Research ArticlePortfolio Optimization with Asset-Liability RatioRegulation Constraints

De-Lei Sheng 1 and Peilong Shen2

1School of Applied Mathematics Shanxi University of Finance and Economics Taiyuan 030006 China2School of Finance Shanxi University of Finance and Economics Taiyuan 030006 China

Correspondence should be addressed to De-Lei Sheng tjhbsdl126com

Received 12 June 2019 Accepted 8 February 2020 Published 23 March 2020

Guest Editor iago Christiano Silva

Copyright copy 2020 De-Lei Sheng and Peilong Shen is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

is paper considers both a top regulation bound and a bottom regulation bound imposed on the asset-liability ratio at theregulatory time T to reduce risks of abnormal high-speed growth of asset price within a short period of time (or high investmentleverage) and to mitigate risks of low assetsrsquo return (or a sharp fall) Applying the stochastic optimal control technique aHamiltonndashJacobindashBellman (HJB) equation is derived en the effective investment strategy and the minimum variance areobtained explicitly by using the Lagrange duality method Moreover some numerical examples are provided to verify the ef-fectiveness of our results

1 Introduction

Taking liabilities into the traditional portfolio modelsSharpe and Tint [1] put forward the asset-liability problemfor pension fund management under the mean-varianceframework Kell and Muller [2] point out that liabilitiesaffect the efficient frontier of this asset-liability problem Butearly research on asset-liability problems was limited to thestandard single-period mean-variance criterion Leippoldet al [3] obtain an analytical optimal strategy and efficientfrontier for asset-liability problems by using embeddingtechnique proposed by Li and Ng [4] under a multiperiodmean-variance framework Leippold et al [3] also considerthe multiperiod mean-variance asset-liability problem andshow that the optimal strategy can be decomposed intoorthogonal sets and the efficient frontier spanned by anorthogonal basis of dynamic returns For the continuous-time asset-liability problem Chiu and Li [5] derive theoptimal policy and the mean-variance efficient frontier byapplying the technique of stochastic linear-quadratic con-trol In comparison with the embedding technique and thestochastic linear-quadratic control method the Lagrangeduality method is more convenient to solve the mean-

variance problem Fu et al [6] consider the continuous-timemean-variance portfolio selection under different borrowinginterest rates and lending interest rates with the Lagrangeduality method Pan and Xiao [7] also investigate thecontinuous-time asset-liability management problem byconsidering the stochastic interest rates and inflation risksConsidering a financial market consists of a risk-free bond astock and a derivative Li et al [8] give the optimal in-vestment strategies of a continuous-time mean-varianceasset-liability management in presence of stochastic vola-tility For other related literature studies you can refer toWei and Wang [9] Zhang et al [10] Duarte et al [11] andso on

As everyone knows the assets scale pursued by invest-ment growing too fast over a relatively short period of timewhich indicates that the growth rate increasing faster thanthe normal rate of growth is itself accompanied by highpotential risks e most typical example in practice is thestock market crash which has happened many times almostin all countries with stock markets e earliest stock marketcrash occurred in 1720 the Mississippi Stock Disaster inFrance and the South Sea Stock Disaster in Britaine shareprice of theMississippi company rose from around 500 livres

HindawiComplexityVolume 2020 Article ID 1435356 13 pageshttpsdoiorg10115520201435356

in May 1719 to nearly 10000 livres in February 1720 but itdeclined to 500 livres in September 1721 e stock prices ofSouth Sea Company soar from pound128 in January pound175 inFebruary pound330 in March and pound550 in May and then theshares leaped to pound1000 per share by August 1720 and finallypeaked at this level but soon it plunged and triggered anavalanche of selling As for the most devastating stock marketcrashes of the United States such as the American stockmarket crash in 1929 and the American stock market crash in1987 all ended in a catastrophic decline after a period ofdrastic growthWhen the China stock market crash broke outin 2007 the Shanghai stock exchange composite index surgedfrom around 2700 points to the peak value 6124 points whichtook only a few months But then it went down sharply to thelowest value 1644 points Another China stock market crashin 2015 the Shanghai stock exchange composite index risefrom around 3000 points to 5178 points on June 12 but onlyin the next two months it plunged to around 2800 points

e sustained high-speed growth of stock prices within ashort period of time is actually accumulating enough de-structive energy which may erupt at any time Sophisticatedinvestors will actively sell their risky stocks in time when thestock returns reach a certain high level instead of blindlypursuing much higher returns so as to avoid the sharp dropbefore these assets are successfully sold is high return leveltargeted by these traders to sell assets actively is a practical topbound used in investment management practice and deter-mined by investment managers initiatively which is actuallythe concrete example for the application of top investmentregulation bound in business us it confirms the real ex-istence of the investment regulation top bound in real in-vestment practice e requirement of investment regulationbottom bound is necessary for the pursuit of much higherinvestment returns Only if the assets scale pursued by in-vestment is larger than liability then it can ensure that theinvestment activities are really valuable and vigorous

ese kinds of investment risks of asset price collapseresulting from the abnormally rapid rise within a limitedtime period have extremely destructive which may even bethe real manifestation of systemic financial risks As forsystemic financial risks Guerra et al [12] and Souza et al[13] conduct in-depth and innovative research studies andpropose a novel methodology to measure systemic risk innetworks composed of financial institutions ey define thebankrsquos probability of default and calculate this probabilityusing Mertonrsquos method inspired by Black and Scholes [14]and Merton [15] Interestingly the probability of defaultdefined by Guerra et al [12] and Souza et al [13] can also beused to research the asset-liability management problemwith unbearable collapse risk resulted from the abnormallyrapid rise of asset price within a limited time period Forinstance you can take the horizon on which the entity (firm)may default to be short enough and then large declines inassets or large increases in liabilities can be researchedsimilar to the probability of default calculated by Mertonrsquosmethod However based on different considerations werealize our ideas about the asset-liability managementproblem with unbearable investment risk using a widelyused mathematical method in this paper

Financial leverage needs to be considered because it isubiquitous in practice and the excessive leverage is an issuetackled in the Basel III requirements In the narrow sensefinancial leverage refers to a measure of operating large-scalebusiness with less money which is widely used in variouseconomic activities For example the transaction relying onfutures margin is commonly used in futures markets and themargin rate usually varies between 5 and 15 of the totalcontract value Only a down payment of 5 of the fullcontract value is required and an investor can carry out thefinancial transaction equivalent to 20 times the amount ofthe margin Another example is the way to invest in realestate with installment repayment and investors only needto pay the down payment usually 10 to 30 but they canleverage the investment scales of the total market valueQuite evidently investors generally use smaller self-ownedfunds to operate businesses of great market value with fi-nancial leverage If these assets appreciate investors canimmediately obtain considerable investment returnsHowever once the value of these assets shrinks the losses aremagnified and the corresponding leverage multiples arecatastrophic and unbearable erefore the essence of fi-nancial leverage lies in magnifying the gains and losses tomeet the needs of investors who have insufficient funds butwant to do large-scale businessesere is a large body in theliterature of financial stability which estimates systemic riskand explicitly takes into account excessive leverage Relatedworks can be referred to the study bySilva et al [16] andmany others

However financial markets themselves also have thefunction of enlarging profits and losses As long as investorsinvest in risky assets in financial markets they are actuallyusing the financial market to enlarge their scale of assetswhich is essentially consistent with the core meaning offinancial leverage in narrow sense erefore in a broadsense a financial market itself can be regarded as a financialleverage In other words the financial leverage has alreadybeen used by investors as long as they invest in financialmarkets in pursuit of higher investment returns so it is alsonecessary to consider leverage regulation even if you onlyinvest in ordinary financial markets

Different from the previous research on the modelingmethod of the asset-liability problem both a bottom reg-ulation bound and a top regulation bound are imposed onthe asset-liability ratio to control the variation range so as toprevent the risks of asset price collapse resulted from anabnormal high-speed growth (such as stock market crash)asset shrink and the collapse of investment leverage edynamic process of asset-liability ratio is defined as an assetprocess being divided by a liability process after eliminatingthe influence of inflation A stochastic optimal control modelis formulated under the framework of mean variance withthe variance being minimized given a determined expec-tation of asset-liability ratio at regulatory time T Using theLagrange multiplier method the original problem istransformed into an unconstrained optimization problemand then a HamiltonndashJacobindashBellman (HJB) equation isestablished by adopting technique of stochastic control Atlast using Lagrange duality between the original problem

2 Complexity

and the unconstrained problem the minimum variance andthe effective investment strategy are obtained

e remainder of this paper is organized as followsSection 2 describes the models of asset-liability ratio and theconstrained control problem In Section 3 a Lagrange un-constrained problem is solved by technique of stochasticoptimal control and the results of the original problem areobtained according to Lagrange duality Meanwhile aspecial case is also solved at the end of this section Someinterpretations of main results are presented in Section 4and numerical examples are illustrated in Section 5 At lastSection 6 gives a conclusion of the research work

2 Formulation of the Model

roughout this paper (ΩF P Ft1113864 11138650le tleT) denotes acomplete probability space satisfying the usual condition Afinite constant Tgt 0 represents the preselected investmentregulatory time Ft is the smallest σminus field generated by allrandom information available until time t and all randomvariables and stochastic processes involved in this article areFt measurable for every t isin [0 T]

21 FinancialMarket Similar to the previous research workthis paper considers an inflation-affected financial market incontinuous time which consists of one risk-free bond oneinflation-linked index bond and one risky stock e in-flation rate P(t) can be regarded as the Consumer PriceIndex which is described by a price level process as follows

P(t) exp 1113946t

01113957μs minus

12

1113957σ2s1113874 1113875ds + 1113946t

01113957σsd 1113957W(s)1113896 1113897 (1)

where 1113957μt is the expected inflation rate at time t 1113957σt gt 0 is thevolatility of inflation rate and 1113957W(t) is a standard Brownianmotion on the probability space (ΩF P)

e price process of the risk-free bond is modeled asB(t) e1113957rt where the constant 1113957rge 0 represents the nominalinterest rate

e inflation-linked index bond I(t) tge 0 has the samerisk source with the price level process and thus it can beexpressed as

I(t) exp 1113946t

0r + 1113957μs( 1113857 minus

12

1113957σ2s1113874 1113875ds + 1113946t

01113957σsd 1113957W(s)1113896 1113897 (2)

where r is the real interest rate at time te price process of risky stock is formulated as

S(t) S0 exp 1113946t

01113954μs minus

12

1113954σ2s1113874 1113875ds + 1113946t

01113954σsd 1113954W(s)1113896 1113897 (3)

where 1113954μt is the investment return rate satisfying 1113954μt gt 1113957r and 1113954σt

is the volatility of the risky asset

22 Asset Process and Liability Process A control vector(π0(t) π1(t) π2(t)) represents the investment strategy needto be found where π0(t) denotes the investment share ofrisk-free bond π1(t) represents the investment share ofinflation-linked index bond and π2(t) signifies the invest-ment share of risky stock and they always satisfy theequation π0(t) + π1(t) + π2(t) 1

e companyrsquos accumulated wealth is allowed to investin the financial market thus the asset process can be de-scribed as an investment process Assuming thatπ (π0(t) π1(t) π2(t))1113864 11138650le tleT is an adaptive control var-iable the dynamics of asset process is governed by thefollowing equation

dXπ(t)

Xπ(t) π0(t)

dB(t)

B(t)+ π1(t)

dI(t)

I(t)+ π2(t)

dS(t)

S(t)

1113957r + π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 11138571113858 1113859dt + π1(t)1113957σtd 1113957W(t) + π2(t)1113954σtd 1113954W(t)

Xπ(0) x0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

To avoid the influence of control variable on the liabilityprocess L(t) we also assume the liability process is governedby a geometric Brownian motion as follows

L(t) L0 exp 1113946t

0μs minus

12σ2s1113874 1113875ds + 1113946

t

0σsdW(s)1113896 1113897 L0 gt 0

(5)

e real liability process 1113954L(t) after eliminating inflationis defined as 1113954L(t) L(t)P(t) It is easy to get the followingform of expression using Itorsquos formula

d1113954L(t)

1113954L(t) μt minus 1113957μt + 1113957σ2t1113872 1113873dt minus 1113957σtd 1113957W(t) + σtdW(t)

1113954L(t) L0 exp 1113946t

0μs minus 1113957μs +

12

1113957σ2s minus12σ2s1113874 1113875ds minus 1113946

t

01113957σsd 1113957W(s)1113896

+ 1113946t

0σsdW(s)1113897

(6)

e asset process after eliminating the influence of in-flation is defined as 1113954X

π(t) Xπ(t)P(t) then we get the

following asset process according to Itorsquos formula

Complexity 3

d 1113954Xπ(t)

1113954Xπ(t)

1113957r minus 1113957μt + 1113957σ2t + π1(t) 1113957μt + r minus 1113957r minus 1113957σ2t1113872 11138731113960

+ π2(t) 1113954μt minus 1113957r( 11138571113859dt + π1(t) minus 1( 11138571113957σtd 1113957W(t)

+ π2(t)1113954σtd 1113954W(t)

(7)

23 Asset-Liability Problem with Regulation Constraintse asset-liability ratio is defined as Zπ(t) 1113954X

π(t)1113954L(t)

which describes the ratio between the size of asset andthe size of liability at time t According to Itorsquos formulathe dynamic process of the asset-liability ratio can be givenbydZπ(t)

Zπ(t) 1113957r minus μt minus σ2t1113872 1113873dt minus σtdW(t) + π1(t) 1113957μt + r minus 1113957r( 1113857dt(

+ 1113957σtd 1113957W(t)1113857 + π2(t) 1113954μt minus 1113957r( 1113857dt + 1113954σtd 1113954W(t)1113872 1113873

(8)

In practice the regulation only occurs at certain fixedtime which is called regulatory time e regulatory time isusually selected in advance which provides a time period ofappropriate length but the length may decrease if theregulatory frequency increased In accordance with theneed of practice this paper considers the asset-liabilityproblem with the constraints imposed at the regulatory timeT to find the optimal investment strategy in order to reducethe risk of abnormal high-speed growth of asset price withina short period of time or high investment leverage and alsoto lessen the risk of too low return rate or a sharp fall eirmathematical descriptions of the regulation constraints atthe regulatory time T are formulated as the followinginequality

αltZπ(T)lt β (9)

where the bottom regulation bound αge 1 and the top reg-ulation bound β are two appropriate constants satisfying therequirement βgt α e bottom bound α should not be toosmall otherwise it cannot cover the liability which meansthe investment has failede top bound β should not be toolarge or else a much larger value of β may lead to excessiveleverage multiples and too much investment risks beingpulled in the company In practice it is highly technical todetermine the values of α and β which requires abundant

practical experience and rigorous mathematical calculationand only can be completed by well-trained and experiencedinvestment managers based on the real market quotationand their practical experiences

Let Π denote the set of all admissible strategies π einvestor aims to find an optimal portfolio π isinΠ of thefollowing original problem

minπisin1113937

Var Zπ(T)[ ] E Zπ(T) minus (α + ρ(β minus α))1113858 11138592

st E Zπ(T)[ ] α + ρ(β minus α)gt 0

Zπ(t)satisfy(8) ρ isin (0 1) for π isin 1113937

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)

where ρ is a constant Only if ρ takes an unequivocal valuethen the expectation of Zπ(T) can be fixed at a certain valuebetween the top regulation bound β and the bottom regu-lation bound α

For a determined value of expectation α+ ρ(β minus α) ifthere exists at least one admissible pair (Z(middot) π(middot)) satis-fying E[Zπ(T)] α+ ρ(β minus α) then the problem (10) iscalled feasible Given α+ ρ(β minus α) the optimal strategy πlowastof (10) is called an efficient strategy e pair(Zπlowast(T)Var[Zπlowast(T)]) under this optimal strategy πlowast iscalled an efficient point e set of all efficient points iscalled the efficient frontier Obviously problem (10) is adynamic quadratic convex optimization problem andthus it has an unique solution

3 Solutions of the Problem

In this section an unconstrained optimization problem isderived by the Lagrange multiplier method and then aHamiltonndashJacobindashBellman equation is deduced to solvethe unconstrained problem Next solution of the originalproblem is obtained according to Lagrange duality be-tween the unconstrained problem and the original prob-lem At the end solution of a special case is given forcomparison

31 Solution of an Unconstrained Problem For this convexoptimization problem (10) the equality constraint E[Zπ(T)] α+ ρ(β minus α) can be eliminated by using theLagrange multiplier method so we get a Lagrange dualproblem

minπisin1113937

E Zπ(T) minus (α + ρ(β minus α))1113858 11138592

+ 2λE Zπ(T) minus (α + ρ(β minus α))1113858 1113859

st Zπ(T)satisfy(8) ρ isin (0 1) for π isin1113937

⎧⎪⎨

⎪⎩(11)

where λ isinR is a Lagrange multiplierAfter a simple calculation it yields

E Zπ(T) minus (α + ρ(β minus α))1113858 1113859

2+ 2λE Z

π(T) minus (α + ρ(β minus α))1113858 1113859

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 1113859

2minus λ2

(12)

4 Complexity

Remark 1 e link between problems (10) and (11) isprovided by the Lagrange duality theorem which can bereferred in the study by Luenberger [17] that

minπisin1113937

Var Zπ(T)1113858 1113859

maxλisinR

minπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 1113859

2minus λ21113966 1113967

(13)

us for the fixed constants λ and u problem (11) isequivalent to

minπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 11138592

st Zπ(t) satisfy(8) ρ isin (0 1) for π isin1113937

⎧⎪⎨

⎪⎩(14)

erefore in order to solve problem (11) we only needto solve problem (14) Fortunately problem (14) can besolved by using the stochastic optimal control technology

To solve problem (14) a truncated form beginning attime t is considered here and the corresponding valuefunction is defined as

V(t z) infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

211138681113868111386811138681113868 Zπ(t)z1113876 1113877

(15)

with the boundary condition V(T z) (z minus ((α+

ρ(β minus α)) minus λ))2e controlled infinitesimal generator for any test

function ϕ(t x) and any admissible control π is deduced asfollows

Aπϕ(t x) ϕt + xϕx 1113957r minus μt minus σ2t + π1(t) 1113957μt + r minus 1113957r( 11138571113960

+ π2(t) 1113954μt minus 1113957r( 11138571113859 +x2

2ϕxx σ2t + π2

1(t)1113957σ2t + π22(t)1113954σ2t1113960 1113961

(16)

for any real valued function ϕ(t x) isinC12([0 T] R) andadmissible strategy π where

C12([0 T]R) ψ(t x)|ψ(t middot) is once continuously differentiable on [0 T] andψ1113864

(middot x) is twice continuously differentiable almost surely onR(17)

If V(t z) isinC12([0 T]timesR) then V(t z) satisfies thefollowing HJB equation

infπisin1113937

AπV(t z) 0 (18)

which can be given more specifically as follows

infπisin1113937

Vt + zVz 1113957r minus μt minus σ2t1113872 1113873 +z2

2Vzzσ

2t + zVz π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 11138571113858 1113859 +

z2

2Vzz π2

1(t)1113957σ2t + π22(t)1113954σ2t1113872 1113873 01113896 1113897 (19)

For convenience some simple notations are introducedas follows

δ ≜1113957μt + r minus 1113957r( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t

ϱ(t)≜e(tminus T) δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2t

ω≜(α + ρ(β minus α))

(1 minus 1113954ϱ)

1113954ϱ ≜ ϱ(0) eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2t

ε≜eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)

(20)

Solving equation (19) gives the following importanttheorem

Theorem 1 For the asset-liability management with in-vestment regulation and eliminating inflation the efficientinvestment strategy πlowast (πlowast0 πlowast1 πlowast2 ) corresponding toproblem (11) and problem (14) is given by

πlowast0 1 + δ 1 minus1zξβα(t)1113874 1113875

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ξβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ξβα(t)

zminus 11113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

and the value function is

Complexity 5

V(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( )

minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )

+((α + ρ(β minus α)) minus λ)2ϱ(t)

(22)

where ξβα(t) e(tminus T)(1113957rminus μt)((α + ρ(β minus α)) minus λ)

Proof According to the first-order necessary condition ofextremum equation (19) yields

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

z1113957σ2t

Vz

Vzz

πlowast2(t) minus1113954μt minus 1113957r( 1113857

z1113954σ2t

Vz

Vzz

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)

Plugging the above expressions of πlowast1(t) and πlowast2(t) into(19) the HJB equation becomes

Vt + zVz 1113957r minus μt minus σ2t1113872 1113873 +z2

2Vzzσ

2t minus δ

V2z

2Vzz

0 (24)

Based on the boundary condition V(T z)

(z minus ((α + ρ(β minus α)) minus λ))2 we try a conjecture

W(t z) f(t)z2

+ g(t)z + h(t) (25)

for equation (24) which satisfies

f(T) 1

g(T) minus 2((α + ρ(β minus α)) minus λ)

h(T) ((α + ρ(β minus α)) minus λ)2

(26)

e derivatives of this conjecture W(t z) are given asfollows

Wt fprime(t)z2

+ gprime(t)z + hprime(t)

Wz 2f(t)z + g(t)

Wzz 2f(t)

(27)

Plugging the expressions of Wz and Wzz into (23) theoptimal strategy becomes

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

z1113957σ2tz +

g(t)

2f(t)1113888 1113889

πlowast2(t) minus1113954μt minus 1113957r( 1113857

z1113954σ2tz +

g(t)

2f(t)1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

Substituting these derivatives of W(t z) into (24) theHJB equation becomes

fprime(t)z2

+ gprime(t)z + hprime(t) + 2f(t)z2

+ g(t)z1113872 1113873 1113957r minus μt minus σ2t1113872 1113873

+ z2f(t)σ2t minus δ f(t)z

2+ g(t)z +

g2(t)

4f(t)1113888 1113889 0

(29)

Equation (29) can be split into three ordinary differentialequations as follows

fprime(t) + 2f(t) 1113957r minus μt( 1113857 minus f(t)σ2t minus δf(t) 0

f(T) 1

gprime(t) + g(t) 1113957r minus μt minus σ2t( 1113857 minus δg(t) 0

g(T) minus 2((α + ρ(β minus α)) minus λ)

hprime(t) minusδ4

g(t)2

f(t) 0

h(T) ((α + ρ(β minus α)) minus λ)2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

Solving these ordinary differential equations in (30)yields

f(t) e(tminus T) minus 21113957r+δ+2μt+σ2t( )

g(t) minus 2e(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

h(t) ((α + ρ(β minus α)) minus λ)2ϱ(t)

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)

Plugging the explicit expression of f(t) and g(t) into (28)and using the equality πlowast0(t) + πlowast1(t) + πlowast2(t) 1 the opti-mal strategy is obtained as follows

πlowast0(t) 1 + δ 1 minus e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

z1113888 1113889

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2te

(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2te

(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

6 Complexity

Denoting

ξβα(t)≜e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ) (33)

the optimal strategy can be simplified into

πlowast0(t) 1 + δ 1 minus1zξβα(t)1113874 1113875

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tξβα(t)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tξβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

Meanwhile substituting the expressions of f(t) g(t) andh(t) into W(t z) the explicit solution of HJB equation isobtained as follows

W(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( ) +((α + ρ(β minus α)) minus λ)2

ϱ(t) minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

(35)

e following verification theorem shows the obtainedresults are exactly the optimal strategy and the optimal valuefunction as required

Theorem 2 (Verification Theorem) If W(t z) is a solutionof HJB equation (19) and satisfies the boundary conditionW(T z) (z minus ((α + ρ(β minus α)) minus λ))2 then for all admissi-ble strategies π isin Π V(t z)geW(t z) If πlowast satisfies

πlowastisin arg infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

2|Z

π(t) z1113960 1113961

(36)

then V(t z)W(t z) and πlowast is the optimal investmentstrategy of problem (14)

Proof Proof is similar to that given by Pham [18] Chang[19] and so on so the detail is omitted

32 Solution of the Original Problem e optimal valuefunction of problem (11) is defined as

1113954V 0 z0( 1113857 infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

21113960 1113961 minus λ2

(37)

where z0 Z(0)In this section the Lagrange duality method is used to

find a solution of original optimization problem (10) basedon the obtained results of the unconstrained problem

Theorem 3 For original problem (10) the efficient strategyπlowast (πlowast0 πlowast1 πlowast2 ) is given by

πlowast0 1 minus δζβα(t)

zminus 11113888 1113889

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)

and the optimal value function is given by

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +(ω minus ε)21113954ϱ

minus 2eminus T minus 1113957r+δ+μt+σ2t( )(ω minus ε)z0 minus (ε minus 1113954ϱω)

2

(39)

where

ζβα(t) e(tminus T) 1113957rminus μt( )(ω minus ε) (40)

Proof e optimal value function 1113954V(0 z0) can be ob-tained with the equivalence between problem (11) andproblem (14) By taking z0 Z(0) and setting t 0 inV(t z) ityields

1113954V 0 z0( 1113857 V 0 z0( 1113857 minus λ2 f(0)z20 + g(0)z0 + h(0) minus λ2

eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +((α + ρ(β minus α)) minus λ)

2

1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)z0 minus λ2

(41)

Since (δ + σ2t )gt 0 then eminus T(δ+σ2t ) lt 1 and it yields

eminus T δ+σ2t( )δ + σ2t1113874 1113875lt δ + σ2t (42)

and then

1113954ϱ eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tlt 1 (43)

Note that the optimal value function 1113954V(0 z0) is aquadratic function with respect to λ and 1113954ϱ minus 1 is the coef-ficient of quadratic term

1113954ϱ minus 1 eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tminus 1lt 0 (44)

us the finite maximum value of 1113954V(0 z0) can beobtained at a specific value λlowast and

λlowast eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ) (45)

Applying the Lagrangian duality substituting λlowast into1113954V(0 z0) the minimum variance corresponding to an ar-bitrarily given expectation (α + ρ(β minus α)) is obtained asfollows

Complexity 7

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +

(α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

middot 1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( ) (α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠z0 minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

(46)

Substitute (45) into (21) the optimal investment strategyof problem (10) is given by

πlowast0 1 minus δ1zζβα(t) minus 11113874 1113875

πlowast1 minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tζβα(t)

πlowast2 minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tζβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where

ζβα(t) e(tminus T) 1113957rminus μt( ) (α + ρ(β minus α))

1 minus 1113954ϱminus

eminus T minus 1113957r+δ+μt+σ2t( )z0

1 minus 1113954ϱ⎛⎝ ⎞⎠

(48)

Remark 2 However in the obtained results (47) and (46)above the expected value (α + ρ(β minus α)) of Zπ(t) is not fixedat a determined position but can take any value between αand β depending on the value of ρ isin (0 1) erefore theobtained minimum variance depends on the parametervalue ρ isin (0 1) in the expression of expectation(α + ρ(β minus α)) Next let us turn our attention to the pa-rameter ρ and determine the optimal parameter value ρlowast forminimizing the variance with respect to ρ as well eoptimal ρlowast to achieve the minimum of Var[Zπlowast(T)] is givenas follows

ρlowast eminus T minus 1113957r+δ+μt+σ2t( )z01113872 11138731113954ϱ1113872 1113873 minus α

(β minus α) (49)

and the expected value of Zπ(t) corresponding to the optimalparameter value ρlowast is given by

α + ρlowast(β minus α) z01113954ϱ

eminus T minus 1113957r+δ+μt+σ2t( ) (50)

Substituting ρlowast into (47) we can get the optimal strategycorresponding to the optimal value α+ ρlowast minus α) of expecta-tion Moreover plugging the expression of ρlowast into theformula of Var[Zπlowast(T)] the optimal minimum varianceVar[Zπlowast ρlowast(T)] can also be determined corresponding to theuniquely optimal value (z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t ) of theexpectation

33 Special Case e investment regulation is also im-portant for an investment process without liabilities Letμt σt 0 the original problem degenerates to the case of noliability and the process Zπ(t) becomes the dynamics process1113954Xπ(t) as follows

d 1113954Xπ(t) 1113957r + π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 1113857 minus π1(t)1113957σ2t minus 1113957μt1113960 1113961

middot 1113954Xπ(t)dt + π1(t) minus 1( 11138571113957σt

1113954Xπ(t)d 1113957W(t) + π2(t)1113954σt

middot 1113954Xπ(t)d 1113954W(t)

(51)

Meanwhile the regulation bound constraints also de-generate into a restriction on the investment dynamicsprocess after eliminating inflation

αlt 1113954Xπ(T)lt β (52)

Proposition 1 For the special case μt σt 0 the minimumvariance of Zπ(t) is

Var Zlowast(T)1113858 1113859 e

minus T1113957δ (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

minus 2eminus T minus r+1113957δ+1113957σ

2t( 1113857 (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠1113954x0

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0 minus eminus T1113957δ(α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

+ eminus T minus 2r+1113957δ+21113957σ

2t( 11138571113954x

20

(53)

and the optimal strategy of the special case is given as follows

8 Complexity

πlowast1(t) 1 minus1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

1113957σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast0(t) 1113957δ 1 minuse(tminus T) minus r+1113957δ+1113957σ

2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(54)

where

1113957δ 1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t (55)

Proof e results can be obtained by the same calculationprocess as the original problem so the details are omittedhere

Remark 3 e expected value (α + ρ(β minus α)) is not fixed ata determined position in the above proposition but it cantake any value between α and β depending on the value ofρ isin (0 1) erefore the obtained minimum variancedepends on the parameter value ρ isin (0 1) Using the first-order necessary condition of extremum value for Var[Zlowast(T)] with respect to the parameter ρ the optimal valueof parameter ρ is given by

ρlowast eminus T minus r+1113957σ

2t( 11138571113954x0 minus α

(β minus α) (56)

and the corresponding optimal expectation

α + ρlowast(β minus α)( 1113857 eminus T minus r+1113957σ

2t( 11138571113954x0 (57)

Substituting ρlowast into the optimal strategy (πlowast0(t)

πlowast1(t) πlowast2(t)) the optimal strategy corresponding to theoptimal value α+ ρlowast(β minus α) of expectation can be obtainedMeanwhile plugging the expression of ρlowast into Var[Zπlowast(t)]the optimal minimum variance Var[Zπlowast ρlowast(T)] can also bedetermined It is worth mentioning that the optimalminimum variance for this special case with no liabilityis zero under the uniquely optimal expectation(z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t )

4 Interpretations of the Main Results

e expressions of the obtained results are so abstract that itis necessary to give some interpretations for the main resultsto clarify their specific meaning especially for the invest-ment share π1(t) of the inflation-linked index bond and theinvestment share π2(t) of the risky stock

Scrutinizing features of the expressions π1(t) and π2(t)

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

(58)

where ζβα(t) e(tminus T)(1113957rminus μt)(ω minus ε) ω (α + ρ(β minus α))(1 minus 1113954ϱ) and ε (eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) it is easy to findthat understanding implications of these simplified nota-tions δ 1113954ϱω ε and ζβα(t) is the key point Subsequently theinterpretations will start from descriptions of these sim-plified notations

In notation δ ((1113957μt + r minus 1113957r)21113957σ2t ) + ((1113954μt minus 1113957r)21113954σ2t ) theone part (1113957μt + r minus 1113957r)21113957σ2t is the risk compensation rate ofinflation-linked index bond and the other part (1113954μt minus 1113957r)21113954σ2t isthe risk compensation rate of risky stock us δ synthet-ically reflects the risk compensation of two risky assets

Obviously eminus T(δ+σ2t ) lt 1 so the notation1113954ϱ (eminus T(δ+σ2t )δ + σ2t )(δ + σ2t ) is less than 1 Notation 1113954ϱ givesa ratio of one sum with the risk compensation rate δ beingconverted by eminus T(δ+σ2t ) divided by the other sum in which therisk compensation rate δ has no conversion

e notation ω (α + ρ(β minus α))(1 minus 1113954ϱ) is an expressionof top regulation bound β and bottom regulation bound αwhere α+ ρ(β minus α) is a value of the expectation E[Zπ(T)]Because ρ takes a determined value the changes of regu-lation bounds directly result in the changes of ω ereforeω is the key point reflecting the influences of both regulationbounds on investment strategy

In notation ε the parameter z0 has the most obviouseconomic connotation since it represents the initial value ofasset-liability ratio at time 0 e expression(eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) gives a value resulted from theinitial asset-liability ratio z0 influenced by several differentfactors (return rates and volatilities) and then converted atthe constant relative rate 1113954ϱ

e following expression

ζβα(t)

z

e(tminus T) 1113957rminus μt( )

(1 minus 1113954ϱ)middot

(α + ρ(β minus α)) minus eminus T minus 1113957r+δ+μt+σ2t( )z01113872 1113873

z

(59)

is key to understand the optimal strategy (πlowast0 πlowast1 πlowast2 ) edenominator z of this fraction ζβα(t)z represents the currentlevel of asset-liability ratio at time t

Complexity 9

But the numerator ζβα(t) of this fraction is morecomplicated e one multiplier ((α + ρ(β minus α))minus

z0eminus T(minus 1113957r+δ+μ+1113957σ

2t )) is a difference between the expected value

(α + ρ(β minus α)) of Z(T) and the converted valueeminus T(minus 1113957r+δ+μt+σ2t )z0 of the initial value z0 which characterizesthe distance between initial value affected by various eco-nomic factors and the expectation of Z(T) e othermultiplier e(tminus T)(1113957rminus μt)(1 minus 1113954ϱ) is a converted value compre-hensively influenced by several parameters embodying thestates of financial market

In addition it is easy to see that if the relative rate ζβα(t)zis larger than 1 πlowast1(t) ((1113957μt + r minus 1113957r)1113957σ2t )((ζβα(t)z) minus 1) andπlowast2(t) ((1113954μt minus 1113957r)1113954σ2t )(ζβα(t)z minus 1) are positive otherwiseboth will be negative Since πlowast0(t) 1 minus πlowast1(t) minus πlowast2(t) itindicates that πlowast0(t) 1 + δ(1 minus (1z)ζβα(t)) can be deter-mined as long as the investment shares of risky assets havebeen determined so the adjustment of risk-free asset isusually passive If the relative rate ζβα(t)z is greater than 1the investment shares on risk-free asset can be ensuredwithin a much better range between 0 and 1 ese ob-servations can serve as an important reference for deter-mining both the top regulation bound β and the bottomregulation bound α

5 Numerical Examples

is section discusses the variation tendency of optimalstrategies πlowast1 and πlowast2 varying with some important parame-ters mainly focusing on the current value z of asset-liabilityratio the top regulation bound β the bottom regulationbound α and the parameter ρ e values of parameters aregiven in Table 1 unless otherwise specified other values forthe same variable can be consulted from the correspondinggraphic legend

First let top regulation bound β be fixed if the bottomregulation bound takes much higher values then the in-vestment shares of both inflation-linked index bond and riskystock increase significantly ese variety trends can be ob-served intuitively from Figures 1 and 2 respectively Mean-while Figures 1 and 2 also show that the appropriate bottomregulation bound should be taken at one value between 1 and2 for an economic environment same as this exampleOtherwise the investment shares on risky assets may increasetoomuch but the investment share on risk-free asset becomesnegative which may result in leverage risk increases rapidly

Next let bottom regulation bound α be fixed if the topregulation bound β takes much higher value then the invest-ment shares of both inflation-linked index bond and risky stockshould also increase significantly e variety trends can beperceived from Figures 3 and 4 respectively MeanwhileFigures 3 and 4 also show that the appropriate top regulationbound should be taken at one value between 6 and 7 for aneconomic environment same as this example Otherwise theinvestment shares on risky assets increase too much but theinvestment share on risk-free asset becomes negative whichalso result in leverage risk increases rapidly

As can be seen from Figures 3ndash6 taking the larger valueof either the top regulation bound or the bottom regulation

bound means allowing a wider volatility range of investmentreturns which may lead to the result that assets with greaterinvestment risk are allowed to invest or greater market risksare incorporated into the wealth process In addition theabove numerical analysis roughly gives a valuable referencefor selecting an appropriate range of top regulation boundand bottom regulation bound according to the relationshipbetween the reasonable values of investment strategy andregulation bounds for an economic environment set byvalues of parameters

Table 1 Values of the parameters

1113957μ 008 r 001 1113957σ 02 α 1 β 7 z0 3 z 51113957r 005 T 6 1113954μ 015 1113954σ 06 μ 003 σ 01 ρ 05

02

04

06

08

10

(π1)

lowast

π1lowast changes with α when top bound β fixed

2 3 4 5 61t

α = 1α = 18α = 26

Figure 1 Values of πlowast1 change with t for different bottom bound α

(π2)

lowast

π2lowast changes with α when top bound β fixed

005

010

015

020

2 3 4 5 61t

α = 1α = 2α = 3

Figure 2 Values of πlowast2 change with t for different bottom bound α

10 Complexity

In fact the regulatory bound cannot be determinedarbitrarily Both the top regulatory bound and bottomregulatory bound determine the fluctuation range that de-cision-making needs to control Too narrow fluctuationrange of investment returns results in loss of great invest-ment opportunities and makes investment in the financialmarket meaningless If the regulatory bound is too large thepotential huge risks cannot be dealt with effectively when theinvestment return soars fast which is almost equivalent tosituations without regulatory bounds and also lose thesignificance of using regulatory bounds It is hard to de-termine the regulatory bounds only by quantitative calcu-lation Once there is a model error it may result in seriousdeviation between theory and practice in actual manage-ment thus the determination of upper and lower limitsshould not be purely theoretical Nevertheless it should be

noted that advanced statistical analysis and maturelypractical experience of personnel are essential and of greatadvantage for determining an exact value of α or β

Assuming that both top regulation bound β and bottomregulation bound α are determined the impact of asset-li-ability ratio z at time t on investment strategy can be ob-served in Figures 5 and 6 If the current asset-liability ratio zat time t takes much greater value then the investmentshares of both inflation-linked index bond and risky stockshould be much lower which shows that increasing shares ofinvestment on risk-free asset is an important measure toreduce investment risk For the numerical examples shownin Figures 5 and 6 the regulation bounds being fixed at α 1and β 7 the most appropriate asset-liability ratio shouldtake values around 5 which is much more advantageous forthe company

e impact of parameter ρ on investment strategy can beobserved in Figures 7 and 8 If the parameter ρ takes a muchgreater value then the investment shares of risky assets mustincrease correspondingly Since the top regulation bound βand the bottom regulation bound α are predetermined theexpected value α+ ρ(β minus α) of asset-liability ratio entirelydepends on the value of parameter ρerefore the variationtendency of investment shares πlowast1(t) and πlowast2(t) with pa-rameter ρ actually illustrates the trend of investment shareschanging with the expectation of asset-liability ratio Z(T) Inone word the greater the expectation of Z(T) the muchhigher the shares of the companyrsquos wealth to be invested onrisky assets

6 Conclusion

Some empirical studies in the banking literature deal withthe effects of prolonged periods of low interest rates in theeconomy on risk-taking real effects on the real sector andalso on financial stability and so on as given by Chaudron[20] Bikker and Vervliet [21] However this paper uses

π1lowast changes with β when bottom bound α fixed

02

04

06

08

2 3 4 5 61t

β = 6β = 7β = 8

(π1)

lowast

Figure 3 Values of πlowast1 change with t for different top bound β

π2lowast changes with β when bottom bound α fixed

01

02

03

04

2 3 4 5 61t

β = 6β = 8β = 10

(π2)

lowast

Figure 4 Values of πlowast2 change with t for different top bound β

π1lowast changes with t for different z

02

04

06

08

2 3 4 5 61t

z = 4z = 5z = 6

(π1)

lowast

Figure 5 Values of πlowast1 change with t for different current asset-liability ratio z

Complexity 11

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

and the unconstrained problem the minimum variance andthe effective investment strategy are obtained

e remainder of this paper is organized as followsSection 2 describes the models of asset-liability ratio and theconstrained control problem In Section 3 a Lagrange un-constrained problem is solved by technique of stochasticoptimal control and the results of the original problem areobtained according to Lagrange duality Meanwhile aspecial case is also solved at the end of this section Someinterpretations of main results are presented in Section 4and numerical examples are illustrated in Section 5 At lastSection 6 gives a conclusion of the research work

2 Formulation of the Model

roughout this paper (ΩF P Ft1113864 11138650le tleT) denotes acomplete probability space satisfying the usual condition Afinite constant Tgt 0 represents the preselected investmentregulatory time Ft is the smallest σminus field generated by allrandom information available until time t and all randomvariables and stochastic processes involved in this article areFt measurable for every t isin [0 T]

21 FinancialMarket Similar to the previous research workthis paper considers an inflation-affected financial market incontinuous time which consists of one risk-free bond oneinflation-linked index bond and one risky stock e in-flation rate P(t) can be regarded as the Consumer PriceIndex which is described by a price level process as follows

P(t) exp 1113946t

01113957μs minus

12

1113957σ2s1113874 1113875ds + 1113946t

01113957σsd 1113957W(s)1113896 1113897 (1)

where 1113957μt is the expected inflation rate at time t 1113957σt gt 0 is thevolatility of inflation rate and 1113957W(t) is a standard Brownianmotion on the probability space (ΩF P)

e price process of the risk-free bond is modeled asB(t) e1113957rt where the constant 1113957rge 0 represents the nominalinterest rate

e inflation-linked index bond I(t) tge 0 has the samerisk source with the price level process and thus it can beexpressed as

I(t) exp 1113946t

0r + 1113957μs( 1113857 minus

12

1113957σ2s1113874 1113875ds + 1113946t

01113957σsd 1113957W(s)1113896 1113897 (2)

where r is the real interest rate at time te price process of risky stock is formulated as

S(t) S0 exp 1113946t

01113954μs minus

12

1113954σ2s1113874 1113875ds + 1113946t

01113954σsd 1113954W(s)1113896 1113897 (3)

where 1113954μt is the investment return rate satisfying 1113954μt gt 1113957r and 1113954σt

is the volatility of the risky asset

22 Asset Process and Liability Process A control vector(π0(t) π1(t) π2(t)) represents the investment strategy needto be found where π0(t) denotes the investment share ofrisk-free bond π1(t) represents the investment share ofinflation-linked index bond and π2(t) signifies the invest-ment share of risky stock and they always satisfy theequation π0(t) + π1(t) + π2(t) 1

e companyrsquos accumulated wealth is allowed to investin the financial market thus the asset process can be de-scribed as an investment process Assuming thatπ (π0(t) π1(t) π2(t))1113864 11138650le tleT is an adaptive control var-iable the dynamics of asset process is governed by thefollowing equation

dXπ(t)

Xπ(t) π0(t)

dB(t)

B(t)+ π1(t)

dI(t)

I(t)+ π2(t)

dS(t)

S(t)

1113957r + π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 11138571113858 1113859dt + π1(t)1113957σtd 1113957W(t) + π2(t)1113954σtd 1113954W(t)

Xπ(0) x0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

To avoid the influence of control variable on the liabilityprocess L(t) we also assume the liability process is governedby a geometric Brownian motion as follows

L(t) L0 exp 1113946t

0μs minus

12σ2s1113874 1113875ds + 1113946

t

0σsdW(s)1113896 1113897 L0 gt 0

(5)

e real liability process 1113954L(t) after eliminating inflationis defined as 1113954L(t) L(t)P(t) It is easy to get the followingform of expression using Itorsquos formula

d1113954L(t)

1113954L(t) μt minus 1113957μt + 1113957σ2t1113872 1113873dt minus 1113957σtd 1113957W(t) + σtdW(t)

1113954L(t) L0 exp 1113946t

0μs minus 1113957μs +

12

1113957σ2s minus12σ2s1113874 1113875ds minus 1113946

t

01113957σsd 1113957W(s)1113896

+ 1113946t

0σsdW(s)1113897

(6)

e asset process after eliminating the influence of in-flation is defined as 1113954X

π(t) Xπ(t)P(t) then we get the

following asset process according to Itorsquos formula

Complexity 3

d 1113954Xπ(t)

1113954Xπ(t)

1113957r minus 1113957μt + 1113957σ2t + π1(t) 1113957μt + r minus 1113957r minus 1113957σ2t1113872 11138731113960

+ π2(t) 1113954μt minus 1113957r( 11138571113859dt + π1(t) minus 1( 11138571113957σtd 1113957W(t)

+ π2(t)1113954σtd 1113954W(t)

(7)

23 Asset-Liability Problem with Regulation Constraintse asset-liability ratio is defined as Zπ(t) 1113954X

π(t)1113954L(t)

which describes the ratio between the size of asset andthe size of liability at time t According to Itorsquos formulathe dynamic process of the asset-liability ratio can be givenbydZπ(t)

Zπ(t) 1113957r minus μt minus σ2t1113872 1113873dt minus σtdW(t) + π1(t) 1113957μt + r minus 1113957r( 1113857dt(

+ 1113957σtd 1113957W(t)1113857 + π2(t) 1113954μt minus 1113957r( 1113857dt + 1113954σtd 1113954W(t)1113872 1113873

(8)

In practice the regulation only occurs at certain fixedtime which is called regulatory time e regulatory time isusually selected in advance which provides a time period ofappropriate length but the length may decrease if theregulatory frequency increased In accordance with theneed of practice this paper considers the asset-liabilityproblem with the constraints imposed at the regulatory timeT to find the optimal investment strategy in order to reducethe risk of abnormal high-speed growth of asset price withina short period of time or high investment leverage and alsoto lessen the risk of too low return rate or a sharp fall eirmathematical descriptions of the regulation constraints atthe regulatory time T are formulated as the followinginequality

αltZπ(T)lt β (9)

where the bottom regulation bound αge 1 and the top reg-ulation bound β are two appropriate constants satisfying therequirement βgt α e bottom bound α should not be toosmall otherwise it cannot cover the liability which meansthe investment has failede top bound β should not be toolarge or else a much larger value of β may lead to excessiveleverage multiples and too much investment risks beingpulled in the company In practice it is highly technical todetermine the values of α and β which requires abundant

practical experience and rigorous mathematical calculationand only can be completed by well-trained and experiencedinvestment managers based on the real market quotationand their practical experiences

Let Π denote the set of all admissible strategies π einvestor aims to find an optimal portfolio π isinΠ of thefollowing original problem

minπisin1113937

Var Zπ(T)[ ] E Zπ(T) minus (α + ρ(β minus α))1113858 11138592

st E Zπ(T)[ ] α + ρ(β minus α)gt 0

Zπ(t)satisfy(8) ρ isin (0 1) for π isin 1113937

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)

where ρ is a constant Only if ρ takes an unequivocal valuethen the expectation of Zπ(T) can be fixed at a certain valuebetween the top regulation bound β and the bottom regu-lation bound α

For a determined value of expectation α+ ρ(β minus α) ifthere exists at least one admissible pair (Z(middot) π(middot)) satis-fying E[Zπ(T)] α+ ρ(β minus α) then the problem (10) iscalled feasible Given α+ ρ(β minus α) the optimal strategy πlowastof (10) is called an efficient strategy e pair(Zπlowast(T)Var[Zπlowast(T)]) under this optimal strategy πlowast iscalled an efficient point e set of all efficient points iscalled the efficient frontier Obviously problem (10) is adynamic quadratic convex optimization problem andthus it has an unique solution

3 Solutions of the Problem

In this section an unconstrained optimization problem isderived by the Lagrange multiplier method and then aHamiltonndashJacobindashBellman equation is deduced to solvethe unconstrained problem Next solution of the originalproblem is obtained according to Lagrange duality be-tween the unconstrained problem and the original prob-lem At the end solution of a special case is given forcomparison

31 Solution of an Unconstrained Problem For this convexoptimization problem (10) the equality constraint E[Zπ(T)] α+ ρ(β minus α) can be eliminated by using theLagrange multiplier method so we get a Lagrange dualproblem

minπisin1113937

E Zπ(T) minus (α + ρ(β minus α))1113858 11138592

+ 2λE Zπ(T) minus (α + ρ(β minus α))1113858 1113859

st Zπ(T)satisfy(8) ρ isin (0 1) for π isin1113937

⎧⎪⎨

⎪⎩(11)

where λ isinR is a Lagrange multiplierAfter a simple calculation it yields

E Zπ(T) minus (α + ρ(β minus α))1113858 1113859

2+ 2λE Z

π(T) minus (α + ρ(β minus α))1113858 1113859

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 1113859

2minus λ2

(12)

4 Complexity

Remark 1 e link between problems (10) and (11) isprovided by the Lagrange duality theorem which can bereferred in the study by Luenberger [17] that

minπisin1113937

Var Zπ(T)1113858 1113859

maxλisinR

minπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 1113859

2minus λ21113966 1113967

(13)

us for the fixed constants λ and u problem (11) isequivalent to

minπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 11138592

st Zπ(t) satisfy(8) ρ isin (0 1) for π isin1113937

⎧⎪⎨

⎪⎩(14)

erefore in order to solve problem (11) we only needto solve problem (14) Fortunately problem (14) can besolved by using the stochastic optimal control technology

To solve problem (14) a truncated form beginning attime t is considered here and the corresponding valuefunction is defined as

V(t z) infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

211138681113868111386811138681113868 Zπ(t)z1113876 1113877

(15)

with the boundary condition V(T z) (z minus ((α+

ρ(β minus α)) minus λ))2e controlled infinitesimal generator for any test

function ϕ(t x) and any admissible control π is deduced asfollows

Aπϕ(t x) ϕt + xϕx 1113957r minus μt minus σ2t + π1(t) 1113957μt + r minus 1113957r( 11138571113960

+ π2(t) 1113954μt minus 1113957r( 11138571113859 +x2

2ϕxx σ2t + π2

1(t)1113957σ2t + π22(t)1113954σ2t1113960 1113961

(16)

for any real valued function ϕ(t x) isinC12([0 T] R) andadmissible strategy π where

C12([0 T]R) ψ(t x)|ψ(t middot) is once continuously differentiable on [0 T] andψ1113864

(middot x) is twice continuously differentiable almost surely onR(17)

If V(t z) isinC12([0 T]timesR) then V(t z) satisfies thefollowing HJB equation

infπisin1113937

AπV(t z) 0 (18)

which can be given more specifically as follows

infπisin1113937

Vt + zVz 1113957r minus μt minus σ2t1113872 1113873 +z2

2Vzzσ

2t + zVz π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 11138571113858 1113859 +

z2

2Vzz π2

1(t)1113957σ2t + π22(t)1113954σ2t1113872 1113873 01113896 1113897 (19)

For convenience some simple notations are introducedas follows

δ ≜1113957μt + r minus 1113957r( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t

ϱ(t)≜e(tminus T) δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2t

ω≜(α + ρ(β minus α))

(1 minus 1113954ϱ)

1113954ϱ ≜ ϱ(0) eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2t

ε≜eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)

(20)

Solving equation (19) gives the following importanttheorem

Theorem 1 For the asset-liability management with in-vestment regulation and eliminating inflation the efficientinvestment strategy πlowast (πlowast0 πlowast1 πlowast2 ) corresponding toproblem (11) and problem (14) is given by

πlowast0 1 + δ 1 minus1zξβα(t)1113874 1113875

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ξβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ξβα(t)

zminus 11113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

and the value function is

Complexity 5

V(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( )

minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )

+((α + ρ(β minus α)) minus λ)2ϱ(t)

(22)

where ξβα(t) e(tminus T)(1113957rminus μt)((α + ρ(β minus α)) minus λ)

Proof According to the first-order necessary condition ofextremum equation (19) yields

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

z1113957σ2t

Vz

Vzz

πlowast2(t) minus1113954μt minus 1113957r( 1113857

z1113954σ2t

Vz

Vzz

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)

Plugging the above expressions of πlowast1(t) and πlowast2(t) into(19) the HJB equation becomes

Vt + zVz 1113957r minus μt minus σ2t1113872 1113873 +z2

2Vzzσ

2t minus δ

V2z

2Vzz

0 (24)

Based on the boundary condition V(T z)

(z minus ((α + ρ(β minus α)) minus λ))2 we try a conjecture

W(t z) f(t)z2

+ g(t)z + h(t) (25)

for equation (24) which satisfies

f(T) 1

g(T) minus 2((α + ρ(β minus α)) minus λ)

h(T) ((α + ρ(β minus α)) minus λ)2

(26)

e derivatives of this conjecture W(t z) are given asfollows

Wt fprime(t)z2

+ gprime(t)z + hprime(t)

Wz 2f(t)z + g(t)

Wzz 2f(t)

(27)

Plugging the expressions of Wz and Wzz into (23) theoptimal strategy becomes

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

z1113957σ2tz +

g(t)

2f(t)1113888 1113889

πlowast2(t) minus1113954μt minus 1113957r( 1113857

z1113954σ2tz +

g(t)

2f(t)1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

Substituting these derivatives of W(t z) into (24) theHJB equation becomes

fprime(t)z2

+ gprime(t)z + hprime(t) + 2f(t)z2

+ g(t)z1113872 1113873 1113957r minus μt minus σ2t1113872 1113873

+ z2f(t)σ2t minus δ f(t)z

2+ g(t)z +

g2(t)

4f(t)1113888 1113889 0

(29)

Equation (29) can be split into three ordinary differentialequations as follows

fprime(t) + 2f(t) 1113957r minus μt( 1113857 minus f(t)σ2t minus δf(t) 0

f(T) 1

gprime(t) + g(t) 1113957r minus μt minus σ2t( 1113857 minus δg(t) 0

g(T) minus 2((α + ρ(β minus α)) minus λ)

hprime(t) minusδ4

g(t)2

f(t) 0

h(T) ((α + ρ(β minus α)) minus λ)2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

Solving these ordinary differential equations in (30)yields

f(t) e(tminus T) minus 21113957r+δ+2μt+σ2t( )

g(t) minus 2e(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

h(t) ((α + ρ(β minus α)) minus λ)2ϱ(t)

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)

Plugging the explicit expression of f(t) and g(t) into (28)and using the equality πlowast0(t) + πlowast1(t) + πlowast2(t) 1 the opti-mal strategy is obtained as follows

πlowast0(t) 1 + δ 1 minus e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

z1113888 1113889

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2te

(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2te

(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

6 Complexity

Denoting

ξβα(t)≜e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ) (33)

the optimal strategy can be simplified into

πlowast0(t) 1 + δ 1 minus1zξβα(t)1113874 1113875

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tξβα(t)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tξβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

Meanwhile substituting the expressions of f(t) g(t) andh(t) into W(t z) the explicit solution of HJB equation isobtained as follows

W(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( ) +((α + ρ(β minus α)) minus λ)2

ϱ(t) minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

(35)

e following verification theorem shows the obtainedresults are exactly the optimal strategy and the optimal valuefunction as required

Theorem 2 (Verification Theorem) If W(t z) is a solutionof HJB equation (19) and satisfies the boundary conditionW(T z) (z minus ((α + ρ(β minus α)) minus λ))2 then for all admissi-ble strategies π isin Π V(t z)geW(t z) If πlowast satisfies

πlowastisin arg infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

2|Z

π(t) z1113960 1113961

(36)

then V(t z)W(t z) and πlowast is the optimal investmentstrategy of problem (14)

Proof Proof is similar to that given by Pham [18] Chang[19] and so on so the detail is omitted

32 Solution of the Original Problem e optimal valuefunction of problem (11) is defined as

1113954V 0 z0( 1113857 infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

21113960 1113961 minus λ2

(37)

where z0 Z(0)In this section the Lagrange duality method is used to

find a solution of original optimization problem (10) basedon the obtained results of the unconstrained problem

Theorem 3 For original problem (10) the efficient strategyπlowast (πlowast0 πlowast1 πlowast2 ) is given by

πlowast0 1 minus δζβα(t)

zminus 11113888 1113889

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)

and the optimal value function is given by

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +(ω minus ε)21113954ϱ

minus 2eminus T minus 1113957r+δ+μt+σ2t( )(ω minus ε)z0 minus (ε minus 1113954ϱω)

2

(39)

where

ζβα(t) e(tminus T) 1113957rminus μt( )(ω minus ε) (40)

Proof e optimal value function 1113954V(0 z0) can be ob-tained with the equivalence between problem (11) andproblem (14) By taking z0 Z(0) and setting t 0 inV(t z) ityields

1113954V 0 z0( 1113857 V 0 z0( 1113857 minus λ2 f(0)z20 + g(0)z0 + h(0) minus λ2

eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +((α + ρ(β minus α)) minus λ)

2

1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)z0 minus λ2

(41)

Since (δ + σ2t )gt 0 then eminus T(δ+σ2t ) lt 1 and it yields

eminus T δ+σ2t( )δ + σ2t1113874 1113875lt δ + σ2t (42)

and then

1113954ϱ eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tlt 1 (43)

Note that the optimal value function 1113954V(0 z0) is aquadratic function with respect to λ and 1113954ϱ minus 1 is the coef-ficient of quadratic term

1113954ϱ minus 1 eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tminus 1lt 0 (44)

us the finite maximum value of 1113954V(0 z0) can beobtained at a specific value λlowast and

λlowast eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ) (45)

Applying the Lagrangian duality substituting λlowast into1113954V(0 z0) the minimum variance corresponding to an ar-bitrarily given expectation (α + ρ(β minus α)) is obtained asfollows

Complexity 7

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +

(α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

middot 1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( ) (α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠z0 minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

(46)

Substitute (45) into (21) the optimal investment strategyof problem (10) is given by

πlowast0 1 minus δ1zζβα(t) minus 11113874 1113875

πlowast1 minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tζβα(t)

πlowast2 minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tζβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where

ζβα(t) e(tminus T) 1113957rminus μt( ) (α + ρ(β minus α))

1 minus 1113954ϱminus

eminus T minus 1113957r+δ+μt+σ2t( )z0

1 minus 1113954ϱ⎛⎝ ⎞⎠

(48)

Remark 2 However in the obtained results (47) and (46)above the expected value (α + ρ(β minus α)) of Zπ(t) is not fixedat a determined position but can take any value between αand β depending on the value of ρ isin (0 1) erefore theobtained minimum variance depends on the parametervalue ρ isin (0 1) in the expression of expectation(α + ρ(β minus α)) Next let us turn our attention to the pa-rameter ρ and determine the optimal parameter value ρlowast forminimizing the variance with respect to ρ as well eoptimal ρlowast to achieve the minimum of Var[Zπlowast(T)] is givenas follows

ρlowast eminus T minus 1113957r+δ+μt+σ2t( )z01113872 11138731113954ϱ1113872 1113873 minus α

(β minus α) (49)

and the expected value of Zπ(t) corresponding to the optimalparameter value ρlowast is given by

α + ρlowast(β minus α) z01113954ϱ

eminus T minus 1113957r+δ+μt+σ2t( ) (50)

Substituting ρlowast into (47) we can get the optimal strategycorresponding to the optimal value α+ ρlowast minus α) of expecta-tion Moreover plugging the expression of ρlowast into theformula of Var[Zπlowast(T)] the optimal minimum varianceVar[Zπlowast ρlowast(T)] can also be determined corresponding to theuniquely optimal value (z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t ) of theexpectation

33 Special Case e investment regulation is also im-portant for an investment process without liabilities Letμt σt 0 the original problem degenerates to the case of noliability and the process Zπ(t) becomes the dynamics process1113954Xπ(t) as follows

d 1113954Xπ(t) 1113957r + π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 1113857 minus π1(t)1113957σ2t minus 1113957μt1113960 1113961

middot 1113954Xπ(t)dt + π1(t) minus 1( 11138571113957σt

1113954Xπ(t)d 1113957W(t) + π2(t)1113954σt

middot 1113954Xπ(t)d 1113954W(t)

(51)

Meanwhile the regulation bound constraints also de-generate into a restriction on the investment dynamicsprocess after eliminating inflation

αlt 1113954Xπ(T)lt β (52)

Proposition 1 For the special case μt σt 0 the minimumvariance of Zπ(t) is

Var Zlowast(T)1113858 1113859 e

minus T1113957δ (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

minus 2eminus T minus r+1113957δ+1113957σ

2t( 1113857 (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠1113954x0

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0 minus eminus T1113957δ(α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

+ eminus T minus 2r+1113957δ+21113957σ

2t( 11138571113954x

20

(53)

and the optimal strategy of the special case is given as follows

8 Complexity

πlowast1(t) 1 minus1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

1113957σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast0(t) 1113957δ 1 minuse(tminus T) minus r+1113957δ+1113957σ

2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(54)

where

1113957δ 1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t (55)

Proof e results can be obtained by the same calculationprocess as the original problem so the details are omittedhere

Remark 3 e expected value (α + ρ(β minus α)) is not fixed ata determined position in the above proposition but it cantake any value between α and β depending on the value ofρ isin (0 1) erefore the obtained minimum variancedepends on the parameter value ρ isin (0 1) Using the first-order necessary condition of extremum value for Var[Zlowast(T)] with respect to the parameter ρ the optimal valueof parameter ρ is given by

ρlowast eminus T minus r+1113957σ

2t( 11138571113954x0 minus α

(β minus α) (56)

and the corresponding optimal expectation

α + ρlowast(β minus α)( 1113857 eminus T minus r+1113957σ

2t( 11138571113954x0 (57)

Substituting ρlowast into the optimal strategy (πlowast0(t)

πlowast1(t) πlowast2(t)) the optimal strategy corresponding to theoptimal value α+ ρlowast(β minus α) of expectation can be obtainedMeanwhile plugging the expression of ρlowast into Var[Zπlowast(t)]the optimal minimum variance Var[Zπlowast ρlowast(T)] can also bedetermined It is worth mentioning that the optimalminimum variance for this special case with no liabilityis zero under the uniquely optimal expectation(z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t )

4 Interpretations of the Main Results

e expressions of the obtained results are so abstract that itis necessary to give some interpretations for the main resultsto clarify their specific meaning especially for the invest-ment share π1(t) of the inflation-linked index bond and theinvestment share π2(t) of the risky stock

Scrutinizing features of the expressions π1(t) and π2(t)

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

(58)

where ζβα(t) e(tminus T)(1113957rminus μt)(ω minus ε) ω (α + ρ(β minus α))(1 minus 1113954ϱ) and ε (eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) it is easy to findthat understanding implications of these simplified nota-tions δ 1113954ϱω ε and ζβα(t) is the key point Subsequently theinterpretations will start from descriptions of these sim-plified notations

In notation δ ((1113957μt + r minus 1113957r)21113957σ2t ) + ((1113954μt minus 1113957r)21113954σ2t ) theone part (1113957μt + r minus 1113957r)21113957σ2t is the risk compensation rate ofinflation-linked index bond and the other part (1113954μt minus 1113957r)21113954σ2t isthe risk compensation rate of risky stock us δ synthet-ically reflects the risk compensation of two risky assets

Obviously eminus T(δ+σ2t ) lt 1 so the notation1113954ϱ (eminus T(δ+σ2t )δ + σ2t )(δ + σ2t ) is less than 1 Notation 1113954ϱ givesa ratio of one sum with the risk compensation rate δ beingconverted by eminus T(δ+σ2t ) divided by the other sum in which therisk compensation rate δ has no conversion

e notation ω (α + ρ(β minus α))(1 minus 1113954ϱ) is an expressionof top regulation bound β and bottom regulation bound αwhere α+ ρ(β minus α) is a value of the expectation E[Zπ(T)]Because ρ takes a determined value the changes of regu-lation bounds directly result in the changes of ω ereforeω is the key point reflecting the influences of both regulationbounds on investment strategy

In notation ε the parameter z0 has the most obviouseconomic connotation since it represents the initial value ofasset-liability ratio at time 0 e expression(eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) gives a value resulted from theinitial asset-liability ratio z0 influenced by several differentfactors (return rates and volatilities) and then converted atthe constant relative rate 1113954ϱ

e following expression

ζβα(t)

z

e(tminus T) 1113957rminus μt( )

(1 minus 1113954ϱ)middot

(α + ρ(β minus α)) minus eminus T minus 1113957r+δ+μt+σ2t( )z01113872 1113873

z

(59)

is key to understand the optimal strategy (πlowast0 πlowast1 πlowast2 ) edenominator z of this fraction ζβα(t)z represents the currentlevel of asset-liability ratio at time t

Complexity 9

But the numerator ζβα(t) of this fraction is morecomplicated e one multiplier ((α + ρ(β minus α))minus

z0eminus T(minus 1113957r+δ+μ+1113957σ

2t )) is a difference between the expected value

(α + ρ(β minus α)) of Z(T) and the converted valueeminus T(minus 1113957r+δ+μt+σ2t )z0 of the initial value z0 which characterizesthe distance between initial value affected by various eco-nomic factors and the expectation of Z(T) e othermultiplier e(tminus T)(1113957rminus μt)(1 minus 1113954ϱ) is a converted value compre-hensively influenced by several parameters embodying thestates of financial market

In addition it is easy to see that if the relative rate ζβα(t)zis larger than 1 πlowast1(t) ((1113957μt + r minus 1113957r)1113957σ2t )((ζβα(t)z) minus 1) andπlowast2(t) ((1113954μt minus 1113957r)1113954σ2t )(ζβα(t)z minus 1) are positive otherwiseboth will be negative Since πlowast0(t) 1 minus πlowast1(t) minus πlowast2(t) itindicates that πlowast0(t) 1 + δ(1 minus (1z)ζβα(t)) can be deter-mined as long as the investment shares of risky assets havebeen determined so the adjustment of risk-free asset isusually passive If the relative rate ζβα(t)z is greater than 1the investment shares on risk-free asset can be ensuredwithin a much better range between 0 and 1 ese ob-servations can serve as an important reference for deter-mining both the top regulation bound β and the bottomregulation bound α

5 Numerical Examples

is section discusses the variation tendency of optimalstrategies πlowast1 and πlowast2 varying with some important parame-ters mainly focusing on the current value z of asset-liabilityratio the top regulation bound β the bottom regulationbound α and the parameter ρ e values of parameters aregiven in Table 1 unless otherwise specified other values forthe same variable can be consulted from the correspondinggraphic legend

First let top regulation bound β be fixed if the bottomregulation bound takes much higher values then the in-vestment shares of both inflation-linked index bond and riskystock increase significantly ese variety trends can be ob-served intuitively from Figures 1 and 2 respectively Mean-while Figures 1 and 2 also show that the appropriate bottomregulation bound should be taken at one value between 1 and2 for an economic environment same as this exampleOtherwise the investment shares on risky assets may increasetoomuch but the investment share on risk-free asset becomesnegative which may result in leverage risk increases rapidly

Next let bottom regulation bound α be fixed if the topregulation bound β takes much higher value then the invest-ment shares of both inflation-linked index bond and risky stockshould also increase significantly e variety trends can beperceived from Figures 3 and 4 respectively MeanwhileFigures 3 and 4 also show that the appropriate top regulationbound should be taken at one value between 6 and 7 for aneconomic environment same as this example Otherwise theinvestment shares on risky assets increase too much but theinvestment share on risk-free asset becomes negative whichalso result in leverage risk increases rapidly

As can be seen from Figures 3ndash6 taking the larger valueof either the top regulation bound or the bottom regulation

bound means allowing a wider volatility range of investmentreturns which may lead to the result that assets with greaterinvestment risk are allowed to invest or greater market risksare incorporated into the wealth process In addition theabove numerical analysis roughly gives a valuable referencefor selecting an appropriate range of top regulation boundand bottom regulation bound according to the relationshipbetween the reasonable values of investment strategy andregulation bounds for an economic environment set byvalues of parameters

Table 1 Values of the parameters

1113957μ 008 r 001 1113957σ 02 α 1 β 7 z0 3 z 51113957r 005 T 6 1113954μ 015 1113954σ 06 μ 003 σ 01 ρ 05

02

04

06

08

10

(π1)

lowast

π1lowast changes with α when top bound β fixed

2 3 4 5 61t

α = 1α = 18α = 26

Figure 1 Values of πlowast1 change with t for different bottom bound α

(π2)

lowast

π2lowast changes with α when top bound β fixed

005

010

015

020

2 3 4 5 61t

α = 1α = 2α = 3

Figure 2 Values of πlowast2 change with t for different bottom bound α

10 Complexity

In fact the regulatory bound cannot be determinedarbitrarily Both the top regulatory bound and bottomregulatory bound determine the fluctuation range that de-cision-making needs to control Too narrow fluctuationrange of investment returns results in loss of great invest-ment opportunities and makes investment in the financialmarket meaningless If the regulatory bound is too large thepotential huge risks cannot be dealt with effectively when theinvestment return soars fast which is almost equivalent tosituations without regulatory bounds and also lose thesignificance of using regulatory bounds It is hard to de-termine the regulatory bounds only by quantitative calcu-lation Once there is a model error it may result in seriousdeviation between theory and practice in actual manage-ment thus the determination of upper and lower limitsshould not be purely theoretical Nevertheless it should be

noted that advanced statistical analysis and maturelypractical experience of personnel are essential and of greatadvantage for determining an exact value of α or β

Assuming that both top regulation bound β and bottomregulation bound α are determined the impact of asset-li-ability ratio z at time t on investment strategy can be ob-served in Figures 5 and 6 If the current asset-liability ratio zat time t takes much greater value then the investmentshares of both inflation-linked index bond and risky stockshould be much lower which shows that increasing shares ofinvestment on risk-free asset is an important measure toreduce investment risk For the numerical examples shownin Figures 5 and 6 the regulation bounds being fixed at α 1and β 7 the most appropriate asset-liability ratio shouldtake values around 5 which is much more advantageous forthe company

e impact of parameter ρ on investment strategy can beobserved in Figures 7 and 8 If the parameter ρ takes a muchgreater value then the investment shares of risky assets mustincrease correspondingly Since the top regulation bound βand the bottom regulation bound α are predetermined theexpected value α+ ρ(β minus α) of asset-liability ratio entirelydepends on the value of parameter ρerefore the variationtendency of investment shares πlowast1(t) and πlowast2(t) with pa-rameter ρ actually illustrates the trend of investment shareschanging with the expectation of asset-liability ratio Z(T) Inone word the greater the expectation of Z(T) the muchhigher the shares of the companyrsquos wealth to be invested onrisky assets

6 Conclusion

Some empirical studies in the banking literature deal withthe effects of prolonged periods of low interest rates in theeconomy on risk-taking real effects on the real sector andalso on financial stability and so on as given by Chaudron[20] Bikker and Vervliet [21] However this paper uses

π1lowast changes with β when bottom bound α fixed

02

04

06

08

2 3 4 5 61t

β = 6β = 7β = 8

(π1)

lowast

Figure 3 Values of πlowast1 change with t for different top bound β

π2lowast changes with β when bottom bound α fixed

01

02

03

04

2 3 4 5 61t

β = 6β = 8β = 10

(π2)

lowast

Figure 4 Values of πlowast2 change with t for different top bound β

π1lowast changes with t for different z

02

04

06

08

2 3 4 5 61t

z = 4z = 5z = 6

(π1)

lowast

Figure 5 Values of πlowast1 change with t for different current asset-liability ratio z

Complexity 11

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

d 1113954Xπ(t)

1113954Xπ(t)

1113957r minus 1113957μt + 1113957σ2t + π1(t) 1113957μt + r minus 1113957r minus 1113957σ2t1113872 11138731113960

+ π2(t) 1113954μt minus 1113957r( 11138571113859dt + π1(t) minus 1( 11138571113957σtd 1113957W(t)

+ π2(t)1113954σtd 1113954W(t)

(7)

23 Asset-Liability Problem with Regulation Constraintse asset-liability ratio is defined as Zπ(t) 1113954X

π(t)1113954L(t)

which describes the ratio between the size of asset andthe size of liability at time t According to Itorsquos formulathe dynamic process of the asset-liability ratio can be givenbydZπ(t)

Zπ(t) 1113957r minus μt minus σ2t1113872 1113873dt minus σtdW(t) + π1(t) 1113957μt + r minus 1113957r( 1113857dt(

+ 1113957σtd 1113957W(t)1113857 + π2(t) 1113954μt minus 1113957r( 1113857dt + 1113954σtd 1113954W(t)1113872 1113873

(8)

In practice the regulation only occurs at certain fixedtime which is called regulatory time e regulatory time isusually selected in advance which provides a time period ofappropriate length but the length may decrease if theregulatory frequency increased In accordance with theneed of practice this paper considers the asset-liabilityproblem with the constraints imposed at the regulatory timeT to find the optimal investment strategy in order to reducethe risk of abnormal high-speed growth of asset price withina short period of time or high investment leverage and alsoto lessen the risk of too low return rate or a sharp fall eirmathematical descriptions of the regulation constraints atthe regulatory time T are formulated as the followinginequality

αltZπ(T)lt β (9)

where the bottom regulation bound αge 1 and the top reg-ulation bound β are two appropriate constants satisfying therequirement βgt α e bottom bound α should not be toosmall otherwise it cannot cover the liability which meansthe investment has failede top bound β should not be toolarge or else a much larger value of β may lead to excessiveleverage multiples and too much investment risks beingpulled in the company In practice it is highly technical todetermine the values of α and β which requires abundant

practical experience and rigorous mathematical calculationand only can be completed by well-trained and experiencedinvestment managers based on the real market quotationand their practical experiences

Let Π denote the set of all admissible strategies π einvestor aims to find an optimal portfolio π isinΠ of thefollowing original problem

minπisin1113937

Var Zπ(T)[ ] E Zπ(T) minus (α + ρ(β minus α))1113858 11138592

st E Zπ(T)[ ] α + ρ(β minus α)gt 0

Zπ(t)satisfy(8) ρ isin (0 1) for π isin 1113937

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)

where ρ is a constant Only if ρ takes an unequivocal valuethen the expectation of Zπ(T) can be fixed at a certain valuebetween the top regulation bound β and the bottom regu-lation bound α

For a determined value of expectation α+ ρ(β minus α) ifthere exists at least one admissible pair (Z(middot) π(middot)) satis-fying E[Zπ(T)] α+ ρ(β minus α) then the problem (10) iscalled feasible Given α+ ρ(β minus α) the optimal strategy πlowastof (10) is called an efficient strategy e pair(Zπlowast(T)Var[Zπlowast(T)]) under this optimal strategy πlowast iscalled an efficient point e set of all efficient points iscalled the efficient frontier Obviously problem (10) is adynamic quadratic convex optimization problem andthus it has an unique solution

3 Solutions of the Problem

In this section an unconstrained optimization problem isderived by the Lagrange multiplier method and then aHamiltonndashJacobindashBellman equation is deduced to solvethe unconstrained problem Next solution of the originalproblem is obtained according to Lagrange duality be-tween the unconstrained problem and the original prob-lem At the end solution of a special case is given forcomparison

31 Solution of an Unconstrained Problem For this convexoptimization problem (10) the equality constraint E[Zπ(T)] α+ ρ(β minus α) can be eliminated by using theLagrange multiplier method so we get a Lagrange dualproblem

minπisin1113937

E Zπ(T) minus (α + ρ(β minus α))1113858 11138592

+ 2λE Zπ(T) minus (α + ρ(β minus α))1113858 1113859

st Zπ(T)satisfy(8) ρ isin (0 1) for π isin1113937

⎧⎪⎨

⎪⎩(11)

where λ isinR is a Lagrange multiplierAfter a simple calculation it yields

E Zπ(T) minus (α + ρ(β minus α))1113858 1113859

2+ 2λE Z

π(T) minus (α + ρ(β minus α))1113858 1113859

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 1113859

2minus λ2

(12)

4 Complexity

Remark 1 e link between problems (10) and (11) isprovided by the Lagrange duality theorem which can bereferred in the study by Luenberger [17] that

minπisin1113937

Var Zπ(T)1113858 1113859

maxλisinR

minπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 1113859

2minus λ21113966 1113967

(13)

us for the fixed constants λ and u problem (11) isequivalent to

minπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 11138592

st Zπ(t) satisfy(8) ρ isin (0 1) for π isin1113937

⎧⎪⎨

⎪⎩(14)

erefore in order to solve problem (11) we only needto solve problem (14) Fortunately problem (14) can besolved by using the stochastic optimal control technology

To solve problem (14) a truncated form beginning attime t is considered here and the corresponding valuefunction is defined as

V(t z) infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

211138681113868111386811138681113868 Zπ(t)z1113876 1113877

(15)

with the boundary condition V(T z) (z minus ((α+

ρ(β minus α)) minus λ))2e controlled infinitesimal generator for any test

function ϕ(t x) and any admissible control π is deduced asfollows

Aπϕ(t x) ϕt + xϕx 1113957r minus μt minus σ2t + π1(t) 1113957μt + r minus 1113957r( 11138571113960

+ π2(t) 1113954μt minus 1113957r( 11138571113859 +x2

2ϕxx σ2t + π2

1(t)1113957σ2t + π22(t)1113954σ2t1113960 1113961

(16)

for any real valued function ϕ(t x) isinC12([0 T] R) andadmissible strategy π where

C12([0 T]R) ψ(t x)|ψ(t middot) is once continuously differentiable on [0 T] andψ1113864

(middot x) is twice continuously differentiable almost surely onR(17)

If V(t z) isinC12([0 T]timesR) then V(t z) satisfies thefollowing HJB equation

infπisin1113937

AπV(t z) 0 (18)

which can be given more specifically as follows

infπisin1113937

Vt + zVz 1113957r minus μt minus σ2t1113872 1113873 +z2

2Vzzσ

2t + zVz π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 11138571113858 1113859 +

z2

2Vzz π2

1(t)1113957σ2t + π22(t)1113954σ2t1113872 1113873 01113896 1113897 (19)

For convenience some simple notations are introducedas follows

δ ≜1113957μt + r minus 1113957r( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t

ϱ(t)≜e(tminus T) δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2t

ω≜(α + ρ(β minus α))

(1 minus 1113954ϱ)

1113954ϱ ≜ ϱ(0) eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2t

ε≜eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)

(20)

Solving equation (19) gives the following importanttheorem

Theorem 1 For the asset-liability management with in-vestment regulation and eliminating inflation the efficientinvestment strategy πlowast (πlowast0 πlowast1 πlowast2 ) corresponding toproblem (11) and problem (14) is given by

πlowast0 1 + δ 1 minus1zξβα(t)1113874 1113875

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ξβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ξβα(t)

zminus 11113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

and the value function is

Complexity 5

V(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( )

minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )

+((α + ρ(β minus α)) minus λ)2ϱ(t)

(22)

where ξβα(t) e(tminus T)(1113957rminus μt)((α + ρ(β minus α)) minus λ)

Proof According to the first-order necessary condition ofextremum equation (19) yields

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

z1113957σ2t

Vz

Vzz

πlowast2(t) minus1113954μt minus 1113957r( 1113857

z1113954σ2t

Vz

Vzz

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)

Plugging the above expressions of πlowast1(t) and πlowast2(t) into(19) the HJB equation becomes

Vt + zVz 1113957r minus μt minus σ2t1113872 1113873 +z2

2Vzzσ

2t minus δ

V2z

2Vzz

0 (24)

Based on the boundary condition V(T z)

(z minus ((α + ρ(β minus α)) minus λ))2 we try a conjecture

W(t z) f(t)z2

+ g(t)z + h(t) (25)

for equation (24) which satisfies

f(T) 1

g(T) minus 2((α + ρ(β minus α)) minus λ)

h(T) ((α + ρ(β minus α)) minus λ)2

(26)

e derivatives of this conjecture W(t z) are given asfollows

Wt fprime(t)z2

+ gprime(t)z + hprime(t)

Wz 2f(t)z + g(t)

Wzz 2f(t)

(27)

Plugging the expressions of Wz and Wzz into (23) theoptimal strategy becomes

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

z1113957σ2tz +

g(t)

2f(t)1113888 1113889

πlowast2(t) minus1113954μt minus 1113957r( 1113857

z1113954σ2tz +

g(t)

2f(t)1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

Substituting these derivatives of W(t z) into (24) theHJB equation becomes

fprime(t)z2

+ gprime(t)z + hprime(t) + 2f(t)z2

+ g(t)z1113872 1113873 1113957r minus μt minus σ2t1113872 1113873

+ z2f(t)σ2t minus δ f(t)z

2+ g(t)z +

g2(t)

4f(t)1113888 1113889 0

(29)

Equation (29) can be split into three ordinary differentialequations as follows

fprime(t) + 2f(t) 1113957r minus μt( 1113857 minus f(t)σ2t minus δf(t) 0

f(T) 1

gprime(t) + g(t) 1113957r minus μt minus σ2t( 1113857 minus δg(t) 0

g(T) minus 2((α + ρ(β minus α)) minus λ)

hprime(t) minusδ4

g(t)2

f(t) 0

h(T) ((α + ρ(β minus α)) minus λ)2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

Solving these ordinary differential equations in (30)yields

f(t) e(tminus T) minus 21113957r+δ+2μt+σ2t( )

g(t) minus 2e(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

h(t) ((α + ρ(β minus α)) minus λ)2ϱ(t)

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)

Plugging the explicit expression of f(t) and g(t) into (28)and using the equality πlowast0(t) + πlowast1(t) + πlowast2(t) 1 the opti-mal strategy is obtained as follows

πlowast0(t) 1 + δ 1 minus e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

z1113888 1113889

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2te

(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2te

(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

6 Complexity

Denoting

ξβα(t)≜e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ) (33)

the optimal strategy can be simplified into

πlowast0(t) 1 + δ 1 minus1zξβα(t)1113874 1113875

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tξβα(t)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tξβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

Meanwhile substituting the expressions of f(t) g(t) andh(t) into W(t z) the explicit solution of HJB equation isobtained as follows

W(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( ) +((α + ρ(β minus α)) minus λ)2

ϱ(t) minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

(35)

e following verification theorem shows the obtainedresults are exactly the optimal strategy and the optimal valuefunction as required

Theorem 2 (Verification Theorem) If W(t z) is a solutionof HJB equation (19) and satisfies the boundary conditionW(T z) (z minus ((α + ρ(β minus α)) minus λ))2 then for all admissi-ble strategies π isin Π V(t z)geW(t z) If πlowast satisfies

πlowastisin arg infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

2|Z

π(t) z1113960 1113961

(36)

then V(t z)W(t z) and πlowast is the optimal investmentstrategy of problem (14)

Proof Proof is similar to that given by Pham [18] Chang[19] and so on so the detail is omitted

32 Solution of the Original Problem e optimal valuefunction of problem (11) is defined as

1113954V 0 z0( 1113857 infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

21113960 1113961 minus λ2

(37)

where z0 Z(0)In this section the Lagrange duality method is used to

find a solution of original optimization problem (10) basedon the obtained results of the unconstrained problem

Theorem 3 For original problem (10) the efficient strategyπlowast (πlowast0 πlowast1 πlowast2 ) is given by

πlowast0 1 minus δζβα(t)

zminus 11113888 1113889

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)

and the optimal value function is given by

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +(ω minus ε)21113954ϱ

minus 2eminus T minus 1113957r+δ+μt+σ2t( )(ω minus ε)z0 minus (ε minus 1113954ϱω)

2

(39)

where

ζβα(t) e(tminus T) 1113957rminus μt( )(ω minus ε) (40)

Proof e optimal value function 1113954V(0 z0) can be ob-tained with the equivalence between problem (11) andproblem (14) By taking z0 Z(0) and setting t 0 inV(t z) ityields

1113954V 0 z0( 1113857 V 0 z0( 1113857 minus λ2 f(0)z20 + g(0)z0 + h(0) minus λ2

eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +((α + ρ(β minus α)) minus λ)

2

1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)z0 minus λ2

(41)

Since (δ + σ2t )gt 0 then eminus T(δ+σ2t ) lt 1 and it yields

eminus T δ+σ2t( )δ + σ2t1113874 1113875lt δ + σ2t (42)

and then

1113954ϱ eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tlt 1 (43)

Note that the optimal value function 1113954V(0 z0) is aquadratic function with respect to λ and 1113954ϱ minus 1 is the coef-ficient of quadratic term

1113954ϱ minus 1 eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tminus 1lt 0 (44)

us the finite maximum value of 1113954V(0 z0) can beobtained at a specific value λlowast and

λlowast eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ) (45)

Applying the Lagrangian duality substituting λlowast into1113954V(0 z0) the minimum variance corresponding to an ar-bitrarily given expectation (α + ρ(β minus α)) is obtained asfollows

Complexity 7

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +

(α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

middot 1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( ) (α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠z0 minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

(46)

Substitute (45) into (21) the optimal investment strategyof problem (10) is given by

πlowast0 1 minus δ1zζβα(t) minus 11113874 1113875

πlowast1 minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tζβα(t)

πlowast2 minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tζβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where

ζβα(t) e(tminus T) 1113957rminus μt( ) (α + ρ(β minus α))

1 minus 1113954ϱminus

eminus T minus 1113957r+δ+μt+σ2t( )z0

1 minus 1113954ϱ⎛⎝ ⎞⎠

(48)

Remark 2 However in the obtained results (47) and (46)above the expected value (α + ρ(β minus α)) of Zπ(t) is not fixedat a determined position but can take any value between αand β depending on the value of ρ isin (0 1) erefore theobtained minimum variance depends on the parametervalue ρ isin (0 1) in the expression of expectation(α + ρ(β minus α)) Next let us turn our attention to the pa-rameter ρ and determine the optimal parameter value ρlowast forminimizing the variance with respect to ρ as well eoptimal ρlowast to achieve the minimum of Var[Zπlowast(T)] is givenas follows

ρlowast eminus T minus 1113957r+δ+μt+σ2t( )z01113872 11138731113954ϱ1113872 1113873 minus α

(β minus α) (49)

and the expected value of Zπ(t) corresponding to the optimalparameter value ρlowast is given by

α + ρlowast(β minus α) z01113954ϱ

eminus T minus 1113957r+δ+μt+σ2t( ) (50)

Substituting ρlowast into (47) we can get the optimal strategycorresponding to the optimal value α+ ρlowast minus α) of expecta-tion Moreover plugging the expression of ρlowast into theformula of Var[Zπlowast(T)] the optimal minimum varianceVar[Zπlowast ρlowast(T)] can also be determined corresponding to theuniquely optimal value (z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t ) of theexpectation

33 Special Case e investment regulation is also im-portant for an investment process without liabilities Letμt σt 0 the original problem degenerates to the case of noliability and the process Zπ(t) becomes the dynamics process1113954Xπ(t) as follows

d 1113954Xπ(t) 1113957r + π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 1113857 minus π1(t)1113957σ2t minus 1113957μt1113960 1113961

middot 1113954Xπ(t)dt + π1(t) minus 1( 11138571113957σt

1113954Xπ(t)d 1113957W(t) + π2(t)1113954σt

middot 1113954Xπ(t)d 1113954W(t)

(51)

Meanwhile the regulation bound constraints also de-generate into a restriction on the investment dynamicsprocess after eliminating inflation

αlt 1113954Xπ(T)lt β (52)

Proposition 1 For the special case μt σt 0 the minimumvariance of Zπ(t) is

Var Zlowast(T)1113858 1113859 e

minus T1113957δ (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

minus 2eminus T minus r+1113957δ+1113957σ

2t( 1113857 (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠1113954x0

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0 minus eminus T1113957δ(α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

+ eminus T minus 2r+1113957δ+21113957σ

2t( 11138571113954x

20

(53)

and the optimal strategy of the special case is given as follows

8 Complexity

πlowast1(t) 1 minus1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

1113957σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast0(t) 1113957δ 1 minuse(tminus T) minus r+1113957δ+1113957σ

2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(54)

where

1113957δ 1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t (55)

Proof e results can be obtained by the same calculationprocess as the original problem so the details are omittedhere

Remark 3 e expected value (α + ρ(β minus α)) is not fixed ata determined position in the above proposition but it cantake any value between α and β depending on the value ofρ isin (0 1) erefore the obtained minimum variancedepends on the parameter value ρ isin (0 1) Using the first-order necessary condition of extremum value for Var[Zlowast(T)] with respect to the parameter ρ the optimal valueof parameter ρ is given by

ρlowast eminus T minus r+1113957σ

2t( 11138571113954x0 minus α

(β minus α) (56)

and the corresponding optimal expectation

α + ρlowast(β minus α)( 1113857 eminus T minus r+1113957σ

2t( 11138571113954x0 (57)

Substituting ρlowast into the optimal strategy (πlowast0(t)

πlowast1(t) πlowast2(t)) the optimal strategy corresponding to theoptimal value α+ ρlowast(β minus α) of expectation can be obtainedMeanwhile plugging the expression of ρlowast into Var[Zπlowast(t)]the optimal minimum variance Var[Zπlowast ρlowast(T)] can also bedetermined It is worth mentioning that the optimalminimum variance for this special case with no liabilityis zero under the uniquely optimal expectation(z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t )

4 Interpretations of the Main Results

e expressions of the obtained results are so abstract that itis necessary to give some interpretations for the main resultsto clarify their specific meaning especially for the invest-ment share π1(t) of the inflation-linked index bond and theinvestment share π2(t) of the risky stock

Scrutinizing features of the expressions π1(t) and π2(t)

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

(58)

where ζβα(t) e(tminus T)(1113957rminus μt)(ω minus ε) ω (α + ρ(β minus α))(1 minus 1113954ϱ) and ε (eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) it is easy to findthat understanding implications of these simplified nota-tions δ 1113954ϱω ε and ζβα(t) is the key point Subsequently theinterpretations will start from descriptions of these sim-plified notations

In notation δ ((1113957μt + r minus 1113957r)21113957σ2t ) + ((1113954μt minus 1113957r)21113954σ2t ) theone part (1113957μt + r minus 1113957r)21113957σ2t is the risk compensation rate ofinflation-linked index bond and the other part (1113954μt minus 1113957r)21113954σ2t isthe risk compensation rate of risky stock us δ synthet-ically reflects the risk compensation of two risky assets

Obviously eminus T(δ+σ2t ) lt 1 so the notation1113954ϱ (eminus T(δ+σ2t )δ + σ2t )(δ + σ2t ) is less than 1 Notation 1113954ϱ givesa ratio of one sum with the risk compensation rate δ beingconverted by eminus T(δ+σ2t ) divided by the other sum in which therisk compensation rate δ has no conversion

e notation ω (α + ρ(β minus α))(1 minus 1113954ϱ) is an expressionof top regulation bound β and bottom regulation bound αwhere α+ ρ(β minus α) is a value of the expectation E[Zπ(T)]Because ρ takes a determined value the changes of regu-lation bounds directly result in the changes of ω ereforeω is the key point reflecting the influences of both regulationbounds on investment strategy

In notation ε the parameter z0 has the most obviouseconomic connotation since it represents the initial value ofasset-liability ratio at time 0 e expression(eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) gives a value resulted from theinitial asset-liability ratio z0 influenced by several differentfactors (return rates and volatilities) and then converted atthe constant relative rate 1113954ϱ

e following expression

ζβα(t)

z

e(tminus T) 1113957rminus μt( )

(1 minus 1113954ϱ)middot

(α + ρ(β minus α)) minus eminus T minus 1113957r+δ+μt+σ2t( )z01113872 1113873

z

(59)

is key to understand the optimal strategy (πlowast0 πlowast1 πlowast2 ) edenominator z of this fraction ζβα(t)z represents the currentlevel of asset-liability ratio at time t

Complexity 9

But the numerator ζβα(t) of this fraction is morecomplicated e one multiplier ((α + ρ(β minus α))minus

z0eminus T(minus 1113957r+δ+μ+1113957σ

2t )) is a difference between the expected value

(α + ρ(β minus α)) of Z(T) and the converted valueeminus T(minus 1113957r+δ+μt+σ2t )z0 of the initial value z0 which characterizesthe distance between initial value affected by various eco-nomic factors and the expectation of Z(T) e othermultiplier e(tminus T)(1113957rminus μt)(1 minus 1113954ϱ) is a converted value compre-hensively influenced by several parameters embodying thestates of financial market

In addition it is easy to see that if the relative rate ζβα(t)zis larger than 1 πlowast1(t) ((1113957μt + r minus 1113957r)1113957σ2t )((ζβα(t)z) minus 1) andπlowast2(t) ((1113954μt minus 1113957r)1113954σ2t )(ζβα(t)z minus 1) are positive otherwiseboth will be negative Since πlowast0(t) 1 minus πlowast1(t) minus πlowast2(t) itindicates that πlowast0(t) 1 + δ(1 minus (1z)ζβα(t)) can be deter-mined as long as the investment shares of risky assets havebeen determined so the adjustment of risk-free asset isusually passive If the relative rate ζβα(t)z is greater than 1the investment shares on risk-free asset can be ensuredwithin a much better range between 0 and 1 ese ob-servations can serve as an important reference for deter-mining both the top regulation bound β and the bottomregulation bound α

5 Numerical Examples

is section discusses the variation tendency of optimalstrategies πlowast1 and πlowast2 varying with some important parame-ters mainly focusing on the current value z of asset-liabilityratio the top regulation bound β the bottom regulationbound α and the parameter ρ e values of parameters aregiven in Table 1 unless otherwise specified other values forthe same variable can be consulted from the correspondinggraphic legend

First let top regulation bound β be fixed if the bottomregulation bound takes much higher values then the in-vestment shares of both inflation-linked index bond and riskystock increase significantly ese variety trends can be ob-served intuitively from Figures 1 and 2 respectively Mean-while Figures 1 and 2 also show that the appropriate bottomregulation bound should be taken at one value between 1 and2 for an economic environment same as this exampleOtherwise the investment shares on risky assets may increasetoomuch but the investment share on risk-free asset becomesnegative which may result in leverage risk increases rapidly

Next let bottom regulation bound α be fixed if the topregulation bound β takes much higher value then the invest-ment shares of both inflation-linked index bond and risky stockshould also increase significantly e variety trends can beperceived from Figures 3 and 4 respectively MeanwhileFigures 3 and 4 also show that the appropriate top regulationbound should be taken at one value between 6 and 7 for aneconomic environment same as this example Otherwise theinvestment shares on risky assets increase too much but theinvestment share on risk-free asset becomes negative whichalso result in leverage risk increases rapidly

As can be seen from Figures 3ndash6 taking the larger valueof either the top regulation bound or the bottom regulation

bound means allowing a wider volatility range of investmentreturns which may lead to the result that assets with greaterinvestment risk are allowed to invest or greater market risksare incorporated into the wealth process In addition theabove numerical analysis roughly gives a valuable referencefor selecting an appropriate range of top regulation boundand bottom regulation bound according to the relationshipbetween the reasonable values of investment strategy andregulation bounds for an economic environment set byvalues of parameters

Table 1 Values of the parameters

1113957μ 008 r 001 1113957σ 02 α 1 β 7 z0 3 z 51113957r 005 T 6 1113954μ 015 1113954σ 06 μ 003 σ 01 ρ 05

02

04

06

08

10

(π1)

lowast

π1lowast changes with α when top bound β fixed

2 3 4 5 61t

α = 1α = 18α = 26

Figure 1 Values of πlowast1 change with t for different bottom bound α

(π2)

lowast

π2lowast changes with α when top bound β fixed

005

010

015

020

2 3 4 5 61t

α = 1α = 2α = 3

Figure 2 Values of πlowast2 change with t for different bottom bound α

10 Complexity

In fact the regulatory bound cannot be determinedarbitrarily Both the top regulatory bound and bottomregulatory bound determine the fluctuation range that de-cision-making needs to control Too narrow fluctuationrange of investment returns results in loss of great invest-ment opportunities and makes investment in the financialmarket meaningless If the regulatory bound is too large thepotential huge risks cannot be dealt with effectively when theinvestment return soars fast which is almost equivalent tosituations without regulatory bounds and also lose thesignificance of using regulatory bounds It is hard to de-termine the regulatory bounds only by quantitative calcu-lation Once there is a model error it may result in seriousdeviation between theory and practice in actual manage-ment thus the determination of upper and lower limitsshould not be purely theoretical Nevertheless it should be

noted that advanced statistical analysis and maturelypractical experience of personnel are essential and of greatadvantage for determining an exact value of α or β

Assuming that both top regulation bound β and bottomregulation bound α are determined the impact of asset-li-ability ratio z at time t on investment strategy can be ob-served in Figures 5 and 6 If the current asset-liability ratio zat time t takes much greater value then the investmentshares of both inflation-linked index bond and risky stockshould be much lower which shows that increasing shares ofinvestment on risk-free asset is an important measure toreduce investment risk For the numerical examples shownin Figures 5 and 6 the regulation bounds being fixed at α 1and β 7 the most appropriate asset-liability ratio shouldtake values around 5 which is much more advantageous forthe company

e impact of parameter ρ on investment strategy can beobserved in Figures 7 and 8 If the parameter ρ takes a muchgreater value then the investment shares of risky assets mustincrease correspondingly Since the top regulation bound βand the bottom regulation bound α are predetermined theexpected value α+ ρ(β minus α) of asset-liability ratio entirelydepends on the value of parameter ρerefore the variationtendency of investment shares πlowast1(t) and πlowast2(t) with pa-rameter ρ actually illustrates the trend of investment shareschanging with the expectation of asset-liability ratio Z(T) Inone word the greater the expectation of Z(T) the muchhigher the shares of the companyrsquos wealth to be invested onrisky assets

6 Conclusion

Some empirical studies in the banking literature deal withthe effects of prolonged periods of low interest rates in theeconomy on risk-taking real effects on the real sector andalso on financial stability and so on as given by Chaudron[20] Bikker and Vervliet [21] However this paper uses

π1lowast changes with β when bottom bound α fixed

02

04

06

08

2 3 4 5 61t

β = 6β = 7β = 8

(π1)

lowast

Figure 3 Values of πlowast1 change with t for different top bound β

π2lowast changes with β when bottom bound α fixed

01

02

03

04

2 3 4 5 61t

β = 6β = 8β = 10

(π2)

lowast

Figure 4 Values of πlowast2 change with t for different top bound β

π1lowast changes with t for different z

02

04

06

08

2 3 4 5 61t

z = 4z = 5z = 6

(π1)

lowast

Figure 5 Values of πlowast1 change with t for different current asset-liability ratio z

Complexity 11

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

Remark 1 e link between problems (10) and (11) isprovided by the Lagrange duality theorem which can bereferred in the study by Luenberger [17] that

minπisin1113937

Var Zπ(T)1113858 1113859

maxλisinR

minπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 1113859

2minus λ21113966 1113967

(13)

us for the fixed constants λ and u problem (11) isequivalent to

minπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)1113858 11138592

st Zπ(t) satisfy(8) ρ isin (0 1) for π isin1113937

⎧⎪⎨

⎪⎩(14)

erefore in order to solve problem (11) we only needto solve problem (14) Fortunately problem (14) can besolved by using the stochastic optimal control technology

To solve problem (14) a truncated form beginning attime t is considered here and the corresponding valuefunction is defined as

V(t z) infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

211138681113868111386811138681113868 Zπ(t)z1113876 1113877

(15)

with the boundary condition V(T z) (z minus ((α+

ρ(β minus α)) minus λ))2e controlled infinitesimal generator for any test

function ϕ(t x) and any admissible control π is deduced asfollows

Aπϕ(t x) ϕt + xϕx 1113957r minus μt minus σ2t + π1(t) 1113957μt + r minus 1113957r( 11138571113960

+ π2(t) 1113954μt minus 1113957r( 11138571113859 +x2

2ϕxx σ2t + π2

1(t)1113957σ2t + π22(t)1113954σ2t1113960 1113961

(16)

for any real valued function ϕ(t x) isinC12([0 T] R) andadmissible strategy π where

C12([0 T]R) ψ(t x)|ψ(t middot) is once continuously differentiable on [0 T] andψ1113864

(middot x) is twice continuously differentiable almost surely onR(17)

If V(t z) isinC12([0 T]timesR) then V(t z) satisfies thefollowing HJB equation

infπisin1113937

AπV(t z) 0 (18)

which can be given more specifically as follows

infπisin1113937

Vt + zVz 1113957r minus μt minus σ2t1113872 1113873 +z2

2Vzzσ

2t + zVz π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 11138571113858 1113859 +

z2

2Vzz π2

1(t)1113957σ2t + π22(t)1113954σ2t1113872 1113873 01113896 1113897 (19)

For convenience some simple notations are introducedas follows

δ ≜1113957μt + r minus 1113957r( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t

ϱ(t)≜e(tminus T) δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2t

ω≜(α + ρ(β minus α))

(1 minus 1113954ϱ)

1113954ϱ ≜ ϱ(0) eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2t

ε≜eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)

(20)

Solving equation (19) gives the following importanttheorem

Theorem 1 For the asset-liability management with in-vestment regulation and eliminating inflation the efficientinvestment strategy πlowast (πlowast0 πlowast1 πlowast2 ) corresponding toproblem (11) and problem (14) is given by

πlowast0 1 + δ 1 minus1zξβα(t)1113874 1113875

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ξβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ξβα(t)

zminus 11113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

and the value function is

Complexity 5

V(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( )

minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )

+((α + ρ(β minus α)) minus λ)2ϱ(t)

(22)

where ξβα(t) e(tminus T)(1113957rminus μt)((α + ρ(β minus α)) minus λ)

Proof According to the first-order necessary condition ofextremum equation (19) yields

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

z1113957σ2t

Vz

Vzz

πlowast2(t) minus1113954μt minus 1113957r( 1113857

z1113954σ2t

Vz

Vzz

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)

Plugging the above expressions of πlowast1(t) and πlowast2(t) into(19) the HJB equation becomes

Vt + zVz 1113957r minus μt minus σ2t1113872 1113873 +z2

2Vzzσ

2t minus δ

V2z

2Vzz

0 (24)

Based on the boundary condition V(T z)

(z minus ((α + ρ(β minus α)) minus λ))2 we try a conjecture

W(t z) f(t)z2

+ g(t)z + h(t) (25)

for equation (24) which satisfies

f(T) 1

g(T) minus 2((α + ρ(β minus α)) minus λ)

h(T) ((α + ρ(β minus α)) minus λ)2

(26)

e derivatives of this conjecture W(t z) are given asfollows

Wt fprime(t)z2

+ gprime(t)z + hprime(t)

Wz 2f(t)z + g(t)

Wzz 2f(t)

(27)

Plugging the expressions of Wz and Wzz into (23) theoptimal strategy becomes

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

z1113957σ2tz +

g(t)

2f(t)1113888 1113889

πlowast2(t) minus1113954μt minus 1113957r( 1113857

z1113954σ2tz +

g(t)

2f(t)1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

Substituting these derivatives of W(t z) into (24) theHJB equation becomes

fprime(t)z2

+ gprime(t)z + hprime(t) + 2f(t)z2

+ g(t)z1113872 1113873 1113957r minus μt minus σ2t1113872 1113873

+ z2f(t)σ2t minus δ f(t)z

2+ g(t)z +

g2(t)

4f(t)1113888 1113889 0

(29)

Equation (29) can be split into three ordinary differentialequations as follows

fprime(t) + 2f(t) 1113957r minus μt( 1113857 minus f(t)σ2t minus δf(t) 0

f(T) 1

gprime(t) + g(t) 1113957r minus μt minus σ2t( 1113857 minus δg(t) 0

g(T) minus 2((α + ρ(β minus α)) minus λ)

hprime(t) minusδ4

g(t)2

f(t) 0

h(T) ((α + ρ(β minus α)) minus λ)2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

Solving these ordinary differential equations in (30)yields

f(t) e(tminus T) minus 21113957r+δ+2μt+σ2t( )

g(t) minus 2e(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

h(t) ((α + ρ(β minus α)) minus λ)2ϱ(t)

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)

Plugging the explicit expression of f(t) and g(t) into (28)and using the equality πlowast0(t) + πlowast1(t) + πlowast2(t) 1 the opti-mal strategy is obtained as follows

πlowast0(t) 1 + δ 1 minus e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

z1113888 1113889

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2te

(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2te

(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

6 Complexity

Denoting

ξβα(t)≜e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ) (33)

the optimal strategy can be simplified into

πlowast0(t) 1 + δ 1 minus1zξβα(t)1113874 1113875

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tξβα(t)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tξβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

Meanwhile substituting the expressions of f(t) g(t) andh(t) into W(t z) the explicit solution of HJB equation isobtained as follows

W(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( ) +((α + ρ(β minus α)) minus λ)2

ϱ(t) minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

(35)

e following verification theorem shows the obtainedresults are exactly the optimal strategy and the optimal valuefunction as required

Theorem 2 (Verification Theorem) If W(t z) is a solutionof HJB equation (19) and satisfies the boundary conditionW(T z) (z minus ((α + ρ(β minus α)) minus λ))2 then for all admissi-ble strategies π isin Π V(t z)geW(t z) If πlowast satisfies

πlowastisin arg infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

2|Z

π(t) z1113960 1113961

(36)

then V(t z)W(t z) and πlowast is the optimal investmentstrategy of problem (14)

Proof Proof is similar to that given by Pham [18] Chang[19] and so on so the detail is omitted

32 Solution of the Original Problem e optimal valuefunction of problem (11) is defined as

1113954V 0 z0( 1113857 infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

21113960 1113961 minus λ2

(37)

where z0 Z(0)In this section the Lagrange duality method is used to

find a solution of original optimization problem (10) basedon the obtained results of the unconstrained problem

Theorem 3 For original problem (10) the efficient strategyπlowast (πlowast0 πlowast1 πlowast2 ) is given by

πlowast0 1 minus δζβα(t)

zminus 11113888 1113889

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)

and the optimal value function is given by

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +(ω minus ε)21113954ϱ

minus 2eminus T minus 1113957r+δ+μt+σ2t( )(ω minus ε)z0 minus (ε minus 1113954ϱω)

2

(39)

where

ζβα(t) e(tminus T) 1113957rminus μt( )(ω minus ε) (40)

Proof e optimal value function 1113954V(0 z0) can be ob-tained with the equivalence between problem (11) andproblem (14) By taking z0 Z(0) and setting t 0 inV(t z) ityields

1113954V 0 z0( 1113857 V 0 z0( 1113857 minus λ2 f(0)z20 + g(0)z0 + h(0) minus λ2

eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +((α + ρ(β minus α)) minus λ)

2

1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)z0 minus λ2

(41)

Since (δ + σ2t )gt 0 then eminus T(δ+σ2t ) lt 1 and it yields

eminus T δ+σ2t( )δ + σ2t1113874 1113875lt δ + σ2t (42)

and then

1113954ϱ eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tlt 1 (43)

Note that the optimal value function 1113954V(0 z0) is aquadratic function with respect to λ and 1113954ϱ minus 1 is the coef-ficient of quadratic term

1113954ϱ minus 1 eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tminus 1lt 0 (44)

us the finite maximum value of 1113954V(0 z0) can beobtained at a specific value λlowast and

λlowast eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ) (45)

Applying the Lagrangian duality substituting λlowast into1113954V(0 z0) the minimum variance corresponding to an ar-bitrarily given expectation (α + ρ(β minus α)) is obtained asfollows

Complexity 7

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +

(α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

middot 1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( ) (α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠z0 minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

(46)

Substitute (45) into (21) the optimal investment strategyof problem (10) is given by

πlowast0 1 minus δ1zζβα(t) minus 11113874 1113875

πlowast1 minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tζβα(t)

πlowast2 minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tζβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where

ζβα(t) e(tminus T) 1113957rminus μt( ) (α + ρ(β minus α))

1 minus 1113954ϱminus

eminus T minus 1113957r+δ+μt+σ2t( )z0

1 minus 1113954ϱ⎛⎝ ⎞⎠

(48)

Remark 2 However in the obtained results (47) and (46)above the expected value (α + ρ(β minus α)) of Zπ(t) is not fixedat a determined position but can take any value between αand β depending on the value of ρ isin (0 1) erefore theobtained minimum variance depends on the parametervalue ρ isin (0 1) in the expression of expectation(α + ρ(β minus α)) Next let us turn our attention to the pa-rameter ρ and determine the optimal parameter value ρlowast forminimizing the variance with respect to ρ as well eoptimal ρlowast to achieve the minimum of Var[Zπlowast(T)] is givenas follows

ρlowast eminus T minus 1113957r+δ+μt+σ2t( )z01113872 11138731113954ϱ1113872 1113873 minus α

(β minus α) (49)

and the expected value of Zπ(t) corresponding to the optimalparameter value ρlowast is given by

α + ρlowast(β minus α) z01113954ϱ

eminus T minus 1113957r+δ+μt+σ2t( ) (50)

Substituting ρlowast into (47) we can get the optimal strategycorresponding to the optimal value α+ ρlowast minus α) of expecta-tion Moreover plugging the expression of ρlowast into theformula of Var[Zπlowast(T)] the optimal minimum varianceVar[Zπlowast ρlowast(T)] can also be determined corresponding to theuniquely optimal value (z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t ) of theexpectation

33 Special Case e investment regulation is also im-portant for an investment process without liabilities Letμt σt 0 the original problem degenerates to the case of noliability and the process Zπ(t) becomes the dynamics process1113954Xπ(t) as follows

d 1113954Xπ(t) 1113957r + π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 1113857 minus π1(t)1113957σ2t minus 1113957μt1113960 1113961

middot 1113954Xπ(t)dt + π1(t) minus 1( 11138571113957σt

1113954Xπ(t)d 1113957W(t) + π2(t)1113954σt

middot 1113954Xπ(t)d 1113954W(t)

(51)

Meanwhile the regulation bound constraints also de-generate into a restriction on the investment dynamicsprocess after eliminating inflation

αlt 1113954Xπ(T)lt β (52)

Proposition 1 For the special case μt σt 0 the minimumvariance of Zπ(t) is

Var Zlowast(T)1113858 1113859 e

minus T1113957δ (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

minus 2eminus T minus r+1113957δ+1113957σ

2t( 1113857 (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠1113954x0

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0 minus eminus T1113957δ(α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

+ eminus T minus 2r+1113957δ+21113957σ

2t( 11138571113954x

20

(53)

and the optimal strategy of the special case is given as follows

8 Complexity

πlowast1(t) 1 minus1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

1113957σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast0(t) 1113957δ 1 minuse(tminus T) minus r+1113957δ+1113957σ

2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(54)

where

1113957δ 1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t (55)

Proof e results can be obtained by the same calculationprocess as the original problem so the details are omittedhere

Remark 3 e expected value (α + ρ(β minus α)) is not fixed ata determined position in the above proposition but it cantake any value between α and β depending on the value ofρ isin (0 1) erefore the obtained minimum variancedepends on the parameter value ρ isin (0 1) Using the first-order necessary condition of extremum value for Var[Zlowast(T)] with respect to the parameter ρ the optimal valueof parameter ρ is given by

ρlowast eminus T minus r+1113957σ

2t( 11138571113954x0 minus α

(β minus α) (56)

and the corresponding optimal expectation

α + ρlowast(β minus α)( 1113857 eminus T minus r+1113957σ

2t( 11138571113954x0 (57)

Substituting ρlowast into the optimal strategy (πlowast0(t)

πlowast1(t) πlowast2(t)) the optimal strategy corresponding to theoptimal value α+ ρlowast(β minus α) of expectation can be obtainedMeanwhile plugging the expression of ρlowast into Var[Zπlowast(t)]the optimal minimum variance Var[Zπlowast ρlowast(T)] can also bedetermined It is worth mentioning that the optimalminimum variance for this special case with no liabilityis zero under the uniquely optimal expectation(z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t )

4 Interpretations of the Main Results

e expressions of the obtained results are so abstract that itis necessary to give some interpretations for the main resultsto clarify their specific meaning especially for the invest-ment share π1(t) of the inflation-linked index bond and theinvestment share π2(t) of the risky stock

Scrutinizing features of the expressions π1(t) and π2(t)

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

(58)

where ζβα(t) e(tminus T)(1113957rminus μt)(ω minus ε) ω (α + ρ(β minus α))(1 minus 1113954ϱ) and ε (eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) it is easy to findthat understanding implications of these simplified nota-tions δ 1113954ϱω ε and ζβα(t) is the key point Subsequently theinterpretations will start from descriptions of these sim-plified notations

In notation δ ((1113957μt + r minus 1113957r)21113957σ2t ) + ((1113954μt minus 1113957r)21113954σ2t ) theone part (1113957μt + r minus 1113957r)21113957σ2t is the risk compensation rate ofinflation-linked index bond and the other part (1113954μt minus 1113957r)21113954σ2t isthe risk compensation rate of risky stock us δ synthet-ically reflects the risk compensation of two risky assets

Obviously eminus T(δ+σ2t ) lt 1 so the notation1113954ϱ (eminus T(δ+σ2t )δ + σ2t )(δ + σ2t ) is less than 1 Notation 1113954ϱ givesa ratio of one sum with the risk compensation rate δ beingconverted by eminus T(δ+σ2t ) divided by the other sum in which therisk compensation rate δ has no conversion

e notation ω (α + ρ(β minus α))(1 minus 1113954ϱ) is an expressionof top regulation bound β and bottom regulation bound αwhere α+ ρ(β minus α) is a value of the expectation E[Zπ(T)]Because ρ takes a determined value the changes of regu-lation bounds directly result in the changes of ω ereforeω is the key point reflecting the influences of both regulationbounds on investment strategy

In notation ε the parameter z0 has the most obviouseconomic connotation since it represents the initial value ofasset-liability ratio at time 0 e expression(eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) gives a value resulted from theinitial asset-liability ratio z0 influenced by several differentfactors (return rates and volatilities) and then converted atthe constant relative rate 1113954ϱ

e following expression

ζβα(t)

z

e(tminus T) 1113957rminus μt( )

(1 minus 1113954ϱ)middot

(α + ρ(β minus α)) minus eminus T minus 1113957r+δ+μt+σ2t( )z01113872 1113873

z

(59)

is key to understand the optimal strategy (πlowast0 πlowast1 πlowast2 ) edenominator z of this fraction ζβα(t)z represents the currentlevel of asset-liability ratio at time t

Complexity 9

But the numerator ζβα(t) of this fraction is morecomplicated e one multiplier ((α + ρ(β minus α))minus

z0eminus T(minus 1113957r+δ+μ+1113957σ

2t )) is a difference between the expected value

(α + ρ(β minus α)) of Z(T) and the converted valueeminus T(minus 1113957r+δ+μt+σ2t )z0 of the initial value z0 which characterizesthe distance between initial value affected by various eco-nomic factors and the expectation of Z(T) e othermultiplier e(tminus T)(1113957rminus μt)(1 minus 1113954ϱ) is a converted value compre-hensively influenced by several parameters embodying thestates of financial market

In addition it is easy to see that if the relative rate ζβα(t)zis larger than 1 πlowast1(t) ((1113957μt + r minus 1113957r)1113957σ2t )((ζβα(t)z) minus 1) andπlowast2(t) ((1113954μt minus 1113957r)1113954σ2t )(ζβα(t)z minus 1) are positive otherwiseboth will be negative Since πlowast0(t) 1 minus πlowast1(t) minus πlowast2(t) itindicates that πlowast0(t) 1 + δ(1 minus (1z)ζβα(t)) can be deter-mined as long as the investment shares of risky assets havebeen determined so the adjustment of risk-free asset isusually passive If the relative rate ζβα(t)z is greater than 1the investment shares on risk-free asset can be ensuredwithin a much better range between 0 and 1 ese ob-servations can serve as an important reference for deter-mining both the top regulation bound β and the bottomregulation bound α

5 Numerical Examples

is section discusses the variation tendency of optimalstrategies πlowast1 and πlowast2 varying with some important parame-ters mainly focusing on the current value z of asset-liabilityratio the top regulation bound β the bottom regulationbound α and the parameter ρ e values of parameters aregiven in Table 1 unless otherwise specified other values forthe same variable can be consulted from the correspondinggraphic legend

First let top regulation bound β be fixed if the bottomregulation bound takes much higher values then the in-vestment shares of both inflation-linked index bond and riskystock increase significantly ese variety trends can be ob-served intuitively from Figures 1 and 2 respectively Mean-while Figures 1 and 2 also show that the appropriate bottomregulation bound should be taken at one value between 1 and2 for an economic environment same as this exampleOtherwise the investment shares on risky assets may increasetoomuch but the investment share on risk-free asset becomesnegative which may result in leverage risk increases rapidly

Next let bottom regulation bound α be fixed if the topregulation bound β takes much higher value then the invest-ment shares of both inflation-linked index bond and risky stockshould also increase significantly e variety trends can beperceived from Figures 3 and 4 respectively MeanwhileFigures 3 and 4 also show that the appropriate top regulationbound should be taken at one value between 6 and 7 for aneconomic environment same as this example Otherwise theinvestment shares on risky assets increase too much but theinvestment share on risk-free asset becomes negative whichalso result in leverage risk increases rapidly

As can be seen from Figures 3ndash6 taking the larger valueof either the top regulation bound or the bottom regulation

bound means allowing a wider volatility range of investmentreturns which may lead to the result that assets with greaterinvestment risk are allowed to invest or greater market risksare incorporated into the wealth process In addition theabove numerical analysis roughly gives a valuable referencefor selecting an appropriate range of top regulation boundand bottom regulation bound according to the relationshipbetween the reasonable values of investment strategy andregulation bounds for an economic environment set byvalues of parameters

Table 1 Values of the parameters

1113957μ 008 r 001 1113957σ 02 α 1 β 7 z0 3 z 51113957r 005 T 6 1113954μ 015 1113954σ 06 μ 003 σ 01 ρ 05

02

04

06

08

10

(π1)

lowast

π1lowast changes with α when top bound β fixed

2 3 4 5 61t

α = 1α = 18α = 26

Figure 1 Values of πlowast1 change with t for different bottom bound α

(π2)

lowast

π2lowast changes with α when top bound β fixed

005

010

015

020

2 3 4 5 61t

α = 1α = 2α = 3

Figure 2 Values of πlowast2 change with t for different bottom bound α

10 Complexity

In fact the regulatory bound cannot be determinedarbitrarily Both the top regulatory bound and bottomregulatory bound determine the fluctuation range that de-cision-making needs to control Too narrow fluctuationrange of investment returns results in loss of great invest-ment opportunities and makes investment in the financialmarket meaningless If the regulatory bound is too large thepotential huge risks cannot be dealt with effectively when theinvestment return soars fast which is almost equivalent tosituations without regulatory bounds and also lose thesignificance of using regulatory bounds It is hard to de-termine the regulatory bounds only by quantitative calcu-lation Once there is a model error it may result in seriousdeviation between theory and practice in actual manage-ment thus the determination of upper and lower limitsshould not be purely theoretical Nevertheless it should be

noted that advanced statistical analysis and maturelypractical experience of personnel are essential and of greatadvantage for determining an exact value of α or β

Assuming that both top regulation bound β and bottomregulation bound α are determined the impact of asset-li-ability ratio z at time t on investment strategy can be ob-served in Figures 5 and 6 If the current asset-liability ratio zat time t takes much greater value then the investmentshares of both inflation-linked index bond and risky stockshould be much lower which shows that increasing shares ofinvestment on risk-free asset is an important measure toreduce investment risk For the numerical examples shownin Figures 5 and 6 the regulation bounds being fixed at α 1and β 7 the most appropriate asset-liability ratio shouldtake values around 5 which is much more advantageous forthe company

e impact of parameter ρ on investment strategy can beobserved in Figures 7 and 8 If the parameter ρ takes a muchgreater value then the investment shares of risky assets mustincrease correspondingly Since the top regulation bound βand the bottom regulation bound α are predetermined theexpected value α+ ρ(β minus α) of asset-liability ratio entirelydepends on the value of parameter ρerefore the variationtendency of investment shares πlowast1(t) and πlowast2(t) with pa-rameter ρ actually illustrates the trend of investment shareschanging with the expectation of asset-liability ratio Z(T) Inone word the greater the expectation of Z(T) the muchhigher the shares of the companyrsquos wealth to be invested onrisky assets

6 Conclusion

Some empirical studies in the banking literature deal withthe effects of prolonged periods of low interest rates in theeconomy on risk-taking real effects on the real sector andalso on financial stability and so on as given by Chaudron[20] Bikker and Vervliet [21] However this paper uses

π1lowast changes with β when bottom bound α fixed

02

04

06

08

2 3 4 5 61t

β = 6β = 7β = 8

(π1)

lowast

Figure 3 Values of πlowast1 change with t for different top bound β

π2lowast changes with β when bottom bound α fixed

01

02

03

04

2 3 4 5 61t

β = 6β = 8β = 10

(π2)

lowast

Figure 4 Values of πlowast2 change with t for different top bound β

π1lowast changes with t for different z

02

04

06

08

2 3 4 5 61t

z = 4z = 5z = 6

(π1)

lowast

Figure 5 Values of πlowast1 change with t for different current asset-liability ratio z

Complexity 11

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

V(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( )

minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )

+((α + ρ(β minus α)) minus λ)2ϱ(t)

(22)

where ξβα(t) e(tminus T)(1113957rminus μt)((α + ρ(β minus α)) minus λ)

Proof According to the first-order necessary condition ofextremum equation (19) yields

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

z1113957σ2t

Vz

Vzz

πlowast2(t) minus1113954μt minus 1113957r( 1113857

z1113954σ2t

Vz

Vzz

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(23)

Plugging the above expressions of πlowast1(t) and πlowast2(t) into(19) the HJB equation becomes

Vt + zVz 1113957r minus μt minus σ2t1113872 1113873 +z2

2Vzzσ

2t minus δ

V2z

2Vzz

0 (24)

Based on the boundary condition V(T z)

(z minus ((α + ρ(β minus α)) minus λ))2 we try a conjecture

W(t z) f(t)z2

+ g(t)z + h(t) (25)

for equation (24) which satisfies

f(T) 1

g(T) minus 2((α + ρ(β minus α)) minus λ)

h(T) ((α + ρ(β minus α)) minus λ)2

(26)

e derivatives of this conjecture W(t z) are given asfollows

Wt fprime(t)z2

+ gprime(t)z + hprime(t)

Wz 2f(t)z + g(t)

Wzz 2f(t)

(27)

Plugging the expressions of Wz and Wzz into (23) theoptimal strategy becomes

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

z1113957σ2tz +

g(t)

2f(t)1113888 1113889

πlowast2(t) minus1113954μt minus 1113957r( 1113857

z1113954σ2tz +

g(t)

2f(t)1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

Substituting these derivatives of W(t z) into (24) theHJB equation becomes

fprime(t)z2

+ gprime(t)z + hprime(t) + 2f(t)z2

+ g(t)z1113872 1113873 1113957r minus μt minus σ2t1113872 1113873

+ z2f(t)σ2t minus δ f(t)z

2+ g(t)z +

g2(t)

4f(t)1113888 1113889 0

(29)

Equation (29) can be split into three ordinary differentialequations as follows

fprime(t) + 2f(t) 1113957r minus μt( 1113857 minus f(t)σ2t minus δf(t) 0

f(T) 1

gprime(t) + g(t) 1113957r minus μt minus σ2t( 1113857 minus δg(t) 0

g(T) minus 2((α + ρ(β minus α)) minus λ)

hprime(t) minusδ4

g(t)2

f(t) 0

h(T) ((α + ρ(β minus α)) minus λ)2

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

Solving these ordinary differential equations in (30)yields

f(t) e(tminus T) minus 21113957r+δ+2μt+σ2t( )

g(t) minus 2e(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

h(t) ((α + ρ(β minus α)) minus λ)2ϱ(t)

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(31)

Plugging the explicit expression of f(t) and g(t) into (28)and using the equality πlowast0(t) + πlowast1(t) + πlowast2(t) 1 the opti-mal strategy is obtained as follows

πlowast0(t) 1 + δ 1 minus e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

z1113888 1113889

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2te

(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2te

(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)

6 Complexity

Denoting

ξβα(t)≜e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ) (33)

the optimal strategy can be simplified into

πlowast0(t) 1 + δ 1 minus1zξβα(t)1113874 1113875

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tξβα(t)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tξβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

Meanwhile substituting the expressions of f(t) g(t) andh(t) into W(t z) the explicit solution of HJB equation isobtained as follows

W(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( ) +((α + ρ(β minus α)) minus λ)2

ϱ(t) minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

(35)

e following verification theorem shows the obtainedresults are exactly the optimal strategy and the optimal valuefunction as required

Theorem 2 (Verification Theorem) If W(t z) is a solutionof HJB equation (19) and satisfies the boundary conditionW(T z) (z minus ((α + ρ(β minus α)) minus λ))2 then for all admissi-ble strategies π isin Π V(t z)geW(t z) If πlowast satisfies

πlowastisin arg infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

2|Z

π(t) z1113960 1113961

(36)

then V(t z)W(t z) and πlowast is the optimal investmentstrategy of problem (14)

Proof Proof is similar to that given by Pham [18] Chang[19] and so on so the detail is omitted

32 Solution of the Original Problem e optimal valuefunction of problem (11) is defined as

1113954V 0 z0( 1113857 infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

21113960 1113961 minus λ2

(37)

where z0 Z(0)In this section the Lagrange duality method is used to

find a solution of original optimization problem (10) basedon the obtained results of the unconstrained problem

Theorem 3 For original problem (10) the efficient strategyπlowast (πlowast0 πlowast1 πlowast2 ) is given by

πlowast0 1 minus δζβα(t)

zminus 11113888 1113889

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)

and the optimal value function is given by

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +(ω minus ε)21113954ϱ

minus 2eminus T minus 1113957r+δ+μt+σ2t( )(ω minus ε)z0 minus (ε minus 1113954ϱω)

2

(39)

where

ζβα(t) e(tminus T) 1113957rminus μt( )(ω minus ε) (40)

Proof e optimal value function 1113954V(0 z0) can be ob-tained with the equivalence between problem (11) andproblem (14) By taking z0 Z(0) and setting t 0 inV(t z) ityields

1113954V 0 z0( 1113857 V 0 z0( 1113857 minus λ2 f(0)z20 + g(0)z0 + h(0) minus λ2

eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +((α + ρ(β minus α)) minus λ)

2

1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)z0 minus λ2

(41)

Since (δ + σ2t )gt 0 then eminus T(δ+σ2t ) lt 1 and it yields

eminus T δ+σ2t( )δ + σ2t1113874 1113875lt δ + σ2t (42)

and then

1113954ϱ eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tlt 1 (43)

Note that the optimal value function 1113954V(0 z0) is aquadratic function with respect to λ and 1113954ϱ minus 1 is the coef-ficient of quadratic term

1113954ϱ minus 1 eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tminus 1lt 0 (44)

us the finite maximum value of 1113954V(0 z0) can beobtained at a specific value λlowast and

λlowast eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ) (45)

Applying the Lagrangian duality substituting λlowast into1113954V(0 z0) the minimum variance corresponding to an ar-bitrarily given expectation (α + ρ(β minus α)) is obtained asfollows

Complexity 7

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +

(α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

middot 1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( ) (α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠z0 minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

(46)

Substitute (45) into (21) the optimal investment strategyof problem (10) is given by

πlowast0 1 minus δ1zζβα(t) minus 11113874 1113875

πlowast1 minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tζβα(t)

πlowast2 minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tζβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where

ζβα(t) e(tminus T) 1113957rminus μt( ) (α + ρ(β minus α))

1 minus 1113954ϱminus

eminus T minus 1113957r+δ+μt+σ2t( )z0

1 minus 1113954ϱ⎛⎝ ⎞⎠

(48)

Remark 2 However in the obtained results (47) and (46)above the expected value (α + ρ(β minus α)) of Zπ(t) is not fixedat a determined position but can take any value between αand β depending on the value of ρ isin (0 1) erefore theobtained minimum variance depends on the parametervalue ρ isin (0 1) in the expression of expectation(α + ρ(β minus α)) Next let us turn our attention to the pa-rameter ρ and determine the optimal parameter value ρlowast forminimizing the variance with respect to ρ as well eoptimal ρlowast to achieve the minimum of Var[Zπlowast(T)] is givenas follows

ρlowast eminus T minus 1113957r+δ+μt+σ2t( )z01113872 11138731113954ϱ1113872 1113873 minus α

(β minus α) (49)

and the expected value of Zπ(t) corresponding to the optimalparameter value ρlowast is given by

α + ρlowast(β minus α) z01113954ϱ

eminus T minus 1113957r+δ+μt+σ2t( ) (50)

Substituting ρlowast into (47) we can get the optimal strategycorresponding to the optimal value α+ ρlowast minus α) of expecta-tion Moreover plugging the expression of ρlowast into theformula of Var[Zπlowast(T)] the optimal minimum varianceVar[Zπlowast ρlowast(T)] can also be determined corresponding to theuniquely optimal value (z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t ) of theexpectation

33 Special Case e investment regulation is also im-portant for an investment process without liabilities Letμt σt 0 the original problem degenerates to the case of noliability and the process Zπ(t) becomes the dynamics process1113954Xπ(t) as follows

d 1113954Xπ(t) 1113957r + π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 1113857 minus π1(t)1113957σ2t minus 1113957μt1113960 1113961

middot 1113954Xπ(t)dt + π1(t) minus 1( 11138571113957σt

1113954Xπ(t)d 1113957W(t) + π2(t)1113954σt

middot 1113954Xπ(t)d 1113954W(t)

(51)

Meanwhile the regulation bound constraints also de-generate into a restriction on the investment dynamicsprocess after eliminating inflation

αlt 1113954Xπ(T)lt β (52)

Proposition 1 For the special case μt σt 0 the minimumvariance of Zπ(t) is

Var Zlowast(T)1113858 1113859 e

minus T1113957δ (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

minus 2eminus T minus r+1113957δ+1113957σ

2t( 1113857 (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠1113954x0

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0 minus eminus T1113957δ(α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

+ eminus T minus 2r+1113957δ+21113957σ

2t( 11138571113954x

20

(53)

and the optimal strategy of the special case is given as follows

8 Complexity

πlowast1(t) 1 minus1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

1113957σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast0(t) 1113957δ 1 minuse(tminus T) minus r+1113957δ+1113957σ

2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(54)

where

1113957δ 1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t (55)

Proof e results can be obtained by the same calculationprocess as the original problem so the details are omittedhere

Remark 3 e expected value (α + ρ(β minus α)) is not fixed ata determined position in the above proposition but it cantake any value between α and β depending on the value ofρ isin (0 1) erefore the obtained minimum variancedepends on the parameter value ρ isin (0 1) Using the first-order necessary condition of extremum value for Var[Zlowast(T)] with respect to the parameter ρ the optimal valueof parameter ρ is given by

ρlowast eminus T minus r+1113957σ

2t( 11138571113954x0 minus α

(β minus α) (56)

and the corresponding optimal expectation

α + ρlowast(β minus α)( 1113857 eminus T minus r+1113957σ

2t( 11138571113954x0 (57)

Substituting ρlowast into the optimal strategy (πlowast0(t)

πlowast1(t) πlowast2(t)) the optimal strategy corresponding to theoptimal value α+ ρlowast(β minus α) of expectation can be obtainedMeanwhile plugging the expression of ρlowast into Var[Zπlowast(t)]the optimal minimum variance Var[Zπlowast ρlowast(T)] can also bedetermined It is worth mentioning that the optimalminimum variance for this special case with no liabilityis zero under the uniquely optimal expectation(z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t )

4 Interpretations of the Main Results

e expressions of the obtained results are so abstract that itis necessary to give some interpretations for the main resultsto clarify their specific meaning especially for the invest-ment share π1(t) of the inflation-linked index bond and theinvestment share π2(t) of the risky stock

Scrutinizing features of the expressions π1(t) and π2(t)

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

(58)

where ζβα(t) e(tminus T)(1113957rminus μt)(ω minus ε) ω (α + ρ(β minus α))(1 minus 1113954ϱ) and ε (eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) it is easy to findthat understanding implications of these simplified nota-tions δ 1113954ϱω ε and ζβα(t) is the key point Subsequently theinterpretations will start from descriptions of these sim-plified notations

In notation δ ((1113957μt + r minus 1113957r)21113957σ2t ) + ((1113954μt minus 1113957r)21113954σ2t ) theone part (1113957μt + r minus 1113957r)21113957σ2t is the risk compensation rate ofinflation-linked index bond and the other part (1113954μt minus 1113957r)21113954σ2t isthe risk compensation rate of risky stock us δ synthet-ically reflects the risk compensation of two risky assets

Obviously eminus T(δ+σ2t ) lt 1 so the notation1113954ϱ (eminus T(δ+σ2t )δ + σ2t )(δ + σ2t ) is less than 1 Notation 1113954ϱ givesa ratio of one sum with the risk compensation rate δ beingconverted by eminus T(δ+σ2t ) divided by the other sum in which therisk compensation rate δ has no conversion

e notation ω (α + ρ(β minus α))(1 minus 1113954ϱ) is an expressionof top regulation bound β and bottom regulation bound αwhere α+ ρ(β minus α) is a value of the expectation E[Zπ(T)]Because ρ takes a determined value the changes of regu-lation bounds directly result in the changes of ω ereforeω is the key point reflecting the influences of both regulationbounds on investment strategy

In notation ε the parameter z0 has the most obviouseconomic connotation since it represents the initial value ofasset-liability ratio at time 0 e expression(eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) gives a value resulted from theinitial asset-liability ratio z0 influenced by several differentfactors (return rates and volatilities) and then converted atthe constant relative rate 1113954ϱ

e following expression

ζβα(t)

z

e(tminus T) 1113957rminus μt( )

(1 minus 1113954ϱ)middot

(α + ρ(β minus α)) minus eminus T minus 1113957r+δ+μt+σ2t( )z01113872 1113873

z

(59)

is key to understand the optimal strategy (πlowast0 πlowast1 πlowast2 ) edenominator z of this fraction ζβα(t)z represents the currentlevel of asset-liability ratio at time t

Complexity 9

But the numerator ζβα(t) of this fraction is morecomplicated e one multiplier ((α + ρ(β minus α))minus

z0eminus T(minus 1113957r+δ+μ+1113957σ

2t )) is a difference between the expected value

(α + ρ(β minus α)) of Z(T) and the converted valueeminus T(minus 1113957r+δ+μt+σ2t )z0 of the initial value z0 which characterizesthe distance between initial value affected by various eco-nomic factors and the expectation of Z(T) e othermultiplier e(tminus T)(1113957rminus μt)(1 minus 1113954ϱ) is a converted value compre-hensively influenced by several parameters embodying thestates of financial market

In addition it is easy to see that if the relative rate ζβα(t)zis larger than 1 πlowast1(t) ((1113957μt + r minus 1113957r)1113957σ2t )((ζβα(t)z) minus 1) andπlowast2(t) ((1113954μt minus 1113957r)1113954σ2t )(ζβα(t)z minus 1) are positive otherwiseboth will be negative Since πlowast0(t) 1 minus πlowast1(t) minus πlowast2(t) itindicates that πlowast0(t) 1 + δ(1 minus (1z)ζβα(t)) can be deter-mined as long as the investment shares of risky assets havebeen determined so the adjustment of risk-free asset isusually passive If the relative rate ζβα(t)z is greater than 1the investment shares on risk-free asset can be ensuredwithin a much better range between 0 and 1 ese ob-servations can serve as an important reference for deter-mining both the top regulation bound β and the bottomregulation bound α

5 Numerical Examples

is section discusses the variation tendency of optimalstrategies πlowast1 and πlowast2 varying with some important parame-ters mainly focusing on the current value z of asset-liabilityratio the top regulation bound β the bottom regulationbound α and the parameter ρ e values of parameters aregiven in Table 1 unless otherwise specified other values forthe same variable can be consulted from the correspondinggraphic legend

First let top regulation bound β be fixed if the bottomregulation bound takes much higher values then the in-vestment shares of both inflation-linked index bond and riskystock increase significantly ese variety trends can be ob-served intuitively from Figures 1 and 2 respectively Mean-while Figures 1 and 2 also show that the appropriate bottomregulation bound should be taken at one value between 1 and2 for an economic environment same as this exampleOtherwise the investment shares on risky assets may increasetoomuch but the investment share on risk-free asset becomesnegative which may result in leverage risk increases rapidly

Next let bottom regulation bound α be fixed if the topregulation bound β takes much higher value then the invest-ment shares of both inflation-linked index bond and risky stockshould also increase significantly e variety trends can beperceived from Figures 3 and 4 respectively MeanwhileFigures 3 and 4 also show that the appropriate top regulationbound should be taken at one value between 6 and 7 for aneconomic environment same as this example Otherwise theinvestment shares on risky assets increase too much but theinvestment share on risk-free asset becomes negative whichalso result in leverage risk increases rapidly

As can be seen from Figures 3ndash6 taking the larger valueof either the top regulation bound or the bottom regulation

bound means allowing a wider volatility range of investmentreturns which may lead to the result that assets with greaterinvestment risk are allowed to invest or greater market risksare incorporated into the wealth process In addition theabove numerical analysis roughly gives a valuable referencefor selecting an appropriate range of top regulation boundand bottom regulation bound according to the relationshipbetween the reasonable values of investment strategy andregulation bounds for an economic environment set byvalues of parameters

Table 1 Values of the parameters

1113957μ 008 r 001 1113957σ 02 α 1 β 7 z0 3 z 51113957r 005 T 6 1113954μ 015 1113954σ 06 μ 003 σ 01 ρ 05

02

04

06

08

10

(π1)

lowast

π1lowast changes with α when top bound β fixed

2 3 4 5 61t

α = 1α = 18α = 26

Figure 1 Values of πlowast1 change with t for different bottom bound α

(π2)

lowast

π2lowast changes with α when top bound β fixed

005

010

015

020

2 3 4 5 61t

α = 1α = 2α = 3

Figure 2 Values of πlowast2 change with t for different bottom bound α

10 Complexity

In fact the regulatory bound cannot be determinedarbitrarily Both the top regulatory bound and bottomregulatory bound determine the fluctuation range that de-cision-making needs to control Too narrow fluctuationrange of investment returns results in loss of great invest-ment opportunities and makes investment in the financialmarket meaningless If the regulatory bound is too large thepotential huge risks cannot be dealt with effectively when theinvestment return soars fast which is almost equivalent tosituations without regulatory bounds and also lose thesignificance of using regulatory bounds It is hard to de-termine the regulatory bounds only by quantitative calcu-lation Once there is a model error it may result in seriousdeviation between theory and practice in actual manage-ment thus the determination of upper and lower limitsshould not be purely theoretical Nevertheless it should be

noted that advanced statistical analysis and maturelypractical experience of personnel are essential and of greatadvantage for determining an exact value of α or β

Assuming that both top regulation bound β and bottomregulation bound α are determined the impact of asset-li-ability ratio z at time t on investment strategy can be ob-served in Figures 5 and 6 If the current asset-liability ratio zat time t takes much greater value then the investmentshares of both inflation-linked index bond and risky stockshould be much lower which shows that increasing shares ofinvestment on risk-free asset is an important measure toreduce investment risk For the numerical examples shownin Figures 5 and 6 the regulation bounds being fixed at α 1and β 7 the most appropriate asset-liability ratio shouldtake values around 5 which is much more advantageous forthe company

e impact of parameter ρ on investment strategy can beobserved in Figures 7 and 8 If the parameter ρ takes a muchgreater value then the investment shares of risky assets mustincrease correspondingly Since the top regulation bound βand the bottom regulation bound α are predetermined theexpected value α+ ρ(β minus α) of asset-liability ratio entirelydepends on the value of parameter ρerefore the variationtendency of investment shares πlowast1(t) and πlowast2(t) with pa-rameter ρ actually illustrates the trend of investment shareschanging with the expectation of asset-liability ratio Z(T) Inone word the greater the expectation of Z(T) the muchhigher the shares of the companyrsquos wealth to be invested onrisky assets

6 Conclusion

Some empirical studies in the banking literature deal withthe effects of prolonged periods of low interest rates in theeconomy on risk-taking real effects on the real sector andalso on financial stability and so on as given by Chaudron[20] Bikker and Vervliet [21] However this paper uses

π1lowast changes with β when bottom bound α fixed

02

04

06

08

2 3 4 5 61t

β = 6β = 7β = 8

(π1)

lowast

Figure 3 Values of πlowast1 change with t for different top bound β

π2lowast changes with β when bottom bound α fixed

01

02

03

04

2 3 4 5 61t

β = 6β = 8β = 10

(π2)

lowast

Figure 4 Values of πlowast2 change with t for different top bound β

π1lowast changes with t for different z

02

04

06

08

2 3 4 5 61t

z = 4z = 5z = 6

(π1)

lowast

Figure 5 Values of πlowast1 change with t for different current asset-liability ratio z

Complexity 11

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

Denoting

ξβα(t)≜e(tminus T) 1113957rminus μt( )((α + ρ(β minus α)) minus λ) (33)

the optimal strategy can be simplified into

πlowast0(t) 1 + δ 1 minus1zξβα(t)1113874 1113875

πlowast1(t) minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tξβα(t)

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tξβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

Meanwhile substituting the expressions of f(t) g(t) andh(t) into W(t z) the explicit solution of HJB equation isobtained as follows

W(t z) z2e

(tminus T) minus 21113957r+δ+2μt+σ2t( ) +((α + ρ(β minus α)) minus λ)2

ϱ(t) minus 2ze(tminus T) minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)

(35)

e following verification theorem shows the obtainedresults are exactly the optimal strategy and the optimal valuefunction as required

Theorem 2 (Verification Theorem) If W(t z) is a solutionof HJB equation (19) and satisfies the boundary conditionW(T z) (z minus ((α + ρ(β minus α)) minus λ))2 then for all admissi-ble strategies π isin Π V(t z)geW(t z) If πlowast satisfies

πlowastisin arg infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

2|Z

π(t) z1113960 1113961

(36)

then V(t z)W(t z) and πlowast is the optimal investmentstrategy of problem (14)

Proof Proof is similar to that given by Pham [18] Chang[19] and so on so the detail is omitted

32 Solution of the Original Problem e optimal valuefunction of problem (11) is defined as

1113954V 0 z0( 1113857 infπisin1113937

E Zπ(T) minus ((α + ρ(β minus α)) minus λ)( 1113857

21113960 1113961 minus λ2

(37)

where z0 Z(0)In this section the Lagrange duality method is used to

find a solution of original optimization problem (10) basedon the obtained results of the unconstrained problem

Theorem 3 For original problem (10) the efficient strategyπlowast (πlowast0 πlowast1 πlowast2 ) is given by

πlowast0 1 minus δζβα(t)

zminus 11113888 1113889

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)

and the optimal value function is given by

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +(ω minus ε)21113954ϱ

minus 2eminus T minus 1113957r+δ+μt+σ2t( )(ω minus ε)z0 minus (ε minus 1113954ϱω)

2

(39)

where

ζβα(t) e(tminus T) 1113957rminus μt( )(ω minus ε) (40)

Proof e optimal value function 1113954V(0 z0) can be ob-tained with the equivalence between problem (11) andproblem (14) By taking z0 Z(0) and setting t 0 inV(t z) ityields

1113954V 0 z0( 1113857 V 0 z0( 1113857 minus λ2 f(0)z20 + g(0)z0 + h(0) minus λ2

eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +((α + ρ(β minus α)) minus λ)

2

1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( )((α + ρ(β minus α)) minus λ)z0 minus λ2

(41)

Since (δ + σ2t )gt 0 then eminus T(δ+σ2t ) lt 1 and it yields

eminus T δ+σ2t( )δ + σ2t1113874 1113875lt δ + σ2t (42)

and then

1113954ϱ eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tlt 1 (43)

Note that the optimal value function 1113954V(0 z0) is aquadratic function with respect to λ and 1113954ϱ minus 1 is the coef-ficient of quadratic term

1113954ϱ minus 1 eminus T δ+σ2t( )δ + σ2t1113872 1113873

δ + σ2tminus 1lt 0 (44)

us the finite maximum value of 1113954V(0 z0) can beobtained at a specific value λlowast and

λlowast eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ) (45)

Applying the Lagrangian duality substituting λlowast into1113954V(0 z0) the minimum variance corresponding to an ar-bitrarily given expectation (α + ρ(β minus α)) is obtained asfollows

Complexity 7

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +

(α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

middot 1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( ) (α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠z0 minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

(46)

Substitute (45) into (21) the optimal investment strategyof problem (10) is given by

πlowast0 1 minus δ1zζβα(t) minus 11113874 1113875

πlowast1 minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tζβα(t)

πlowast2 minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tζβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where

ζβα(t) e(tminus T) 1113957rminus μt( ) (α + ρ(β minus α))

1 minus 1113954ϱminus

eminus T minus 1113957r+δ+μt+σ2t( )z0

1 minus 1113954ϱ⎛⎝ ⎞⎠

(48)

Remark 2 However in the obtained results (47) and (46)above the expected value (α + ρ(β minus α)) of Zπ(t) is not fixedat a determined position but can take any value between αand β depending on the value of ρ isin (0 1) erefore theobtained minimum variance depends on the parametervalue ρ isin (0 1) in the expression of expectation(α + ρ(β minus α)) Next let us turn our attention to the pa-rameter ρ and determine the optimal parameter value ρlowast forminimizing the variance with respect to ρ as well eoptimal ρlowast to achieve the minimum of Var[Zπlowast(T)] is givenas follows

ρlowast eminus T minus 1113957r+δ+μt+σ2t( )z01113872 11138731113954ϱ1113872 1113873 minus α

(β minus α) (49)

and the expected value of Zπ(t) corresponding to the optimalparameter value ρlowast is given by

α + ρlowast(β minus α) z01113954ϱ

eminus T minus 1113957r+δ+μt+σ2t( ) (50)

Substituting ρlowast into (47) we can get the optimal strategycorresponding to the optimal value α+ ρlowast minus α) of expecta-tion Moreover plugging the expression of ρlowast into theformula of Var[Zπlowast(T)] the optimal minimum varianceVar[Zπlowast ρlowast(T)] can also be determined corresponding to theuniquely optimal value (z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t ) of theexpectation

33 Special Case e investment regulation is also im-portant for an investment process without liabilities Letμt σt 0 the original problem degenerates to the case of noliability and the process Zπ(t) becomes the dynamics process1113954Xπ(t) as follows

d 1113954Xπ(t) 1113957r + π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 1113857 minus π1(t)1113957σ2t minus 1113957μt1113960 1113961

middot 1113954Xπ(t)dt + π1(t) minus 1( 11138571113957σt

1113954Xπ(t)d 1113957W(t) + π2(t)1113954σt

middot 1113954Xπ(t)d 1113954W(t)

(51)

Meanwhile the regulation bound constraints also de-generate into a restriction on the investment dynamicsprocess after eliminating inflation

αlt 1113954Xπ(T)lt β (52)

Proposition 1 For the special case μt σt 0 the minimumvariance of Zπ(t) is

Var Zlowast(T)1113858 1113859 e

minus T1113957δ (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

minus 2eminus T minus r+1113957δ+1113957σ

2t( 1113857 (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠1113954x0

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0 minus eminus T1113957δ(α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

+ eminus T minus 2r+1113957δ+21113957σ

2t( 11138571113954x

20

(53)

and the optimal strategy of the special case is given as follows

8 Complexity

πlowast1(t) 1 minus1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

1113957σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast0(t) 1113957δ 1 minuse(tminus T) minus r+1113957δ+1113957σ

2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(54)

where

1113957δ 1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t (55)

Proof e results can be obtained by the same calculationprocess as the original problem so the details are omittedhere

Remark 3 e expected value (α + ρ(β minus α)) is not fixed ata determined position in the above proposition but it cantake any value between α and β depending on the value ofρ isin (0 1) erefore the obtained minimum variancedepends on the parameter value ρ isin (0 1) Using the first-order necessary condition of extremum value for Var[Zlowast(T)] with respect to the parameter ρ the optimal valueof parameter ρ is given by

ρlowast eminus T minus r+1113957σ

2t( 11138571113954x0 minus α

(β minus α) (56)

and the corresponding optimal expectation

α + ρlowast(β minus α)( 1113857 eminus T minus r+1113957σ

2t( 11138571113954x0 (57)

Substituting ρlowast into the optimal strategy (πlowast0(t)

πlowast1(t) πlowast2(t)) the optimal strategy corresponding to theoptimal value α+ ρlowast(β minus α) of expectation can be obtainedMeanwhile plugging the expression of ρlowast into Var[Zπlowast(t)]the optimal minimum variance Var[Zπlowast ρlowast(T)] can also bedetermined It is worth mentioning that the optimalminimum variance for this special case with no liabilityis zero under the uniquely optimal expectation(z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t )

4 Interpretations of the Main Results

e expressions of the obtained results are so abstract that itis necessary to give some interpretations for the main resultsto clarify their specific meaning especially for the invest-ment share π1(t) of the inflation-linked index bond and theinvestment share π2(t) of the risky stock

Scrutinizing features of the expressions π1(t) and π2(t)

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

(58)

where ζβα(t) e(tminus T)(1113957rminus μt)(ω minus ε) ω (α + ρ(β minus α))(1 minus 1113954ϱ) and ε (eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) it is easy to findthat understanding implications of these simplified nota-tions δ 1113954ϱω ε and ζβα(t) is the key point Subsequently theinterpretations will start from descriptions of these sim-plified notations

In notation δ ((1113957μt + r minus 1113957r)21113957σ2t ) + ((1113954μt minus 1113957r)21113954σ2t ) theone part (1113957μt + r minus 1113957r)21113957σ2t is the risk compensation rate ofinflation-linked index bond and the other part (1113954μt minus 1113957r)21113954σ2t isthe risk compensation rate of risky stock us δ synthet-ically reflects the risk compensation of two risky assets

Obviously eminus T(δ+σ2t ) lt 1 so the notation1113954ϱ (eminus T(δ+σ2t )δ + σ2t )(δ + σ2t ) is less than 1 Notation 1113954ϱ givesa ratio of one sum with the risk compensation rate δ beingconverted by eminus T(δ+σ2t ) divided by the other sum in which therisk compensation rate δ has no conversion

e notation ω (α + ρ(β minus α))(1 minus 1113954ϱ) is an expressionof top regulation bound β and bottom regulation bound αwhere α+ ρ(β minus α) is a value of the expectation E[Zπ(T)]Because ρ takes a determined value the changes of regu-lation bounds directly result in the changes of ω ereforeω is the key point reflecting the influences of both regulationbounds on investment strategy

In notation ε the parameter z0 has the most obviouseconomic connotation since it represents the initial value ofasset-liability ratio at time 0 e expression(eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) gives a value resulted from theinitial asset-liability ratio z0 influenced by several differentfactors (return rates and volatilities) and then converted atthe constant relative rate 1113954ϱ

e following expression

ζβα(t)

z

e(tminus T) 1113957rminus μt( )

(1 minus 1113954ϱ)middot

(α + ρ(β minus α)) minus eminus T minus 1113957r+δ+μt+σ2t( )z01113872 1113873

z

(59)

is key to understand the optimal strategy (πlowast0 πlowast1 πlowast2 ) edenominator z of this fraction ζβα(t)z represents the currentlevel of asset-liability ratio at time t

Complexity 9

But the numerator ζβα(t) of this fraction is morecomplicated e one multiplier ((α + ρ(β minus α))minus

z0eminus T(minus 1113957r+δ+μ+1113957σ

2t )) is a difference between the expected value

(α + ρ(β minus α)) of Z(T) and the converted valueeminus T(minus 1113957r+δ+μt+σ2t )z0 of the initial value z0 which characterizesthe distance between initial value affected by various eco-nomic factors and the expectation of Z(T) e othermultiplier e(tminus T)(1113957rminus μt)(1 minus 1113954ϱ) is a converted value compre-hensively influenced by several parameters embodying thestates of financial market

In addition it is easy to see that if the relative rate ζβα(t)zis larger than 1 πlowast1(t) ((1113957μt + r minus 1113957r)1113957σ2t )((ζβα(t)z) minus 1) andπlowast2(t) ((1113954μt minus 1113957r)1113954σ2t )(ζβα(t)z minus 1) are positive otherwiseboth will be negative Since πlowast0(t) 1 minus πlowast1(t) minus πlowast2(t) itindicates that πlowast0(t) 1 + δ(1 minus (1z)ζβα(t)) can be deter-mined as long as the investment shares of risky assets havebeen determined so the adjustment of risk-free asset isusually passive If the relative rate ζβα(t)z is greater than 1the investment shares on risk-free asset can be ensuredwithin a much better range between 0 and 1 ese ob-servations can serve as an important reference for deter-mining both the top regulation bound β and the bottomregulation bound α

5 Numerical Examples

is section discusses the variation tendency of optimalstrategies πlowast1 and πlowast2 varying with some important parame-ters mainly focusing on the current value z of asset-liabilityratio the top regulation bound β the bottom regulationbound α and the parameter ρ e values of parameters aregiven in Table 1 unless otherwise specified other values forthe same variable can be consulted from the correspondinggraphic legend

First let top regulation bound β be fixed if the bottomregulation bound takes much higher values then the in-vestment shares of both inflation-linked index bond and riskystock increase significantly ese variety trends can be ob-served intuitively from Figures 1 and 2 respectively Mean-while Figures 1 and 2 also show that the appropriate bottomregulation bound should be taken at one value between 1 and2 for an economic environment same as this exampleOtherwise the investment shares on risky assets may increasetoomuch but the investment share on risk-free asset becomesnegative which may result in leverage risk increases rapidly

Next let bottom regulation bound α be fixed if the topregulation bound β takes much higher value then the invest-ment shares of both inflation-linked index bond and risky stockshould also increase significantly e variety trends can beperceived from Figures 3 and 4 respectively MeanwhileFigures 3 and 4 also show that the appropriate top regulationbound should be taken at one value between 6 and 7 for aneconomic environment same as this example Otherwise theinvestment shares on risky assets increase too much but theinvestment share on risk-free asset becomes negative whichalso result in leverage risk increases rapidly

As can be seen from Figures 3ndash6 taking the larger valueof either the top regulation bound or the bottom regulation

bound means allowing a wider volatility range of investmentreturns which may lead to the result that assets with greaterinvestment risk are allowed to invest or greater market risksare incorporated into the wealth process In addition theabove numerical analysis roughly gives a valuable referencefor selecting an appropriate range of top regulation boundand bottom regulation bound according to the relationshipbetween the reasonable values of investment strategy andregulation bounds for an economic environment set byvalues of parameters

Table 1 Values of the parameters

1113957μ 008 r 001 1113957σ 02 α 1 β 7 z0 3 z 51113957r 005 T 6 1113954μ 015 1113954σ 06 μ 003 σ 01 ρ 05

02

04

06

08

10

(π1)

lowast

π1lowast changes with α when top bound β fixed

2 3 4 5 61t

α = 1α = 18α = 26

Figure 1 Values of πlowast1 change with t for different bottom bound α

(π2)

lowast

π2lowast changes with α when top bound β fixed

005

010

015

020

2 3 4 5 61t

α = 1α = 2α = 3

Figure 2 Values of πlowast2 change with t for different bottom bound α

10 Complexity

In fact the regulatory bound cannot be determinedarbitrarily Both the top regulatory bound and bottomregulatory bound determine the fluctuation range that de-cision-making needs to control Too narrow fluctuationrange of investment returns results in loss of great invest-ment opportunities and makes investment in the financialmarket meaningless If the regulatory bound is too large thepotential huge risks cannot be dealt with effectively when theinvestment return soars fast which is almost equivalent tosituations without regulatory bounds and also lose thesignificance of using regulatory bounds It is hard to de-termine the regulatory bounds only by quantitative calcu-lation Once there is a model error it may result in seriousdeviation between theory and practice in actual manage-ment thus the determination of upper and lower limitsshould not be purely theoretical Nevertheless it should be

noted that advanced statistical analysis and maturelypractical experience of personnel are essential and of greatadvantage for determining an exact value of α or β

Assuming that both top regulation bound β and bottomregulation bound α are determined the impact of asset-li-ability ratio z at time t on investment strategy can be ob-served in Figures 5 and 6 If the current asset-liability ratio zat time t takes much greater value then the investmentshares of both inflation-linked index bond and risky stockshould be much lower which shows that increasing shares ofinvestment on risk-free asset is an important measure toreduce investment risk For the numerical examples shownin Figures 5 and 6 the regulation bounds being fixed at α 1and β 7 the most appropriate asset-liability ratio shouldtake values around 5 which is much more advantageous forthe company

e impact of parameter ρ on investment strategy can beobserved in Figures 7 and 8 If the parameter ρ takes a muchgreater value then the investment shares of risky assets mustincrease correspondingly Since the top regulation bound βand the bottom regulation bound α are predetermined theexpected value α+ ρ(β minus α) of asset-liability ratio entirelydepends on the value of parameter ρerefore the variationtendency of investment shares πlowast1(t) and πlowast2(t) with pa-rameter ρ actually illustrates the trend of investment shareschanging with the expectation of asset-liability ratio Z(T) Inone word the greater the expectation of Z(T) the muchhigher the shares of the companyrsquos wealth to be invested onrisky assets

6 Conclusion

Some empirical studies in the banking literature deal withthe effects of prolonged periods of low interest rates in theeconomy on risk-taking real effects on the real sector andalso on financial stability and so on as given by Chaudron[20] Bikker and Vervliet [21] However this paper uses

π1lowast changes with β when bottom bound α fixed

02

04

06

08

2 3 4 5 61t

β = 6β = 7β = 8

(π1)

lowast

Figure 3 Values of πlowast1 change with t for different top bound β

π2lowast changes with β when bottom bound α fixed

01

02

03

04

2 3 4 5 61t

β = 6β = 8β = 10

(π2)

lowast

Figure 4 Values of πlowast2 change with t for different top bound β

π1lowast changes with t for different z

02

04

06

08

2 3 4 5 61t

z = 4z = 5z = 6

(π1)

lowast

Figure 5 Values of πlowast1 change with t for different current asset-liability ratio z

Complexity 11

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

Var Zπlowast

(T)1113960 1113961 eminus T minus 21113957r+δ+2μt+σ2t( )z

20 +

(α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

middot 1113954ϱ minus 2eminus T minus 1113957r+δ+μt+σ2t( ) (α + ρ(β minus α))

(1 minus 1113954ϱ)minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)⎛⎝ ⎞⎠z0 minus

eminus T minus 1113957r+δ+μt+σ2t( )z0

(1 minus 1113954ϱ)minus

1113954ϱ(α + ρ(β minus α))

(1 minus 1113954ϱ)⎛⎝ ⎞⎠

2

(46)

Substitute (45) into (21) the optimal investment strategyof problem (10) is given by

πlowast0 1 minus δ1zζβα(t) minus 11113874 1113875

πlowast1 minus1113957μt + r minus 1113957r( 1113857

1113957σ2t+

1113957μt + r minus 1113957r( 1113857

z1113957σ2tζβα(t)

πlowast2 minus1113954μt minus 1113957r( 1113857

1113954σ2t+

1113954μt minus 1113957r( 1113857

z1113954σ2tζβα(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(47)

where

ζβα(t) e(tminus T) 1113957rminus μt( ) (α + ρ(β minus α))

1 minus 1113954ϱminus

eminus T minus 1113957r+δ+μt+σ2t( )z0

1 minus 1113954ϱ⎛⎝ ⎞⎠

(48)

Remark 2 However in the obtained results (47) and (46)above the expected value (α + ρ(β minus α)) of Zπ(t) is not fixedat a determined position but can take any value between αand β depending on the value of ρ isin (0 1) erefore theobtained minimum variance depends on the parametervalue ρ isin (0 1) in the expression of expectation(α + ρ(β minus α)) Next let us turn our attention to the pa-rameter ρ and determine the optimal parameter value ρlowast forminimizing the variance with respect to ρ as well eoptimal ρlowast to achieve the minimum of Var[Zπlowast(T)] is givenas follows

ρlowast eminus T minus 1113957r+δ+μt+σ2t( )z01113872 11138731113954ϱ1113872 1113873 minus α

(β minus α) (49)

and the expected value of Zπ(t) corresponding to the optimalparameter value ρlowast is given by

α + ρlowast(β minus α) z01113954ϱ

eminus T minus 1113957r+δ+μt+σ2t( ) (50)

Substituting ρlowast into (47) we can get the optimal strategycorresponding to the optimal value α+ ρlowast minus α) of expecta-tion Moreover plugging the expression of ρlowast into theformula of Var[Zπlowast(T)] the optimal minimum varianceVar[Zπlowast ρlowast(T)] can also be determined corresponding to theuniquely optimal value (z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t ) of theexpectation

33 Special Case e investment regulation is also im-portant for an investment process without liabilities Letμt σt 0 the original problem degenerates to the case of noliability and the process Zπ(t) becomes the dynamics process1113954Xπ(t) as follows

d 1113954Xπ(t) 1113957r + π1(t) 1113957μt + r minus 1113957r( 1113857 + π2(t) 1113954μt minus 1113957r( 1113857 minus π1(t)1113957σ2t minus 1113957μt1113960 1113961

middot 1113954Xπ(t)dt + π1(t) minus 1( 11138571113957σt

1113954Xπ(t)d 1113957W(t) + π2(t)1113954σt

middot 1113954Xπ(t)d 1113954W(t)

(51)

Meanwhile the regulation bound constraints also de-generate into a restriction on the investment dynamicsprocess after eliminating inflation

αlt 1113954Xπ(T)lt β (52)

Proposition 1 For the special case μt σt 0 the minimumvariance of Zπ(t) is

Var Zlowast(T)1113858 1113859 e

minus T1113957δ (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

minus 2eminus T minus r+1113957δ+1113957σ

2t( 1113857 (α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠1113954x0

minuseminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0 minus eminus T1113957δ(α + ρ(β minus α))

1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

2

+ eminus T minus 2r+1113957δ+21113957σ

2t( 11138571113954x

20

(53)

and the optimal strategy of the special case is given as follows

8 Complexity

πlowast1(t) 1 minus1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

1113957σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast0(t) 1113957δ 1 minuse(tminus T) minus r+1113957δ+1113957σ

2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(54)

where

1113957δ 1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t (55)

Proof e results can be obtained by the same calculationprocess as the original problem so the details are omittedhere

Remark 3 e expected value (α + ρ(β minus α)) is not fixed ata determined position in the above proposition but it cantake any value between α and β depending on the value ofρ isin (0 1) erefore the obtained minimum variancedepends on the parameter value ρ isin (0 1) Using the first-order necessary condition of extremum value for Var[Zlowast(T)] with respect to the parameter ρ the optimal valueof parameter ρ is given by

ρlowast eminus T minus r+1113957σ

2t( 11138571113954x0 minus α

(β minus α) (56)

and the corresponding optimal expectation

α + ρlowast(β minus α)( 1113857 eminus T minus r+1113957σ

2t( 11138571113954x0 (57)

Substituting ρlowast into the optimal strategy (πlowast0(t)

πlowast1(t) πlowast2(t)) the optimal strategy corresponding to theoptimal value α+ ρlowast(β minus α) of expectation can be obtainedMeanwhile plugging the expression of ρlowast into Var[Zπlowast(t)]the optimal minimum variance Var[Zπlowast ρlowast(T)] can also bedetermined It is worth mentioning that the optimalminimum variance for this special case with no liabilityis zero under the uniquely optimal expectation(z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t )

4 Interpretations of the Main Results

e expressions of the obtained results are so abstract that itis necessary to give some interpretations for the main resultsto clarify their specific meaning especially for the invest-ment share π1(t) of the inflation-linked index bond and theinvestment share π2(t) of the risky stock

Scrutinizing features of the expressions π1(t) and π2(t)

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

(58)

where ζβα(t) e(tminus T)(1113957rminus μt)(ω minus ε) ω (α + ρ(β minus α))(1 minus 1113954ϱ) and ε (eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) it is easy to findthat understanding implications of these simplified nota-tions δ 1113954ϱω ε and ζβα(t) is the key point Subsequently theinterpretations will start from descriptions of these sim-plified notations

In notation δ ((1113957μt + r minus 1113957r)21113957σ2t ) + ((1113954μt minus 1113957r)21113954σ2t ) theone part (1113957μt + r minus 1113957r)21113957σ2t is the risk compensation rate ofinflation-linked index bond and the other part (1113954μt minus 1113957r)21113954σ2t isthe risk compensation rate of risky stock us δ synthet-ically reflects the risk compensation of two risky assets

Obviously eminus T(δ+σ2t ) lt 1 so the notation1113954ϱ (eminus T(δ+σ2t )δ + σ2t )(δ + σ2t ) is less than 1 Notation 1113954ϱ givesa ratio of one sum with the risk compensation rate δ beingconverted by eminus T(δ+σ2t ) divided by the other sum in which therisk compensation rate δ has no conversion

e notation ω (α + ρ(β minus α))(1 minus 1113954ϱ) is an expressionof top regulation bound β and bottom regulation bound αwhere α+ ρ(β minus α) is a value of the expectation E[Zπ(T)]Because ρ takes a determined value the changes of regu-lation bounds directly result in the changes of ω ereforeω is the key point reflecting the influences of both regulationbounds on investment strategy

In notation ε the parameter z0 has the most obviouseconomic connotation since it represents the initial value ofasset-liability ratio at time 0 e expression(eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) gives a value resulted from theinitial asset-liability ratio z0 influenced by several differentfactors (return rates and volatilities) and then converted atthe constant relative rate 1113954ϱ

e following expression

ζβα(t)

z

e(tminus T) 1113957rminus μt( )

(1 minus 1113954ϱ)middot

(α + ρ(β minus α)) minus eminus T minus 1113957r+δ+μt+σ2t( )z01113872 1113873

z

(59)

is key to understand the optimal strategy (πlowast0 πlowast1 πlowast2 ) edenominator z of this fraction ζβα(t)z represents the currentlevel of asset-liability ratio at time t

Complexity 9

But the numerator ζβα(t) of this fraction is morecomplicated e one multiplier ((α + ρ(β minus α))minus

z0eminus T(minus 1113957r+δ+μ+1113957σ

2t )) is a difference between the expected value

(α + ρ(β minus α)) of Z(T) and the converted valueeminus T(minus 1113957r+δ+μt+σ2t )z0 of the initial value z0 which characterizesthe distance between initial value affected by various eco-nomic factors and the expectation of Z(T) e othermultiplier e(tminus T)(1113957rminus μt)(1 minus 1113954ϱ) is a converted value compre-hensively influenced by several parameters embodying thestates of financial market

In addition it is easy to see that if the relative rate ζβα(t)zis larger than 1 πlowast1(t) ((1113957μt + r minus 1113957r)1113957σ2t )((ζβα(t)z) minus 1) andπlowast2(t) ((1113954μt minus 1113957r)1113954σ2t )(ζβα(t)z minus 1) are positive otherwiseboth will be negative Since πlowast0(t) 1 minus πlowast1(t) minus πlowast2(t) itindicates that πlowast0(t) 1 + δ(1 minus (1z)ζβα(t)) can be deter-mined as long as the investment shares of risky assets havebeen determined so the adjustment of risk-free asset isusually passive If the relative rate ζβα(t)z is greater than 1the investment shares on risk-free asset can be ensuredwithin a much better range between 0 and 1 ese ob-servations can serve as an important reference for deter-mining both the top regulation bound β and the bottomregulation bound α

5 Numerical Examples

is section discusses the variation tendency of optimalstrategies πlowast1 and πlowast2 varying with some important parame-ters mainly focusing on the current value z of asset-liabilityratio the top regulation bound β the bottom regulationbound α and the parameter ρ e values of parameters aregiven in Table 1 unless otherwise specified other values forthe same variable can be consulted from the correspondinggraphic legend

First let top regulation bound β be fixed if the bottomregulation bound takes much higher values then the in-vestment shares of both inflation-linked index bond and riskystock increase significantly ese variety trends can be ob-served intuitively from Figures 1 and 2 respectively Mean-while Figures 1 and 2 also show that the appropriate bottomregulation bound should be taken at one value between 1 and2 for an economic environment same as this exampleOtherwise the investment shares on risky assets may increasetoomuch but the investment share on risk-free asset becomesnegative which may result in leverage risk increases rapidly

Next let bottom regulation bound α be fixed if the topregulation bound β takes much higher value then the invest-ment shares of both inflation-linked index bond and risky stockshould also increase significantly e variety trends can beperceived from Figures 3 and 4 respectively MeanwhileFigures 3 and 4 also show that the appropriate top regulationbound should be taken at one value between 6 and 7 for aneconomic environment same as this example Otherwise theinvestment shares on risky assets increase too much but theinvestment share on risk-free asset becomes negative whichalso result in leverage risk increases rapidly

As can be seen from Figures 3ndash6 taking the larger valueof either the top regulation bound or the bottom regulation

bound means allowing a wider volatility range of investmentreturns which may lead to the result that assets with greaterinvestment risk are allowed to invest or greater market risksare incorporated into the wealth process In addition theabove numerical analysis roughly gives a valuable referencefor selecting an appropriate range of top regulation boundand bottom regulation bound according to the relationshipbetween the reasonable values of investment strategy andregulation bounds for an economic environment set byvalues of parameters

Table 1 Values of the parameters

1113957μ 008 r 001 1113957σ 02 α 1 β 7 z0 3 z 51113957r 005 T 6 1113954μ 015 1113954σ 06 μ 003 σ 01 ρ 05

02

04

06

08

10

(π1)

lowast

π1lowast changes with α when top bound β fixed

2 3 4 5 61t

α = 1α = 18α = 26

Figure 1 Values of πlowast1 change with t for different bottom bound α

(π2)

lowast

π2lowast changes with α when top bound β fixed

005

010

015

020

2 3 4 5 61t

α = 1α = 2α = 3

Figure 2 Values of πlowast2 change with t for different bottom bound α

10 Complexity

In fact the regulatory bound cannot be determinedarbitrarily Both the top regulatory bound and bottomregulatory bound determine the fluctuation range that de-cision-making needs to control Too narrow fluctuationrange of investment returns results in loss of great invest-ment opportunities and makes investment in the financialmarket meaningless If the regulatory bound is too large thepotential huge risks cannot be dealt with effectively when theinvestment return soars fast which is almost equivalent tosituations without regulatory bounds and also lose thesignificance of using regulatory bounds It is hard to de-termine the regulatory bounds only by quantitative calcu-lation Once there is a model error it may result in seriousdeviation between theory and practice in actual manage-ment thus the determination of upper and lower limitsshould not be purely theoretical Nevertheless it should be

noted that advanced statistical analysis and maturelypractical experience of personnel are essential and of greatadvantage for determining an exact value of α or β

Assuming that both top regulation bound β and bottomregulation bound α are determined the impact of asset-li-ability ratio z at time t on investment strategy can be ob-served in Figures 5 and 6 If the current asset-liability ratio zat time t takes much greater value then the investmentshares of both inflation-linked index bond and risky stockshould be much lower which shows that increasing shares ofinvestment on risk-free asset is an important measure toreduce investment risk For the numerical examples shownin Figures 5 and 6 the regulation bounds being fixed at α 1and β 7 the most appropriate asset-liability ratio shouldtake values around 5 which is much more advantageous forthe company

e impact of parameter ρ on investment strategy can beobserved in Figures 7 and 8 If the parameter ρ takes a muchgreater value then the investment shares of risky assets mustincrease correspondingly Since the top regulation bound βand the bottom regulation bound α are predetermined theexpected value α+ ρ(β minus α) of asset-liability ratio entirelydepends on the value of parameter ρerefore the variationtendency of investment shares πlowast1(t) and πlowast2(t) with pa-rameter ρ actually illustrates the trend of investment shareschanging with the expectation of asset-liability ratio Z(T) Inone word the greater the expectation of Z(T) the muchhigher the shares of the companyrsquos wealth to be invested onrisky assets

6 Conclusion

Some empirical studies in the banking literature deal withthe effects of prolonged periods of low interest rates in theeconomy on risk-taking real effects on the real sector andalso on financial stability and so on as given by Chaudron[20] Bikker and Vervliet [21] However this paper uses

π1lowast changes with β when bottom bound α fixed

02

04

06

08

2 3 4 5 61t

β = 6β = 7β = 8

(π1)

lowast

Figure 3 Values of πlowast1 change with t for different top bound β

π2lowast changes with β when bottom bound α fixed

01

02

03

04

2 3 4 5 61t

β = 6β = 8β = 10

(π2)

lowast

Figure 4 Values of πlowast2 change with t for different top bound β

π1lowast changes with t for different z

02

04

06

08

2 3 4 5 61t

z = 4z = 5z = 6

(π1)

lowast

Figure 5 Values of πlowast1 change with t for different current asset-liability ratio z

Complexity 11

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

πlowast1(t) 1 minus1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

1113957σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast2(t) minus1113954μt minus 1113957r( 1113857

1113954σ2t1 minus

e(tminus T) minus r+1113957δ+1113957σ2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

πlowast0(t) 1113957δ 1 minuse(tminus T) minus r+1113957δ+1113957σ

2t( 1113857(α + ρ(β minus α)) minus eminus T minus r+1113957δ+1113957σ

2t( 11138571113954x0

1113954xe(tminus T) minus 2r+1113957δ+21113957σ2t( 1113857 1 minus eminus T1113957δ1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(54)

where

1113957δ 1113957μt + r minus 1113957r minus 1113957σ2t( 1113857

2

1113957σ2t+

1113954μt minus 1113957r( 11138572

1113954σ2t (55)

Proof e results can be obtained by the same calculationprocess as the original problem so the details are omittedhere

Remark 3 e expected value (α + ρ(β minus α)) is not fixed ata determined position in the above proposition but it cantake any value between α and β depending on the value ofρ isin (0 1) erefore the obtained minimum variancedepends on the parameter value ρ isin (0 1) Using the first-order necessary condition of extremum value for Var[Zlowast(T)] with respect to the parameter ρ the optimal valueof parameter ρ is given by

ρlowast eminus T minus r+1113957σ

2t( 11138571113954x0 minus α

(β minus α) (56)

and the corresponding optimal expectation

α + ρlowast(β minus α)( 1113857 eminus T minus r+1113957σ

2t( 11138571113954x0 (57)

Substituting ρlowast into the optimal strategy (πlowast0(t)

πlowast1(t) πlowast2(t)) the optimal strategy corresponding to theoptimal value α+ ρlowast(β minus α) of expectation can be obtainedMeanwhile plugging the expression of ρlowast into Var[Zπlowast(t)]the optimal minimum variance Var[Zπlowast ρlowast(T)] can also bedetermined It is worth mentioning that the optimalminimum variance for this special case with no liabilityis zero under the uniquely optimal expectation(z01113954ϱ)eminus T(minus 1113957r+δ+μt+σ2t )

4 Interpretations of the Main Results

e expressions of the obtained results are so abstract that itis necessary to give some interpretations for the main resultsto clarify their specific meaning especially for the invest-ment share π1(t) of the inflation-linked index bond and theinvestment share π2(t) of the risky stock

Scrutinizing features of the expressions π1(t) and π2(t)

πlowast1 1113957μt + r minus 1113957r( 1113857

1113957σ2t

ζβα(t)

zminus 11113888 1113889

πlowast2 1113954μt minus 1113957r( 1113857

1113954σ2t

ζβα(t)

zminus 11113888 1113889

(58)

where ζβα(t) e(tminus T)(1113957rminus μt)(ω minus ε) ω (α + ρ(β minus α))(1 minus 1113954ϱ) and ε (eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) it is easy to findthat understanding implications of these simplified nota-tions δ 1113954ϱω ε and ζβα(t) is the key point Subsequently theinterpretations will start from descriptions of these sim-plified notations

In notation δ ((1113957μt + r minus 1113957r)21113957σ2t ) + ((1113954μt minus 1113957r)21113954σ2t ) theone part (1113957μt + r minus 1113957r)21113957σ2t is the risk compensation rate ofinflation-linked index bond and the other part (1113954μt minus 1113957r)21113954σ2t isthe risk compensation rate of risky stock us δ synthet-ically reflects the risk compensation of two risky assets

Obviously eminus T(δ+σ2t ) lt 1 so the notation1113954ϱ (eminus T(δ+σ2t )δ + σ2t )(δ + σ2t ) is less than 1 Notation 1113954ϱ givesa ratio of one sum with the risk compensation rate δ beingconverted by eminus T(δ+σ2t ) divided by the other sum in which therisk compensation rate δ has no conversion

e notation ω (α + ρ(β minus α))(1 minus 1113954ϱ) is an expressionof top regulation bound β and bottom regulation bound αwhere α+ ρ(β minus α) is a value of the expectation E[Zπ(T)]Because ρ takes a determined value the changes of regu-lation bounds directly result in the changes of ω ereforeω is the key point reflecting the influences of both regulationbounds on investment strategy

In notation ε the parameter z0 has the most obviouseconomic connotation since it represents the initial value ofasset-liability ratio at time 0 e expression(eminus T(minus 1113957r+δ+μt+σ2t )z0)(1 minus 1113954ϱ) gives a value resulted from theinitial asset-liability ratio z0 influenced by several differentfactors (return rates and volatilities) and then converted atthe constant relative rate 1113954ϱ

e following expression

ζβα(t)

z

e(tminus T) 1113957rminus μt( )

(1 minus 1113954ϱ)middot

(α + ρ(β minus α)) minus eminus T minus 1113957r+δ+μt+σ2t( )z01113872 1113873

z

(59)

is key to understand the optimal strategy (πlowast0 πlowast1 πlowast2 ) edenominator z of this fraction ζβα(t)z represents the currentlevel of asset-liability ratio at time t

Complexity 9

But the numerator ζβα(t) of this fraction is morecomplicated e one multiplier ((α + ρ(β minus α))minus

z0eminus T(minus 1113957r+δ+μ+1113957σ

2t )) is a difference between the expected value

(α + ρ(β minus α)) of Z(T) and the converted valueeminus T(minus 1113957r+δ+μt+σ2t )z0 of the initial value z0 which characterizesthe distance between initial value affected by various eco-nomic factors and the expectation of Z(T) e othermultiplier e(tminus T)(1113957rminus μt)(1 minus 1113954ϱ) is a converted value compre-hensively influenced by several parameters embodying thestates of financial market

In addition it is easy to see that if the relative rate ζβα(t)zis larger than 1 πlowast1(t) ((1113957μt + r minus 1113957r)1113957σ2t )((ζβα(t)z) minus 1) andπlowast2(t) ((1113954μt minus 1113957r)1113954σ2t )(ζβα(t)z minus 1) are positive otherwiseboth will be negative Since πlowast0(t) 1 minus πlowast1(t) minus πlowast2(t) itindicates that πlowast0(t) 1 + δ(1 minus (1z)ζβα(t)) can be deter-mined as long as the investment shares of risky assets havebeen determined so the adjustment of risk-free asset isusually passive If the relative rate ζβα(t)z is greater than 1the investment shares on risk-free asset can be ensuredwithin a much better range between 0 and 1 ese ob-servations can serve as an important reference for deter-mining both the top regulation bound β and the bottomregulation bound α

5 Numerical Examples

is section discusses the variation tendency of optimalstrategies πlowast1 and πlowast2 varying with some important parame-ters mainly focusing on the current value z of asset-liabilityratio the top regulation bound β the bottom regulationbound α and the parameter ρ e values of parameters aregiven in Table 1 unless otherwise specified other values forthe same variable can be consulted from the correspondinggraphic legend

First let top regulation bound β be fixed if the bottomregulation bound takes much higher values then the in-vestment shares of both inflation-linked index bond and riskystock increase significantly ese variety trends can be ob-served intuitively from Figures 1 and 2 respectively Mean-while Figures 1 and 2 also show that the appropriate bottomregulation bound should be taken at one value between 1 and2 for an economic environment same as this exampleOtherwise the investment shares on risky assets may increasetoomuch but the investment share on risk-free asset becomesnegative which may result in leverage risk increases rapidly

Next let bottom regulation bound α be fixed if the topregulation bound β takes much higher value then the invest-ment shares of both inflation-linked index bond and risky stockshould also increase significantly e variety trends can beperceived from Figures 3 and 4 respectively MeanwhileFigures 3 and 4 also show that the appropriate top regulationbound should be taken at one value between 6 and 7 for aneconomic environment same as this example Otherwise theinvestment shares on risky assets increase too much but theinvestment share on risk-free asset becomes negative whichalso result in leverage risk increases rapidly

As can be seen from Figures 3ndash6 taking the larger valueof either the top regulation bound or the bottom regulation

bound means allowing a wider volatility range of investmentreturns which may lead to the result that assets with greaterinvestment risk are allowed to invest or greater market risksare incorporated into the wealth process In addition theabove numerical analysis roughly gives a valuable referencefor selecting an appropriate range of top regulation boundand bottom regulation bound according to the relationshipbetween the reasonable values of investment strategy andregulation bounds for an economic environment set byvalues of parameters

Table 1 Values of the parameters

1113957μ 008 r 001 1113957σ 02 α 1 β 7 z0 3 z 51113957r 005 T 6 1113954μ 015 1113954σ 06 μ 003 σ 01 ρ 05

02

04

06

08

10

(π1)

lowast

π1lowast changes with α when top bound β fixed

2 3 4 5 61t

α = 1α = 18α = 26

Figure 1 Values of πlowast1 change with t for different bottom bound α

(π2)

lowast

π2lowast changes with α when top bound β fixed

005

010

015

020

2 3 4 5 61t

α = 1α = 2α = 3

Figure 2 Values of πlowast2 change with t for different bottom bound α

10 Complexity

In fact the regulatory bound cannot be determinedarbitrarily Both the top regulatory bound and bottomregulatory bound determine the fluctuation range that de-cision-making needs to control Too narrow fluctuationrange of investment returns results in loss of great invest-ment opportunities and makes investment in the financialmarket meaningless If the regulatory bound is too large thepotential huge risks cannot be dealt with effectively when theinvestment return soars fast which is almost equivalent tosituations without regulatory bounds and also lose thesignificance of using regulatory bounds It is hard to de-termine the regulatory bounds only by quantitative calcu-lation Once there is a model error it may result in seriousdeviation between theory and practice in actual manage-ment thus the determination of upper and lower limitsshould not be purely theoretical Nevertheless it should be

noted that advanced statistical analysis and maturelypractical experience of personnel are essential and of greatadvantage for determining an exact value of α or β

Assuming that both top regulation bound β and bottomregulation bound α are determined the impact of asset-li-ability ratio z at time t on investment strategy can be ob-served in Figures 5 and 6 If the current asset-liability ratio zat time t takes much greater value then the investmentshares of both inflation-linked index bond and risky stockshould be much lower which shows that increasing shares ofinvestment on risk-free asset is an important measure toreduce investment risk For the numerical examples shownin Figures 5 and 6 the regulation bounds being fixed at α 1and β 7 the most appropriate asset-liability ratio shouldtake values around 5 which is much more advantageous forthe company

e impact of parameter ρ on investment strategy can beobserved in Figures 7 and 8 If the parameter ρ takes a muchgreater value then the investment shares of risky assets mustincrease correspondingly Since the top regulation bound βand the bottom regulation bound α are predetermined theexpected value α+ ρ(β minus α) of asset-liability ratio entirelydepends on the value of parameter ρerefore the variationtendency of investment shares πlowast1(t) and πlowast2(t) with pa-rameter ρ actually illustrates the trend of investment shareschanging with the expectation of asset-liability ratio Z(T) Inone word the greater the expectation of Z(T) the muchhigher the shares of the companyrsquos wealth to be invested onrisky assets

6 Conclusion

Some empirical studies in the banking literature deal withthe effects of prolonged periods of low interest rates in theeconomy on risk-taking real effects on the real sector andalso on financial stability and so on as given by Chaudron[20] Bikker and Vervliet [21] However this paper uses

π1lowast changes with β when bottom bound α fixed

02

04

06

08

2 3 4 5 61t

β = 6β = 7β = 8

(π1)

lowast

Figure 3 Values of πlowast1 change with t for different top bound β

π2lowast changes with β when bottom bound α fixed

01

02

03

04

2 3 4 5 61t

β = 6β = 8β = 10

(π2)

lowast

Figure 4 Values of πlowast2 change with t for different top bound β

π1lowast changes with t for different z

02

04

06

08

2 3 4 5 61t

z = 4z = 5z = 6

(π1)

lowast

Figure 5 Values of πlowast1 change with t for different current asset-liability ratio z

Complexity 11

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

But the numerator ζβα(t) of this fraction is morecomplicated e one multiplier ((α + ρ(β minus α))minus

z0eminus T(minus 1113957r+δ+μ+1113957σ

2t )) is a difference between the expected value

(α + ρ(β minus α)) of Z(T) and the converted valueeminus T(minus 1113957r+δ+μt+σ2t )z0 of the initial value z0 which characterizesthe distance between initial value affected by various eco-nomic factors and the expectation of Z(T) e othermultiplier e(tminus T)(1113957rminus μt)(1 minus 1113954ϱ) is a converted value compre-hensively influenced by several parameters embodying thestates of financial market

In addition it is easy to see that if the relative rate ζβα(t)zis larger than 1 πlowast1(t) ((1113957μt + r minus 1113957r)1113957σ2t )((ζβα(t)z) minus 1) andπlowast2(t) ((1113954μt minus 1113957r)1113954σ2t )(ζβα(t)z minus 1) are positive otherwiseboth will be negative Since πlowast0(t) 1 minus πlowast1(t) minus πlowast2(t) itindicates that πlowast0(t) 1 + δ(1 minus (1z)ζβα(t)) can be deter-mined as long as the investment shares of risky assets havebeen determined so the adjustment of risk-free asset isusually passive If the relative rate ζβα(t)z is greater than 1the investment shares on risk-free asset can be ensuredwithin a much better range between 0 and 1 ese ob-servations can serve as an important reference for deter-mining both the top regulation bound β and the bottomregulation bound α

5 Numerical Examples

is section discusses the variation tendency of optimalstrategies πlowast1 and πlowast2 varying with some important parame-ters mainly focusing on the current value z of asset-liabilityratio the top regulation bound β the bottom regulationbound α and the parameter ρ e values of parameters aregiven in Table 1 unless otherwise specified other values forthe same variable can be consulted from the correspondinggraphic legend

First let top regulation bound β be fixed if the bottomregulation bound takes much higher values then the in-vestment shares of both inflation-linked index bond and riskystock increase significantly ese variety trends can be ob-served intuitively from Figures 1 and 2 respectively Mean-while Figures 1 and 2 also show that the appropriate bottomregulation bound should be taken at one value between 1 and2 for an economic environment same as this exampleOtherwise the investment shares on risky assets may increasetoomuch but the investment share on risk-free asset becomesnegative which may result in leverage risk increases rapidly

Next let bottom regulation bound α be fixed if the topregulation bound β takes much higher value then the invest-ment shares of both inflation-linked index bond and risky stockshould also increase significantly e variety trends can beperceived from Figures 3 and 4 respectively MeanwhileFigures 3 and 4 also show that the appropriate top regulationbound should be taken at one value between 6 and 7 for aneconomic environment same as this example Otherwise theinvestment shares on risky assets increase too much but theinvestment share on risk-free asset becomes negative whichalso result in leverage risk increases rapidly

As can be seen from Figures 3ndash6 taking the larger valueof either the top regulation bound or the bottom regulation

bound means allowing a wider volatility range of investmentreturns which may lead to the result that assets with greaterinvestment risk are allowed to invest or greater market risksare incorporated into the wealth process In addition theabove numerical analysis roughly gives a valuable referencefor selecting an appropriate range of top regulation boundand bottom regulation bound according to the relationshipbetween the reasonable values of investment strategy andregulation bounds for an economic environment set byvalues of parameters

Table 1 Values of the parameters

1113957μ 008 r 001 1113957σ 02 α 1 β 7 z0 3 z 51113957r 005 T 6 1113954μ 015 1113954σ 06 μ 003 σ 01 ρ 05

02

04

06

08

10

(π1)

lowast

π1lowast changes with α when top bound β fixed

2 3 4 5 61t

α = 1α = 18α = 26

Figure 1 Values of πlowast1 change with t for different bottom bound α

(π2)

lowast

π2lowast changes with α when top bound β fixed

005

010

015

020

2 3 4 5 61t

α = 1α = 2α = 3

Figure 2 Values of πlowast2 change with t for different bottom bound α

10 Complexity

In fact the regulatory bound cannot be determinedarbitrarily Both the top regulatory bound and bottomregulatory bound determine the fluctuation range that de-cision-making needs to control Too narrow fluctuationrange of investment returns results in loss of great invest-ment opportunities and makes investment in the financialmarket meaningless If the regulatory bound is too large thepotential huge risks cannot be dealt with effectively when theinvestment return soars fast which is almost equivalent tosituations without regulatory bounds and also lose thesignificance of using regulatory bounds It is hard to de-termine the regulatory bounds only by quantitative calcu-lation Once there is a model error it may result in seriousdeviation between theory and practice in actual manage-ment thus the determination of upper and lower limitsshould not be purely theoretical Nevertheless it should be

noted that advanced statistical analysis and maturelypractical experience of personnel are essential and of greatadvantage for determining an exact value of α or β

Assuming that both top regulation bound β and bottomregulation bound α are determined the impact of asset-li-ability ratio z at time t on investment strategy can be ob-served in Figures 5 and 6 If the current asset-liability ratio zat time t takes much greater value then the investmentshares of both inflation-linked index bond and risky stockshould be much lower which shows that increasing shares ofinvestment on risk-free asset is an important measure toreduce investment risk For the numerical examples shownin Figures 5 and 6 the regulation bounds being fixed at α 1and β 7 the most appropriate asset-liability ratio shouldtake values around 5 which is much more advantageous forthe company

e impact of parameter ρ on investment strategy can beobserved in Figures 7 and 8 If the parameter ρ takes a muchgreater value then the investment shares of risky assets mustincrease correspondingly Since the top regulation bound βand the bottom regulation bound α are predetermined theexpected value α+ ρ(β minus α) of asset-liability ratio entirelydepends on the value of parameter ρerefore the variationtendency of investment shares πlowast1(t) and πlowast2(t) with pa-rameter ρ actually illustrates the trend of investment shareschanging with the expectation of asset-liability ratio Z(T) Inone word the greater the expectation of Z(T) the muchhigher the shares of the companyrsquos wealth to be invested onrisky assets

6 Conclusion

Some empirical studies in the banking literature deal withthe effects of prolonged periods of low interest rates in theeconomy on risk-taking real effects on the real sector andalso on financial stability and so on as given by Chaudron[20] Bikker and Vervliet [21] However this paper uses

π1lowast changes with β when bottom bound α fixed

02

04

06

08

2 3 4 5 61t

β = 6β = 7β = 8

(π1)

lowast

Figure 3 Values of πlowast1 change with t for different top bound β

π2lowast changes with β when bottom bound α fixed

01

02

03

04

2 3 4 5 61t

β = 6β = 8β = 10

(π2)

lowast

Figure 4 Values of πlowast2 change with t for different top bound β

π1lowast changes with t for different z

02

04

06

08

2 3 4 5 61t

z = 4z = 5z = 6

(π1)

lowast

Figure 5 Values of πlowast1 change with t for different current asset-liability ratio z

Complexity 11

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

In fact the regulatory bound cannot be determinedarbitrarily Both the top regulatory bound and bottomregulatory bound determine the fluctuation range that de-cision-making needs to control Too narrow fluctuationrange of investment returns results in loss of great invest-ment opportunities and makes investment in the financialmarket meaningless If the regulatory bound is too large thepotential huge risks cannot be dealt with effectively when theinvestment return soars fast which is almost equivalent tosituations without regulatory bounds and also lose thesignificance of using regulatory bounds It is hard to de-termine the regulatory bounds only by quantitative calcu-lation Once there is a model error it may result in seriousdeviation between theory and practice in actual manage-ment thus the determination of upper and lower limitsshould not be purely theoretical Nevertheless it should be

noted that advanced statistical analysis and maturelypractical experience of personnel are essential and of greatadvantage for determining an exact value of α or β

Assuming that both top regulation bound β and bottomregulation bound α are determined the impact of asset-li-ability ratio z at time t on investment strategy can be ob-served in Figures 5 and 6 If the current asset-liability ratio zat time t takes much greater value then the investmentshares of both inflation-linked index bond and risky stockshould be much lower which shows that increasing shares ofinvestment on risk-free asset is an important measure toreduce investment risk For the numerical examples shownin Figures 5 and 6 the regulation bounds being fixed at α 1and β 7 the most appropriate asset-liability ratio shouldtake values around 5 which is much more advantageous forthe company

e impact of parameter ρ on investment strategy can beobserved in Figures 7 and 8 If the parameter ρ takes a muchgreater value then the investment shares of risky assets mustincrease correspondingly Since the top regulation bound βand the bottom regulation bound α are predetermined theexpected value α+ ρ(β minus α) of asset-liability ratio entirelydepends on the value of parameter ρerefore the variationtendency of investment shares πlowast1(t) and πlowast2(t) with pa-rameter ρ actually illustrates the trend of investment shareschanging with the expectation of asset-liability ratio Z(T) Inone word the greater the expectation of Z(T) the muchhigher the shares of the companyrsquos wealth to be invested onrisky assets

6 Conclusion

Some empirical studies in the banking literature deal withthe effects of prolonged periods of low interest rates in theeconomy on risk-taking real effects on the real sector andalso on financial stability and so on as given by Chaudron[20] Bikker and Vervliet [21] However this paper uses

π1lowast changes with β when bottom bound α fixed

02

04

06

08

2 3 4 5 61t

β = 6β = 7β = 8

(π1)

lowast

Figure 3 Values of πlowast1 change with t for different top bound β

π2lowast changes with β when bottom bound α fixed

01

02

03

04

2 3 4 5 61t

β = 6β = 8β = 10

(π2)

lowast

Figure 4 Values of πlowast2 change with t for different top bound β

π1lowast changes with t for different z

02

04

06

08

2 3 4 5 61t

z = 4z = 5z = 6

(π1)

lowast

Figure 5 Values of πlowast1 change with t for different current asset-liability ratio z

Complexity 11

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

mathematical models and methods to study a problem ofasset-liability management with financial market risks risksof too low return rate risks of the abnormal rapid growth ofreturn rate and risks of investment leverage A model ofquantitative regulation (both the top regulation bound andthe bottom regulation bound being imposed on the asset-liability ratio) is put forward e efficient strategy and ef-ficient frontier are obtained under the objective of varianceminimization with regulation constraints at the regulatorytime T But the most important value of a theoretical re-search lies in its guidance and reference for practicerough numerical examples it is found that the obtainedexplicit optimal strategy can also provide reference fordetermination of regulation bounds which reinforces thetheoretical significance of this research work

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was supported by China Postdoctoral ScienceFoundation funded project (Grant no 2017M611192)Youth Science Fund of Shanxi University of Finance andEconomics (Grant no QN-2017019) and Youth ScienceFoundation of National Natural Science Fund (Grant no11801179)

References

[1] W F Sharpe and L G Tint ldquoLiabilities- a new approachrdquoLeJournal of Portfolio Management vol 16 no 2 pp 5ndash10 1990

[2] A Kell and H Muller ldquoEfficient portfolios in the asset liabilitycontextrdquo ASTIN Bulletin vol 25 no 1 pp 33ndash48 1995

[3] M Leippold F Trojani and P Vanini ldquoA geometric approachto multiperiod mean variance optimization of assets and li-abilitiesrdquo Journal of Economic Dynamics and Control vol 28no 6 pp 1079ndash1113 2004

[4] D Li and W-L Ng ldquoOptimal dynamic portfolio selectionmultiperiod mean-variance formulationrdquo Mathematical Fi-nance vol 10 no 3 pp 387ndash406 2000

[5] M C Chiu and D Li ldquoAsset and liability management undera continuous-time mean-variance optimization frameworkrdquoInsurance Mathematics and Economics vol 39 no 3pp 330ndash355 2006

[6] C Fu A Lari-Lavassani and X Li ldquoDynamic mean-varianceportfolio selection with borrowing constraintrdquo EuropeanJournal of Operational Research vol 200 no 1 pp 312ndash3192010

π2lowast changes with t for different z

005

010

015

020

025

2 3 4 5 61t

z = 3z = 4z = 5

(π2)

lowast

Figure 6 Values of πlowast2 change with t for different current asset-liability ratio z

π1lowast changes with t for different ρ

02

04

06

08

2 3 4 5 61t

ρ = 04ρ = 05ρ = 06

(π1)

lowast

Figure 7 Values of πlowast1 change with t for different ρ

π2lowast changes with t for different ρ

01

02

03

04

2 3 4 5 61t

ρ = 04ρ = 06ρ = 08

(π2)

lowast

Figure 8 Values of πlowast2 change with t for different ρ

12 Complexity

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13

[7] J Pan and Q Xiao ldquoOptimal mean-variance asset-liabilitymanagement with stochastic interest rates and inflation risksrdquoMathematical Methods of Operations Research vol 85 no 3pp 491ndash519 2017

[8] D Li Y Shen and Y Zeng ldquoDynamic derivative-based in-vestment strategy for mean-variance asset-liability manage-ment with stochastic volatilityrdquo Insurance Mathematics andEconomics vol 78 pp 72ndash86 2018

[9] J Wei and T Wang ldquoTime-consistent mean-variance asset-liability management with random coefficientsrdquo InsuranceMathematics and Economics vol 77 pp 84ndash96 2017

[10] Y Zhang Y Wu S Li and B Wiwatanapataphee ldquoMean-variance asset liability management with state-dependent riskaversionrdquo North American Actuarial Journal vol 21 no 1pp 87ndash106 2017

[11] T B Duarte D M Valladatildeo and A Veiga ldquoAsset liabilitymanagement for open pension schemes using multistagestochastic programming under solvency-II-based regulatoryconstraintsrdquo Insurance Mathematics and Economics vol 77pp 177ndash188 2017

[12] S M Guerra T C Silva B M Tabak R A S Penaloza andR C C Miranda ldquoSystemic risk measuresrdquo Physica Avol 442 pp 329ndash342 2015

[13] S R S D Souza T C Silva B M Tabak and S M GuerraldquoEvaluating systemic risk using bank default probabilities infinancial networksrdquo Journal of Economic Dynamics andControl vol 66 pp 54ndash75 2016

[14] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[15] R C Merton ldquoOn the pricing of corporate debt the riskstructure of interest ratesrdquo Le Journal of Finance vol 29no 2 pp 449ndash470 1974

[16] T C Silva S R S Souza and B M Tabak ldquoMonitoringvulnerability and impact diffusion in financial networksrdquoJournal of Economic Dynamics and Control vol 76pp 109ndash135 2017

[17] D G Luenberger Optimization by Vector Space MethodsWiley New York NY USA 1968

[18] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer Berlin Germany2009

[19] H Chang ldquoDynamic mean-variance portfolio selection withliability and stochastic interest raterdquo Economic Modellingvol 51 pp 172ndash182 2015

[20] R Chaudron Bank Profitability and Risk Taking in a Pro-longed Environment of Low Interest Rates A Study of InterestRate Risk inLe Banking Book of Dutch Banks DNBWorkingPapers Amsterdam Netherlands 2016

[21] J A Bikker and T M Vervliet ldquoBank profitability and risk-taking under low interest ratesrdquo International Journal ofFinance amp Economics vol 23 no 1 pp 3ndash18 2018

Complexity 13