheCEVmodel Gao [19] on the extendedCEVmodel and theLegendre transform-dual-asymptotic solutions for annuitycontracts and Jung and Kim [20] on the optimal investmentstrategies for the hyperbolic absolute risk aversion (HARA)utility function under the CEV model
In our contribution the issues of asset portfolio and capi-tal adequacymanagement for Basel III compliant commercialbanks are addressed in a continuous-time setting Morespecifically our goal is to maximize an expected utility of acommercial bankrsquos asset portfolio at a future date We assumethat the interest rate in the market is stochastic and that the
Journal of Applied Mathematics 3
asset portfolio consists of a treasury security a marketablesecurity and a loan We take the stochastic optimal controlapproach and arrive at a nonlinear second-order PDE for thevalue function for which it is difficult to find a solution Ourchoice of asset portfolio is based on the portfolio of Gao [17]In a pension fund setting Gao [17] solved the optimizationproblem in question using the Legendre transform and dualtheory With our optimization problem similar to that in[17] we evoke the results of Gao Based on the investmentstrategy that solves our optimization problem we derive amodel for the Total Capital Ratio or CAR of the bank in termsof the optimal amounts of bank capital invested in the assetsFurthermore we assume that the bank follows the optimalinvestment strategy while simultaneously maintaining itsCAR at the fixed level of 8 We obtain a formula forthe bankrsquos asset portfolio into which is incorporated therestriction on the CAR By means of numerical simulationswe characterize the optimal investment strategy by presentinggraphically the optimal proportions of capital invested inthe bankrsquos assets In addition we simulate and graphicallyillustrate the behaviours of the CAR and the modified assetportfolio subject to the optimal investment strategy To ourknowledge a study as described above has not been done fora commercial Basel III compliant bank
The layout of the rest of our paper is as follows InSection 2 we present some general banking theory Morespecifically we describe the bankrsquos assets liabilities andcapital all of which comprise its stylized balance sheet Weintroduce the financial market in Section 3 Here we specifymodels for the stochastic interest rate in the market andthe assets and asset portfolio of the bank In Section 4 weformulate the optimization problem and present the optimalsolution In this section we also present the bank capitalmodel relevant to the optimization problem The formulafor the CAR is derived in Section 5 We simulate graphicallythe optimal investment proportions of capital invested in theassets as well as the dynamics of theCAR Section 6 is devotedto the derivation of the asset portfolio pertaining to a constantCAR In this section we also provide a simulation of themodified asset portfolio obtained via the optimal investmentstrategy and restricted CAR We conclude the paper withSection 7
2 The Commercial Banking Model
We consider a complete and frictionless financial marketwhich is continuously open over a fixed time interval [0 119879]Throughout we assume that we are working with a proba-bility space (ΩFP) where P is the real world probabilityWe assume that the Brownian motions that appear in thedynamics of the bank items are defined on the probabilityspace (ΩFP) and that the filtration (F(119905))
119905ge0satisfies the
usual conditionsTo understand the operation and management of com-
mercial banks we study its stylized balance sheet whichrecords the assets (uses of funds) and liabilities (sources offunds) of the bank
Bank capital fulfills the role of balancing the assets andliabilities of the bank A useful way for our analysis ofrepresenting the balance sheet of the bank is as follows
119877 (119905) + 119878 (119905) + 119871 (119905) = 119863 (119905) + 119861 (119905) + 119862 (119905) (2)
where 119877 119878 119871 119863 119861 and 119862 represent the values of treasuriessecurities loans deposits borrowings and capital respec-tively Each of the variables above is regarded as a functionΩ timesR
+rarr R+
In order for a commercial bank to make a profit itis important that the bank manages the asset side of itsbalance sheet properly The latter is determined by twofactors namely the amount of capital available to invest ithas and the attitude it has toward risk and return The bankmust therefore allocate its capital optimally among its assetsBelow we explain each of the items on the balance sheet of acommercial bank
The term reserves refers to the vault cash of the bank plusthe compulsory amount of its money deposited at the centralbank Reserves are used to meet the day-to-day currencywithdrawals by its customers Securities consist of treasurysecurities (treasuries) and marketable securities (securities)Treasuries are bonds issued by national treasuries in mostcountries as a means of borrowing money to meet govern-ment expenditures not covered by tax revenues Securities onthe other hand are stocks and bonds that can be convertedto cash quickly and easily Loans granted by the bank includebusiness loans mortgage loans (land loans) and consumerloans Consumer loans include credit extended by the bankfor credit card purchases Mortgages are long term loans usedto buy a house or land where the house or land acts ascollateral Business loans are taken out by firms that borrowfunds to finance their inventories which act as collateral forthe loan A loan which has collateral (a secured loan) has alower interest rate associatedwith it compared to a loanwhichhas no collateral (unsecured loans)
Bank capital is raised by selling new equity retainingearnings and by issuing debt or building up loan-lossreserves The bankrsquos risk management department is usuallyresponsible for calculating its capital requirements Calcu-lated risk capital is then approved by the bankrsquos top executivemanagement Furthermore the structure of bank capitalis proposed by the Finance Department and subsequentlyapproved by the bankrsquos top executive management Thedynamics of bank capital is stochastic in nature becauseit depends in part on the uncertainty related to debt andshareholder contributions In theory the bank can decide onthe rate at which debt and equity are raised
Under the Basel III Accord (see [21]) the bankrsquos capital 119862also referred to as total bank capital has the form
119862 (119905) = 119862T1 (119905) + 119862T2 (119905) (3)
where 119862T1(119905) and 119862T2 are Tier 1 and Tier 2 capital respec-tively
Tier 1 capital consists of the sum of Common Equity Tier1 capital and Additional Tier 1 capital Common Equity Tier1 capital is defined as the sum of common shares issued bythe bank that meet the criteria for classification as common
4 Journal of Applied Mathematics
shares for regulatory purposes stock surplus resulting fromthe issue of instruments included in Common Equity Tier 1capital retained earnings other accumulated comprehensiveincome and other disclosed reserves common shares issuedby consolidated subsidiaries of the bank and held by thirdparties thatmeet the criteria for inclusion in CommonEquityTier 1 capital and regulatory adjustments applied in thecalculation of Common Equity Tier 1 capital Additional Tier1 capital is the sum of instruments issued by the bank thatmeet the criteria for inclusion in Additional Tier 1 capital(and are not included inCommonEquity Tier 1 capital) stocksurplus resulting from the issue of instruments included inAdditional Tier 1 capital instruments issued by consolidatedsubsidiaries of the bank and held by third parties that meetthe criteria for inclusion in Additional Tier 1 capital and arenot included inCommonEquity Tier 1 capital and regulatoryadjustments applied in the calculation of Additional Tier 1capital
Tier 2 capital is defined as the sum of the followingelements instruments issued by the bank that meet thecriteria for inclusion in Tier 2 capital (and are not includedin Tier 1 capital) stock surplus resulting from the issue ofinstruments included in Tier 2 capital instruments issued byconsolidated subsidiaries of the bank andheld by third partiesthat meet the criteria for inclusion in Tier 2 capital and arenot included in Tier 1 capital certain loan-loss provisionsand regulatory adjustments applied in the calculation of Tier2 capital
Deposits refer to the money that the bankrsquos customersplace in the banking institution for safekeeping Depositsare made to deposit accounts at a banking institution suchas savings accounts checking accounts and money marketaccounts The account holder has the right to withdraw anydeposited funds as set forth in the terms and conditions ofthe account Deposits are considered to be themain liabilitiesof the bank
The term borrowings refers to the funds that commercialbanks borrow from other banks and the central bank
3 The Financial Market Setting
We assume that it is possible for the bank to continuouslyraise small levels of capital at a rate 119889119862(119905) The bank isassumed to invest its capital in a market consisting of atleast two types of assets (treasury and security) which can bebought and sold without incurring any transaction costs orrestriction on short sales It is assumed that the bank investsin a third asset namely a loan
The first asset in the financial market is a riskless treasuryWe denote its price at time 119905 by 119878
0(119905) and assume that its
dynamics evolve according to the equation
1198891198780(119905)
1198780(119905)
= 119903 (119905) 119889119905 1198780(0) = 1 (4)
The dynamics of the short rate process 119903(119905) is given bythe stochastic differential equation (SDE)
119889119903 (119905) = (119886 minus 119887119903 (119905)) 119889119905 minus 120590119903119889119882119903(119905) (5)
for 119905 ge 0 and where 120590119903= radic1198961119903(119905) + 119896
2 The coefficients 119886 119887
1198961 and 119896
2 as well as the initial value 119903(0) are all positive real
constants The above dynamics recover as a special case theVasicek [22] (resp Cox et al [23]) dynamics when 119896
1(resp
1198962) is equal to zero The term structure of the interest rates is
affine under the aforementioned dynamicsThe second asset in the market is a risky security whose
price is denoted by 119878(119905) 119905 ge 0 Its dynamics are given by theequation
119889119878 (119905)
119878 (119905)= 119903 (119905) 119889119905 + 120590
1(119889119882119904(119905) + 120582
1119889119905)
+ 1205902120590119903(119889119882119903(119905) + 120582
2120590119903119889119905)
(6)
with 119878(0) = 1 and 1205821 1205822(resp 120590
1 1205902) being constants (resp
positive constants) as in Deelstra et al [24]The third asset is a loan to be amortized over a period
[0 119879]whose price at time 119905 ge 0 is denoted by 119871(119905) We assumethat its dynamics can be described by the SDE
119889119871 (119905)
119871 (119905)= 119903 (119905) 119889119905 + 120590
119871(119879 minus 119905 119903 (119905)) (119889119882
119903(119905) + 120582
2120590119903119889119905) (7)
We now model the asset portfolio of the bank Let 119883(119905)
denote the value of the asset portfolio at time 119905 isin [0 119879] Thedynamics of the asset portfolio are described by
119889119883 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+ 120579119878(119905)
119889119878 (119905)
119878 (119905)
+ 120579119871(119905)
119889119871 (119905)
119871 (119905)+ 119889119862 (119905)
(8)
where 120579119878(119905) 120579119871(119905) and 120579
119903(119905) denote the amounts of capital
invested in the two risky assets (security and loan) and inthe riskless asset (treasury) respectively Making use of themodels for 119878
0(119905) 119878(119905) and 119871(119905) we can rewrite (8) as
119889119883 (119905) = 119883 (119905) 119903 (119905) + 120579119878(119905) [12058211205901+ 120582212059021205902
119903]
+120579119871(119905) 1205822120590119871(119879 minus 119905 119903 (119905)) 120590
119903 119889119905
+ 120579119878(119905) 1205901119889119882119878(119905)+(120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
+120579119878(119905) 1205902120590119903) 119889119882119903(119905)+119889119862 (119905)
(9)
4 The Portfolio Problem andOptimal Solution
In this section we formulate the optimal control problemassociated with an influx of capital at a rate
119889119862 (119905) = 119888 (119905) 119889119905 119862 (0) gt 0 (10)
We derive theHJB equation for the value function and wepresent the optimal solution from the methodology of Gao[17]
We wish to choose a portfolio strategy in order tomaximize the expected utility of the bankrsquos asset portfolio ata future date 119879 gt 0 Mathematically the stochastic optimalcontrol problem can be stated as follows
Journal of Applied Mathematics 5
Problem 1 Our objective is to maximize the expected utilityof the bankrsquos asset portfolio at a future date 119879 gt 0 Thus wemust
maximize 119869 (120579119878 120579119871) = E (119906 (119883 (119879)))
subject to 119889119903 (119905) = (119886 minus 119887119903 (119905)) 119889119905 minus 120590119903119889119882119903(119905)
119889119883 (119905) = [119883 (119905) 119903 (119905) + 120579119878(119905)
times (12058211205901+ 120582212059021205902
119903)
+ 120579119871(119905) 1205822120590119871(119879 minus 119905 119903 (119905)) 120590
119903
+ 119888 (119905)] 119889119905 + 120579119878(119905) 1205901119889119882119878(119905)
+ (120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
+ 120579119878(119905) 1205902120590119903) 119889119882119903(119905)
119883 (0) = 1199090 119903 (0) = 119903
0
(11)
with 0 le 119905 le 119879 and where 119883(0) = 1199090and 119903(0) = 119903
0denote
the initial conditions of the optimal control problem
In this paper we describe the bankrsquos objective with alogarithmic utility function that is
119906 (119909) = ln119909 with 119909 gt 0 (12)
Wenote that the utility function 119906(sdot) is strictly concave up andsatisfies the Inada conditions 1199061015840(+infin) = 0 and 119906
1015840(0) = +infin
By using the classical tools of stochastic optimal control wedefine the value function
119867(119905 119903 119909) = sup120579119878120579119871
E (119906 (119883 (119879) | 119903 (119905) = 119903 119883 (119905) = 119909))
0 lt 119905 lt 119879
(13)
The value function can be considered as a kind of utilityfunction The marginal utility of the value function is aconstant while the marginal utility of the original utilityfunction 119906(sdot) decreases to zero as 119909 rarr infin (see Kramkovand Schachermayer [25])The value function also inherits theconvexity of the utility function (see Jonsson and Sircar [26])Moreover it is strictly convex for 119905 lt 119879 even if 119906(sdot) is not
The maximum principle leads to the HJB equation (seeDuffie [27])
119867119905+ 119886 (119887 minus 119903)119867
119903
+ [119909119903 + (12058211205901+ 120582212059021205902
119903) 120579119878+ 1205822120590119871120590119903120579119871+ 119888]119867
119909
+1
2[1205902
11205792
119878+ (120590119871120579119871+ 1205902120590119903120579119878)2
]119867119909119909
+1205902
119903
2119867119903119903
minus (120590119871120590119903120579119871+ 12059021205902
119903120579119878)119867119903119909
= 0
(14)
where we have suppressed the time variable 119905 In the HJBequation above119867
119905119867119903119867119909119867119903119903119867119909119909 and119867
119903119909denote partial
derivatives of first and second orders with respect to timeinterest rate and asset portfolio
The first-order maximizing conditions for the optimalstrategies 120579
119878and 120579119871are
120579119878= minus
1205821
1205901
119867119909
119867119909119909
120579119871=
120590119903(12058211205902minus 12058221205901)119867119909+ 1205901120590119903119867119903119909
1205901120590119871119867119909119909
(15)
If we put (15) into (14) we obtain a PDE for the value function119867
119867119905+ 119886 (119887 minus 119903)119867
119903+
1205902
119903
2119867119903119903+ (119909119903 + 119888)119867
119909
minus1205822
1
2
1198672
119909
119867119909119909
minus(1205822120590119903119867119909minus 120590119903119867119903119909)2
2119867119909119909
= 0
(16)
The problem is to solve (16) for the value function 119867 andreplace it in (15) in order to obtain the optimal investmentstrategy The above equation is a nonlinear second-orderPDE which is very difficult to solve The solution to thisparticular problem was derived in a pension fund setting byGao [17] who did so by employing the Legendre transformand dual theory We present the optimal solution from Gao[17] in Remark 2 below
At this point we shall specify a particular candidate for thefunction 120590
119871appearing in (7) We assume 120590
119871to take the form
120590119871(119879 minus 119905 119903 (119905)) = ℎ (119879 minus 119905) 120590
119903(17)
with
ℎ (119905) =2 (119890119898119905
minus 1)
119898 minus (119887 minus 11989611205822) + 119890119898119905 (119898 + 119887 minus 119896
11205822)
119898 = radic(119887 minus 11989611205822)2
+ 21198961
(18)
Remark 2 The solution of Problem 1 is obtained by solving(16) in terms of 119867 and substituting the result into (15) Theoptimal solution is as follows
The optimal amount of bank capital invested in the loanis
120579119871=
120590119903(12058221205901minus 12058211205902) 119909
1205901120590119871
minus 120590119903119888 (119905) [
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(19)
6 Journal of Applied Mathematics
or if we denote the optimal proportion of capital invested inthe loan by 120578
119871 then we can write
120578119871=
120590119903(12058221205901minus 12058211205902)
1205901120590119871
minus120590119903119888 (119905)
119909
[
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(20)
Furthermore the optimal amount of capital invested in thesecurity is given by
120579119878=
1205821
1205901
(119909 + 119888 (119905) 119886119879minus119905|119903119905
) (21)
or if 120578119878denotes the optimal proportion of capital invested in
the security then
120578119878=
1205821
1205901
+
119888 (119905) 119886119879minus119905|119903119905
119909 (22)
According to the above models we may write the optimalamount of capital invested in the treasury as
120579119903= 119909 minus 120579
119871minus 120579119878
(23)
and hence
120578119903= 1 minus 120578
119871minus 120578119878 (24)
where 120578119903is the optimal proportion of capital invested in the
treasuryThe expression 119886119879minus119905
denotes a continuous annuity ofduration 119879 minus 119905
5 The Total Capital Ratio
We now proceed to derive the dynamics of the Total CapitalRatio (CAR) In order to do so we first derive the dynamics ofthe TRWAs of the bank In addition we simulate the optimalproportions of bank capital invested in the assets and theCARat the end of this section The dynamics of the TRWAs andCAR are respectively derived in the remark and propositionbelowWe still assume that the bank capital evolves accordingto (10)
Remark 3 Suppose that at time 119905 the TRWAs can bedescribed by the SDE
119889119886rw (119905) = 0 times 120579119903(119905)
1198891198780(119905)
1198780(119905)
+ 02 times 120579119878(119905)
119889119878 (119905)
119878 (119905)
+ 05 times 120579119871(119905)
119889119871 (119905)
119871 (119905)+ 119888 (119905) 119889119905
(25)
where 0 02 and 05 are the risk-weights associated withrespectively the treasury security and loan under the Basel
III Accord By simplifying (25) the expression for 119889119886rw(119905)reduces to the following
119889119886rw (119905)
= [02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)] 119889119905
+ 02120579119878(119905) 1205901119889119882119878(119905)
+ [02120579119878(119905) 1205902120590119903
+05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))] 119889119882
119903(119905)
(26)
In the proof of the proposition below wemake use of ItorsquosLemma and Itorsquos Product Rule for which we refer to the book[28] by Oslashksendal
Proposition 4 With the dynamics of the total bank capital119862(119905) given by the ODE in (10) and with the dynamics of theTRWAs 119886
119903119908 given by (26) we can write the dynamics of the
Total Capital Ratio or CAR at time 119905 as
119889Λ (119905) = Ψ1119889119905 +
119862 (119905)
1198862119903119908
(119905)[(minusΨ
2+
Ψ3
119886119903119908
(119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(27)
where
Ψ1=
119888 (119905)
119886119903119908
(119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(28)
Proof Wemainly use Itorsquos general formula to derive (27) LetΦ(119886rw(119905)) = 1(119886rw(119905)) Then by Itorsquos Lemma
119889Φ (119886rw (119905))
= Φ (119886rw (119905)) 119889119905 + Φ1015840(119886rw (119905)) 119889119886rw (119905)
+1
2Φ10158401015840(119886rw (119905)) [119889119886rw (119905)]
2
= 0119889119905 minus119889119886rw (119905)
1198862rw (119905)+
[119889119886rw (119905)]2
1198863rw (119905)
Journal of Applied Mathematics 7
= minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
+ 119888 (119905)] +1
1198863rw (119905)[(02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905)))2] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))) 119889119882
119903(119905)]
(29)
Let Λ(119905) denote the CAR at time 119905 for 119905 isin [0 119879] Then bydefinition (see (1)) we can write Λ(119905) as
Λ (119905) =119862 (119905)
119886rw (119905)= 119862 (119905)Φ (119886rw (119905)) (30)
We apply Itorsquos Product Rule to Λ(119905) = 119862(119905)Φ(119886rw(119905)) to findan expression for 119889Λ(119905)
119889Λ (119905)
= 119889119862 (119905)Φ (119886rw (119905)) + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905 + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905
+ 119862 (119905) minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905))
times 1205822120590119903)+119888 (119905)]+
1
1198863rw (119905)
times[(02120579119878(119905) 1205901)2
+(02120579119878(119905) 1205902120590119903+ 05120579
119871
times(119905)120590119871(119879minus 119905 119903 (119905)))
2
] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))) 119889119882119903(119905)]
= Ψ1119889119905 + 119862 (119905) [(minus
1
1198862rw (119905)Ψ2+
1
1198863rw (119905)Ψ3)119889119905 minus
1
1198862rw (119905)
times (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
= Ψ1119889119905 +
119862 (119905)
1198862rw (119905)[(minusΨ
2+
Ψ3
119886rw (119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(31)
where we have defined
Ψ1=
119888 (119905)
119886rw (119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(32)
This concludes the proof
We provide a numerical simulation in order to charac-terize the behaviour of the CAR Λ We assume that theinterest rate follows the CIR dynamics (119896
2= 0) and that the
financial market consists of a treasury a security and a loanFurthermore we consider an investment horizon of 119879 = 10
years and that capital is raised at the fixed rate of 119888 = 013The rest of the parameters of the simulation are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(33)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119886rw (0) = 14 Λ (0) = 008
(34)
In Figure 1 we characterize the optimal investmentstrategy that solves Problem 1 by simulating the optimalproportions of capital invested in the assets The optimalinvestment strategy depicted in Figure 1 leads to the CARin Figure 4 By diversifying its asset portfolio according tothe optimal strategy illustrated by Figure 1 we note that thebank maintains its CAR in such a manner that it is abovethe minimum required level of 8 Since the bank complieswith the minimum required ratio of total capital to TRWAs
8 Journal of Applied Mathematics
0 2 4 6 8 10
TreasurySecurity
Loan
minus04
minus02
0
02
04
06
08
1
Time (years)
Figure 1 A simulation of the optimal proportions 120578119903(119905) 120578119878(119905) and
120578119871(119905) of bank capital invested in the assets given a constant stream
of capital inflow
as imposed by the Basel III Accord it is guaranteed theability to absorb reasonable levels of losses before becominginsolvent over the 10-year period We also simulate theoptimal asset portfolio and TRWAs in Figures 2 and 3respectively An interesting observation is that the TRWAsattain itsmaximum at the end of the investment period underconsideration This is due to the optimal amounts of bankcapital 120579
119878and 120579
119871 invested respectively in the security and
loan being embedded in the formula for the TRWAs in (25)The volatile natures of the quantities simulated above areconsistent with the stochastic variables used throughout thepaper
6 The Asset Portfolio for a Constant CAR
We now set out to modify the asset portfolio of Problem 1 insuch a way as to maintain the Total Capital Ratio or CAR ata constant rate of 8 To this end we need to have the bankcapital 119862(119905) to be stochastic We assume that the stochasticterm is sufficiently small in order to use the solution ofProblem 1 as a reasonable approximation The actual formof 119862(119905) is deduced from the identity 119862(119905) = 008119886rwThe formula for the asset portfolio is given in Remark 5below
Remark 5 At time 119905 the dynamics of the asset portfolio119884(119905) of the bank investing its capital according to theoptimal investment strategy from Problem 1 and in additionmaintaining its CAR at 8 can be written as
119889119884 (119905) = 1205941119889119905 + 120594
2119889119882119878(119905) + 120594
3119889119882119903(119905) (35)
0 2 4 6 8 101
2
3
4
5
6
7
8
Optimal asset portfolio
Time (years)
Figure 2 A simulation of 119883(119905) the optimal asset portfolio given aconstant stream of capital inflow
0 2 4 6 8 10
TRWAs
Time (years)
14
16
18
2
22
24
26
28
3
32
Figure 3 A simulation of the total risk-weighted assets 119886rw(119905) givena constant stream of capital inflow
where
1205941= (119883 (119905) minus 120579
119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
1205942=
117
115120579119878(119905) 1205901
1205943=
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(36)
Journal of Applied Mathematics 9
0 2 4 6 8 10008
009
01
011
012
013
014
015
016
Total Capital Ratio
Time (years)
Figure 4 A simulation of Λ(119905) the Total Capital Ratio given aconstant stream of capital inflow
The argument for this remark goes as follows From thecondition Λ(119905) = 119862(119905)119886rw(119905) = 008 we have 119889119862(119905)008 =
119889119886rw(119905) By substituting 119889119862(119905)008 for the left hand side of(25) and replacing the 119888(119905)119889119905 of (25) by 119889119862(119905) we obtain
(1
008minus 1) 119889119862 (119905) = 02120579
119878(119905)
119889119878 (119905)
119878 (119905)+ 05120579
119871(119905)
119889119871 (119905)
119871 (119905)
(37)
Solving 119889119862(119905) in terms of 120579119878(119905)(119889119878(119905)119878(119905)) and 120579
119871(119905)(119889119871(119905)
119871(119905)) yields
119889119862 (119905) =2
115120579119878(119905)
119889119878 (119905)
119878 (119905)+
1
23120579119871(119905)
119889119871 (119905)
119871 (119905) (38)
Substituting the above expression as the 119889119862(119905) of (8) gives
119889119884 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+117
115120579119878(119905)
119889119878 (119905)
119878 (119905)
+24
23120579119871(119905)
119889119871 (119905)
119871 (119905)
(39)
By simplifying this expression in terms of the models of1198891198780(119905)1198780(119905) 119889119878(119905)119878(119905) and 119889119871(119905)119871(119905) and by recalling that
120579119903(119905) = 119883(119905) minus 120579
119871(119905) minus 120579
119878(119905) we can eventually write 119889119884(119905) as
119889119884 (119905) = [(119883 (119905) minus 120579119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)] 119889119905
+117
115120579119878(119905) 1205901119889119882119878+ [
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))] 119889119882119903
(40)
By writing the coefficients of 119889119905 119889119882119878(119905) and 119889119882
119903(119905) of the
above expression respectively as 1205941 1205942 and 120594
3 the asserted
expression for 119889119884(119905) emergesWe now characterize the behaviour of the modified asset
portfolio given by (35)We consider an interest rate followingthe CIR dynamics (119896
2= 0) and an investment horizon of 119879 =
10 years A capital contribution rate of 119888 = 0 is consideredwhile the rest of the simulation parameters are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(41)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119884 (0) = 15
(42)
For a bank that diversifies its asset portfolio accordingto the solution of Problem 1 and maintains its CAR at aconstant value of 8 we simulate the modified version ofits asset portfolio in Figure 5 In this simulation we controlthe bank capital 119862(119905) as in Figure 6 during the 10-yearinvestment period We note that the growth of the modifiedasset portfolio given by (35) is much slower compared tothe original asset portfolio of Problem 1 For the specificsimulation shown the modified asset portfolio is maximal attime 119879 = 10 years
7 Conclusion
In this paper we consider asset portfolio and capital adequacymanagement in banking In particular our objective isto maximize an expected utility of a Basel III compliantcommercial bankrsquos asset portfolio at a future date119879 gt 0 Fromthe optimal investment strategy that solves our investmentproblem we derive a dynamic formula for the Total CapitalRatio or Capital Adequacy Ratio (CAR) In addition wederive a modified version of the bankrsquos asset portfolio thatwould be obtained should the bank simultaneously followthe optimal investment strategy and maintain its CAR at aconstant level of 8 This modified portfolio is extremelyimportant since it takes maximum advantage by keepingthe Total Capital Ratio exactly at the threshold This meansthat all the funding available for investment is indeed beinginvested
10 Journal of Applied Mathematics
0 2 4 6 8 10
Modified asset portfolio
Time (years)
1
15
2
25
3
35
4
45
5
55
6
Figure 5 A simulation of119884(119905) themodified asset portfolio requiredto maintain the Total Capital Ratio at 8
0 2 4 6 8 100995
1
1005
101
1015
102
1025
103
1035
Time (years)
Controlled bank capital
Figure 6 A simulation of the bank capital 119862(119905) required tomaintain the Total Capital Ratio at 8
We present a simulation study in which we illustratethe optimal investment strategy and characterize the cor-responding CAR and modified asset portfolio The mainresults of our study are as follows The optimal investmentstrategy for the bank is to diversify its asset portfolio awayfrom the risky assets that is the security and loan andtowards the riskless treasury This is in accordance with thebanking literature on optimal asset management in whichit is reported that banks diversify their asset portfolios inan attempt to lower risk (see Dangl and Lehar [29] andDecamps et al [30] for instance) In doing so the bankmaintains its CAR above the Basel III prescribed minimum
level of 8 Of course in line with Basel III standards thebank is able to absorb reasonable levels of unexpected lossesbefore becoming insolvent Furthermore in order to derivethe asset portfolio at constant (minimum) CAR value thelevel of bank capital needs to be controlled This is donemodifying the deterministic capital model to be stochasticWe find that the asset portfolio at constant (minimum) CARvalue grows considerably slower than the asset portfolioof the original investment problem but that this modifiedportfolio is still maximal at the end of the investment periodunder consideration An interesting observation is that ourbank raises capital while at the same time lowering risk Apossible explanation for this is that the bank holds low capitalConservation Buffers which it is trying to rebuild (see Heidet al [31]) In contrast banks with large buffersmaintain theircapital buffer by increasing risk when capital increases [31]
In future research it would be interesting to go beyond thesimulated data approach and model the CAR and bank assetportfolio at constant (minimum) CAR value using real datasourced from for example the US Federal Deposit InsuranceCorporation (FDIC)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors acknowledge financial support from theNationalResearch Foundation of South Africa
References
[1] Basel Committee on Banking Supervision International Con-vergence of Capital Measurements and Capital Standards ARevised Framework Bank for International Settlements 2004
[2] A V Thakor ldquoCapital requirements monetary policy andaggregate bank lending theory and empirical evidencerdquo Journalof Finance vol 51 no 1 pp 279ndash324 1996
[3] E-L vonThadden ldquoBank capital adequacy regulation under thenew Basel accordrdquo Journal of Financial Intermediation vol 13no 2 pp 90ndash95 2004
[4] C H Fouche J Mukuddem-Petersen and M A PetersenldquoContinuous-time stochastic modelling of capital adequacyratios for banksrdquo Applied Stochastic Models in Business andIndustry vol 22 no 1 pp 41ndash71 2006
[5] J Mukuddem-Petersen and M A Petersen ldquoOptimizing assetand capital adequacy management in bankingrdquo Journal ofOptimization Theory and Applications vol 137 no 1 pp 205ndash230 2008
[6] M P Mulaudzi M A Petersen and I M Schoeman ldquoOptimalallocation between bank loans and treasuries with regretrdquoOptimization Letters vol 2 no 4 pp 555ndash566 2008
[7] M Jayadev ldquoBasel III implementation issues and challenges forIndian banksrdquo IIMB Management Review vol 25 pp 115ndash1302013
[8] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
that requires banks to adhere to minimum levels of capitalrequirements By complying with minimum capital levelsbanks are guaranteed the ability to absorb reasonable levelsof losses before becoming insolvent Thus CARs ensure thesafety and stability of the banking system Under the BaselII Accord banks were required to maintain a CAR thatis a ratio of total bank capital to total risk-weighted assets(TRWAs) with a minimum value of 8
In response to the 2007-2008 financial crisis the BaselCommittee on Banking Supervision (BCBS) introduced acomprehensive set of reform measures known as the BaselIII Accord The Basel III Accord is aimed at improving theregulation supervision and risk management within thebanking sector and is part of the continuous effort madeby the BCBS to enhance the banking regulatory frameworkThe Basel III Accord builds on the Basel I and II documentsand seeks to improve the banking sectorrsquos ability to deal withfinancial and economic stress improve riskmanagement andstrengthen the banksrsquo transparencyThe focus of the Basel IIIAccord is to foster greater resilience at the individual banklevel in order to reduce the risk of system wide shocks In thisregard the Basel III Accord contains changes in the followingareas (i) augmentation in the level and quality of capital(ii) introduction of liquidity standards (iii) modifications inprovisioning norms (iv) introduction of a leverage ratio Fora detailed discussion on the aforementioned enhancementsin the Basel III Accord over the Basel II Accord see the paper[7] of Jayadev or the book [8] of Petersen and Mukuddem-Petersen
In our paper we are concerned with the level of bankcapital Thus we study the Total Capital Ratio or CAR whichif denoted by Λ is defined as
Λ =119862
119886rw (1)
In the expression for the CAR above 119862 denotes the totalbank capital and 119886rw the TRWAs of the bank Under BaselIII rules the minimum prescribed value of the bankrsquos CARremains unchanged at 8 However banks are now requiredto hold a Capital Conservation Buffer (CCB) of 25 andCountercyclical Buffer Capital (CBC) in the range of 0ndash25 The CCB ensures that banks can absorb losses withoutbreaching the minimum capital requirement and are able tocarry on business even in a downturn without deleveragingThe CBC is a preemptive measure that requires banks tobuild up capital gradually as imbalances in the credit marketdevelop It may be in the range of 0ndash25 of risk-weightedassets which could be imposed on banks during periods ofexcess credit growth There is also a provision for a highercapital surcharge on systematically important banks (see [78])
In finance the stochastic optimal control method is apopular optimization technique and was used for the firsttime by Merton [9 10] This technique involves solvingthe Hamilton-Jacobi-Bellman (HJB) equation arising fromdynamic programming under the real world probabilitymeasure Examples of the application of this technique inthe banking literature can be the papers by Mukuddem-Petersen and Petersen [11] Mukuddem-Petersen et al [12]
Mukuddem-Petersen et al [13] and Gideon et al [14] forinstance In their paper Mukuddem-Petersen and Petersen[11] studied a problem related to the optimal risk manage-ment of banks in a stochastic setting The aforementionedauthors minimized market and capital adequacy risk thatrespectively involves the safety of the securities held andthe stability of sources of funds In this regard the authorssuggested an optimal portfolio choice and rate of bank capitalinflow that will keep the loan level as close as possible toan actuarially determined reference process In the paper[12] by Mukuddem-Petersen et al the authors solved astochastic maximization problem that is related to depositoryconsumption and banking profit on a random time intervalIn particular they showed that it is possible for a bankto maximize its expected utility of discounted depositoryconsumption on a random time interval [119905 120591] and its finalprofit at time 120591 In the paper [13] of Mukuddem-Petersenet al (see also the book of Petersen et al [15]) the authorsanalyzed the mortgage loan securitization process whichwas the root cause of the subprime mortgage crisis (SMC)of 2008 The authors considered a bank that has the cashoutflow rate for financing a portfolio of mortgage-backedsecurities (MBSs) and studied an optimal securitizationproblemwith the investment in theMBSs as controls Gideonet al [14] quantitatively validated the new Basel III liquiditystandards as encapsulated by the net stable funding ratio Byconsidering the inverse net stable funding ratio as a measureto quantify the bankrsquos prospects for a stable funding overthe period of a year they solved an optimal control problemfor a continuous-time inverse net stable funding ratio Morespecifically the authors made optimal choices for the inversenet stable funding targets in order to formulate its cost Thelatter was achieved by finding an analytical solution for thevalue function
In theoretical physics the Legendre transform is com-monly used in areas such as classical mechanics statisticalmechanics and thermodynamics The Legendre transformalso has its uses in mathematics of finance For instance inpension fund optimization problems the Legendre transformis generally applied to transform nonlinear HJB PDEs arisingfrom the optimal control technique to linear PDEs for whichit is easier to find a closed form solution Authors who haveapplied the Legendre transform in pension funds are Xiao etal [16] on the constant elasticity of variance (CEV) modelfor a defined contribution pension plan and the Legendretransform-dual solutions for annuity contracts Gao [17] onthe stochastic optimal control of pension funds Gao [18] onthe optimal investment strategy for annuity contracts undertheCEVmodel Gao [19] on the extendedCEVmodel and theLegendre transform-dual-asymptotic solutions for annuitycontracts and Jung and Kim [20] on the optimal investmentstrategies for the hyperbolic absolute risk aversion (HARA)utility function under the CEV model
In our contribution the issues of asset portfolio and capi-tal adequacymanagement for Basel III compliant commercialbanks are addressed in a continuous-time setting Morespecifically our goal is to maximize an expected utility of acommercial bankrsquos asset portfolio at a future date We assumethat the interest rate in the market is stochastic and that the
Journal of Applied Mathematics 3
asset portfolio consists of a treasury security a marketablesecurity and a loan We take the stochastic optimal controlapproach and arrive at a nonlinear second-order PDE for thevalue function for which it is difficult to find a solution Ourchoice of asset portfolio is based on the portfolio of Gao [17]In a pension fund setting Gao [17] solved the optimizationproblem in question using the Legendre transform and dualtheory With our optimization problem similar to that in[17] we evoke the results of Gao Based on the investmentstrategy that solves our optimization problem we derive amodel for the Total Capital Ratio or CAR of the bank in termsof the optimal amounts of bank capital invested in the assetsFurthermore we assume that the bank follows the optimalinvestment strategy while simultaneously maintaining itsCAR at the fixed level of 8 We obtain a formula forthe bankrsquos asset portfolio into which is incorporated therestriction on the CAR By means of numerical simulationswe characterize the optimal investment strategy by presentinggraphically the optimal proportions of capital invested inthe bankrsquos assets In addition we simulate and graphicallyillustrate the behaviours of the CAR and the modified assetportfolio subject to the optimal investment strategy To ourknowledge a study as described above has not been done fora commercial Basel III compliant bank
The layout of the rest of our paper is as follows InSection 2 we present some general banking theory Morespecifically we describe the bankrsquos assets liabilities andcapital all of which comprise its stylized balance sheet Weintroduce the financial market in Section 3 Here we specifymodels for the stochastic interest rate in the market andthe assets and asset portfolio of the bank In Section 4 weformulate the optimization problem and present the optimalsolution In this section we also present the bank capitalmodel relevant to the optimization problem The formulafor the CAR is derived in Section 5 We simulate graphicallythe optimal investment proportions of capital invested in theassets as well as the dynamics of theCAR Section 6 is devotedto the derivation of the asset portfolio pertaining to a constantCAR In this section we also provide a simulation of themodified asset portfolio obtained via the optimal investmentstrategy and restricted CAR We conclude the paper withSection 7
2 The Commercial Banking Model
We consider a complete and frictionless financial marketwhich is continuously open over a fixed time interval [0 119879]Throughout we assume that we are working with a proba-bility space (ΩFP) where P is the real world probabilityWe assume that the Brownian motions that appear in thedynamics of the bank items are defined on the probabilityspace (ΩFP) and that the filtration (F(119905))
119905ge0satisfies the
usual conditionsTo understand the operation and management of com-
mercial banks we study its stylized balance sheet whichrecords the assets (uses of funds) and liabilities (sources offunds) of the bank
Bank capital fulfills the role of balancing the assets andliabilities of the bank A useful way for our analysis ofrepresenting the balance sheet of the bank is as follows
119877 (119905) + 119878 (119905) + 119871 (119905) = 119863 (119905) + 119861 (119905) + 119862 (119905) (2)
where 119877 119878 119871 119863 119861 and 119862 represent the values of treasuriessecurities loans deposits borrowings and capital respec-tively Each of the variables above is regarded as a functionΩ timesR
+rarr R+
In order for a commercial bank to make a profit itis important that the bank manages the asset side of itsbalance sheet properly The latter is determined by twofactors namely the amount of capital available to invest ithas and the attitude it has toward risk and return The bankmust therefore allocate its capital optimally among its assetsBelow we explain each of the items on the balance sheet of acommercial bank
The term reserves refers to the vault cash of the bank plusthe compulsory amount of its money deposited at the centralbank Reserves are used to meet the day-to-day currencywithdrawals by its customers Securities consist of treasurysecurities (treasuries) and marketable securities (securities)Treasuries are bonds issued by national treasuries in mostcountries as a means of borrowing money to meet govern-ment expenditures not covered by tax revenues Securities onthe other hand are stocks and bonds that can be convertedto cash quickly and easily Loans granted by the bank includebusiness loans mortgage loans (land loans) and consumerloans Consumer loans include credit extended by the bankfor credit card purchases Mortgages are long term loans usedto buy a house or land where the house or land acts ascollateral Business loans are taken out by firms that borrowfunds to finance their inventories which act as collateral forthe loan A loan which has collateral (a secured loan) has alower interest rate associatedwith it compared to a loanwhichhas no collateral (unsecured loans)
Bank capital is raised by selling new equity retainingearnings and by issuing debt or building up loan-lossreserves The bankrsquos risk management department is usuallyresponsible for calculating its capital requirements Calcu-lated risk capital is then approved by the bankrsquos top executivemanagement Furthermore the structure of bank capitalis proposed by the Finance Department and subsequentlyapproved by the bankrsquos top executive management Thedynamics of bank capital is stochastic in nature becauseit depends in part on the uncertainty related to debt andshareholder contributions In theory the bank can decide onthe rate at which debt and equity are raised
Under the Basel III Accord (see [21]) the bankrsquos capital 119862also referred to as total bank capital has the form
119862 (119905) = 119862T1 (119905) + 119862T2 (119905) (3)
where 119862T1(119905) and 119862T2 are Tier 1 and Tier 2 capital respec-tively
Tier 1 capital consists of the sum of Common Equity Tier1 capital and Additional Tier 1 capital Common Equity Tier1 capital is defined as the sum of common shares issued bythe bank that meet the criteria for classification as common
4 Journal of Applied Mathematics
shares for regulatory purposes stock surplus resulting fromthe issue of instruments included in Common Equity Tier 1capital retained earnings other accumulated comprehensiveincome and other disclosed reserves common shares issuedby consolidated subsidiaries of the bank and held by thirdparties thatmeet the criteria for inclusion in CommonEquityTier 1 capital and regulatory adjustments applied in thecalculation of Common Equity Tier 1 capital Additional Tier1 capital is the sum of instruments issued by the bank thatmeet the criteria for inclusion in Additional Tier 1 capital(and are not included inCommonEquity Tier 1 capital) stocksurplus resulting from the issue of instruments included inAdditional Tier 1 capital instruments issued by consolidatedsubsidiaries of the bank and held by third parties that meetthe criteria for inclusion in Additional Tier 1 capital and arenot included inCommonEquity Tier 1 capital and regulatoryadjustments applied in the calculation of Additional Tier 1capital
Tier 2 capital is defined as the sum of the followingelements instruments issued by the bank that meet thecriteria for inclusion in Tier 2 capital (and are not includedin Tier 1 capital) stock surplus resulting from the issue ofinstruments included in Tier 2 capital instruments issued byconsolidated subsidiaries of the bank andheld by third partiesthat meet the criteria for inclusion in Tier 2 capital and arenot included in Tier 1 capital certain loan-loss provisionsand regulatory adjustments applied in the calculation of Tier2 capital
Deposits refer to the money that the bankrsquos customersplace in the banking institution for safekeeping Depositsare made to deposit accounts at a banking institution suchas savings accounts checking accounts and money marketaccounts The account holder has the right to withdraw anydeposited funds as set forth in the terms and conditions ofthe account Deposits are considered to be themain liabilitiesof the bank
The term borrowings refers to the funds that commercialbanks borrow from other banks and the central bank
3 The Financial Market Setting
We assume that it is possible for the bank to continuouslyraise small levels of capital at a rate 119889119862(119905) The bank isassumed to invest its capital in a market consisting of atleast two types of assets (treasury and security) which can bebought and sold without incurring any transaction costs orrestriction on short sales It is assumed that the bank investsin a third asset namely a loan
The first asset in the financial market is a riskless treasuryWe denote its price at time 119905 by 119878
0(119905) and assume that its
dynamics evolve according to the equation
1198891198780(119905)
1198780(119905)
= 119903 (119905) 119889119905 1198780(0) = 1 (4)
The dynamics of the short rate process 119903(119905) is given bythe stochastic differential equation (SDE)
119889119903 (119905) = (119886 minus 119887119903 (119905)) 119889119905 minus 120590119903119889119882119903(119905) (5)
for 119905 ge 0 and where 120590119903= radic1198961119903(119905) + 119896
2 The coefficients 119886 119887
1198961 and 119896
2 as well as the initial value 119903(0) are all positive real
constants The above dynamics recover as a special case theVasicek [22] (resp Cox et al [23]) dynamics when 119896
1(resp
1198962) is equal to zero The term structure of the interest rates is
affine under the aforementioned dynamicsThe second asset in the market is a risky security whose
price is denoted by 119878(119905) 119905 ge 0 Its dynamics are given by theequation
119889119878 (119905)
119878 (119905)= 119903 (119905) 119889119905 + 120590
1(119889119882119904(119905) + 120582
1119889119905)
+ 1205902120590119903(119889119882119903(119905) + 120582
2120590119903119889119905)
(6)
with 119878(0) = 1 and 1205821 1205822(resp 120590
1 1205902) being constants (resp
positive constants) as in Deelstra et al [24]The third asset is a loan to be amortized over a period
[0 119879]whose price at time 119905 ge 0 is denoted by 119871(119905) We assumethat its dynamics can be described by the SDE
119889119871 (119905)
119871 (119905)= 119903 (119905) 119889119905 + 120590
119871(119879 minus 119905 119903 (119905)) (119889119882
119903(119905) + 120582
2120590119903119889119905) (7)
We now model the asset portfolio of the bank Let 119883(119905)
denote the value of the asset portfolio at time 119905 isin [0 119879] Thedynamics of the asset portfolio are described by
119889119883 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+ 120579119878(119905)
119889119878 (119905)
119878 (119905)
+ 120579119871(119905)
119889119871 (119905)
119871 (119905)+ 119889119862 (119905)
(8)
where 120579119878(119905) 120579119871(119905) and 120579
119903(119905) denote the amounts of capital
invested in the two risky assets (security and loan) and inthe riskless asset (treasury) respectively Making use of themodels for 119878
0(119905) 119878(119905) and 119871(119905) we can rewrite (8) as
119889119883 (119905) = 119883 (119905) 119903 (119905) + 120579119878(119905) [12058211205901+ 120582212059021205902
119903]
+120579119871(119905) 1205822120590119871(119879 minus 119905 119903 (119905)) 120590
119903 119889119905
+ 120579119878(119905) 1205901119889119882119878(119905)+(120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
+120579119878(119905) 1205902120590119903) 119889119882119903(119905)+119889119862 (119905)
(9)
4 The Portfolio Problem andOptimal Solution
In this section we formulate the optimal control problemassociated with an influx of capital at a rate
119889119862 (119905) = 119888 (119905) 119889119905 119862 (0) gt 0 (10)
We derive theHJB equation for the value function and wepresent the optimal solution from the methodology of Gao[17]
We wish to choose a portfolio strategy in order tomaximize the expected utility of the bankrsquos asset portfolio ata future date 119879 gt 0 Mathematically the stochastic optimalcontrol problem can be stated as follows
Journal of Applied Mathematics 5
Problem 1 Our objective is to maximize the expected utilityof the bankrsquos asset portfolio at a future date 119879 gt 0 Thus wemust
maximize 119869 (120579119878 120579119871) = E (119906 (119883 (119879)))
subject to 119889119903 (119905) = (119886 minus 119887119903 (119905)) 119889119905 minus 120590119903119889119882119903(119905)
119889119883 (119905) = [119883 (119905) 119903 (119905) + 120579119878(119905)
times (12058211205901+ 120582212059021205902
119903)
+ 120579119871(119905) 1205822120590119871(119879 minus 119905 119903 (119905)) 120590
119903
+ 119888 (119905)] 119889119905 + 120579119878(119905) 1205901119889119882119878(119905)
+ (120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
+ 120579119878(119905) 1205902120590119903) 119889119882119903(119905)
119883 (0) = 1199090 119903 (0) = 119903
0
(11)
with 0 le 119905 le 119879 and where 119883(0) = 1199090and 119903(0) = 119903
0denote
the initial conditions of the optimal control problem
In this paper we describe the bankrsquos objective with alogarithmic utility function that is
119906 (119909) = ln119909 with 119909 gt 0 (12)
Wenote that the utility function 119906(sdot) is strictly concave up andsatisfies the Inada conditions 1199061015840(+infin) = 0 and 119906
1015840(0) = +infin
By using the classical tools of stochastic optimal control wedefine the value function
119867(119905 119903 119909) = sup120579119878120579119871
E (119906 (119883 (119879) | 119903 (119905) = 119903 119883 (119905) = 119909))
0 lt 119905 lt 119879
(13)
The value function can be considered as a kind of utilityfunction The marginal utility of the value function is aconstant while the marginal utility of the original utilityfunction 119906(sdot) decreases to zero as 119909 rarr infin (see Kramkovand Schachermayer [25])The value function also inherits theconvexity of the utility function (see Jonsson and Sircar [26])Moreover it is strictly convex for 119905 lt 119879 even if 119906(sdot) is not
The maximum principle leads to the HJB equation (seeDuffie [27])
119867119905+ 119886 (119887 minus 119903)119867
119903
+ [119909119903 + (12058211205901+ 120582212059021205902
119903) 120579119878+ 1205822120590119871120590119903120579119871+ 119888]119867
119909
+1
2[1205902
11205792
119878+ (120590119871120579119871+ 1205902120590119903120579119878)2
]119867119909119909
+1205902
119903
2119867119903119903
minus (120590119871120590119903120579119871+ 12059021205902
119903120579119878)119867119903119909
= 0
(14)
where we have suppressed the time variable 119905 In the HJBequation above119867
119905119867119903119867119909119867119903119903119867119909119909 and119867
119903119909denote partial
derivatives of first and second orders with respect to timeinterest rate and asset portfolio
The first-order maximizing conditions for the optimalstrategies 120579
119878and 120579119871are
120579119878= minus
1205821
1205901
119867119909
119867119909119909
120579119871=
120590119903(12058211205902minus 12058221205901)119867119909+ 1205901120590119903119867119903119909
1205901120590119871119867119909119909
(15)
If we put (15) into (14) we obtain a PDE for the value function119867
119867119905+ 119886 (119887 minus 119903)119867
119903+
1205902
119903
2119867119903119903+ (119909119903 + 119888)119867
119909
minus1205822
1
2
1198672
119909
119867119909119909
minus(1205822120590119903119867119909minus 120590119903119867119903119909)2
2119867119909119909
= 0
(16)
The problem is to solve (16) for the value function 119867 andreplace it in (15) in order to obtain the optimal investmentstrategy The above equation is a nonlinear second-orderPDE which is very difficult to solve The solution to thisparticular problem was derived in a pension fund setting byGao [17] who did so by employing the Legendre transformand dual theory We present the optimal solution from Gao[17] in Remark 2 below
At this point we shall specify a particular candidate for thefunction 120590
119871appearing in (7) We assume 120590
119871to take the form
120590119871(119879 minus 119905 119903 (119905)) = ℎ (119879 minus 119905) 120590
119903(17)
with
ℎ (119905) =2 (119890119898119905
minus 1)
119898 minus (119887 minus 11989611205822) + 119890119898119905 (119898 + 119887 minus 119896
11205822)
119898 = radic(119887 minus 11989611205822)2
+ 21198961
(18)
Remark 2 The solution of Problem 1 is obtained by solving(16) in terms of 119867 and substituting the result into (15) Theoptimal solution is as follows
The optimal amount of bank capital invested in the loanis
120579119871=
120590119903(12058221205901minus 12058211205902) 119909
1205901120590119871
minus 120590119903119888 (119905) [
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(19)
6 Journal of Applied Mathematics
or if we denote the optimal proportion of capital invested inthe loan by 120578
119871 then we can write
120578119871=
120590119903(12058221205901minus 12058211205902)
1205901120590119871
minus120590119903119888 (119905)
119909
[
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(20)
Furthermore the optimal amount of capital invested in thesecurity is given by
120579119878=
1205821
1205901
(119909 + 119888 (119905) 119886119879minus119905|119903119905
) (21)
or if 120578119878denotes the optimal proportion of capital invested in
the security then
120578119878=
1205821
1205901
+
119888 (119905) 119886119879minus119905|119903119905
119909 (22)
According to the above models we may write the optimalamount of capital invested in the treasury as
120579119903= 119909 minus 120579
119871minus 120579119878
(23)
and hence
120578119903= 1 minus 120578
119871minus 120578119878 (24)
where 120578119903is the optimal proportion of capital invested in the
treasuryThe expression 119886119879minus119905
denotes a continuous annuity ofduration 119879 minus 119905
5 The Total Capital Ratio
We now proceed to derive the dynamics of the Total CapitalRatio (CAR) In order to do so we first derive the dynamics ofthe TRWAs of the bank In addition we simulate the optimalproportions of bank capital invested in the assets and theCARat the end of this section The dynamics of the TRWAs andCAR are respectively derived in the remark and propositionbelowWe still assume that the bank capital evolves accordingto (10)
Remark 3 Suppose that at time 119905 the TRWAs can bedescribed by the SDE
119889119886rw (119905) = 0 times 120579119903(119905)
1198891198780(119905)
1198780(119905)
+ 02 times 120579119878(119905)
119889119878 (119905)
119878 (119905)
+ 05 times 120579119871(119905)
119889119871 (119905)
119871 (119905)+ 119888 (119905) 119889119905
(25)
where 0 02 and 05 are the risk-weights associated withrespectively the treasury security and loan under the Basel
III Accord By simplifying (25) the expression for 119889119886rw(119905)reduces to the following
119889119886rw (119905)
= [02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)] 119889119905
+ 02120579119878(119905) 1205901119889119882119878(119905)
+ [02120579119878(119905) 1205902120590119903
+05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))] 119889119882
119903(119905)
(26)
In the proof of the proposition below wemake use of ItorsquosLemma and Itorsquos Product Rule for which we refer to the book[28] by Oslashksendal
Proposition 4 With the dynamics of the total bank capital119862(119905) given by the ODE in (10) and with the dynamics of theTRWAs 119886
119903119908 given by (26) we can write the dynamics of the
Total Capital Ratio or CAR at time 119905 as
119889Λ (119905) = Ψ1119889119905 +
119862 (119905)
1198862119903119908
(119905)[(minusΨ
2+
Ψ3
119886119903119908
(119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(27)
where
Ψ1=
119888 (119905)
119886119903119908
(119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(28)
Proof Wemainly use Itorsquos general formula to derive (27) LetΦ(119886rw(119905)) = 1(119886rw(119905)) Then by Itorsquos Lemma
119889Φ (119886rw (119905))
= Φ (119886rw (119905)) 119889119905 + Φ1015840(119886rw (119905)) 119889119886rw (119905)
+1
2Φ10158401015840(119886rw (119905)) [119889119886rw (119905)]
2
= 0119889119905 minus119889119886rw (119905)
1198862rw (119905)+
[119889119886rw (119905)]2
1198863rw (119905)
Journal of Applied Mathematics 7
= minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
+ 119888 (119905)] +1
1198863rw (119905)[(02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905)))2] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))) 119889119882
119903(119905)]
(29)
Let Λ(119905) denote the CAR at time 119905 for 119905 isin [0 119879] Then bydefinition (see (1)) we can write Λ(119905) as
Λ (119905) =119862 (119905)
119886rw (119905)= 119862 (119905)Φ (119886rw (119905)) (30)
We apply Itorsquos Product Rule to Λ(119905) = 119862(119905)Φ(119886rw(119905)) to findan expression for 119889Λ(119905)
119889Λ (119905)
= 119889119862 (119905)Φ (119886rw (119905)) + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905 + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905
+ 119862 (119905) minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905))
times 1205822120590119903)+119888 (119905)]+
1
1198863rw (119905)
times[(02120579119878(119905) 1205901)2
+(02120579119878(119905) 1205902120590119903+ 05120579
119871
times(119905)120590119871(119879minus 119905 119903 (119905)))
2
] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))) 119889119882119903(119905)]
= Ψ1119889119905 + 119862 (119905) [(minus
1
1198862rw (119905)Ψ2+
1
1198863rw (119905)Ψ3)119889119905 minus
1
1198862rw (119905)
times (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
= Ψ1119889119905 +
119862 (119905)
1198862rw (119905)[(minusΨ
2+
Ψ3
119886rw (119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(31)
where we have defined
Ψ1=
119888 (119905)
119886rw (119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(32)
This concludes the proof
We provide a numerical simulation in order to charac-terize the behaviour of the CAR Λ We assume that theinterest rate follows the CIR dynamics (119896
2= 0) and that the
financial market consists of a treasury a security and a loanFurthermore we consider an investment horizon of 119879 = 10
years and that capital is raised at the fixed rate of 119888 = 013The rest of the parameters of the simulation are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(33)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119886rw (0) = 14 Λ (0) = 008
(34)
In Figure 1 we characterize the optimal investmentstrategy that solves Problem 1 by simulating the optimalproportions of capital invested in the assets The optimalinvestment strategy depicted in Figure 1 leads to the CARin Figure 4 By diversifying its asset portfolio according tothe optimal strategy illustrated by Figure 1 we note that thebank maintains its CAR in such a manner that it is abovethe minimum required level of 8 Since the bank complieswith the minimum required ratio of total capital to TRWAs
8 Journal of Applied Mathematics
0 2 4 6 8 10
TreasurySecurity
Loan
minus04
minus02
0
02
04
06
08
1
Time (years)
Figure 1 A simulation of the optimal proportions 120578119903(119905) 120578119878(119905) and
120578119871(119905) of bank capital invested in the assets given a constant stream
of capital inflow
as imposed by the Basel III Accord it is guaranteed theability to absorb reasonable levels of losses before becominginsolvent over the 10-year period We also simulate theoptimal asset portfolio and TRWAs in Figures 2 and 3respectively An interesting observation is that the TRWAsattain itsmaximum at the end of the investment period underconsideration This is due to the optimal amounts of bankcapital 120579
119878and 120579
119871 invested respectively in the security and
loan being embedded in the formula for the TRWAs in (25)The volatile natures of the quantities simulated above areconsistent with the stochastic variables used throughout thepaper
6 The Asset Portfolio for a Constant CAR
We now set out to modify the asset portfolio of Problem 1 insuch a way as to maintain the Total Capital Ratio or CAR ata constant rate of 8 To this end we need to have the bankcapital 119862(119905) to be stochastic We assume that the stochasticterm is sufficiently small in order to use the solution ofProblem 1 as a reasonable approximation The actual formof 119862(119905) is deduced from the identity 119862(119905) = 008119886rwThe formula for the asset portfolio is given in Remark 5below
Remark 5 At time 119905 the dynamics of the asset portfolio119884(119905) of the bank investing its capital according to theoptimal investment strategy from Problem 1 and in additionmaintaining its CAR at 8 can be written as
119889119884 (119905) = 1205941119889119905 + 120594
2119889119882119878(119905) + 120594
3119889119882119903(119905) (35)
0 2 4 6 8 101
2
3
4
5
6
7
8
Optimal asset portfolio
Time (years)
Figure 2 A simulation of 119883(119905) the optimal asset portfolio given aconstant stream of capital inflow
0 2 4 6 8 10
TRWAs
Time (years)
14
16
18
2
22
24
26
28
3
32
Figure 3 A simulation of the total risk-weighted assets 119886rw(119905) givena constant stream of capital inflow
where
1205941= (119883 (119905) minus 120579
119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
1205942=
117
115120579119878(119905) 1205901
1205943=
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(36)
Journal of Applied Mathematics 9
0 2 4 6 8 10008
009
01
011
012
013
014
015
016
Total Capital Ratio
Time (years)
Figure 4 A simulation of Λ(119905) the Total Capital Ratio given aconstant stream of capital inflow
The argument for this remark goes as follows From thecondition Λ(119905) = 119862(119905)119886rw(119905) = 008 we have 119889119862(119905)008 =
119889119886rw(119905) By substituting 119889119862(119905)008 for the left hand side of(25) and replacing the 119888(119905)119889119905 of (25) by 119889119862(119905) we obtain
(1
008minus 1) 119889119862 (119905) = 02120579
119878(119905)
119889119878 (119905)
119878 (119905)+ 05120579
119871(119905)
119889119871 (119905)
119871 (119905)
(37)
Solving 119889119862(119905) in terms of 120579119878(119905)(119889119878(119905)119878(119905)) and 120579
119871(119905)(119889119871(119905)
119871(119905)) yields
119889119862 (119905) =2
115120579119878(119905)
119889119878 (119905)
119878 (119905)+
1
23120579119871(119905)
119889119871 (119905)
119871 (119905) (38)
Substituting the above expression as the 119889119862(119905) of (8) gives
119889119884 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+117
115120579119878(119905)
119889119878 (119905)
119878 (119905)
+24
23120579119871(119905)
119889119871 (119905)
119871 (119905)
(39)
By simplifying this expression in terms of the models of1198891198780(119905)1198780(119905) 119889119878(119905)119878(119905) and 119889119871(119905)119871(119905) and by recalling that
120579119903(119905) = 119883(119905) minus 120579
119871(119905) minus 120579
119878(119905) we can eventually write 119889119884(119905) as
119889119884 (119905) = [(119883 (119905) minus 120579119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)] 119889119905
+117
115120579119878(119905) 1205901119889119882119878+ [
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))] 119889119882119903
(40)
By writing the coefficients of 119889119905 119889119882119878(119905) and 119889119882
119903(119905) of the
above expression respectively as 1205941 1205942 and 120594
3 the asserted
expression for 119889119884(119905) emergesWe now characterize the behaviour of the modified asset
portfolio given by (35)We consider an interest rate followingthe CIR dynamics (119896
2= 0) and an investment horizon of 119879 =
10 years A capital contribution rate of 119888 = 0 is consideredwhile the rest of the simulation parameters are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(41)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119884 (0) = 15
(42)
For a bank that diversifies its asset portfolio accordingto the solution of Problem 1 and maintains its CAR at aconstant value of 8 we simulate the modified version ofits asset portfolio in Figure 5 In this simulation we controlthe bank capital 119862(119905) as in Figure 6 during the 10-yearinvestment period We note that the growth of the modifiedasset portfolio given by (35) is much slower compared tothe original asset portfolio of Problem 1 For the specificsimulation shown the modified asset portfolio is maximal attime 119879 = 10 years
7 Conclusion
In this paper we consider asset portfolio and capital adequacymanagement in banking In particular our objective isto maximize an expected utility of a Basel III compliantcommercial bankrsquos asset portfolio at a future date119879 gt 0 Fromthe optimal investment strategy that solves our investmentproblem we derive a dynamic formula for the Total CapitalRatio or Capital Adequacy Ratio (CAR) In addition wederive a modified version of the bankrsquos asset portfolio thatwould be obtained should the bank simultaneously followthe optimal investment strategy and maintain its CAR at aconstant level of 8 This modified portfolio is extremelyimportant since it takes maximum advantage by keepingthe Total Capital Ratio exactly at the threshold This meansthat all the funding available for investment is indeed beinginvested
10 Journal of Applied Mathematics
0 2 4 6 8 10
Modified asset portfolio
Time (years)
1
15
2
25
3
35
4
45
5
55
6
Figure 5 A simulation of119884(119905) themodified asset portfolio requiredto maintain the Total Capital Ratio at 8
0 2 4 6 8 100995
1
1005
101
1015
102
1025
103
1035
Time (years)
Controlled bank capital
Figure 6 A simulation of the bank capital 119862(119905) required tomaintain the Total Capital Ratio at 8
We present a simulation study in which we illustratethe optimal investment strategy and characterize the cor-responding CAR and modified asset portfolio The mainresults of our study are as follows The optimal investmentstrategy for the bank is to diversify its asset portfolio awayfrom the risky assets that is the security and loan andtowards the riskless treasury This is in accordance with thebanking literature on optimal asset management in whichit is reported that banks diversify their asset portfolios inan attempt to lower risk (see Dangl and Lehar [29] andDecamps et al [30] for instance) In doing so the bankmaintains its CAR above the Basel III prescribed minimum
level of 8 Of course in line with Basel III standards thebank is able to absorb reasonable levels of unexpected lossesbefore becoming insolvent Furthermore in order to derivethe asset portfolio at constant (minimum) CAR value thelevel of bank capital needs to be controlled This is donemodifying the deterministic capital model to be stochasticWe find that the asset portfolio at constant (minimum) CARvalue grows considerably slower than the asset portfolioof the original investment problem but that this modifiedportfolio is still maximal at the end of the investment periodunder consideration An interesting observation is that ourbank raises capital while at the same time lowering risk Apossible explanation for this is that the bank holds low capitalConservation Buffers which it is trying to rebuild (see Heidet al [31]) In contrast banks with large buffersmaintain theircapital buffer by increasing risk when capital increases [31]
In future research it would be interesting to go beyond thesimulated data approach and model the CAR and bank assetportfolio at constant (minimum) CAR value using real datasourced from for example the US Federal Deposit InsuranceCorporation (FDIC)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors acknowledge financial support from theNationalResearch Foundation of South Africa
References
[1] Basel Committee on Banking Supervision International Con-vergence of Capital Measurements and Capital Standards ARevised Framework Bank for International Settlements 2004
[2] A V Thakor ldquoCapital requirements monetary policy andaggregate bank lending theory and empirical evidencerdquo Journalof Finance vol 51 no 1 pp 279ndash324 1996
[3] E-L vonThadden ldquoBank capital adequacy regulation under thenew Basel accordrdquo Journal of Financial Intermediation vol 13no 2 pp 90ndash95 2004
[4] C H Fouche J Mukuddem-Petersen and M A PetersenldquoContinuous-time stochastic modelling of capital adequacyratios for banksrdquo Applied Stochastic Models in Business andIndustry vol 22 no 1 pp 41ndash71 2006
[5] J Mukuddem-Petersen and M A Petersen ldquoOptimizing assetand capital adequacy management in bankingrdquo Journal ofOptimization Theory and Applications vol 137 no 1 pp 205ndash230 2008
[6] M P Mulaudzi M A Petersen and I M Schoeman ldquoOptimalallocation between bank loans and treasuries with regretrdquoOptimization Letters vol 2 no 4 pp 555ndash566 2008
[7] M Jayadev ldquoBasel III implementation issues and challenges forIndian banksrdquo IIMB Management Review vol 25 pp 115ndash1302013
[8] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
asset portfolio consists of a treasury security a marketablesecurity and a loan We take the stochastic optimal controlapproach and arrive at a nonlinear second-order PDE for thevalue function for which it is difficult to find a solution Ourchoice of asset portfolio is based on the portfolio of Gao [17]In a pension fund setting Gao [17] solved the optimizationproblem in question using the Legendre transform and dualtheory With our optimization problem similar to that in[17] we evoke the results of Gao Based on the investmentstrategy that solves our optimization problem we derive amodel for the Total Capital Ratio or CAR of the bank in termsof the optimal amounts of bank capital invested in the assetsFurthermore we assume that the bank follows the optimalinvestment strategy while simultaneously maintaining itsCAR at the fixed level of 8 We obtain a formula forthe bankrsquos asset portfolio into which is incorporated therestriction on the CAR By means of numerical simulationswe characterize the optimal investment strategy by presentinggraphically the optimal proportions of capital invested inthe bankrsquos assets In addition we simulate and graphicallyillustrate the behaviours of the CAR and the modified assetportfolio subject to the optimal investment strategy To ourknowledge a study as described above has not been done fora commercial Basel III compliant bank
The layout of the rest of our paper is as follows InSection 2 we present some general banking theory Morespecifically we describe the bankrsquos assets liabilities andcapital all of which comprise its stylized balance sheet Weintroduce the financial market in Section 3 Here we specifymodels for the stochastic interest rate in the market andthe assets and asset portfolio of the bank In Section 4 weformulate the optimization problem and present the optimalsolution In this section we also present the bank capitalmodel relevant to the optimization problem The formulafor the CAR is derived in Section 5 We simulate graphicallythe optimal investment proportions of capital invested in theassets as well as the dynamics of theCAR Section 6 is devotedto the derivation of the asset portfolio pertaining to a constantCAR In this section we also provide a simulation of themodified asset portfolio obtained via the optimal investmentstrategy and restricted CAR We conclude the paper withSection 7
2 The Commercial Banking Model
We consider a complete and frictionless financial marketwhich is continuously open over a fixed time interval [0 119879]Throughout we assume that we are working with a proba-bility space (ΩFP) where P is the real world probabilityWe assume that the Brownian motions that appear in thedynamics of the bank items are defined on the probabilityspace (ΩFP) and that the filtration (F(119905))
119905ge0satisfies the
usual conditionsTo understand the operation and management of com-
mercial banks we study its stylized balance sheet whichrecords the assets (uses of funds) and liabilities (sources offunds) of the bank
Bank capital fulfills the role of balancing the assets andliabilities of the bank A useful way for our analysis ofrepresenting the balance sheet of the bank is as follows
119877 (119905) + 119878 (119905) + 119871 (119905) = 119863 (119905) + 119861 (119905) + 119862 (119905) (2)
where 119877 119878 119871 119863 119861 and 119862 represent the values of treasuriessecurities loans deposits borrowings and capital respec-tively Each of the variables above is regarded as a functionΩ timesR
+rarr R+
In order for a commercial bank to make a profit itis important that the bank manages the asset side of itsbalance sheet properly The latter is determined by twofactors namely the amount of capital available to invest ithas and the attitude it has toward risk and return The bankmust therefore allocate its capital optimally among its assetsBelow we explain each of the items on the balance sheet of acommercial bank
The term reserves refers to the vault cash of the bank plusthe compulsory amount of its money deposited at the centralbank Reserves are used to meet the day-to-day currencywithdrawals by its customers Securities consist of treasurysecurities (treasuries) and marketable securities (securities)Treasuries are bonds issued by national treasuries in mostcountries as a means of borrowing money to meet govern-ment expenditures not covered by tax revenues Securities onthe other hand are stocks and bonds that can be convertedto cash quickly and easily Loans granted by the bank includebusiness loans mortgage loans (land loans) and consumerloans Consumer loans include credit extended by the bankfor credit card purchases Mortgages are long term loans usedto buy a house or land where the house or land acts ascollateral Business loans are taken out by firms that borrowfunds to finance their inventories which act as collateral forthe loan A loan which has collateral (a secured loan) has alower interest rate associatedwith it compared to a loanwhichhas no collateral (unsecured loans)
Bank capital is raised by selling new equity retainingearnings and by issuing debt or building up loan-lossreserves The bankrsquos risk management department is usuallyresponsible for calculating its capital requirements Calcu-lated risk capital is then approved by the bankrsquos top executivemanagement Furthermore the structure of bank capitalis proposed by the Finance Department and subsequentlyapproved by the bankrsquos top executive management Thedynamics of bank capital is stochastic in nature becauseit depends in part on the uncertainty related to debt andshareholder contributions In theory the bank can decide onthe rate at which debt and equity are raised
Under the Basel III Accord (see [21]) the bankrsquos capital 119862also referred to as total bank capital has the form
119862 (119905) = 119862T1 (119905) + 119862T2 (119905) (3)
where 119862T1(119905) and 119862T2 are Tier 1 and Tier 2 capital respec-tively
Tier 1 capital consists of the sum of Common Equity Tier1 capital and Additional Tier 1 capital Common Equity Tier1 capital is defined as the sum of common shares issued bythe bank that meet the criteria for classification as common
4 Journal of Applied Mathematics
shares for regulatory purposes stock surplus resulting fromthe issue of instruments included in Common Equity Tier 1capital retained earnings other accumulated comprehensiveincome and other disclosed reserves common shares issuedby consolidated subsidiaries of the bank and held by thirdparties thatmeet the criteria for inclusion in CommonEquityTier 1 capital and regulatory adjustments applied in thecalculation of Common Equity Tier 1 capital Additional Tier1 capital is the sum of instruments issued by the bank thatmeet the criteria for inclusion in Additional Tier 1 capital(and are not included inCommonEquity Tier 1 capital) stocksurplus resulting from the issue of instruments included inAdditional Tier 1 capital instruments issued by consolidatedsubsidiaries of the bank and held by third parties that meetthe criteria for inclusion in Additional Tier 1 capital and arenot included inCommonEquity Tier 1 capital and regulatoryadjustments applied in the calculation of Additional Tier 1capital
Tier 2 capital is defined as the sum of the followingelements instruments issued by the bank that meet thecriteria for inclusion in Tier 2 capital (and are not includedin Tier 1 capital) stock surplus resulting from the issue ofinstruments included in Tier 2 capital instruments issued byconsolidated subsidiaries of the bank andheld by third partiesthat meet the criteria for inclusion in Tier 2 capital and arenot included in Tier 1 capital certain loan-loss provisionsand regulatory adjustments applied in the calculation of Tier2 capital
Deposits refer to the money that the bankrsquos customersplace in the banking institution for safekeeping Depositsare made to deposit accounts at a banking institution suchas savings accounts checking accounts and money marketaccounts The account holder has the right to withdraw anydeposited funds as set forth in the terms and conditions ofthe account Deposits are considered to be themain liabilitiesof the bank
The term borrowings refers to the funds that commercialbanks borrow from other banks and the central bank
3 The Financial Market Setting
We assume that it is possible for the bank to continuouslyraise small levels of capital at a rate 119889119862(119905) The bank isassumed to invest its capital in a market consisting of atleast two types of assets (treasury and security) which can bebought and sold without incurring any transaction costs orrestriction on short sales It is assumed that the bank investsin a third asset namely a loan
The first asset in the financial market is a riskless treasuryWe denote its price at time 119905 by 119878
0(119905) and assume that its
dynamics evolve according to the equation
1198891198780(119905)
1198780(119905)
= 119903 (119905) 119889119905 1198780(0) = 1 (4)
The dynamics of the short rate process 119903(119905) is given bythe stochastic differential equation (SDE)
119889119903 (119905) = (119886 minus 119887119903 (119905)) 119889119905 minus 120590119903119889119882119903(119905) (5)
for 119905 ge 0 and where 120590119903= radic1198961119903(119905) + 119896
2 The coefficients 119886 119887
1198961 and 119896
2 as well as the initial value 119903(0) are all positive real
constants The above dynamics recover as a special case theVasicek [22] (resp Cox et al [23]) dynamics when 119896
1(resp
1198962) is equal to zero The term structure of the interest rates is
affine under the aforementioned dynamicsThe second asset in the market is a risky security whose
price is denoted by 119878(119905) 119905 ge 0 Its dynamics are given by theequation
119889119878 (119905)
119878 (119905)= 119903 (119905) 119889119905 + 120590
1(119889119882119904(119905) + 120582
1119889119905)
+ 1205902120590119903(119889119882119903(119905) + 120582
2120590119903119889119905)
(6)
with 119878(0) = 1 and 1205821 1205822(resp 120590
1 1205902) being constants (resp
positive constants) as in Deelstra et al [24]The third asset is a loan to be amortized over a period
[0 119879]whose price at time 119905 ge 0 is denoted by 119871(119905) We assumethat its dynamics can be described by the SDE
119889119871 (119905)
119871 (119905)= 119903 (119905) 119889119905 + 120590
119871(119879 minus 119905 119903 (119905)) (119889119882
119903(119905) + 120582
2120590119903119889119905) (7)
We now model the asset portfolio of the bank Let 119883(119905)
denote the value of the asset portfolio at time 119905 isin [0 119879] Thedynamics of the asset portfolio are described by
119889119883 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+ 120579119878(119905)
119889119878 (119905)
119878 (119905)
+ 120579119871(119905)
119889119871 (119905)
119871 (119905)+ 119889119862 (119905)
(8)
where 120579119878(119905) 120579119871(119905) and 120579
119903(119905) denote the amounts of capital
invested in the two risky assets (security and loan) and inthe riskless asset (treasury) respectively Making use of themodels for 119878
0(119905) 119878(119905) and 119871(119905) we can rewrite (8) as
119889119883 (119905) = 119883 (119905) 119903 (119905) + 120579119878(119905) [12058211205901+ 120582212059021205902
119903]
+120579119871(119905) 1205822120590119871(119879 minus 119905 119903 (119905)) 120590
119903 119889119905
+ 120579119878(119905) 1205901119889119882119878(119905)+(120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
+120579119878(119905) 1205902120590119903) 119889119882119903(119905)+119889119862 (119905)
(9)
4 The Portfolio Problem andOptimal Solution
In this section we formulate the optimal control problemassociated with an influx of capital at a rate
119889119862 (119905) = 119888 (119905) 119889119905 119862 (0) gt 0 (10)
We derive theHJB equation for the value function and wepresent the optimal solution from the methodology of Gao[17]
We wish to choose a portfolio strategy in order tomaximize the expected utility of the bankrsquos asset portfolio ata future date 119879 gt 0 Mathematically the stochastic optimalcontrol problem can be stated as follows
Journal of Applied Mathematics 5
Problem 1 Our objective is to maximize the expected utilityof the bankrsquos asset portfolio at a future date 119879 gt 0 Thus wemust
maximize 119869 (120579119878 120579119871) = E (119906 (119883 (119879)))
subject to 119889119903 (119905) = (119886 minus 119887119903 (119905)) 119889119905 minus 120590119903119889119882119903(119905)
119889119883 (119905) = [119883 (119905) 119903 (119905) + 120579119878(119905)
times (12058211205901+ 120582212059021205902
119903)
+ 120579119871(119905) 1205822120590119871(119879 minus 119905 119903 (119905)) 120590
119903
+ 119888 (119905)] 119889119905 + 120579119878(119905) 1205901119889119882119878(119905)
+ (120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
+ 120579119878(119905) 1205902120590119903) 119889119882119903(119905)
119883 (0) = 1199090 119903 (0) = 119903
0
(11)
with 0 le 119905 le 119879 and where 119883(0) = 1199090and 119903(0) = 119903
0denote
the initial conditions of the optimal control problem
In this paper we describe the bankrsquos objective with alogarithmic utility function that is
119906 (119909) = ln119909 with 119909 gt 0 (12)
Wenote that the utility function 119906(sdot) is strictly concave up andsatisfies the Inada conditions 1199061015840(+infin) = 0 and 119906
1015840(0) = +infin
By using the classical tools of stochastic optimal control wedefine the value function
119867(119905 119903 119909) = sup120579119878120579119871
E (119906 (119883 (119879) | 119903 (119905) = 119903 119883 (119905) = 119909))
0 lt 119905 lt 119879
(13)
The value function can be considered as a kind of utilityfunction The marginal utility of the value function is aconstant while the marginal utility of the original utilityfunction 119906(sdot) decreases to zero as 119909 rarr infin (see Kramkovand Schachermayer [25])The value function also inherits theconvexity of the utility function (see Jonsson and Sircar [26])Moreover it is strictly convex for 119905 lt 119879 even if 119906(sdot) is not
The maximum principle leads to the HJB equation (seeDuffie [27])
119867119905+ 119886 (119887 minus 119903)119867
119903
+ [119909119903 + (12058211205901+ 120582212059021205902
119903) 120579119878+ 1205822120590119871120590119903120579119871+ 119888]119867
119909
+1
2[1205902
11205792
119878+ (120590119871120579119871+ 1205902120590119903120579119878)2
]119867119909119909
+1205902
119903
2119867119903119903
minus (120590119871120590119903120579119871+ 12059021205902
119903120579119878)119867119903119909
= 0
(14)
where we have suppressed the time variable 119905 In the HJBequation above119867
119905119867119903119867119909119867119903119903119867119909119909 and119867
119903119909denote partial
derivatives of first and second orders with respect to timeinterest rate and asset portfolio
The first-order maximizing conditions for the optimalstrategies 120579
119878and 120579119871are
120579119878= minus
1205821
1205901
119867119909
119867119909119909
120579119871=
120590119903(12058211205902minus 12058221205901)119867119909+ 1205901120590119903119867119903119909
1205901120590119871119867119909119909
(15)
If we put (15) into (14) we obtain a PDE for the value function119867
119867119905+ 119886 (119887 minus 119903)119867
119903+
1205902
119903
2119867119903119903+ (119909119903 + 119888)119867
119909
minus1205822
1
2
1198672
119909
119867119909119909
minus(1205822120590119903119867119909minus 120590119903119867119903119909)2
2119867119909119909
= 0
(16)
The problem is to solve (16) for the value function 119867 andreplace it in (15) in order to obtain the optimal investmentstrategy The above equation is a nonlinear second-orderPDE which is very difficult to solve The solution to thisparticular problem was derived in a pension fund setting byGao [17] who did so by employing the Legendre transformand dual theory We present the optimal solution from Gao[17] in Remark 2 below
At this point we shall specify a particular candidate for thefunction 120590
119871appearing in (7) We assume 120590
119871to take the form
120590119871(119879 minus 119905 119903 (119905)) = ℎ (119879 minus 119905) 120590
119903(17)
with
ℎ (119905) =2 (119890119898119905
minus 1)
119898 minus (119887 minus 11989611205822) + 119890119898119905 (119898 + 119887 minus 119896
11205822)
119898 = radic(119887 minus 11989611205822)2
+ 21198961
(18)
Remark 2 The solution of Problem 1 is obtained by solving(16) in terms of 119867 and substituting the result into (15) Theoptimal solution is as follows
The optimal amount of bank capital invested in the loanis
120579119871=
120590119903(12058221205901minus 12058211205902) 119909
1205901120590119871
minus 120590119903119888 (119905) [
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(19)
6 Journal of Applied Mathematics
or if we denote the optimal proportion of capital invested inthe loan by 120578
119871 then we can write
120578119871=
120590119903(12058221205901minus 12058211205902)
1205901120590119871
minus120590119903119888 (119905)
119909
[
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(20)
Furthermore the optimal amount of capital invested in thesecurity is given by
120579119878=
1205821
1205901
(119909 + 119888 (119905) 119886119879minus119905|119903119905
) (21)
or if 120578119878denotes the optimal proportion of capital invested in
the security then
120578119878=
1205821
1205901
+
119888 (119905) 119886119879minus119905|119903119905
119909 (22)
According to the above models we may write the optimalamount of capital invested in the treasury as
120579119903= 119909 minus 120579
119871minus 120579119878
(23)
and hence
120578119903= 1 minus 120578
119871minus 120578119878 (24)
where 120578119903is the optimal proportion of capital invested in the
treasuryThe expression 119886119879minus119905
denotes a continuous annuity ofduration 119879 minus 119905
5 The Total Capital Ratio
We now proceed to derive the dynamics of the Total CapitalRatio (CAR) In order to do so we first derive the dynamics ofthe TRWAs of the bank In addition we simulate the optimalproportions of bank capital invested in the assets and theCARat the end of this section The dynamics of the TRWAs andCAR are respectively derived in the remark and propositionbelowWe still assume that the bank capital evolves accordingto (10)
Remark 3 Suppose that at time 119905 the TRWAs can bedescribed by the SDE
119889119886rw (119905) = 0 times 120579119903(119905)
1198891198780(119905)
1198780(119905)
+ 02 times 120579119878(119905)
119889119878 (119905)
119878 (119905)
+ 05 times 120579119871(119905)
119889119871 (119905)
119871 (119905)+ 119888 (119905) 119889119905
(25)
where 0 02 and 05 are the risk-weights associated withrespectively the treasury security and loan under the Basel
III Accord By simplifying (25) the expression for 119889119886rw(119905)reduces to the following
119889119886rw (119905)
= [02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)] 119889119905
+ 02120579119878(119905) 1205901119889119882119878(119905)
+ [02120579119878(119905) 1205902120590119903
+05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))] 119889119882
119903(119905)
(26)
In the proof of the proposition below wemake use of ItorsquosLemma and Itorsquos Product Rule for which we refer to the book[28] by Oslashksendal
Proposition 4 With the dynamics of the total bank capital119862(119905) given by the ODE in (10) and with the dynamics of theTRWAs 119886
119903119908 given by (26) we can write the dynamics of the
Total Capital Ratio or CAR at time 119905 as
119889Λ (119905) = Ψ1119889119905 +
119862 (119905)
1198862119903119908
(119905)[(minusΨ
2+
Ψ3
119886119903119908
(119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(27)
where
Ψ1=
119888 (119905)
119886119903119908
(119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(28)
Proof Wemainly use Itorsquos general formula to derive (27) LetΦ(119886rw(119905)) = 1(119886rw(119905)) Then by Itorsquos Lemma
119889Φ (119886rw (119905))
= Φ (119886rw (119905)) 119889119905 + Φ1015840(119886rw (119905)) 119889119886rw (119905)
+1
2Φ10158401015840(119886rw (119905)) [119889119886rw (119905)]
2
= 0119889119905 minus119889119886rw (119905)
1198862rw (119905)+
[119889119886rw (119905)]2
1198863rw (119905)
Journal of Applied Mathematics 7
= minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
+ 119888 (119905)] +1
1198863rw (119905)[(02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905)))2] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))) 119889119882
119903(119905)]
(29)
Let Λ(119905) denote the CAR at time 119905 for 119905 isin [0 119879] Then bydefinition (see (1)) we can write Λ(119905) as
Λ (119905) =119862 (119905)
119886rw (119905)= 119862 (119905)Φ (119886rw (119905)) (30)
We apply Itorsquos Product Rule to Λ(119905) = 119862(119905)Φ(119886rw(119905)) to findan expression for 119889Λ(119905)
119889Λ (119905)
= 119889119862 (119905)Φ (119886rw (119905)) + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905 + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905
+ 119862 (119905) minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905))
times 1205822120590119903)+119888 (119905)]+
1
1198863rw (119905)
times[(02120579119878(119905) 1205901)2
+(02120579119878(119905) 1205902120590119903+ 05120579
119871
times(119905)120590119871(119879minus 119905 119903 (119905)))
2
] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))) 119889119882119903(119905)]
= Ψ1119889119905 + 119862 (119905) [(minus
1
1198862rw (119905)Ψ2+
1
1198863rw (119905)Ψ3)119889119905 minus
1
1198862rw (119905)
times (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
= Ψ1119889119905 +
119862 (119905)
1198862rw (119905)[(minusΨ
2+
Ψ3
119886rw (119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(31)
where we have defined
Ψ1=
119888 (119905)
119886rw (119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(32)
This concludes the proof
We provide a numerical simulation in order to charac-terize the behaviour of the CAR Λ We assume that theinterest rate follows the CIR dynamics (119896
2= 0) and that the
financial market consists of a treasury a security and a loanFurthermore we consider an investment horizon of 119879 = 10
years and that capital is raised at the fixed rate of 119888 = 013The rest of the parameters of the simulation are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(33)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119886rw (0) = 14 Λ (0) = 008
(34)
In Figure 1 we characterize the optimal investmentstrategy that solves Problem 1 by simulating the optimalproportions of capital invested in the assets The optimalinvestment strategy depicted in Figure 1 leads to the CARin Figure 4 By diversifying its asset portfolio according tothe optimal strategy illustrated by Figure 1 we note that thebank maintains its CAR in such a manner that it is abovethe minimum required level of 8 Since the bank complieswith the minimum required ratio of total capital to TRWAs
8 Journal of Applied Mathematics
0 2 4 6 8 10
TreasurySecurity
Loan
minus04
minus02
0
02
04
06
08
1
Time (years)
Figure 1 A simulation of the optimal proportions 120578119903(119905) 120578119878(119905) and
120578119871(119905) of bank capital invested in the assets given a constant stream
of capital inflow
as imposed by the Basel III Accord it is guaranteed theability to absorb reasonable levels of losses before becominginsolvent over the 10-year period We also simulate theoptimal asset portfolio and TRWAs in Figures 2 and 3respectively An interesting observation is that the TRWAsattain itsmaximum at the end of the investment period underconsideration This is due to the optimal amounts of bankcapital 120579
119878and 120579
119871 invested respectively in the security and
loan being embedded in the formula for the TRWAs in (25)The volatile natures of the quantities simulated above areconsistent with the stochastic variables used throughout thepaper
6 The Asset Portfolio for a Constant CAR
We now set out to modify the asset portfolio of Problem 1 insuch a way as to maintain the Total Capital Ratio or CAR ata constant rate of 8 To this end we need to have the bankcapital 119862(119905) to be stochastic We assume that the stochasticterm is sufficiently small in order to use the solution ofProblem 1 as a reasonable approximation The actual formof 119862(119905) is deduced from the identity 119862(119905) = 008119886rwThe formula for the asset portfolio is given in Remark 5below
Remark 5 At time 119905 the dynamics of the asset portfolio119884(119905) of the bank investing its capital according to theoptimal investment strategy from Problem 1 and in additionmaintaining its CAR at 8 can be written as
119889119884 (119905) = 1205941119889119905 + 120594
2119889119882119878(119905) + 120594
3119889119882119903(119905) (35)
0 2 4 6 8 101
2
3
4
5
6
7
8
Optimal asset portfolio
Time (years)
Figure 2 A simulation of 119883(119905) the optimal asset portfolio given aconstant stream of capital inflow
0 2 4 6 8 10
TRWAs
Time (years)
14
16
18
2
22
24
26
28
3
32
Figure 3 A simulation of the total risk-weighted assets 119886rw(119905) givena constant stream of capital inflow
where
1205941= (119883 (119905) minus 120579
119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
1205942=
117
115120579119878(119905) 1205901
1205943=
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(36)
Journal of Applied Mathematics 9
0 2 4 6 8 10008
009
01
011
012
013
014
015
016
Total Capital Ratio
Time (years)
Figure 4 A simulation of Λ(119905) the Total Capital Ratio given aconstant stream of capital inflow
The argument for this remark goes as follows From thecondition Λ(119905) = 119862(119905)119886rw(119905) = 008 we have 119889119862(119905)008 =
119889119886rw(119905) By substituting 119889119862(119905)008 for the left hand side of(25) and replacing the 119888(119905)119889119905 of (25) by 119889119862(119905) we obtain
(1
008minus 1) 119889119862 (119905) = 02120579
119878(119905)
119889119878 (119905)
119878 (119905)+ 05120579
119871(119905)
119889119871 (119905)
119871 (119905)
(37)
Solving 119889119862(119905) in terms of 120579119878(119905)(119889119878(119905)119878(119905)) and 120579
119871(119905)(119889119871(119905)
119871(119905)) yields
119889119862 (119905) =2
115120579119878(119905)
119889119878 (119905)
119878 (119905)+
1
23120579119871(119905)
119889119871 (119905)
119871 (119905) (38)
Substituting the above expression as the 119889119862(119905) of (8) gives
119889119884 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+117
115120579119878(119905)
119889119878 (119905)
119878 (119905)
+24
23120579119871(119905)
119889119871 (119905)
119871 (119905)
(39)
By simplifying this expression in terms of the models of1198891198780(119905)1198780(119905) 119889119878(119905)119878(119905) and 119889119871(119905)119871(119905) and by recalling that
120579119903(119905) = 119883(119905) minus 120579
119871(119905) minus 120579
119878(119905) we can eventually write 119889119884(119905) as
119889119884 (119905) = [(119883 (119905) minus 120579119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)] 119889119905
+117
115120579119878(119905) 1205901119889119882119878+ [
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))] 119889119882119903
(40)
By writing the coefficients of 119889119905 119889119882119878(119905) and 119889119882
119903(119905) of the
above expression respectively as 1205941 1205942 and 120594
3 the asserted
expression for 119889119884(119905) emergesWe now characterize the behaviour of the modified asset
portfolio given by (35)We consider an interest rate followingthe CIR dynamics (119896
2= 0) and an investment horizon of 119879 =
10 years A capital contribution rate of 119888 = 0 is consideredwhile the rest of the simulation parameters are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(41)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119884 (0) = 15
(42)
For a bank that diversifies its asset portfolio accordingto the solution of Problem 1 and maintains its CAR at aconstant value of 8 we simulate the modified version ofits asset portfolio in Figure 5 In this simulation we controlthe bank capital 119862(119905) as in Figure 6 during the 10-yearinvestment period We note that the growth of the modifiedasset portfolio given by (35) is much slower compared tothe original asset portfolio of Problem 1 For the specificsimulation shown the modified asset portfolio is maximal attime 119879 = 10 years
7 Conclusion
In this paper we consider asset portfolio and capital adequacymanagement in banking In particular our objective isto maximize an expected utility of a Basel III compliantcommercial bankrsquos asset portfolio at a future date119879 gt 0 Fromthe optimal investment strategy that solves our investmentproblem we derive a dynamic formula for the Total CapitalRatio or Capital Adequacy Ratio (CAR) In addition wederive a modified version of the bankrsquos asset portfolio thatwould be obtained should the bank simultaneously followthe optimal investment strategy and maintain its CAR at aconstant level of 8 This modified portfolio is extremelyimportant since it takes maximum advantage by keepingthe Total Capital Ratio exactly at the threshold This meansthat all the funding available for investment is indeed beinginvested
10 Journal of Applied Mathematics
0 2 4 6 8 10
Modified asset portfolio
Time (years)
1
15
2
25
3
35
4
45
5
55
6
Figure 5 A simulation of119884(119905) themodified asset portfolio requiredto maintain the Total Capital Ratio at 8
0 2 4 6 8 100995
1
1005
101
1015
102
1025
103
1035
Time (years)
Controlled bank capital
Figure 6 A simulation of the bank capital 119862(119905) required tomaintain the Total Capital Ratio at 8
We present a simulation study in which we illustratethe optimal investment strategy and characterize the cor-responding CAR and modified asset portfolio The mainresults of our study are as follows The optimal investmentstrategy for the bank is to diversify its asset portfolio awayfrom the risky assets that is the security and loan andtowards the riskless treasury This is in accordance with thebanking literature on optimal asset management in whichit is reported that banks diversify their asset portfolios inan attempt to lower risk (see Dangl and Lehar [29] andDecamps et al [30] for instance) In doing so the bankmaintains its CAR above the Basel III prescribed minimum
level of 8 Of course in line with Basel III standards thebank is able to absorb reasonable levels of unexpected lossesbefore becoming insolvent Furthermore in order to derivethe asset portfolio at constant (minimum) CAR value thelevel of bank capital needs to be controlled This is donemodifying the deterministic capital model to be stochasticWe find that the asset portfolio at constant (minimum) CARvalue grows considerably slower than the asset portfolioof the original investment problem but that this modifiedportfolio is still maximal at the end of the investment periodunder consideration An interesting observation is that ourbank raises capital while at the same time lowering risk Apossible explanation for this is that the bank holds low capitalConservation Buffers which it is trying to rebuild (see Heidet al [31]) In contrast banks with large buffersmaintain theircapital buffer by increasing risk when capital increases [31]
In future research it would be interesting to go beyond thesimulated data approach and model the CAR and bank assetportfolio at constant (minimum) CAR value using real datasourced from for example the US Federal Deposit InsuranceCorporation (FDIC)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors acknowledge financial support from theNationalResearch Foundation of South Africa
References
[1] Basel Committee on Banking Supervision International Con-vergence of Capital Measurements and Capital Standards ARevised Framework Bank for International Settlements 2004
[2] A V Thakor ldquoCapital requirements monetary policy andaggregate bank lending theory and empirical evidencerdquo Journalof Finance vol 51 no 1 pp 279ndash324 1996
[3] E-L vonThadden ldquoBank capital adequacy regulation under thenew Basel accordrdquo Journal of Financial Intermediation vol 13no 2 pp 90ndash95 2004
[4] C H Fouche J Mukuddem-Petersen and M A PetersenldquoContinuous-time stochastic modelling of capital adequacyratios for banksrdquo Applied Stochastic Models in Business andIndustry vol 22 no 1 pp 41ndash71 2006
[5] J Mukuddem-Petersen and M A Petersen ldquoOptimizing assetand capital adequacy management in bankingrdquo Journal ofOptimization Theory and Applications vol 137 no 1 pp 205ndash230 2008
[6] M P Mulaudzi M A Petersen and I M Schoeman ldquoOptimalallocation between bank loans and treasuries with regretrdquoOptimization Letters vol 2 no 4 pp 555ndash566 2008
[7] M Jayadev ldquoBasel III implementation issues and challenges forIndian banksrdquo IIMB Management Review vol 25 pp 115ndash1302013
[8] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
shares for regulatory purposes stock surplus resulting fromthe issue of instruments included in Common Equity Tier 1capital retained earnings other accumulated comprehensiveincome and other disclosed reserves common shares issuedby consolidated subsidiaries of the bank and held by thirdparties thatmeet the criteria for inclusion in CommonEquityTier 1 capital and regulatory adjustments applied in thecalculation of Common Equity Tier 1 capital Additional Tier1 capital is the sum of instruments issued by the bank thatmeet the criteria for inclusion in Additional Tier 1 capital(and are not included inCommonEquity Tier 1 capital) stocksurplus resulting from the issue of instruments included inAdditional Tier 1 capital instruments issued by consolidatedsubsidiaries of the bank and held by third parties that meetthe criteria for inclusion in Additional Tier 1 capital and arenot included inCommonEquity Tier 1 capital and regulatoryadjustments applied in the calculation of Additional Tier 1capital
Tier 2 capital is defined as the sum of the followingelements instruments issued by the bank that meet thecriteria for inclusion in Tier 2 capital (and are not includedin Tier 1 capital) stock surplus resulting from the issue ofinstruments included in Tier 2 capital instruments issued byconsolidated subsidiaries of the bank andheld by third partiesthat meet the criteria for inclusion in Tier 2 capital and arenot included in Tier 1 capital certain loan-loss provisionsand regulatory adjustments applied in the calculation of Tier2 capital
Deposits refer to the money that the bankrsquos customersplace in the banking institution for safekeeping Depositsare made to deposit accounts at a banking institution suchas savings accounts checking accounts and money marketaccounts The account holder has the right to withdraw anydeposited funds as set forth in the terms and conditions ofthe account Deposits are considered to be themain liabilitiesof the bank
The term borrowings refers to the funds that commercialbanks borrow from other banks and the central bank
3 The Financial Market Setting
We assume that it is possible for the bank to continuouslyraise small levels of capital at a rate 119889119862(119905) The bank isassumed to invest its capital in a market consisting of atleast two types of assets (treasury and security) which can bebought and sold without incurring any transaction costs orrestriction on short sales It is assumed that the bank investsin a third asset namely a loan
The first asset in the financial market is a riskless treasuryWe denote its price at time 119905 by 119878
0(119905) and assume that its
dynamics evolve according to the equation
1198891198780(119905)
1198780(119905)
= 119903 (119905) 119889119905 1198780(0) = 1 (4)
The dynamics of the short rate process 119903(119905) is given bythe stochastic differential equation (SDE)
119889119903 (119905) = (119886 minus 119887119903 (119905)) 119889119905 minus 120590119903119889119882119903(119905) (5)
for 119905 ge 0 and where 120590119903= radic1198961119903(119905) + 119896
2 The coefficients 119886 119887
1198961 and 119896
2 as well as the initial value 119903(0) are all positive real
constants The above dynamics recover as a special case theVasicek [22] (resp Cox et al [23]) dynamics when 119896
1(resp
1198962) is equal to zero The term structure of the interest rates is
affine under the aforementioned dynamicsThe second asset in the market is a risky security whose
price is denoted by 119878(119905) 119905 ge 0 Its dynamics are given by theequation
119889119878 (119905)
119878 (119905)= 119903 (119905) 119889119905 + 120590
1(119889119882119904(119905) + 120582
1119889119905)
+ 1205902120590119903(119889119882119903(119905) + 120582
2120590119903119889119905)
(6)
with 119878(0) = 1 and 1205821 1205822(resp 120590
1 1205902) being constants (resp
positive constants) as in Deelstra et al [24]The third asset is a loan to be amortized over a period
[0 119879]whose price at time 119905 ge 0 is denoted by 119871(119905) We assumethat its dynamics can be described by the SDE
119889119871 (119905)
119871 (119905)= 119903 (119905) 119889119905 + 120590
119871(119879 minus 119905 119903 (119905)) (119889119882
119903(119905) + 120582
2120590119903119889119905) (7)
We now model the asset portfolio of the bank Let 119883(119905)
denote the value of the asset portfolio at time 119905 isin [0 119879] Thedynamics of the asset portfolio are described by
119889119883 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+ 120579119878(119905)
119889119878 (119905)
119878 (119905)
+ 120579119871(119905)
119889119871 (119905)
119871 (119905)+ 119889119862 (119905)
(8)
where 120579119878(119905) 120579119871(119905) and 120579
119903(119905) denote the amounts of capital
invested in the two risky assets (security and loan) and inthe riskless asset (treasury) respectively Making use of themodels for 119878
0(119905) 119878(119905) and 119871(119905) we can rewrite (8) as
119889119883 (119905) = 119883 (119905) 119903 (119905) + 120579119878(119905) [12058211205901+ 120582212059021205902
119903]
+120579119871(119905) 1205822120590119871(119879 minus 119905 119903 (119905)) 120590
119903 119889119905
+ 120579119878(119905) 1205901119889119882119878(119905)+(120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
+120579119878(119905) 1205902120590119903) 119889119882119903(119905)+119889119862 (119905)
(9)
4 The Portfolio Problem andOptimal Solution
In this section we formulate the optimal control problemassociated with an influx of capital at a rate
119889119862 (119905) = 119888 (119905) 119889119905 119862 (0) gt 0 (10)
We derive theHJB equation for the value function and wepresent the optimal solution from the methodology of Gao[17]
We wish to choose a portfolio strategy in order tomaximize the expected utility of the bankrsquos asset portfolio ata future date 119879 gt 0 Mathematically the stochastic optimalcontrol problem can be stated as follows
Journal of Applied Mathematics 5
Problem 1 Our objective is to maximize the expected utilityof the bankrsquos asset portfolio at a future date 119879 gt 0 Thus wemust
maximize 119869 (120579119878 120579119871) = E (119906 (119883 (119879)))
subject to 119889119903 (119905) = (119886 minus 119887119903 (119905)) 119889119905 minus 120590119903119889119882119903(119905)
119889119883 (119905) = [119883 (119905) 119903 (119905) + 120579119878(119905)
times (12058211205901+ 120582212059021205902
119903)
+ 120579119871(119905) 1205822120590119871(119879 minus 119905 119903 (119905)) 120590
119903
+ 119888 (119905)] 119889119905 + 120579119878(119905) 1205901119889119882119878(119905)
+ (120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
+ 120579119878(119905) 1205902120590119903) 119889119882119903(119905)
119883 (0) = 1199090 119903 (0) = 119903
0
(11)
with 0 le 119905 le 119879 and where 119883(0) = 1199090and 119903(0) = 119903
0denote
the initial conditions of the optimal control problem
In this paper we describe the bankrsquos objective with alogarithmic utility function that is
119906 (119909) = ln119909 with 119909 gt 0 (12)
Wenote that the utility function 119906(sdot) is strictly concave up andsatisfies the Inada conditions 1199061015840(+infin) = 0 and 119906
1015840(0) = +infin
By using the classical tools of stochastic optimal control wedefine the value function
119867(119905 119903 119909) = sup120579119878120579119871
E (119906 (119883 (119879) | 119903 (119905) = 119903 119883 (119905) = 119909))
0 lt 119905 lt 119879
(13)
The value function can be considered as a kind of utilityfunction The marginal utility of the value function is aconstant while the marginal utility of the original utilityfunction 119906(sdot) decreases to zero as 119909 rarr infin (see Kramkovand Schachermayer [25])The value function also inherits theconvexity of the utility function (see Jonsson and Sircar [26])Moreover it is strictly convex for 119905 lt 119879 even if 119906(sdot) is not
The maximum principle leads to the HJB equation (seeDuffie [27])
119867119905+ 119886 (119887 minus 119903)119867
119903
+ [119909119903 + (12058211205901+ 120582212059021205902
119903) 120579119878+ 1205822120590119871120590119903120579119871+ 119888]119867
119909
+1
2[1205902
11205792
119878+ (120590119871120579119871+ 1205902120590119903120579119878)2
]119867119909119909
+1205902
119903
2119867119903119903
minus (120590119871120590119903120579119871+ 12059021205902
119903120579119878)119867119903119909
= 0
(14)
where we have suppressed the time variable 119905 In the HJBequation above119867
119905119867119903119867119909119867119903119903119867119909119909 and119867
119903119909denote partial
derivatives of first and second orders with respect to timeinterest rate and asset portfolio
The first-order maximizing conditions for the optimalstrategies 120579
119878and 120579119871are
120579119878= minus
1205821
1205901
119867119909
119867119909119909
120579119871=
120590119903(12058211205902minus 12058221205901)119867119909+ 1205901120590119903119867119903119909
1205901120590119871119867119909119909
(15)
If we put (15) into (14) we obtain a PDE for the value function119867
119867119905+ 119886 (119887 minus 119903)119867
119903+
1205902
119903
2119867119903119903+ (119909119903 + 119888)119867
119909
minus1205822
1
2
1198672
119909
119867119909119909
minus(1205822120590119903119867119909minus 120590119903119867119903119909)2
2119867119909119909
= 0
(16)
The problem is to solve (16) for the value function 119867 andreplace it in (15) in order to obtain the optimal investmentstrategy The above equation is a nonlinear second-orderPDE which is very difficult to solve The solution to thisparticular problem was derived in a pension fund setting byGao [17] who did so by employing the Legendre transformand dual theory We present the optimal solution from Gao[17] in Remark 2 below
At this point we shall specify a particular candidate for thefunction 120590
119871appearing in (7) We assume 120590
119871to take the form
120590119871(119879 minus 119905 119903 (119905)) = ℎ (119879 minus 119905) 120590
119903(17)
with
ℎ (119905) =2 (119890119898119905
minus 1)
119898 minus (119887 minus 11989611205822) + 119890119898119905 (119898 + 119887 minus 119896
11205822)
119898 = radic(119887 minus 11989611205822)2
+ 21198961
(18)
Remark 2 The solution of Problem 1 is obtained by solving(16) in terms of 119867 and substituting the result into (15) Theoptimal solution is as follows
The optimal amount of bank capital invested in the loanis
120579119871=
120590119903(12058221205901minus 12058211205902) 119909
1205901120590119871
minus 120590119903119888 (119905) [
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(19)
6 Journal of Applied Mathematics
or if we denote the optimal proportion of capital invested inthe loan by 120578
119871 then we can write
120578119871=
120590119903(12058221205901minus 12058211205902)
1205901120590119871
minus120590119903119888 (119905)
119909
[
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(20)
Furthermore the optimal amount of capital invested in thesecurity is given by
120579119878=
1205821
1205901
(119909 + 119888 (119905) 119886119879minus119905|119903119905
) (21)
or if 120578119878denotes the optimal proportion of capital invested in
the security then
120578119878=
1205821
1205901
+
119888 (119905) 119886119879minus119905|119903119905
119909 (22)
According to the above models we may write the optimalamount of capital invested in the treasury as
120579119903= 119909 minus 120579
119871minus 120579119878
(23)
and hence
120578119903= 1 minus 120578
119871minus 120578119878 (24)
where 120578119903is the optimal proportion of capital invested in the
treasuryThe expression 119886119879minus119905
denotes a continuous annuity ofduration 119879 minus 119905
5 The Total Capital Ratio
We now proceed to derive the dynamics of the Total CapitalRatio (CAR) In order to do so we first derive the dynamics ofthe TRWAs of the bank In addition we simulate the optimalproportions of bank capital invested in the assets and theCARat the end of this section The dynamics of the TRWAs andCAR are respectively derived in the remark and propositionbelowWe still assume that the bank capital evolves accordingto (10)
Remark 3 Suppose that at time 119905 the TRWAs can bedescribed by the SDE
119889119886rw (119905) = 0 times 120579119903(119905)
1198891198780(119905)
1198780(119905)
+ 02 times 120579119878(119905)
119889119878 (119905)
119878 (119905)
+ 05 times 120579119871(119905)
119889119871 (119905)
119871 (119905)+ 119888 (119905) 119889119905
(25)
where 0 02 and 05 are the risk-weights associated withrespectively the treasury security and loan under the Basel
III Accord By simplifying (25) the expression for 119889119886rw(119905)reduces to the following
119889119886rw (119905)
= [02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)] 119889119905
+ 02120579119878(119905) 1205901119889119882119878(119905)
+ [02120579119878(119905) 1205902120590119903
+05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))] 119889119882
119903(119905)
(26)
In the proof of the proposition below wemake use of ItorsquosLemma and Itorsquos Product Rule for which we refer to the book[28] by Oslashksendal
Proposition 4 With the dynamics of the total bank capital119862(119905) given by the ODE in (10) and with the dynamics of theTRWAs 119886
119903119908 given by (26) we can write the dynamics of the
Total Capital Ratio or CAR at time 119905 as
119889Λ (119905) = Ψ1119889119905 +
119862 (119905)
1198862119903119908
(119905)[(minusΨ
2+
Ψ3
119886119903119908
(119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(27)
where
Ψ1=
119888 (119905)
119886119903119908
(119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(28)
Proof Wemainly use Itorsquos general formula to derive (27) LetΦ(119886rw(119905)) = 1(119886rw(119905)) Then by Itorsquos Lemma
119889Φ (119886rw (119905))
= Φ (119886rw (119905)) 119889119905 + Φ1015840(119886rw (119905)) 119889119886rw (119905)
+1
2Φ10158401015840(119886rw (119905)) [119889119886rw (119905)]
2
= 0119889119905 minus119889119886rw (119905)
1198862rw (119905)+
[119889119886rw (119905)]2
1198863rw (119905)
Journal of Applied Mathematics 7
= minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
+ 119888 (119905)] +1
1198863rw (119905)[(02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905)))2] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))) 119889119882
119903(119905)]
(29)
Let Λ(119905) denote the CAR at time 119905 for 119905 isin [0 119879] Then bydefinition (see (1)) we can write Λ(119905) as
Λ (119905) =119862 (119905)
119886rw (119905)= 119862 (119905)Φ (119886rw (119905)) (30)
We apply Itorsquos Product Rule to Λ(119905) = 119862(119905)Φ(119886rw(119905)) to findan expression for 119889Λ(119905)
119889Λ (119905)
= 119889119862 (119905)Φ (119886rw (119905)) + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905 + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905
+ 119862 (119905) minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905))
times 1205822120590119903)+119888 (119905)]+
1
1198863rw (119905)
times[(02120579119878(119905) 1205901)2
+(02120579119878(119905) 1205902120590119903+ 05120579
119871
times(119905)120590119871(119879minus 119905 119903 (119905)))
2
] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))) 119889119882119903(119905)]
= Ψ1119889119905 + 119862 (119905) [(minus
1
1198862rw (119905)Ψ2+
1
1198863rw (119905)Ψ3)119889119905 minus
1
1198862rw (119905)
times (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
= Ψ1119889119905 +
119862 (119905)
1198862rw (119905)[(minusΨ
2+
Ψ3
119886rw (119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(31)
where we have defined
Ψ1=
119888 (119905)
119886rw (119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(32)
This concludes the proof
We provide a numerical simulation in order to charac-terize the behaviour of the CAR Λ We assume that theinterest rate follows the CIR dynamics (119896
2= 0) and that the
financial market consists of a treasury a security and a loanFurthermore we consider an investment horizon of 119879 = 10
years and that capital is raised at the fixed rate of 119888 = 013The rest of the parameters of the simulation are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(33)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119886rw (0) = 14 Λ (0) = 008
(34)
In Figure 1 we characterize the optimal investmentstrategy that solves Problem 1 by simulating the optimalproportions of capital invested in the assets The optimalinvestment strategy depicted in Figure 1 leads to the CARin Figure 4 By diversifying its asset portfolio according tothe optimal strategy illustrated by Figure 1 we note that thebank maintains its CAR in such a manner that it is abovethe minimum required level of 8 Since the bank complieswith the minimum required ratio of total capital to TRWAs
8 Journal of Applied Mathematics
0 2 4 6 8 10
TreasurySecurity
Loan
minus04
minus02
0
02
04
06
08
1
Time (years)
Figure 1 A simulation of the optimal proportions 120578119903(119905) 120578119878(119905) and
120578119871(119905) of bank capital invested in the assets given a constant stream
of capital inflow
as imposed by the Basel III Accord it is guaranteed theability to absorb reasonable levels of losses before becominginsolvent over the 10-year period We also simulate theoptimal asset portfolio and TRWAs in Figures 2 and 3respectively An interesting observation is that the TRWAsattain itsmaximum at the end of the investment period underconsideration This is due to the optimal amounts of bankcapital 120579
119878and 120579
119871 invested respectively in the security and
loan being embedded in the formula for the TRWAs in (25)The volatile natures of the quantities simulated above areconsistent with the stochastic variables used throughout thepaper
6 The Asset Portfolio for a Constant CAR
We now set out to modify the asset portfolio of Problem 1 insuch a way as to maintain the Total Capital Ratio or CAR ata constant rate of 8 To this end we need to have the bankcapital 119862(119905) to be stochastic We assume that the stochasticterm is sufficiently small in order to use the solution ofProblem 1 as a reasonable approximation The actual formof 119862(119905) is deduced from the identity 119862(119905) = 008119886rwThe formula for the asset portfolio is given in Remark 5below
Remark 5 At time 119905 the dynamics of the asset portfolio119884(119905) of the bank investing its capital according to theoptimal investment strategy from Problem 1 and in additionmaintaining its CAR at 8 can be written as
119889119884 (119905) = 1205941119889119905 + 120594
2119889119882119878(119905) + 120594
3119889119882119903(119905) (35)
0 2 4 6 8 101
2
3
4
5
6
7
8
Optimal asset portfolio
Time (years)
Figure 2 A simulation of 119883(119905) the optimal asset portfolio given aconstant stream of capital inflow
0 2 4 6 8 10
TRWAs
Time (years)
14
16
18
2
22
24
26
28
3
32
Figure 3 A simulation of the total risk-weighted assets 119886rw(119905) givena constant stream of capital inflow
where
1205941= (119883 (119905) minus 120579
119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
1205942=
117
115120579119878(119905) 1205901
1205943=
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(36)
Journal of Applied Mathematics 9
0 2 4 6 8 10008
009
01
011
012
013
014
015
016
Total Capital Ratio
Time (years)
Figure 4 A simulation of Λ(119905) the Total Capital Ratio given aconstant stream of capital inflow
The argument for this remark goes as follows From thecondition Λ(119905) = 119862(119905)119886rw(119905) = 008 we have 119889119862(119905)008 =
119889119886rw(119905) By substituting 119889119862(119905)008 for the left hand side of(25) and replacing the 119888(119905)119889119905 of (25) by 119889119862(119905) we obtain
(1
008minus 1) 119889119862 (119905) = 02120579
119878(119905)
119889119878 (119905)
119878 (119905)+ 05120579
119871(119905)
119889119871 (119905)
119871 (119905)
(37)
Solving 119889119862(119905) in terms of 120579119878(119905)(119889119878(119905)119878(119905)) and 120579
119871(119905)(119889119871(119905)
119871(119905)) yields
119889119862 (119905) =2
115120579119878(119905)
119889119878 (119905)
119878 (119905)+
1
23120579119871(119905)
119889119871 (119905)
119871 (119905) (38)
Substituting the above expression as the 119889119862(119905) of (8) gives
119889119884 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+117
115120579119878(119905)
119889119878 (119905)
119878 (119905)
+24
23120579119871(119905)
119889119871 (119905)
119871 (119905)
(39)
By simplifying this expression in terms of the models of1198891198780(119905)1198780(119905) 119889119878(119905)119878(119905) and 119889119871(119905)119871(119905) and by recalling that
120579119903(119905) = 119883(119905) minus 120579
119871(119905) minus 120579
119878(119905) we can eventually write 119889119884(119905) as
119889119884 (119905) = [(119883 (119905) minus 120579119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)] 119889119905
+117
115120579119878(119905) 1205901119889119882119878+ [
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))] 119889119882119903
(40)
By writing the coefficients of 119889119905 119889119882119878(119905) and 119889119882
119903(119905) of the
above expression respectively as 1205941 1205942 and 120594
3 the asserted
expression for 119889119884(119905) emergesWe now characterize the behaviour of the modified asset
portfolio given by (35)We consider an interest rate followingthe CIR dynamics (119896
2= 0) and an investment horizon of 119879 =
10 years A capital contribution rate of 119888 = 0 is consideredwhile the rest of the simulation parameters are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(41)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119884 (0) = 15
(42)
For a bank that diversifies its asset portfolio accordingto the solution of Problem 1 and maintains its CAR at aconstant value of 8 we simulate the modified version ofits asset portfolio in Figure 5 In this simulation we controlthe bank capital 119862(119905) as in Figure 6 during the 10-yearinvestment period We note that the growth of the modifiedasset portfolio given by (35) is much slower compared tothe original asset portfolio of Problem 1 For the specificsimulation shown the modified asset portfolio is maximal attime 119879 = 10 years
7 Conclusion
In this paper we consider asset portfolio and capital adequacymanagement in banking In particular our objective isto maximize an expected utility of a Basel III compliantcommercial bankrsquos asset portfolio at a future date119879 gt 0 Fromthe optimal investment strategy that solves our investmentproblem we derive a dynamic formula for the Total CapitalRatio or Capital Adequacy Ratio (CAR) In addition wederive a modified version of the bankrsquos asset portfolio thatwould be obtained should the bank simultaneously followthe optimal investment strategy and maintain its CAR at aconstant level of 8 This modified portfolio is extremelyimportant since it takes maximum advantage by keepingthe Total Capital Ratio exactly at the threshold This meansthat all the funding available for investment is indeed beinginvested
10 Journal of Applied Mathematics
0 2 4 6 8 10
Modified asset portfolio
Time (years)
1
15
2
25
3
35
4
45
5
55
6
Figure 5 A simulation of119884(119905) themodified asset portfolio requiredto maintain the Total Capital Ratio at 8
0 2 4 6 8 100995
1
1005
101
1015
102
1025
103
1035
Time (years)
Controlled bank capital
Figure 6 A simulation of the bank capital 119862(119905) required tomaintain the Total Capital Ratio at 8
We present a simulation study in which we illustratethe optimal investment strategy and characterize the cor-responding CAR and modified asset portfolio The mainresults of our study are as follows The optimal investmentstrategy for the bank is to diversify its asset portfolio awayfrom the risky assets that is the security and loan andtowards the riskless treasury This is in accordance with thebanking literature on optimal asset management in whichit is reported that banks diversify their asset portfolios inan attempt to lower risk (see Dangl and Lehar [29] andDecamps et al [30] for instance) In doing so the bankmaintains its CAR above the Basel III prescribed minimum
level of 8 Of course in line with Basel III standards thebank is able to absorb reasonable levels of unexpected lossesbefore becoming insolvent Furthermore in order to derivethe asset portfolio at constant (minimum) CAR value thelevel of bank capital needs to be controlled This is donemodifying the deterministic capital model to be stochasticWe find that the asset portfolio at constant (minimum) CARvalue grows considerably slower than the asset portfolioof the original investment problem but that this modifiedportfolio is still maximal at the end of the investment periodunder consideration An interesting observation is that ourbank raises capital while at the same time lowering risk Apossible explanation for this is that the bank holds low capitalConservation Buffers which it is trying to rebuild (see Heidet al [31]) In contrast banks with large buffersmaintain theircapital buffer by increasing risk when capital increases [31]
In future research it would be interesting to go beyond thesimulated data approach and model the CAR and bank assetportfolio at constant (minimum) CAR value using real datasourced from for example the US Federal Deposit InsuranceCorporation (FDIC)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors acknowledge financial support from theNationalResearch Foundation of South Africa
References
[1] Basel Committee on Banking Supervision International Con-vergence of Capital Measurements and Capital Standards ARevised Framework Bank for International Settlements 2004
[2] A V Thakor ldquoCapital requirements monetary policy andaggregate bank lending theory and empirical evidencerdquo Journalof Finance vol 51 no 1 pp 279ndash324 1996
[3] E-L vonThadden ldquoBank capital adequacy regulation under thenew Basel accordrdquo Journal of Financial Intermediation vol 13no 2 pp 90ndash95 2004
[4] C H Fouche J Mukuddem-Petersen and M A PetersenldquoContinuous-time stochastic modelling of capital adequacyratios for banksrdquo Applied Stochastic Models in Business andIndustry vol 22 no 1 pp 41ndash71 2006
[5] J Mukuddem-Petersen and M A Petersen ldquoOptimizing assetand capital adequacy management in bankingrdquo Journal ofOptimization Theory and Applications vol 137 no 1 pp 205ndash230 2008
[6] M P Mulaudzi M A Petersen and I M Schoeman ldquoOptimalallocation between bank loans and treasuries with regretrdquoOptimization Letters vol 2 no 4 pp 555ndash566 2008
[7] M Jayadev ldquoBasel III implementation issues and challenges forIndian banksrdquo IIMB Management Review vol 25 pp 115ndash1302013
[8] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
Problem 1 Our objective is to maximize the expected utilityof the bankrsquos asset portfolio at a future date 119879 gt 0 Thus wemust
maximize 119869 (120579119878 120579119871) = E (119906 (119883 (119879)))
subject to 119889119903 (119905) = (119886 minus 119887119903 (119905)) 119889119905 minus 120590119903119889119882119903(119905)
119889119883 (119905) = [119883 (119905) 119903 (119905) + 120579119878(119905)
times (12058211205901+ 120582212059021205902
119903)
+ 120579119871(119905) 1205822120590119871(119879 minus 119905 119903 (119905)) 120590
119903
+ 119888 (119905)] 119889119905 + 120579119878(119905) 1205901119889119882119878(119905)
+ (120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
+ 120579119878(119905) 1205902120590119903) 119889119882119903(119905)
119883 (0) = 1199090 119903 (0) = 119903
0
(11)
with 0 le 119905 le 119879 and where 119883(0) = 1199090and 119903(0) = 119903
0denote
the initial conditions of the optimal control problem
In this paper we describe the bankrsquos objective with alogarithmic utility function that is
119906 (119909) = ln119909 with 119909 gt 0 (12)
Wenote that the utility function 119906(sdot) is strictly concave up andsatisfies the Inada conditions 1199061015840(+infin) = 0 and 119906
1015840(0) = +infin
By using the classical tools of stochastic optimal control wedefine the value function
119867(119905 119903 119909) = sup120579119878120579119871
E (119906 (119883 (119879) | 119903 (119905) = 119903 119883 (119905) = 119909))
0 lt 119905 lt 119879
(13)
The value function can be considered as a kind of utilityfunction The marginal utility of the value function is aconstant while the marginal utility of the original utilityfunction 119906(sdot) decreases to zero as 119909 rarr infin (see Kramkovand Schachermayer [25])The value function also inherits theconvexity of the utility function (see Jonsson and Sircar [26])Moreover it is strictly convex for 119905 lt 119879 even if 119906(sdot) is not
The maximum principle leads to the HJB equation (seeDuffie [27])
119867119905+ 119886 (119887 minus 119903)119867
119903
+ [119909119903 + (12058211205901+ 120582212059021205902
119903) 120579119878+ 1205822120590119871120590119903120579119871+ 119888]119867
119909
+1
2[1205902
11205792
119878+ (120590119871120579119871+ 1205902120590119903120579119878)2
]119867119909119909
+1205902
119903
2119867119903119903
minus (120590119871120590119903120579119871+ 12059021205902
119903120579119878)119867119903119909
= 0
(14)
where we have suppressed the time variable 119905 In the HJBequation above119867
119905119867119903119867119909119867119903119903119867119909119909 and119867
119903119909denote partial
derivatives of first and second orders with respect to timeinterest rate and asset portfolio
The first-order maximizing conditions for the optimalstrategies 120579
119878and 120579119871are
120579119878= minus
1205821
1205901
119867119909
119867119909119909
120579119871=
120590119903(12058211205902minus 12058221205901)119867119909+ 1205901120590119903119867119903119909
1205901120590119871119867119909119909
(15)
If we put (15) into (14) we obtain a PDE for the value function119867
119867119905+ 119886 (119887 minus 119903)119867
119903+
1205902
119903
2119867119903119903+ (119909119903 + 119888)119867
119909
minus1205822
1
2
1198672
119909
119867119909119909
minus(1205822120590119903119867119909minus 120590119903119867119903119909)2
2119867119909119909
= 0
(16)
The problem is to solve (16) for the value function 119867 andreplace it in (15) in order to obtain the optimal investmentstrategy The above equation is a nonlinear second-orderPDE which is very difficult to solve The solution to thisparticular problem was derived in a pension fund setting byGao [17] who did so by employing the Legendre transformand dual theory We present the optimal solution from Gao[17] in Remark 2 below
At this point we shall specify a particular candidate for thefunction 120590
119871appearing in (7) We assume 120590
119871to take the form
120590119871(119879 minus 119905 119903 (119905)) = ℎ (119879 minus 119905) 120590
119903(17)
with
ℎ (119905) =2 (119890119898119905
minus 1)
119898 minus (119887 minus 11989611205822) + 119890119898119905 (119898 + 119887 minus 119896
11205822)
119898 = radic(119887 minus 11989611205822)2
+ 21198961
(18)
Remark 2 The solution of Problem 1 is obtained by solving(16) in terms of 119867 and substituting the result into (15) Theoptimal solution is as follows
The optimal amount of bank capital invested in the loanis
120579119871=
120590119903(12058221205901minus 12058211205902) 119909
1205901120590119871
minus 120590119903119888 (119905) [
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(19)
6 Journal of Applied Mathematics
or if we denote the optimal proportion of capital invested inthe loan by 120578
119871 then we can write
120578119871=
120590119903(12058221205901minus 12058211205902)
1205901120590119871
minus120590119903119888 (119905)
119909
[
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(20)
Furthermore the optimal amount of capital invested in thesecurity is given by
120579119878=
1205821
1205901
(119909 + 119888 (119905) 119886119879minus119905|119903119905
) (21)
or if 120578119878denotes the optimal proportion of capital invested in
the security then
120578119878=
1205821
1205901
+
119888 (119905) 119886119879minus119905|119903119905
119909 (22)
According to the above models we may write the optimalamount of capital invested in the treasury as
120579119903= 119909 minus 120579
119871minus 120579119878
(23)
and hence
120578119903= 1 minus 120578
119871minus 120578119878 (24)
where 120578119903is the optimal proportion of capital invested in the
treasuryThe expression 119886119879minus119905
denotes a continuous annuity ofduration 119879 minus 119905
5 The Total Capital Ratio
We now proceed to derive the dynamics of the Total CapitalRatio (CAR) In order to do so we first derive the dynamics ofthe TRWAs of the bank In addition we simulate the optimalproportions of bank capital invested in the assets and theCARat the end of this section The dynamics of the TRWAs andCAR are respectively derived in the remark and propositionbelowWe still assume that the bank capital evolves accordingto (10)
Remark 3 Suppose that at time 119905 the TRWAs can bedescribed by the SDE
119889119886rw (119905) = 0 times 120579119903(119905)
1198891198780(119905)
1198780(119905)
+ 02 times 120579119878(119905)
119889119878 (119905)
119878 (119905)
+ 05 times 120579119871(119905)
119889119871 (119905)
119871 (119905)+ 119888 (119905) 119889119905
(25)
where 0 02 and 05 are the risk-weights associated withrespectively the treasury security and loan under the Basel
III Accord By simplifying (25) the expression for 119889119886rw(119905)reduces to the following
119889119886rw (119905)
= [02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)] 119889119905
+ 02120579119878(119905) 1205901119889119882119878(119905)
+ [02120579119878(119905) 1205902120590119903
+05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))] 119889119882
119903(119905)
(26)
In the proof of the proposition below wemake use of ItorsquosLemma and Itorsquos Product Rule for which we refer to the book[28] by Oslashksendal
Proposition 4 With the dynamics of the total bank capital119862(119905) given by the ODE in (10) and with the dynamics of theTRWAs 119886
119903119908 given by (26) we can write the dynamics of the
Total Capital Ratio or CAR at time 119905 as
119889Λ (119905) = Ψ1119889119905 +
119862 (119905)
1198862119903119908
(119905)[(minusΨ
2+
Ψ3
119886119903119908
(119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(27)
where
Ψ1=
119888 (119905)
119886119903119908
(119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(28)
Proof Wemainly use Itorsquos general formula to derive (27) LetΦ(119886rw(119905)) = 1(119886rw(119905)) Then by Itorsquos Lemma
119889Φ (119886rw (119905))
= Φ (119886rw (119905)) 119889119905 + Φ1015840(119886rw (119905)) 119889119886rw (119905)
+1
2Φ10158401015840(119886rw (119905)) [119889119886rw (119905)]
2
= 0119889119905 minus119889119886rw (119905)
1198862rw (119905)+
[119889119886rw (119905)]2
1198863rw (119905)
Journal of Applied Mathematics 7
= minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
+ 119888 (119905)] +1
1198863rw (119905)[(02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905)))2] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))) 119889119882
119903(119905)]
(29)
Let Λ(119905) denote the CAR at time 119905 for 119905 isin [0 119879] Then bydefinition (see (1)) we can write Λ(119905) as
Λ (119905) =119862 (119905)
119886rw (119905)= 119862 (119905)Φ (119886rw (119905)) (30)
We apply Itorsquos Product Rule to Λ(119905) = 119862(119905)Φ(119886rw(119905)) to findan expression for 119889Λ(119905)
119889Λ (119905)
= 119889119862 (119905)Φ (119886rw (119905)) + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905 + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905
+ 119862 (119905) minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905))
times 1205822120590119903)+119888 (119905)]+
1
1198863rw (119905)
times[(02120579119878(119905) 1205901)2
+(02120579119878(119905) 1205902120590119903+ 05120579
119871
times(119905)120590119871(119879minus 119905 119903 (119905)))
2
] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))) 119889119882119903(119905)]
= Ψ1119889119905 + 119862 (119905) [(minus
1
1198862rw (119905)Ψ2+
1
1198863rw (119905)Ψ3)119889119905 minus
1
1198862rw (119905)
times (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
= Ψ1119889119905 +
119862 (119905)
1198862rw (119905)[(minusΨ
2+
Ψ3
119886rw (119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(31)
where we have defined
Ψ1=
119888 (119905)
119886rw (119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(32)
This concludes the proof
We provide a numerical simulation in order to charac-terize the behaviour of the CAR Λ We assume that theinterest rate follows the CIR dynamics (119896
2= 0) and that the
financial market consists of a treasury a security and a loanFurthermore we consider an investment horizon of 119879 = 10
years and that capital is raised at the fixed rate of 119888 = 013The rest of the parameters of the simulation are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(33)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119886rw (0) = 14 Λ (0) = 008
(34)
In Figure 1 we characterize the optimal investmentstrategy that solves Problem 1 by simulating the optimalproportions of capital invested in the assets The optimalinvestment strategy depicted in Figure 1 leads to the CARin Figure 4 By diversifying its asset portfolio according tothe optimal strategy illustrated by Figure 1 we note that thebank maintains its CAR in such a manner that it is abovethe minimum required level of 8 Since the bank complieswith the minimum required ratio of total capital to TRWAs
8 Journal of Applied Mathematics
0 2 4 6 8 10
TreasurySecurity
Loan
minus04
minus02
0
02
04
06
08
1
Time (years)
Figure 1 A simulation of the optimal proportions 120578119903(119905) 120578119878(119905) and
120578119871(119905) of bank capital invested in the assets given a constant stream
of capital inflow
as imposed by the Basel III Accord it is guaranteed theability to absorb reasonable levels of losses before becominginsolvent over the 10-year period We also simulate theoptimal asset portfolio and TRWAs in Figures 2 and 3respectively An interesting observation is that the TRWAsattain itsmaximum at the end of the investment period underconsideration This is due to the optimal amounts of bankcapital 120579
119878and 120579
119871 invested respectively in the security and
loan being embedded in the formula for the TRWAs in (25)The volatile natures of the quantities simulated above areconsistent with the stochastic variables used throughout thepaper
6 The Asset Portfolio for a Constant CAR
We now set out to modify the asset portfolio of Problem 1 insuch a way as to maintain the Total Capital Ratio or CAR ata constant rate of 8 To this end we need to have the bankcapital 119862(119905) to be stochastic We assume that the stochasticterm is sufficiently small in order to use the solution ofProblem 1 as a reasonable approximation The actual formof 119862(119905) is deduced from the identity 119862(119905) = 008119886rwThe formula for the asset portfolio is given in Remark 5below
Remark 5 At time 119905 the dynamics of the asset portfolio119884(119905) of the bank investing its capital according to theoptimal investment strategy from Problem 1 and in additionmaintaining its CAR at 8 can be written as
119889119884 (119905) = 1205941119889119905 + 120594
2119889119882119878(119905) + 120594
3119889119882119903(119905) (35)
0 2 4 6 8 101
2
3
4
5
6
7
8
Optimal asset portfolio
Time (years)
Figure 2 A simulation of 119883(119905) the optimal asset portfolio given aconstant stream of capital inflow
0 2 4 6 8 10
TRWAs
Time (years)
14
16
18
2
22
24
26
28
3
32
Figure 3 A simulation of the total risk-weighted assets 119886rw(119905) givena constant stream of capital inflow
where
1205941= (119883 (119905) minus 120579
119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
1205942=
117
115120579119878(119905) 1205901
1205943=
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(36)
Journal of Applied Mathematics 9
0 2 4 6 8 10008
009
01
011
012
013
014
015
016
Total Capital Ratio
Time (years)
Figure 4 A simulation of Λ(119905) the Total Capital Ratio given aconstant stream of capital inflow
The argument for this remark goes as follows From thecondition Λ(119905) = 119862(119905)119886rw(119905) = 008 we have 119889119862(119905)008 =
119889119886rw(119905) By substituting 119889119862(119905)008 for the left hand side of(25) and replacing the 119888(119905)119889119905 of (25) by 119889119862(119905) we obtain
(1
008minus 1) 119889119862 (119905) = 02120579
119878(119905)
119889119878 (119905)
119878 (119905)+ 05120579
119871(119905)
119889119871 (119905)
119871 (119905)
(37)
Solving 119889119862(119905) in terms of 120579119878(119905)(119889119878(119905)119878(119905)) and 120579
119871(119905)(119889119871(119905)
119871(119905)) yields
119889119862 (119905) =2
115120579119878(119905)
119889119878 (119905)
119878 (119905)+
1
23120579119871(119905)
119889119871 (119905)
119871 (119905) (38)
Substituting the above expression as the 119889119862(119905) of (8) gives
119889119884 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+117
115120579119878(119905)
119889119878 (119905)
119878 (119905)
+24
23120579119871(119905)
119889119871 (119905)
119871 (119905)
(39)
By simplifying this expression in terms of the models of1198891198780(119905)1198780(119905) 119889119878(119905)119878(119905) and 119889119871(119905)119871(119905) and by recalling that
120579119903(119905) = 119883(119905) minus 120579
119871(119905) minus 120579
119878(119905) we can eventually write 119889119884(119905) as
119889119884 (119905) = [(119883 (119905) minus 120579119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)] 119889119905
+117
115120579119878(119905) 1205901119889119882119878+ [
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))] 119889119882119903
(40)
By writing the coefficients of 119889119905 119889119882119878(119905) and 119889119882
119903(119905) of the
above expression respectively as 1205941 1205942 and 120594
3 the asserted
expression for 119889119884(119905) emergesWe now characterize the behaviour of the modified asset
portfolio given by (35)We consider an interest rate followingthe CIR dynamics (119896
2= 0) and an investment horizon of 119879 =
10 years A capital contribution rate of 119888 = 0 is consideredwhile the rest of the simulation parameters are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(41)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119884 (0) = 15
(42)
For a bank that diversifies its asset portfolio accordingto the solution of Problem 1 and maintains its CAR at aconstant value of 8 we simulate the modified version ofits asset portfolio in Figure 5 In this simulation we controlthe bank capital 119862(119905) as in Figure 6 during the 10-yearinvestment period We note that the growth of the modifiedasset portfolio given by (35) is much slower compared tothe original asset portfolio of Problem 1 For the specificsimulation shown the modified asset portfolio is maximal attime 119879 = 10 years
7 Conclusion
In this paper we consider asset portfolio and capital adequacymanagement in banking In particular our objective isto maximize an expected utility of a Basel III compliantcommercial bankrsquos asset portfolio at a future date119879 gt 0 Fromthe optimal investment strategy that solves our investmentproblem we derive a dynamic formula for the Total CapitalRatio or Capital Adequacy Ratio (CAR) In addition wederive a modified version of the bankrsquos asset portfolio thatwould be obtained should the bank simultaneously followthe optimal investment strategy and maintain its CAR at aconstant level of 8 This modified portfolio is extremelyimportant since it takes maximum advantage by keepingthe Total Capital Ratio exactly at the threshold This meansthat all the funding available for investment is indeed beinginvested
10 Journal of Applied Mathematics
0 2 4 6 8 10
Modified asset portfolio
Time (years)
1
15
2
25
3
35
4
45
5
55
6
Figure 5 A simulation of119884(119905) themodified asset portfolio requiredto maintain the Total Capital Ratio at 8
0 2 4 6 8 100995
1
1005
101
1015
102
1025
103
1035
Time (years)
Controlled bank capital
Figure 6 A simulation of the bank capital 119862(119905) required tomaintain the Total Capital Ratio at 8
We present a simulation study in which we illustratethe optimal investment strategy and characterize the cor-responding CAR and modified asset portfolio The mainresults of our study are as follows The optimal investmentstrategy for the bank is to diversify its asset portfolio awayfrom the risky assets that is the security and loan andtowards the riskless treasury This is in accordance with thebanking literature on optimal asset management in whichit is reported that banks diversify their asset portfolios inan attempt to lower risk (see Dangl and Lehar [29] andDecamps et al [30] for instance) In doing so the bankmaintains its CAR above the Basel III prescribed minimum
level of 8 Of course in line with Basel III standards thebank is able to absorb reasonable levels of unexpected lossesbefore becoming insolvent Furthermore in order to derivethe asset portfolio at constant (minimum) CAR value thelevel of bank capital needs to be controlled This is donemodifying the deterministic capital model to be stochasticWe find that the asset portfolio at constant (minimum) CARvalue grows considerably slower than the asset portfolioof the original investment problem but that this modifiedportfolio is still maximal at the end of the investment periodunder consideration An interesting observation is that ourbank raises capital while at the same time lowering risk Apossible explanation for this is that the bank holds low capitalConservation Buffers which it is trying to rebuild (see Heidet al [31]) In contrast banks with large buffersmaintain theircapital buffer by increasing risk when capital increases [31]
In future research it would be interesting to go beyond thesimulated data approach and model the CAR and bank assetportfolio at constant (minimum) CAR value using real datasourced from for example the US Federal Deposit InsuranceCorporation (FDIC)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors acknowledge financial support from theNationalResearch Foundation of South Africa
References
[1] Basel Committee on Banking Supervision International Con-vergence of Capital Measurements and Capital Standards ARevised Framework Bank for International Settlements 2004
[2] A V Thakor ldquoCapital requirements monetary policy andaggregate bank lending theory and empirical evidencerdquo Journalof Finance vol 51 no 1 pp 279ndash324 1996
[3] E-L vonThadden ldquoBank capital adequacy regulation under thenew Basel accordrdquo Journal of Financial Intermediation vol 13no 2 pp 90ndash95 2004
[4] C H Fouche J Mukuddem-Petersen and M A PetersenldquoContinuous-time stochastic modelling of capital adequacyratios for banksrdquo Applied Stochastic Models in Business andIndustry vol 22 no 1 pp 41ndash71 2006
[5] J Mukuddem-Petersen and M A Petersen ldquoOptimizing assetand capital adequacy management in bankingrdquo Journal ofOptimization Theory and Applications vol 137 no 1 pp 205ndash230 2008
[6] M P Mulaudzi M A Petersen and I M Schoeman ldquoOptimalallocation between bank loans and treasuries with regretrdquoOptimization Letters vol 2 no 4 pp 555ndash566 2008
[7] M Jayadev ldquoBasel III implementation issues and challenges forIndian banksrdquo IIMB Management Review vol 25 pp 115ndash1302013
[8] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
or if we denote the optimal proportion of capital invested inthe loan by 120578
119871 then we can write
120578119871=
120590119903(12058221205901minus 12058211205902)
1205901120590119871
minus120590119903119888 (119905)
119909
[
[
(12058211205902minus 12058221205901) 119886119879minus119905|119903119905
1205901120590119871
+
119886119879minus119905|119903119905
minus (119879 minus 119905) (1 minus 119903119886119879minus119905|119903119905
)
119903120590119871
]
]
(20)
Furthermore the optimal amount of capital invested in thesecurity is given by
120579119878=
1205821
1205901
(119909 + 119888 (119905) 119886119879minus119905|119903119905
) (21)
or if 120578119878denotes the optimal proportion of capital invested in
the security then
120578119878=
1205821
1205901
+
119888 (119905) 119886119879minus119905|119903119905
119909 (22)
According to the above models we may write the optimalamount of capital invested in the treasury as
120579119903= 119909 minus 120579
119871minus 120579119878
(23)
and hence
120578119903= 1 minus 120578
119871minus 120578119878 (24)
where 120578119903is the optimal proportion of capital invested in the
treasuryThe expression 119886119879minus119905
denotes a continuous annuity ofduration 119879 minus 119905
5 The Total Capital Ratio
We now proceed to derive the dynamics of the Total CapitalRatio (CAR) In order to do so we first derive the dynamics ofthe TRWAs of the bank In addition we simulate the optimalproportions of bank capital invested in the assets and theCARat the end of this section The dynamics of the TRWAs andCAR are respectively derived in the remark and propositionbelowWe still assume that the bank capital evolves accordingto (10)
Remark 3 Suppose that at time 119905 the TRWAs can bedescribed by the SDE
119889119886rw (119905) = 0 times 120579119903(119905)
1198891198780(119905)
1198780(119905)
+ 02 times 120579119878(119905)
119889119878 (119905)
119878 (119905)
+ 05 times 120579119871(119905)
119889119871 (119905)
119871 (119905)+ 119888 (119905) 119889119905
(25)
where 0 02 and 05 are the risk-weights associated withrespectively the treasury security and loan under the Basel
III Accord By simplifying (25) the expression for 119889119886rw(119905)reduces to the following
119889119886rw (119905)
= [02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)] 119889119905
+ 02120579119878(119905) 1205901119889119882119878(119905)
+ [02120579119878(119905) 1205902120590119903
+05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))] 119889119882
119903(119905)
(26)
In the proof of the proposition below wemake use of ItorsquosLemma and Itorsquos Product Rule for which we refer to the book[28] by Oslashksendal
Proposition 4 With the dynamics of the total bank capital119862(119905) given by the ODE in (10) and with the dynamics of theTRWAs 119886
119903119908 given by (26) we can write the dynamics of the
Total Capital Ratio or CAR at time 119905 as
119889Λ (119905) = Ψ1119889119905 +
119862 (119905)
1198862119903119908
(119905)[(minusΨ
2+
Ψ3
119886119903119908
(119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(27)
where
Ψ1=
119888 (119905)
119886119903119908
(119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(28)
Proof Wemainly use Itorsquos general formula to derive (27) LetΦ(119886rw(119905)) = 1(119886rw(119905)) Then by Itorsquos Lemma
119889Φ (119886rw (119905))
= Φ (119886rw (119905)) 119889119905 + Φ1015840(119886rw (119905)) 119889119886rw (119905)
+1
2Φ10158401015840(119886rw (119905)) [119889119886rw (119905)]
2
= 0119889119905 minus119889119886rw (119905)
1198862rw (119905)+
[119889119886rw (119905)]2
1198863rw (119905)
Journal of Applied Mathematics 7
= minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
+ 119888 (119905)] +1
1198863rw (119905)[(02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905)))2] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))) 119889119882
119903(119905)]
(29)
Let Λ(119905) denote the CAR at time 119905 for 119905 isin [0 119879] Then bydefinition (see (1)) we can write Λ(119905) as
Λ (119905) =119862 (119905)
119886rw (119905)= 119862 (119905)Φ (119886rw (119905)) (30)
We apply Itorsquos Product Rule to Λ(119905) = 119862(119905)Φ(119886rw(119905)) to findan expression for 119889Λ(119905)
119889Λ (119905)
= 119889119862 (119905)Φ (119886rw (119905)) + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905 + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905
+ 119862 (119905) minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905))
times 1205822120590119903)+119888 (119905)]+
1
1198863rw (119905)
times[(02120579119878(119905) 1205901)2
+(02120579119878(119905) 1205902120590119903+ 05120579
119871
times(119905)120590119871(119879minus 119905 119903 (119905)))
2
] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))) 119889119882119903(119905)]
= Ψ1119889119905 + 119862 (119905) [(minus
1
1198862rw (119905)Ψ2+
1
1198863rw (119905)Ψ3)119889119905 minus
1
1198862rw (119905)
times (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
= Ψ1119889119905 +
119862 (119905)
1198862rw (119905)[(minusΨ
2+
Ψ3
119886rw (119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(31)
where we have defined
Ψ1=
119888 (119905)
119886rw (119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(32)
This concludes the proof
We provide a numerical simulation in order to charac-terize the behaviour of the CAR Λ We assume that theinterest rate follows the CIR dynamics (119896
2= 0) and that the
financial market consists of a treasury a security and a loanFurthermore we consider an investment horizon of 119879 = 10
years and that capital is raised at the fixed rate of 119888 = 013The rest of the parameters of the simulation are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(33)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119886rw (0) = 14 Λ (0) = 008
(34)
In Figure 1 we characterize the optimal investmentstrategy that solves Problem 1 by simulating the optimalproportions of capital invested in the assets The optimalinvestment strategy depicted in Figure 1 leads to the CARin Figure 4 By diversifying its asset portfolio according tothe optimal strategy illustrated by Figure 1 we note that thebank maintains its CAR in such a manner that it is abovethe minimum required level of 8 Since the bank complieswith the minimum required ratio of total capital to TRWAs
8 Journal of Applied Mathematics
0 2 4 6 8 10
TreasurySecurity
Loan
minus04
minus02
0
02
04
06
08
1
Time (years)
Figure 1 A simulation of the optimal proportions 120578119903(119905) 120578119878(119905) and
120578119871(119905) of bank capital invested in the assets given a constant stream
of capital inflow
as imposed by the Basel III Accord it is guaranteed theability to absorb reasonable levels of losses before becominginsolvent over the 10-year period We also simulate theoptimal asset portfolio and TRWAs in Figures 2 and 3respectively An interesting observation is that the TRWAsattain itsmaximum at the end of the investment period underconsideration This is due to the optimal amounts of bankcapital 120579
119878and 120579
119871 invested respectively in the security and
loan being embedded in the formula for the TRWAs in (25)The volatile natures of the quantities simulated above areconsistent with the stochastic variables used throughout thepaper
6 The Asset Portfolio for a Constant CAR
We now set out to modify the asset portfolio of Problem 1 insuch a way as to maintain the Total Capital Ratio or CAR ata constant rate of 8 To this end we need to have the bankcapital 119862(119905) to be stochastic We assume that the stochasticterm is sufficiently small in order to use the solution ofProblem 1 as a reasonable approximation The actual formof 119862(119905) is deduced from the identity 119862(119905) = 008119886rwThe formula for the asset portfolio is given in Remark 5below
Remark 5 At time 119905 the dynamics of the asset portfolio119884(119905) of the bank investing its capital according to theoptimal investment strategy from Problem 1 and in additionmaintaining its CAR at 8 can be written as
119889119884 (119905) = 1205941119889119905 + 120594
2119889119882119878(119905) + 120594
3119889119882119903(119905) (35)
0 2 4 6 8 101
2
3
4
5
6
7
8
Optimal asset portfolio
Time (years)
Figure 2 A simulation of 119883(119905) the optimal asset portfolio given aconstant stream of capital inflow
0 2 4 6 8 10
TRWAs
Time (years)
14
16
18
2
22
24
26
28
3
32
Figure 3 A simulation of the total risk-weighted assets 119886rw(119905) givena constant stream of capital inflow
where
1205941= (119883 (119905) minus 120579
119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
1205942=
117
115120579119878(119905) 1205901
1205943=
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(36)
Journal of Applied Mathematics 9
0 2 4 6 8 10008
009
01
011
012
013
014
015
016
Total Capital Ratio
Time (years)
Figure 4 A simulation of Λ(119905) the Total Capital Ratio given aconstant stream of capital inflow
The argument for this remark goes as follows From thecondition Λ(119905) = 119862(119905)119886rw(119905) = 008 we have 119889119862(119905)008 =
119889119886rw(119905) By substituting 119889119862(119905)008 for the left hand side of(25) and replacing the 119888(119905)119889119905 of (25) by 119889119862(119905) we obtain
(1
008minus 1) 119889119862 (119905) = 02120579
119878(119905)
119889119878 (119905)
119878 (119905)+ 05120579
119871(119905)
119889119871 (119905)
119871 (119905)
(37)
Solving 119889119862(119905) in terms of 120579119878(119905)(119889119878(119905)119878(119905)) and 120579
119871(119905)(119889119871(119905)
119871(119905)) yields
119889119862 (119905) =2
115120579119878(119905)
119889119878 (119905)
119878 (119905)+
1
23120579119871(119905)
119889119871 (119905)
119871 (119905) (38)
Substituting the above expression as the 119889119862(119905) of (8) gives
119889119884 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+117
115120579119878(119905)
119889119878 (119905)
119878 (119905)
+24
23120579119871(119905)
119889119871 (119905)
119871 (119905)
(39)
By simplifying this expression in terms of the models of1198891198780(119905)1198780(119905) 119889119878(119905)119878(119905) and 119889119871(119905)119871(119905) and by recalling that
120579119903(119905) = 119883(119905) minus 120579
119871(119905) minus 120579
119878(119905) we can eventually write 119889119884(119905) as
119889119884 (119905) = [(119883 (119905) minus 120579119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)] 119889119905
+117
115120579119878(119905) 1205901119889119882119878+ [
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))] 119889119882119903
(40)
By writing the coefficients of 119889119905 119889119882119878(119905) and 119889119882
119903(119905) of the
above expression respectively as 1205941 1205942 and 120594
3 the asserted
expression for 119889119884(119905) emergesWe now characterize the behaviour of the modified asset
portfolio given by (35)We consider an interest rate followingthe CIR dynamics (119896
2= 0) and an investment horizon of 119879 =
10 years A capital contribution rate of 119888 = 0 is consideredwhile the rest of the simulation parameters are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(41)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119884 (0) = 15
(42)
For a bank that diversifies its asset portfolio accordingto the solution of Problem 1 and maintains its CAR at aconstant value of 8 we simulate the modified version ofits asset portfolio in Figure 5 In this simulation we controlthe bank capital 119862(119905) as in Figure 6 during the 10-yearinvestment period We note that the growth of the modifiedasset portfolio given by (35) is much slower compared tothe original asset portfolio of Problem 1 For the specificsimulation shown the modified asset portfolio is maximal attime 119879 = 10 years
7 Conclusion
In this paper we consider asset portfolio and capital adequacymanagement in banking In particular our objective isto maximize an expected utility of a Basel III compliantcommercial bankrsquos asset portfolio at a future date119879 gt 0 Fromthe optimal investment strategy that solves our investmentproblem we derive a dynamic formula for the Total CapitalRatio or Capital Adequacy Ratio (CAR) In addition wederive a modified version of the bankrsquos asset portfolio thatwould be obtained should the bank simultaneously followthe optimal investment strategy and maintain its CAR at aconstant level of 8 This modified portfolio is extremelyimportant since it takes maximum advantage by keepingthe Total Capital Ratio exactly at the threshold This meansthat all the funding available for investment is indeed beinginvested
10 Journal of Applied Mathematics
0 2 4 6 8 10
Modified asset portfolio
Time (years)
1
15
2
25
3
35
4
45
5
55
6
Figure 5 A simulation of119884(119905) themodified asset portfolio requiredto maintain the Total Capital Ratio at 8
0 2 4 6 8 100995
1
1005
101
1015
102
1025
103
1035
Time (years)
Controlled bank capital
Figure 6 A simulation of the bank capital 119862(119905) required tomaintain the Total Capital Ratio at 8
We present a simulation study in which we illustratethe optimal investment strategy and characterize the cor-responding CAR and modified asset portfolio The mainresults of our study are as follows The optimal investmentstrategy for the bank is to diversify its asset portfolio awayfrom the risky assets that is the security and loan andtowards the riskless treasury This is in accordance with thebanking literature on optimal asset management in whichit is reported that banks diversify their asset portfolios inan attempt to lower risk (see Dangl and Lehar [29] andDecamps et al [30] for instance) In doing so the bankmaintains its CAR above the Basel III prescribed minimum
level of 8 Of course in line with Basel III standards thebank is able to absorb reasonable levels of unexpected lossesbefore becoming insolvent Furthermore in order to derivethe asset portfolio at constant (minimum) CAR value thelevel of bank capital needs to be controlled This is donemodifying the deterministic capital model to be stochasticWe find that the asset portfolio at constant (minimum) CARvalue grows considerably slower than the asset portfolioof the original investment problem but that this modifiedportfolio is still maximal at the end of the investment periodunder consideration An interesting observation is that ourbank raises capital while at the same time lowering risk Apossible explanation for this is that the bank holds low capitalConservation Buffers which it is trying to rebuild (see Heidet al [31]) In contrast banks with large buffersmaintain theircapital buffer by increasing risk when capital increases [31]
In future research it would be interesting to go beyond thesimulated data approach and model the CAR and bank assetportfolio at constant (minimum) CAR value using real datasourced from for example the US Federal Deposit InsuranceCorporation (FDIC)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors acknowledge financial support from theNationalResearch Foundation of South Africa
References
[1] Basel Committee on Banking Supervision International Con-vergence of Capital Measurements and Capital Standards ARevised Framework Bank for International Settlements 2004
[2] A V Thakor ldquoCapital requirements monetary policy andaggregate bank lending theory and empirical evidencerdquo Journalof Finance vol 51 no 1 pp 279ndash324 1996
[3] E-L vonThadden ldquoBank capital adequacy regulation under thenew Basel accordrdquo Journal of Financial Intermediation vol 13no 2 pp 90ndash95 2004
[4] C H Fouche J Mukuddem-Petersen and M A PetersenldquoContinuous-time stochastic modelling of capital adequacyratios for banksrdquo Applied Stochastic Models in Business andIndustry vol 22 no 1 pp 41ndash71 2006
[5] J Mukuddem-Petersen and M A Petersen ldquoOptimizing assetand capital adequacy management in bankingrdquo Journal ofOptimization Theory and Applications vol 137 no 1 pp 205ndash230 2008
[6] M P Mulaudzi M A Petersen and I M Schoeman ldquoOptimalallocation between bank loans and treasuries with regretrdquoOptimization Letters vol 2 no 4 pp 555ndash566 2008
[7] M Jayadev ldquoBasel III implementation issues and challenges forIndian banksrdquo IIMB Management Review vol 25 pp 115ndash1302013
[8] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
= minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
+ 119888 (119905)] +1
1198863rw (119905)[(02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905)))2] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903
+ 05120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))) 119889119882
119903(119905)]
(29)
Let Λ(119905) denote the CAR at time 119905 for 119905 isin [0 119879] Then bydefinition (see (1)) we can write Λ(119905) as
Λ (119905) =119862 (119905)
119886rw (119905)= 119862 (119905)Φ (119886rw (119905)) (30)
We apply Itorsquos Product Rule to Λ(119905) = 119862(119905)Φ(119886rw(119905)) to findan expression for 119889Λ(119905)
119889Λ (119905)
= 119889119862 (119905)Φ (119886rw (119905)) + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905 + 119862 (119905) 119889Φ (119886rw (119905))
=119888 (119905)
119886rw (119905)119889119905
+ 119862 (119905) minus1
1198862rw (119905)[02120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905))
times 1205822120590119903)+119888 (119905)]+
1
1198863rw (119905)
times[(02120579119878(119905) 1205901)2
+(02120579119878(119905) 1205902120590119903+ 05120579
119871
times(119905)120590119871(119879minus 119905 119903 (119905)))
2
] 119889119905
minus1
1198862rw (119905)[02120579119878(119905) 1205901119889119882119878(119905)
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))) 119889119882119903(119905)]
= Ψ1119889119905 + 119862 (119905) [(minus
1
1198862rw (119905)Ψ2+
1
1198863rw (119905)Ψ3)119889119905 minus
1
1198862rw (119905)
times (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
= Ψ1119889119905 +
119862 (119905)
1198862rw (119905)[(minusΨ
2+
Ψ3
119886rw (119905)) 119889119905
minus (Ξ1119889119882119878(119905) + Ξ
2119889119882119903(119905))]
(31)
where we have defined
Ψ1=
119888 (119905)
119886rw (119905)
Ψ2= 02120579
119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+ 05120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903) + 119888 (119905)
Ψ3= (02120579
119878(119905) 1205901)2
+ (02120579119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905)))
2
Ξ1= 02120579
119878(119905) 1205901
Ξ2= 02120579
119878(119905) 1205902120590119903+ 05120579
119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(32)
This concludes the proof
We provide a numerical simulation in order to charac-terize the behaviour of the CAR Λ We assume that theinterest rate follows the CIR dynamics (119896
2= 0) and that the
financial market consists of a treasury a security and a loanFurthermore we consider an investment horizon of 119879 = 10
years and that capital is raised at the fixed rate of 119888 = 013The rest of the parameters of the simulation are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(33)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119886rw (0) = 14 Λ (0) = 008
(34)
In Figure 1 we characterize the optimal investmentstrategy that solves Problem 1 by simulating the optimalproportions of capital invested in the assets The optimalinvestment strategy depicted in Figure 1 leads to the CARin Figure 4 By diversifying its asset portfolio according tothe optimal strategy illustrated by Figure 1 we note that thebank maintains its CAR in such a manner that it is abovethe minimum required level of 8 Since the bank complieswith the minimum required ratio of total capital to TRWAs
8 Journal of Applied Mathematics
0 2 4 6 8 10
TreasurySecurity
Loan
minus04
minus02
0
02
04
06
08
1
Time (years)
Figure 1 A simulation of the optimal proportions 120578119903(119905) 120578119878(119905) and
120578119871(119905) of bank capital invested in the assets given a constant stream
of capital inflow
as imposed by the Basel III Accord it is guaranteed theability to absorb reasonable levels of losses before becominginsolvent over the 10-year period We also simulate theoptimal asset portfolio and TRWAs in Figures 2 and 3respectively An interesting observation is that the TRWAsattain itsmaximum at the end of the investment period underconsideration This is due to the optimal amounts of bankcapital 120579
119878and 120579
119871 invested respectively in the security and
loan being embedded in the formula for the TRWAs in (25)The volatile natures of the quantities simulated above areconsistent with the stochastic variables used throughout thepaper
6 The Asset Portfolio for a Constant CAR
We now set out to modify the asset portfolio of Problem 1 insuch a way as to maintain the Total Capital Ratio or CAR ata constant rate of 8 To this end we need to have the bankcapital 119862(119905) to be stochastic We assume that the stochasticterm is sufficiently small in order to use the solution ofProblem 1 as a reasonable approximation The actual formof 119862(119905) is deduced from the identity 119862(119905) = 008119886rwThe formula for the asset portfolio is given in Remark 5below
Remark 5 At time 119905 the dynamics of the asset portfolio119884(119905) of the bank investing its capital according to theoptimal investment strategy from Problem 1 and in additionmaintaining its CAR at 8 can be written as
119889119884 (119905) = 1205941119889119905 + 120594
2119889119882119878(119905) + 120594
3119889119882119903(119905) (35)
0 2 4 6 8 101
2
3
4
5
6
7
8
Optimal asset portfolio
Time (years)
Figure 2 A simulation of 119883(119905) the optimal asset portfolio given aconstant stream of capital inflow
0 2 4 6 8 10
TRWAs
Time (years)
14
16
18
2
22
24
26
28
3
32
Figure 3 A simulation of the total risk-weighted assets 119886rw(119905) givena constant stream of capital inflow
where
1205941= (119883 (119905) minus 120579
119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
1205942=
117
115120579119878(119905) 1205901
1205943=
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(36)
Journal of Applied Mathematics 9
0 2 4 6 8 10008
009
01
011
012
013
014
015
016
Total Capital Ratio
Time (years)
Figure 4 A simulation of Λ(119905) the Total Capital Ratio given aconstant stream of capital inflow
The argument for this remark goes as follows From thecondition Λ(119905) = 119862(119905)119886rw(119905) = 008 we have 119889119862(119905)008 =
119889119886rw(119905) By substituting 119889119862(119905)008 for the left hand side of(25) and replacing the 119888(119905)119889119905 of (25) by 119889119862(119905) we obtain
(1
008minus 1) 119889119862 (119905) = 02120579
119878(119905)
119889119878 (119905)
119878 (119905)+ 05120579
119871(119905)
119889119871 (119905)
119871 (119905)
(37)
Solving 119889119862(119905) in terms of 120579119878(119905)(119889119878(119905)119878(119905)) and 120579
119871(119905)(119889119871(119905)
119871(119905)) yields
119889119862 (119905) =2
115120579119878(119905)
119889119878 (119905)
119878 (119905)+
1
23120579119871(119905)
119889119871 (119905)
119871 (119905) (38)
Substituting the above expression as the 119889119862(119905) of (8) gives
119889119884 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+117
115120579119878(119905)
119889119878 (119905)
119878 (119905)
+24
23120579119871(119905)
119889119871 (119905)
119871 (119905)
(39)
By simplifying this expression in terms of the models of1198891198780(119905)1198780(119905) 119889119878(119905)119878(119905) and 119889119871(119905)119871(119905) and by recalling that
120579119903(119905) = 119883(119905) minus 120579
119871(119905) minus 120579
119878(119905) we can eventually write 119889119884(119905) as
119889119884 (119905) = [(119883 (119905) minus 120579119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)] 119889119905
+117
115120579119878(119905) 1205901119889119882119878+ [
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))] 119889119882119903
(40)
By writing the coefficients of 119889119905 119889119882119878(119905) and 119889119882
119903(119905) of the
above expression respectively as 1205941 1205942 and 120594
3 the asserted
expression for 119889119884(119905) emergesWe now characterize the behaviour of the modified asset
portfolio given by (35)We consider an interest rate followingthe CIR dynamics (119896
2= 0) and an investment horizon of 119879 =
10 years A capital contribution rate of 119888 = 0 is consideredwhile the rest of the simulation parameters are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(41)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119884 (0) = 15
(42)
For a bank that diversifies its asset portfolio accordingto the solution of Problem 1 and maintains its CAR at aconstant value of 8 we simulate the modified version ofits asset portfolio in Figure 5 In this simulation we controlthe bank capital 119862(119905) as in Figure 6 during the 10-yearinvestment period We note that the growth of the modifiedasset portfolio given by (35) is much slower compared tothe original asset portfolio of Problem 1 For the specificsimulation shown the modified asset portfolio is maximal attime 119879 = 10 years
7 Conclusion
In this paper we consider asset portfolio and capital adequacymanagement in banking In particular our objective isto maximize an expected utility of a Basel III compliantcommercial bankrsquos asset portfolio at a future date119879 gt 0 Fromthe optimal investment strategy that solves our investmentproblem we derive a dynamic formula for the Total CapitalRatio or Capital Adequacy Ratio (CAR) In addition wederive a modified version of the bankrsquos asset portfolio thatwould be obtained should the bank simultaneously followthe optimal investment strategy and maintain its CAR at aconstant level of 8 This modified portfolio is extremelyimportant since it takes maximum advantage by keepingthe Total Capital Ratio exactly at the threshold This meansthat all the funding available for investment is indeed beinginvested
10 Journal of Applied Mathematics
0 2 4 6 8 10
Modified asset portfolio
Time (years)
1
15
2
25
3
35
4
45
5
55
6
Figure 5 A simulation of119884(119905) themodified asset portfolio requiredto maintain the Total Capital Ratio at 8
0 2 4 6 8 100995
1
1005
101
1015
102
1025
103
1035
Time (years)
Controlled bank capital
Figure 6 A simulation of the bank capital 119862(119905) required tomaintain the Total Capital Ratio at 8
We present a simulation study in which we illustratethe optimal investment strategy and characterize the cor-responding CAR and modified asset portfolio The mainresults of our study are as follows The optimal investmentstrategy for the bank is to diversify its asset portfolio awayfrom the risky assets that is the security and loan andtowards the riskless treasury This is in accordance with thebanking literature on optimal asset management in whichit is reported that banks diversify their asset portfolios inan attempt to lower risk (see Dangl and Lehar [29] andDecamps et al [30] for instance) In doing so the bankmaintains its CAR above the Basel III prescribed minimum
level of 8 Of course in line with Basel III standards thebank is able to absorb reasonable levels of unexpected lossesbefore becoming insolvent Furthermore in order to derivethe asset portfolio at constant (minimum) CAR value thelevel of bank capital needs to be controlled This is donemodifying the deterministic capital model to be stochasticWe find that the asset portfolio at constant (minimum) CARvalue grows considerably slower than the asset portfolioof the original investment problem but that this modifiedportfolio is still maximal at the end of the investment periodunder consideration An interesting observation is that ourbank raises capital while at the same time lowering risk Apossible explanation for this is that the bank holds low capitalConservation Buffers which it is trying to rebuild (see Heidet al [31]) In contrast banks with large buffersmaintain theircapital buffer by increasing risk when capital increases [31]
In future research it would be interesting to go beyond thesimulated data approach and model the CAR and bank assetportfolio at constant (minimum) CAR value using real datasourced from for example the US Federal Deposit InsuranceCorporation (FDIC)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors acknowledge financial support from theNationalResearch Foundation of South Africa
References
[1] Basel Committee on Banking Supervision International Con-vergence of Capital Measurements and Capital Standards ARevised Framework Bank for International Settlements 2004
[2] A V Thakor ldquoCapital requirements monetary policy andaggregate bank lending theory and empirical evidencerdquo Journalof Finance vol 51 no 1 pp 279ndash324 1996
[3] E-L vonThadden ldquoBank capital adequacy regulation under thenew Basel accordrdquo Journal of Financial Intermediation vol 13no 2 pp 90ndash95 2004
[4] C H Fouche J Mukuddem-Petersen and M A PetersenldquoContinuous-time stochastic modelling of capital adequacyratios for banksrdquo Applied Stochastic Models in Business andIndustry vol 22 no 1 pp 41ndash71 2006
[5] J Mukuddem-Petersen and M A Petersen ldquoOptimizing assetand capital adequacy management in bankingrdquo Journal ofOptimization Theory and Applications vol 137 no 1 pp 205ndash230 2008
[6] M P Mulaudzi M A Petersen and I M Schoeman ldquoOptimalallocation between bank loans and treasuries with regretrdquoOptimization Letters vol 2 no 4 pp 555ndash566 2008
[7] M Jayadev ldquoBasel III implementation issues and challenges forIndian banksrdquo IIMB Management Review vol 25 pp 115ndash1302013
[8] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
0 2 4 6 8 10
TreasurySecurity
Loan
minus04
minus02
0
02
04
06
08
1
Time (years)
Figure 1 A simulation of the optimal proportions 120578119903(119905) 120578119878(119905) and
120578119871(119905) of bank capital invested in the assets given a constant stream
of capital inflow
as imposed by the Basel III Accord it is guaranteed theability to absorb reasonable levels of losses before becominginsolvent over the 10-year period We also simulate theoptimal asset portfolio and TRWAs in Figures 2 and 3respectively An interesting observation is that the TRWAsattain itsmaximum at the end of the investment period underconsideration This is due to the optimal amounts of bankcapital 120579
119878and 120579
119871 invested respectively in the security and
loan being embedded in the formula for the TRWAs in (25)The volatile natures of the quantities simulated above areconsistent with the stochastic variables used throughout thepaper
6 The Asset Portfolio for a Constant CAR
We now set out to modify the asset portfolio of Problem 1 insuch a way as to maintain the Total Capital Ratio or CAR ata constant rate of 8 To this end we need to have the bankcapital 119862(119905) to be stochastic We assume that the stochasticterm is sufficiently small in order to use the solution ofProblem 1 as a reasonable approximation The actual formof 119862(119905) is deduced from the identity 119862(119905) = 008119886rwThe formula for the asset portfolio is given in Remark 5below
Remark 5 At time 119905 the dynamics of the asset portfolio119884(119905) of the bank investing its capital according to theoptimal investment strategy from Problem 1 and in additionmaintaining its CAR at 8 can be written as
119889119884 (119905) = 1205941119889119905 + 120594
2119889119882119878(119905) + 120594
3119889119882119903(119905) (35)
0 2 4 6 8 101
2
3
4
5
6
7
8
Optimal asset portfolio
Time (years)
Figure 2 A simulation of 119883(119905) the optimal asset portfolio given aconstant stream of capital inflow
0 2 4 6 8 10
TRWAs
Time (years)
14
16
18
2
22
24
26
28
3
32
Figure 3 A simulation of the total risk-weighted assets 119886rw(119905) givena constant stream of capital inflow
where
1205941= (119883 (119905) minus 120579
119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)
1205942=
117
115120579119878(119905) 1205901
1205943=
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871(119879 minus 119905 119903 (119905))
(36)
Journal of Applied Mathematics 9
0 2 4 6 8 10008
009
01
011
012
013
014
015
016
Total Capital Ratio
Time (years)
Figure 4 A simulation of Λ(119905) the Total Capital Ratio given aconstant stream of capital inflow
The argument for this remark goes as follows From thecondition Λ(119905) = 119862(119905)119886rw(119905) = 008 we have 119889119862(119905)008 =
119889119886rw(119905) By substituting 119889119862(119905)008 for the left hand side of(25) and replacing the 119888(119905)119889119905 of (25) by 119889119862(119905) we obtain
(1
008minus 1) 119889119862 (119905) = 02120579
119878(119905)
119889119878 (119905)
119878 (119905)+ 05120579
119871(119905)
119889119871 (119905)
119871 (119905)
(37)
Solving 119889119862(119905) in terms of 120579119878(119905)(119889119878(119905)119878(119905)) and 120579
119871(119905)(119889119871(119905)
119871(119905)) yields
119889119862 (119905) =2
115120579119878(119905)
119889119878 (119905)
119878 (119905)+
1
23120579119871(119905)
119889119871 (119905)
119871 (119905) (38)
Substituting the above expression as the 119889119862(119905) of (8) gives
119889119884 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+117
115120579119878(119905)
119889119878 (119905)
119878 (119905)
+24
23120579119871(119905)
119889119871 (119905)
119871 (119905)
(39)
By simplifying this expression in terms of the models of1198891198780(119905)1198780(119905) 119889119878(119905)119878(119905) and 119889119871(119905)119871(119905) and by recalling that
120579119903(119905) = 119883(119905) minus 120579
119871(119905) minus 120579
119878(119905) we can eventually write 119889119884(119905) as
119889119884 (119905) = [(119883 (119905) minus 120579119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)] 119889119905
+117
115120579119878(119905) 1205901119889119882119878+ [
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))] 119889119882119903
(40)
By writing the coefficients of 119889119905 119889119882119878(119905) and 119889119882
119903(119905) of the
above expression respectively as 1205941 1205942 and 120594
3 the asserted
expression for 119889119884(119905) emergesWe now characterize the behaviour of the modified asset
portfolio given by (35)We consider an interest rate followingthe CIR dynamics (119896
2= 0) and an investment horizon of 119879 =
10 years A capital contribution rate of 119888 = 0 is consideredwhile the rest of the simulation parameters are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(41)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119884 (0) = 15
(42)
For a bank that diversifies its asset portfolio accordingto the solution of Problem 1 and maintains its CAR at aconstant value of 8 we simulate the modified version ofits asset portfolio in Figure 5 In this simulation we controlthe bank capital 119862(119905) as in Figure 6 during the 10-yearinvestment period We note that the growth of the modifiedasset portfolio given by (35) is much slower compared tothe original asset portfolio of Problem 1 For the specificsimulation shown the modified asset portfolio is maximal attime 119879 = 10 years
7 Conclusion
In this paper we consider asset portfolio and capital adequacymanagement in banking In particular our objective isto maximize an expected utility of a Basel III compliantcommercial bankrsquos asset portfolio at a future date119879 gt 0 Fromthe optimal investment strategy that solves our investmentproblem we derive a dynamic formula for the Total CapitalRatio or Capital Adequacy Ratio (CAR) In addition wederive a modified version of the bankrsquos asset portfolio thatwould be obtained should the bank simultaneously followthe optimal investment strategy and maintain its CAR at aconstant level of 8 This modified portfolio is extremelyimportant since it takes maximum advantage by keepingthe Total Capital Ratio exactly at the threshold This meansthat all the funding available for investment is indeed beinginvested
10 Journal of Applied Mathematics
0 2 4 6 8 10
Modified asset portfolio
Time (years)
1
15
2
25
3
35
4
45
5
55
6
Figure 5 A simulation of119884(119905) themodified asset portfolio requiredto maintain the Total Capital Ratio at 8
0 2 4 6 8 100995
1
1005
101
1015
102
1025
103
1035
Time (years)
Controlled bank capital
Figure 6 A simulation of the bank capital 119862(119905) required tomaintain the Total Capital Ratio at 8
We present a simulation study in which we illustratethe optimal investment strategy and characterize the cor-responding CAR and modified asset portfolio The mainresults of our study are as follows The optimal investmentstrategy for the bank is to diversify its asset portfolio awayfrom the risky assets that is the security and loan andtowards the riskless treasury This is in accordance with thebanking literature on optimal asset management in whichit is reported that banks diversify their asset portfolios inan attempt to lower risk (see Dangl and Lehar [29] andDecamps et al [30] for instance) In doing so the bankmaintains its CAR above the Basel III prescribed minimum
level of 8 Of course in line with Basel III standards thebank is able to absorb reasonable levels of unexpected lossesbefore becoming insolvent Furthermore in order to derivethe asset portfolio at constant (minimum) CAR value thelevel of bank capital needs to be controlled This is donemodifying the deterministic capital model to be stochasticWe find that the asset portfolio at constant (minimum) CARvalue grows considerably slower than the asset portfolioof the original investment problem but that this modifiedportfolio is still maximal at the end of the investment periodunder consideration An interesting observation is that ourbank raises capital while at the same time lowering risk Apossible explanation for this is that the bank holds low capitalConservation Buffers which it is trying to rebuild (see Heidet al [31]) In contrast banks with large buffersmaintain theircapital buffer by increasing risk when capital increases [31]
In future research it would be interesting to go beyond thesimulated data approach and model the CAR and bank assetportfolio at constant (minimum) CAR value using real datasourced from for example the US Federal Deposit InsuranceCorporation (FDIC)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors acknowledge financial support from theNationalResearch Foundation of South Africa
References
[1] Basel Committee on Banking Supervision International Con-vergence of Capital Measurements and Capital Standards ARevised Framework Bank for International Settlements 2004
[2] A V Thakor ldquoCapital requirements monetary policy andaggregate bank lending theory and empirical evidencerdquo Journalof Finance vol 51 no 1 pp 279ndash324 1996
[3] E-L vonThadden ldquoBank capital adequacy regulation under thenew Basel accordrdquo Journal of Financial Intermediation vol 13no 2 pp 90ndash95 2004
[4] C H Fouche J Mukuddem-Petersen and M A PetersenldquoContinuous-time stochastic modelling of capital adequacyratios for banksrdquo Applied Stochastic Models in Business andIndustry vol 22 no 1 pp 41ndash71 2006
[5] J Mukuddem-Petersen and M A Petersen ldquoOptimizing assetand capital adequacy management in bankingrdquo Journal ofOptimization Theory and Applications vol 137 no 1 pp 205ndash230 2008
[6] M P Mulaudzi M A Petersen and I M Schoeman ldquoOptimalallocation between bank loans and treasuries with regretrdquoOptimization Letters vol 2 no 4 pp 555ndash566 2008
[7] M Jayadev ldquoBasel III implementation issues and challenges forIndian banksrdquo IIMB Management Review vol 25 pp 115ndash1302013
[8] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
0 2 4 6 8 10008
009
01
011
012
013
014
015
016
Total Capital Ratio
Time (years)
Figure 4 A simulation of Λ(119905) the Total Capital Ratio given aconstant stream of capital inflow
The argument for this remark goes as follows From thecondition Λ(119905) = 119862(119905)119886rw(119905) = 008 we have 119889119862(119905)008 =
119889119886rw(119905) By substituting 119889119862(119905)008 for the left hand side of(25) and replacing the 119888(119905)119889119905 of (25) by 119889119862(119905) we obtain
(1
008minus 1) 119889119862 (119905) = 02120579
119878(119905)
119889119878 (119905)
119878 (119905)+ 05120579
119871(119905)
119889119871 (119905)
119871 (119905)
(37)
Solving 119889119862(119905) in terms of 120579119878(119905)(119889119878(119905)119878(119905)) and 120579
119871(119905)(119889119871(119905)
119871(119905)) yields
119889119862 (119905) =2
115120579119878(119905)
119889119878 (119905)
119878 (119905)+
1
23120579119871(119905)
119889119871 (119905)
119871 (119905) (38)
Substituting the above expression as the 119889119862(119905) of (8) gives
119889119884 (119905) = 120579119903(119905)
1198891198780(119905)
1198780(119905)
+117
115120579119878(119905)
119889119878 (119905)
119878 (119905)
+24
23120579119871(119905)
119889119871 (119905)
119871 (119905)
(39)
By simplifying this expression in terms of the models of1198891198780(119905)1198780(119905) 119889119878(119905)119878(119905) and 119889119871(119905)119871(119905) and by recalling that
120579119903(119905) = 119883(119905) minus 120579
119871(119905) minus 120579
119878(119905) we can eventually write 119889119884(119905) as
119889119884 (119905) = [(119883 (119905) minus 120579119871(119905) minus 120579
119878(119905)) 119903 (119905)
+117
115120579119878(119905) (119903 (119905) + 120590
11205821+ 12059021205902
1199031205822)
+24
23120579119871(119905) (119903 (119905) + 120590
119871(119879 minus 119905 119903 (119905)) 120582
2120590119903)] 119889119905
+117
115120579119878(119905) 1205901119889119882119878+ [
117
115120579119878(119905) 1205902120590119903+
24
23120579119871(119905) 120590119871
times (119879 minus 119905 119903 (119905))] 119889119882119903
(40)
By writing the coefficients of 119889119905 119889119882119878(119905) and 119889119882
119903(119905) of the
above expression respectively as 1205941 1205942 and 120594
3 the asserted
expression for 119889119884(119905) emergesWe now characterize the behaviour of the modified asset
portfolio given by (35)We consider an interest rate followingthe CIR dynamics (119896
2= 0) and an investment horizon of 119879 =
10 years A capital contribution rate of 119888 = 0 is consideredwhile the rest of the simulation parameters are
119886 = 00118712 119887 = 00339
1198961= 000118712 120590
1= 01475
1205821= 0045 120590
2= 0295 120582
2= 009
(41)
with initial conditions
119862 (0) = 1 119903 (0) = 0075
119883 (0) = 15 119884 (0) = 15
(42)
For a bank that diversifies its asset portfolio accordingto the solution of Problem 1 and maintains its CAR at aconstant value of 8 we simulate the modified version ofits asset portfolio in Figure 5 In this simulation we controlthe bank capital 119862(119905) as in Figure 6 during the 10-yearinvestment period We note that the growth of the modifiedasset portfolio given by (35) is much slower compared tothe original asset portfolio of Problem 1 For the specificsimulation shown the modified asset portfolio is maximal attime 119879 = 10 years
7 Conclusion
In this paper we consider asset portfolio and capital adequacymanagement in banking In particular our objective isto maximize an expected utility of a Basel III compliantcommercial bankrsquos asset portfolio at a future date119879 gt 0 Fromthe optimal investment strategy that solves our investmentproblem we derive a dynamic formula for the Total CapitalRatio or Capital Adequacy Ratio (CAR) In addition wederive a modified version of the bankrsquos asset portfolio thatwould be obtained should the bank simultaneously followthe optimal investment strategy and maintain its CAR at aconstant level of 8 This modified portfolio is extremelyimportant since it takes maximum advantage by keepingthe Total Capital Ratio exactly at the threshold This meansthat all the funding available for investment is indeed beinginvested
10 Journal of Applied Mathematics
0 2 4 6 8 10
Modified asset portfolio
Time (years)
1
15
2
25
3
35
4
45
5
55
6
Figure 5 A simulation of119884(119905) themodified asset portfolio requiredto maintain the Total Capital Ratio at 8
0 2 4 6 8 100995
1
1005
101
1015
102
1025
103
1035
Time (years)
Controlled bank capital
Figure 6 A simulation of the bank capital 119862(119905) required tomaintain the Total Capital Ratio at 8
We present a simulation study in which we illustratethe optimal investment strategy and characterize the cor-responding CAR and modified asset portfolio The mainresults of our study are as follows The optimal investmentstrategy for the bank is to diversify its asset portfolio awayfrom the risky assets that is the security and loan andtowards the riskless treasury This is in accordance with thebanking literature on optimal asset management in whichit is reported that banks diversify their asset portfolios inan attempt to lower risk (see Dangl and Lehar [29] andDecamps et al [30] for instance) In doing so the bankmaintains its CAR above the Basel III prescribed minimum
level of 8 Of course in line with Basel III standards thebank is able to absorb reasonable levels of unexpected lossesbefore becoming insolvent Furthermore in order to derivethe asset portfolio at constant (minimum) CAR value thelevel of bank capital needs to be controlled This is donemodifying the deterministic capital model to be stochasticWe find that the asset portfolio at constant (minimum) CARvalue grows considerably slower than the asset portfolioof the original investment problem but that this modifiedportfolio is still maximal at the end of the investment periodunder consideration An interesting observation is that ourbank raises capital while at the same time lowering risk Apossible explanation for this is that the bank holds low capitalConservation Buffers which it is trying to rebuild (see Heidet al [31]) In contrast banks with large buffersmaintain theircapital buffer by increasing risk when capital increases [31]
In future research it would be interesting to go beyond thesimulated data approach and model the CAR and bank assetportfolio at constant (minimum) CAR value using real datasourced from for example the US Federal Deposit InsuranceCorporation (FDIC)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors acknowledge financial support from theNationalResearch Foundation of South Africa
References
[1] Basel Committee on Banking Supervision International Con-vergence of Capital Measurements and Capital Standards ARevised Framework Bank for International Settlements 2004
[2] A V Thakor ldquoCapital requirements monetary policy andaggregate bank lending theory and empirical evidencerdquo Journalof Finance vol 51 no 1 pp 279ndash324 1996
[3] E-L vonThadden ldquoBank capital adequacy regulation under thenew Basel accordrdquo Journal of Financial Intermediation vol 13no 2 pp 90ndash95 2004
[4] C H Fouche J Mukuddem-Petersen and M A PetersenldquoContinuous-time stochastic modelling of capital adequacyratios for banksrdquo Applied Stochastic Models in Business andIndustry vol 22 no 1 pp 41ndash71 2006
[5] J Mukuddem-Petersen and M A Petersen ldquoOptimizing assetand capital adequacy management in bankingrdquo Journal ofOptimization Theory and Applications vol 137 no 1 pp 205ndash230 2008
[6] M P Mulaudzi M A Petersen and I M Schoeman ldquoOptimalallocation between bank loans and treasuries with regretrdquoOptimization Letters vol 2 no 4 pp 555ndash566 2008
[7] M Jayadev ldquoBasel III implementation issues and challenges forIndian banksrdquo IIMB Management Review vol 25 pp 115ndash1302013
[8] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
0 2 4 6 8 10
Modified asset portfolio
Time (years)
1
15
2
25
3
35
4
45
5
55
6
Figure 5 A simulation of119884(119905) themodified asset portfolio requiredto maintain the Total Capital Ratio at 8
0 2 4 6 8 100995
1
1005
101
1015
102
1025
103
1035
Time (years)
Controlled bank capital
Figure 6 A simulation of the bank capital 119862(119905) required tomaintain the Total Capital Ratio at 8
We present a simulation study in which we illustratethe optimal investment strategy and characterize the cor-responding CAR and modified asset portfolio The mainresults of our study are as follows The optimal investmentstrategy for the bank is to diversify its asset portfolio awayfrom the risky assets that is the security and loan andtowards the riskless treasury This is in accordance with thebanking literature on optimal asset management in whichit is reported that banks diversify their asset portfolios inan attempt to lower risk (see Dangl and Lehar [29] andDecamps et al [30] for instance) In doing so the bankmaintains its CAR above the Basel III prescribed minimum
level of 8 Of course in line with Basel III standards thebank is able to absorb reasonable levels of unexpected lossesbefore becoming insolvent Furthermore in order to derivethe asset portfolio at constant (minimum) CAR value thelevel of bank capital needs to be controlled This is donemodifying the deterministic capital model to be stochasticWe find that the asset portfolio at constant (minimum) CARvalue grows considerably slower than the asset portfolioof the original investment problem but that this modifiedportfolio is still maximal at the end of the investment periodunder consideration An interesting observation is that ourbank raises capital while at the same time lowering risk Apossible explanation for this is that the bank holds low capitalConservation Buffers which it is trying to rebuild (see Heidet al [31]) In contrast banks with large buffersmaintain theircapital buffer by increasing risk when capital increases [31]
In future research it would be interesting to go beyond thesimulated data approach and model the CAR and bank assetportfolio at constant (minimum) CAR value using real datasourced from for example the US Federal Deposit InsuranceCorporation (FDIC)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors acknowledge financial support from theNationalResearch Foundation of South Africa
References
[1] Basel Committee on Banking Supervision International Con-vergence of Capital Measurements and Capital Standards ARevised Framework Bank for International Settlements 2004
[2] A V Thakor ldquoCapital requirements monetary policy andaggregate bank lending theory and empirical evidencerdquo Journalof Finance vol 51 no 1 pp 279ndash324 1996
[3] E-L vonThadden ldquoBank capital adequacy regulation under thenew Basel accordrdquo Journal of Financial Intermediation vol 13no 2 pp 90ndash95 2004
[4] C H Fouche J Mukuddem-Petersen and M A PetersenldquoContinuous-time stochastic modelling of capital adequacyratios for banksrdquo Applied Stochastic Models in Business andIndustry vol 22 no 1 pp 41ndash71 2006
[5] J Mukuddem-Petersen and M A Petersen ldquoOptimizing assetand capital adequacy management in bankingrdquo Journal ofOptimization Theory and Applications vol 137 no 1 pp 205ndash230 2008
[6] M P Mulaudzi M A Petersen and I M Schoeman ldquoOptimalallocation between bank loans and treasuries with regretrdquoOptimization Letters vol 2 no 4 pp 555ndash566 2008
[7] M Jayadev ldquoBasel III implementation issues and challenges forIndian banksrdquo IIMB Management Review vol 25 pp 115ndash1302013
[8] M A Petersen and J Mukuddem-Petersen Basel III LiquidityRegulation and Its Implications Business Expert PressMcGraw-Hill New York NY USA 2013
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 11
[9] R Merton ldquoLifetime portfolio selection under uncertainty thecontinuous-time caserdquo Review of Economics and Statistics vol51 pp 247ndash257 1969
[10] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971
[11] JMukuddem-Petersen andM A Petersen ldquoBankmanagementvia stochastic optimal controlrdquo Automatica vol 42 no 8 pp1395ndash1406 2006
[12] J Mukuddem-Petersen M A Petersen I M Schoeman and BA Tau ldquoMaximizing banking profit on a random time intervalrdquoJournal of Applied Mathematics vol 2007 Article ID 29343 22pages 2007
[13] J Mukuddem-Petersen M P Mulaudzi M A Petersen and IM Schoeman ldquoOptimal mortgage loan securitization and thesubprime crisisrdquo Optimization Letters vol 4 no 1 pp 97ndash1152010
[14] F Gideon M A Petersen J Mukuddem-Petersen and L N PHlatshwayo ldquoBasel III and the net stable funding ratiordquo ISRNApplied Mathematics vol 2013 Article ID 582707 20 pages2013
[15] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2011
[16] J Xiao Z Hong andC Qin ldquoThe constant elasticity of variance(CEV) model and the Legendre transform-dual solution forannuity contractsrdquo Insurance vol 40 no 2 pp 302ndash310 2007
[17] J Gao ldquoStochastic optimal control of DC pension fundsrdquoInsurance vol 42 no 3 pp 1159ndash1164 2008
[18] J Gao ldquoOptimal investment strategy for annuity contractsunder the constant elasticity of variance (CEV) modelrdquo Insur-ance vol 45 no 1 pp 9ndash18 2009
[19] J Gao ldquoAn extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contractsrdquo Insurance vol46 no 3 pp 511ndash530 2010
[20] E J Jung and J H Kim ldquoOptimal investment strategies for theHARA utility under the constant elasticity of variance modelrdquoInsurance vol 51 no 3 pp 667ndash673 2012
[21] Basel Committee on Banking Supervision Basel III A GlobalRegulating Framework for More Resilient Banks and BankingSystems 2010
[22] O E Vasicek ldquoAn equilibrium characterization of the termstructurerdquo Journal of Finance Economics vol 5 pp 177ndash188 1977
[23] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985
[24] G Deelstra M Grasselli and P-F Koehl ldquoOptimal investmentstrategies in the presence of a minimum guaranteerdquo Insurancevol 33 no 1 pp 189ndash207 2003
[25] D Kramkov and W Schachermayer ldquoThe asymptotic elasticityof utility functions and optimal investment in incompletemarketsrdquo The Annals of Applied Probability vol 9 no 3 pp904ndash950 1999
[26] M Jonsson and R Sircar ldquoOptimal investment problems andvolatility homogenization approximationsrdquo inModernMethodsin Scientific Computing and Applications vol 75 of NATOScience Series pp 255ndash281 Kluwer Academic Dodrecht TheNetherlands 2002
[27] D Duffie Dynamic Asset Pricing Theory Princeton UniversityPress 3rd edition 2001
[28] B Oslashksendal Stochastic Differential Equations An IntroductionWith Applications Springer New York NY USA 2000
[29] T Dangl and A Lehar ldquoValue-at-risk vs building block regula-tion in bankingrdquo Journal of Financial Intermediation vol 13 no2 pp 96ndash131 2004
[30] J-P Decamps J-C Rochet and B Roger ldquoThe three pillars ofBasel II optimizing the mixrdquo Journal of Financial Intermedia-tion vol 13 no 2 pp 132ndash155 2004
[31] F Heid D Porath and S Stolz Does Capital Regulation Matterfor Bank Behaviour Evidence for German Savings Banks vol3 of Discussion Paper Series 2 Banking and Financial StudiesDeutsche Bundesbank Research Centre 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
