Research Article Valuing Catastrophe Bonds Involving...

Research ArticleValuing Catastrophe Bonds Involving Credit Risks

Jian Liu1 Jihong Xiao1 Lizhao Yan2 and Fenghua Wen3

1 School of Economics and Management Changsha University of Science and Technology Changsha 410004 China2 Press Hunan Normal University Changsha 410081 China3 Business School Central South University Changsha 410083 China

Correspondence should be addressed to Fenghua Wen wfhamssaccn

Received 12 January 2014 Accepted 1 April 2014 Published 17 April 2014

Academic Editor Wei Chen

Copyright copy 2014 Jian Liu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Catastrophe bonds are the most important products in catastrophe risk securitization market For the operating mechanism CATbonds may have a credit risk so in this paper we consider the influence of the credit risk on CAT bonds pricing that is differentfrom the other literatureWe employ the Jarrow and Turnbull method to model the credit risks and get access to the general pricingformula using the Extreme ValueTheory Furthermore we present an empirical pricing study of the Property Claim Services datawhere the parameters in the loss function distribution are estimated by the MLE method and the default probabilities are deducedby the US financial market data Then we get the catastrophe bonds value by the Monte Carlo method

1 Introduction

The securitization of catastrophe risk springing up in theearly 1990s has created a direct link between the insuranceindustry and capitalmarketThere are a variety of catastropherisk securitization instruments such as options swaps andbonds Catastrophe bonds (CAT bonds) the largest issuedand most successful instrument among them have topped$6 billion in 2013 and are set to be the highest since 2007according to Artemis a deal tracker [1] CAT bonds areusually insurance-linked and meant to raise money in caseof a catastrophic event such as a hurricane or earthquake sothat insurance companies can hedge their exposure by trans-ferring catastrophe risk to a wide pool of willing investorsCompared with other fixed income asset classes CAT bondsoffer high yield as the total returns were almost 95 in 2013according to a Swiss Re index [1] Furthermore CAT bondsare not closely linked with the stock market or economicconditions so they offer significant attractions to investors

The development of CAT bonds market depends on thereasonable prices so the scientific pricing is the key problemto the field of CAT bonds research As a kind of catastropherisk securitization product the value of CAT bonds resultsfrom the probability of the catastrophe risk and the loss inthe catastrophe for CAT bonds have the dual properties of

bonds and options Moreover for the highly skewed propertyof the catastrophe risk distribution valuing CAT bonds hasbecome very complicated

The main pricing models of CAT bonds including Krepsmodel [2] LFC model [3 4] Christofides model [5] andWang two-factor model [6] are all using the quantitativemethods to estimate essential factors of the price and thenprice the CAT bonds with them Zimbidis et al [7] useExtreme Value Theory to get the numerical results of CATbonds prices under stochastic interest rates in an incompletemarket framework Z-G Ma and C-Q Ma [8] derive abond pricing formula under stochastic interest rates withthe losses following a compound nonhomogeneous Poissonprocess and find the numerical solution for the price ofcatastrophe risk bonds Li et al [9] study a representativeagent-pricing model of the multievent CAT bonds by thedata of catastrophic insured property losses of typhoon inChina Based on the idea of layered pricing Xiao and Meng[10] use the extreme value model to discuss the pricingof the excess-of-loss reinsurance premium with differentattachment points Nowak and Romaniuk [11] use MonteCarlo simulation method to price the CAT bonds withdifferent payoff functions However most of the literaturein CAT bonds pricing research does not take the creditrisk influence into account Actually the credit risk has the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 563086 6 pageshttpdxdoiorg1011552014563086

2 Mathematical Problems in Engineering

probability of existence for the operating mechanism of CATbonds So it is necessary to fully consider the credit risk invaluing research which can improve the pricing validity

This paper presents a pricing model of CAT bonds withcredit risks and conducts an empirical analysis of the UScatastrophe market data using the Extreme Value Theorywhere the parameters in the loss function distribution areestimated by the MLE method and the default probabilitiesare deduced by the US financial market data Furthermorebased on the theoretical pricing formula we get the CATbonds value by the Monte Carlo method

2 Operating Mechanism of CAT Bonds

There are four main participants in CAT bonds sponsorspecial purpose vehicle (SPV) investor and trustee Thesponsor usually is an insurance company or a reinsurancecompany which signs a reinsurance contract with SPV andpays the reinsurance premium to SPV SPV connects theinsurance market and the capital market and converts thereinsurance premium into CAT bonds which are issued toinvestors Then the reinsurance premium and the fundscoming from CAT bonds investors are deposited in the trustaccount of the trustee and they are invested in short-termsecuritieswith low riskWhen a catastrophic event takes placeand its loss is higher than the prespecified trigger SPV willprovide compensation which is from the loss of CAT bondsinvestors to the sponsor according to the contract In thiscase investors only receive a part of the principal and interestIf the catastrophic risk event does not occur or trigger duringthe term of the CAT bond bonds investors will receive theirprincipal plus a high-yield compensation for the catastrophicrisk exposure [12 13]

It follows that SPV is the key to the CAT bonds issueand its operation is directly related to the expected effect ofCAT bonds SPV is the intermediary in the securitization ofcatastrophe risk and has a significant role in the risk diversifi-cation and financing funds SPV converts the constant returnof the high security assets into float return using swaps andforwards via the trustees who manage the funds in orderto pay interests to CAT bonds investors However the finalappreciation of the capital depends on the credit rating of thecounterparty If the credit rating of the counterparty is lowthe funds may shrink In this case investors of CAT bondswill not receive the high-yield predetermined compensationIt means that the CAT bonds have the credit risk so in thispaper we consider the influence of the credit risk to CATbonds pricing

3 Valuation Framework

31 Pricing Expression of CAT Bonds Suppose the financialmarket is frictionless and arbitrage-free The uncertaintyin the market is characterized by the probability space(ΩF F

119905119905ge0 119875) where Ω is the state space F is the 120590-

algebra of measurable events and 119875 is the market probabilitymeasure The maturity date of a CAT bond is 119879 that isdivided into 119899 period with continuous trading interval Δ119905

An increasing filtration F119905sub F 119905 isin [0 119879] The investor

receives the predetermined interest at the end of each periodand the interest of the current period and the whole principalin time119879The amount of the interest or the principal dependson whether the loss of the catastrophic event exceeds thetrigger level In this paper when considering the credit riskof the CAT bond the investment income also depends on theprobability of default at each period Suppose the probabilityof default at period 119894 is 120582

119894and the recovery rate is a constant 120579

Let119867119894denote the payment amount of theCATbond at period

119894 which can be expressed as followsWhen 1 le 119894 le 119899 minus 1

119867119894=

119862119894 119868119894le 119872 if not default

120579119862119894 119868119894le 119872 if default

119896119862119894 119868119894gt 119872

(1)

When 119894 = 119899

119867119899=

119862119899+ 119865 119868

119899le 119872 if not default

120579 (119862119899+ 119865) 119868

119899le 119872 if default

119896 (119862119899+ 119865) 119868

119899gt 119872

(2)

where 119865 is the principal of the CAT bond 119862119894is the interest of

period 119894 119896 is the obtained ratio of interest and the principalwhen the CAT bond triggers 119868

119894is the loss amount of the

catastrophic event at period 119894 and 119872 is the amount of theloss trigger

Then the price of the CAT bond at time 0 is

119881 = 119864[

119899

sum119894=1

119890minus119903sdot119894Δ119905

119867(120582119894 119868119894)] (3)

where

119867(120582119894 119868119894) =

119896119862119894sdot 119875 (119868119894gt 119872)

+ [120582119894120579119862119894+ (1 minus 120582

119894) 119862119894]

sdot119875 (119868119894le 119872) 1 le 119894 le 119899 minus 1

119896 (119862119899+ 119865) sdot 119875 (119868

119899gt 119872)

+ [120582119899120579 (119862119899+ 119865)

+ (1 minus 120582119899) (119862119899+ 119865)]

sdot119875 (119868119899le 119872) 119894 = 119899

(4)

Let

119881(119897)=1

119897

119897

sum119895=1

119899

sum119894=1

119890minus119903sdot119894Δ119905

119867(120582(119895)

119894 119868(119895)

119894) (5)

Assuming expression (3) exists we can approximate the priceof the CAT bond as 119881 = lim

119897rarrinfin119881(119897) where the number of

simulation paths is 119897 So the price of the CAT bond at time 0can be calculated by Monte Carlo method We will give moredetail of the method in Section 4 Because 119875(119868

119894gt 119872) = 1 minus

119875(119868119894le 119872) so the key for the CAT bond pricing is to obtain

the probability of default 120582119894and the probability distribution

of loss process 119868119894

Mathematical Problems in Engineering 3

32 Credit Risk This paper uses the methodology of Jarrowand Turnbull [14] to model the credit risk of the CAT bondThe credit risk can be described by the probability of defaultand the recovery value which are deduced by the arbitrage-free valuation techniques This methodology is simple andfits the existing term structure of interest rate well As theabsolute priority rule is often violated and lots of otherfactors affect the payoff of bonds such as the percentageof managerial ownership [14] modeling the actual payoffin default is a complicated problem Therefore we take therecovery rate in the event of default when the CAT bond doesnot trigger as an exogenously given constant The recoveryrate is denoted by 120579 and it is assumed to be the same for allbonds in a given credit ranking Then we can use the risk-free interest rate to discount the cash flow of the CAT bondbut not use the discount rate involving risk premium [15] Infact the probability of default has reflected the credit risk ofthe CAT bond

Suppose the probability of default in time interval [119894minus1 119894]is denoted by 120582

119894 and let 120582lowast

119894= minus(1119894) sum

119894

119905=1ln(1 minus 120582

119905) then we

getprod119894119905=1(1 minus 120582

119905) = 119890minus120582

lowast

119894sdot119894 If 120582

119894is small then we can obtain

120582lowast

119894asympsum119894

119905=1120582119905

119894 (6)

It means that 120582lowast119894is the average intensity of default

The interest rate of the risk bond in time interval [119894 minus 1 119894]is denoted by 119903lowast

119894and the risk-free interest rate is 119903

119894 For the

no-arbitrage principle parameters 119903119894 119903lowast119894 120579 120582lowast

119894meet the

equation 119890minus119903lowast

119894 = [1 sdot (1 minus 120582lowast119894) + 120579 sdot 120582lowast

119894]119890minus119903119894 and then we get

120582lowast

119894=1 minus 119890119903119894minus119903

lowast

119894

1 minus 120579 (7)

Consequently we can obtain the average intensity of default120582lowast119894by (6) and deduce the probability of default in each time

interval [119894 minus 1 119894] which is 120582119894 119894 = 1 2 119899

33 Distribution of Loss Function The loss function 119868119894is the

maximum loss amount of the catastrophic event at period 119894and 119868119894= max119883

1119894 1198832119894 119883

119898119894 where 119898 is the number of

days at each period and 1198831119894 1198832119894 119883

119898119894is the sequence of

loss amount at each period They are independent randomvariables with a common unknown distribution function Ifthere exist sequences of constants 119886

119896 119886119896gt 0 forall119896 isin N

119887119896119896isinN and a nondegenerate distribution function119866(119911) such

that

119875(119868119896minus 119887119896

119886119896

le 119911) 997888rarr 119866 (119911) as 119896 997888rarr infin 119911 isin R (8)

then by Fisher-Tipptt Gnedenko Theorem 119866 is a member ofthe generalized extreme value (GEV) family of distributionsor von Mises Type Extreme Value distribution or the vonMises-Jenkinson type distribution [16] and its distributionfunction form can be denoted as

119866 (119911) = expminus[1 + 120585 (119911 minus 120583

120590)]minus1120585

(9)

Year2015201020052000199519901985

Adju

sted

loss

50000

40000

30000

20000

10000

000

Figure 1 Scatter plot of the annual maximum magnitude ofcatastrophe losses in the USA (1985ndash2011)

It is defined on the set 119911 1 + 120585(119911 minus 120583)120590 gt 0 where thescale parameter satisfies 120590 gt 0 and the trail index satisfiesminusinfin lt 120585 lt infin and location parameter satisfies minusinfin lt 120583 lt

infin If we got the estimated values of the three parametersthe distribution function of the maximum loss amount in acatastrophic event 119868

119894can be gotten [17]

There are different kinds of considerable estimationtechniques for three parameters 120590 120585 120583 such as General-ized Method of Moments the graphical techniques basedon versions of probability plots and maximum likelihoodestimation (MLE) method The MLE method is a classicalestimation technique and it is effective for the estimationsof parameters 120590 120585 120583 studied by many researchers [18ndash20]We employ MLE method to estimate parameters Supposethat the loss amount processes 119868

1 1198682 119868

119899are independent

variables following GEV distribution then the log-likelihoodfunction for parameters 120583 120590 120585 is

119871 (120590 120585 120583) = 119899 log120590 minus (1 + 1

120585)

119899

sum119894=1

log [1 + 120585 (119868119894minus 120583

120590)]

minus

119899

sum119894=1

[1 + 120585 (119868119894minus 120583

120590)]

minus1120585

(10)

where 1 + 120585((119868119894minus 120583)120590) gt 0 as 119894 = 1 2 119899

4 Numerical Analysis

41 Data and Parameter Estimations The analysis is basedon the catastrophe data from Property Claim Services (PSC)The data covers all losses resulting from natural catastrophicevents in the USA over the period 1985ndash2011 with 3628original data We get the series of annual maximum magni-tude of catastrophe losses in the USA In order to facilitatecomparison the data is adjusted for inflation given the timevalue of the capital using the Consumer Price Index (CPI)provided by the US Department of Labor Figure 1 shows the


Table 1 Descriptive statistic of catastrophe losses data (billion dollars)

Sample size (119899) Minimum Maximum Mean Standard deviation Skewness KurtosisStatistic Std error Statistic Std error

27 0168 47337 7046 10697 2539 0448 7193 0872

adjusted annual maximummagnitude of losses As describedin Table 1 the mean of catastrophe losses is 7046 billion thestandard deviation is 10697 billion the skewness is 254 andthe kurtosis is 719 It means that the data is right-skewed andtail-dispersed

Assume that the data is independent random variableswith GEV distribution function By the extRemes programpackage in R software we get the Maximum Likelihoodestimations of parameters in distribution function (10) asfollows

(120583 120590 120585) = (1622 2025 102) (11)

The covariance matrix of the three parameters estimations is

cov = [

[

2340 2599 minus053

2599 3836 014

minus053 014 011

]

]

(12)

Figure 2 shows the various plots for assessing the accuracyof the GEV model fitted to the annual maximummagnitudeof catastrophe losses of US data Each set of plotted pointsis near-line in the probability plot and the quantile plot sothe fitted model is valid The respective estimated curve inthe return level plot is close to a straight line and the corre-sponding density plot seems consistent with the histogramof the data Therefore the estimated GEV distribution fitsthe real data well and the GEV model deduced by the MLEmethod is acceptable and validThen we use the estimationsof the model parameters to calculate the probability of thecatastrophe loss by (9)

Now we consider the credit risk The credit ratings ofmost CAT bonds are BB while a CAT bond rated AA wasissued in 2006 for the first time So we assume that thecredit rating of the CAT bond is BB Furthermore we takethe treasury rates in the US market as the risk-free interestrate and take the corporation bonds rates as the risk ratesSuppose the recovery rate 120579 = 40 when the CAT bonddefaults Then according to the US market data in July2013 the treasury rates of 6 months 2 years 3 years and 5years are 006 039 074 and 161respectively and thecorresponding rates of corporation bonds rated BB are 305524 683and 735 respectively We get the treasuryrates and corporation bonds rates from the risk-free interestrates and the risk interest rates respectively over the timeperiod of 1ndash5 years by the linear interpolation method asTable 2 showsMoreover we get the default rates of risk bondsin each period by expressions (6) and (7) just as Table 3shows

42 PricingAnalysis Consider aCATbondwith the duration119899 = 5 years the face value 119865 = 100 dollars and the couponrate 9The amount of trigger loss is assumed to be themean

Table 2 Risk-free interest rates and risk interest rates

Time 1 2 3 4 5Risk-free 017 039 074 118 161Risk 378 524 683 709 735

Table 3 Default rates of risk bonds

Time period 0-1 (1205821) 1-2 (120582

2) 2-3 (120582

3) 3-4 (120582

4) 4-5 (120582

5)

Default rate 592 986 1377 872 823

of the loss amount that is to say119872 = 7046 billion dollarsThe obtained ratio of the interest and the principal when theCAT bond triggers is 119896 = 08 Then we get the probability oftrigger 119875(119868 gt 119872) = 02404 by expression (9) substituting theabove-mentioned parameters Then by (9) we deduce

119911 = 120583 + [120590 (1 minus (minus log119866 (119911))minus120585)] (13)

As the function 119866(119911) is the distribution probability it meets0 le 119866(119911) le 1 So by generating random numbers onthe interval [0 1] we simulate the variable 119911 employing theMonte Carlomethodwith the number of simulation paths 119897 =10000 For expression (4) we get the pricing result of the CATbond which is 119881 = 11886 dollars by the Matlab software Inthe same way the price of fixed income instrument withoutcredit risk is calculated as 13927 dollars with the same interestratesThe former is much less than the latter that attributes tothe risk of catastrophe and the credit risk For a higher riskthe CAT bond gives a higher yield

5 Conclusions

As catastrophes are small probability and high loss eventsand present high positive correlation among individualsthe traditional insurance company and reinsurance companymay not spread risk completely However the securitizationof catastrophe risk brings an effective solution to transfer andspread the catastrophe risk In order to develop theCATbondmarket it is necessary to make effective pricing to CAT bondwhich is the most mature instrument in catastrophe securi-tization at the present To make the pricing result accuratethe credit risk in the CAT bond should not be ignored Thispaper builds the model of the CAT bond involving the creditriskThis model has the characteristics of an analytic methodandnumericalmethod andhas a goodpractical feasibilityWeemploy the Jarrow and Turnbull method to model the creditrisk and get access to the general pricing formula using theExtreme ValueTheory Furthermore we present an empiricalpricing study of the Property Claim Services data where theparameters in the loss function distribution are estimated bythe MLE method and the default probabilities are deduced


08

06

04

02

00

00 02 04 06 08 10

Empirical

Empi

rical

Probability plot

Model

Mod

el

400

300

200

100

0

0 100 200 300 400 500 600

Quantile plot

0020

0010

0000

0 100 200 300 400 500

z

f(z)

Density plotReturn level plot

01 1 10 100 1000

Return period

0

5000

15000

Retu

rn le

vel

Figure 2 Diagnostic plots for GEV fit to the annual maximum magnitude of catastrophe losses in the USA

by the US financial market data Consequently we get thecatastrophe bonds value by the Monte Carlo method whichis lower than the price of fixed income instruments

Further research is directed to extend the model in morecomplex situations for example given suitable stochasticprocess to describe the trigger amount of loss and it willperfect the model which we have discussed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to express their gratitude to thesupport given by the Natural Science Foundation of China(no 71201013 no 71171024 and no 71371195) the Humanitiesand Social Sciences Project of the Ministry of Education

of China (no 12YJC630118) and the Hunan Social SciencePlanning Project of China (no 11YBA009)

References

[1] httpfinancesinacomcnworldmzjj20131018171517040051shtml

[2] R Kreps ldquoInvestment-equivalent reinsurance pricingrdquo in Pro-ceedings of the Casualty Actuarial Society (PCAS rsquo98) vol 85May 1998

[3] M N Lane Price Risk and Ratings for Insurance-LinkedNotes Evaluating Their Position in Your Portfolio DerivativesQuarterly 1998

[4] M N Lane and O Y Movchan ldquoRisk cubes or price risk andratings (Part II)rdquo Journal of Risk Finance vol 1 no 1 pp 71ndash861999

[5] S Christofides Pricing of Catastrophe Linked Securities ASTINColloquium International Actuarial Association Brussels Bel-gium 2004


[6] S Wang Pricing of Catastrophe Bonds Alternative Risk Strate-gies Risk Press 2002

[7] A A Zimbidis N E Frangos and A A Pantelous ldquoModelingearthquake risk via extreme value theory and pricing therespective catastrophe bondsrdquo Astin Bulletin vol 37 no 1 pp163ndash183 2007

[8] Z-G Ma and C-Q Ma ldquoPricing catastrophe risk bondsa mixed approximation methodrdquo Insurance Mathematics ampEconomics vol 52 no 2 pp 243ndash254 2013

[9] Y Li B Fan and J Liu ldquoDesign and pricing of multi-event CATbonds a case of typhoon bonds in Chinardquo China Soft ScienceMagazine no 3 pp 41ndash48 2012

[10] H Xiao and S Meng ldquoEVT and its application to pricingof catastrophe reinsurancerdquo Journal of Applied Statistics andManagement vol 32 no 2 pp 240ndash246 2013

[11] P Nowak and M Romaniuk ldquoPricing and simulations ofcatastrophe bondsrdquo Insurance Mathematics amp Economics vol52 no 1 pp 18ndash28 2013

[12] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[13] C Huang X Gong X Chen and F Wen ldquoMeasuring andforecasting volatility in Chinese stock market using HAR-CJ-M modelrdquo Abstract and Applied Analysis vol 2013 Article ID143194 13 pages 2013

[14] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 no 1 pp 53ndash85 1995

[15] J Huang J Liu and Y Rao ldquoBinary tree pricing to convertiblebonds with credit risk under stochastic interest ratesrdquo Abstractand Applied Analysis vol 2013 Article ID 270467 8 pages 2013

[16] S Kotz and S Nadarajah Extreme Value Distributions ImperialCollege Press London UK 2000

[17] L Yu SWang FWen and K K Lai ldquoGenetic algorithm-basedmulti-criteria project portfolio selectionrdquo Annals of OperationsResearch vol 197 no 1 pp 71ndash86 2012

[18] J Hosking ldquoAlgorithmAS 215maximum-likelihood estimationof the parameters of the generalized extreme-value distribu-tionrdquo Journal of the Royal Statistical Society vol 34 pp 301ndash3101985

[19] J Liu L Yan and C Ma ldquoPricing options and convertiblebonds based on an actuarial approachrdquoMathematical Problemsin Engineering vol 2013 Article ID 676148 9 pages 2013

[20] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

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probability of existence for the operating mechanism of CATbonds So it is necessary to fully consider the credit risk invaluing research which can improve the pricing validity

This paper presents a pricing model of CAT bonds withcredit risks and conducts an empirical analysis of the UScatastrophe market data using the Extreme Value Theorywhere the parameters in the loss function distribution areestimated by the MLE method and the default probabilitiesare deduced by the US financial market data Furthermorebased on the theoretical pricing formula we get the CATbonds value by the Monte Carlo method

2 Operating Mechanism of CAT Bonds

There are four main participants in CAT bonds sponsorspecial purpose vehicle (SPV) investor and trustee Thesponsor usually is an insurance company or a reinsurancecompany which signs a reinsurance contract with SPV andpays the reinsurance premium to SPV SPV connects theinsurance market and the capital market and converts thereinsurance premium into CAT bonds which are issued toinvestors Then the reinsurance premium and the fundscoming from CAT bonds investors are deposited in the trustaccount of the trustee and they are invested in short-termsecuritieswith low riskWhen a catastrophic event takes placeand its loss is higher than the prespecified trigger SPV willprovide compensation which is from the loss of CAT bondsinvestors to the sponsor according to the contract In thiscase investors only receive a part of the principal and interestIf the catastrophic risk event does not occur or trigger duringthe term of the CAT bond bonds investors will receive theirprincipal plus a high-yield compensation for the catastrophicrisk exposure [12 13]

It follows that SPV is the key to the CAT bonds issueand its operation is directly related to the expected effect ofCAT bonds SPV is the intermediary in the securitization ofcatastrophe risk and has a significant role in the risk diversifi-cation and financing funds SPV converts the constant returnof the high security assets into float return using swaps andforwards via the trustees who manage the funds in orderto pay interests to CAT bonds investors However the finalappreciation of the capital depends on the credit rating of thecounterparty If the credit rating of the counterparty is lowthe funds may shrink In this case investors of CAT bondswill not receive the high-yield predetermined compensationIt means that the CAT bonds have the credit risk so in thispaper we consider the influence of the credit risk to CATbonds pricing

3 Valuation Framework

31 Pricing Expression of CAT Bonds Suppose the financialmarket is frictionless and arbitrage-free The uncertaintyin the market is characterized by the probability space(ΩF F

119905119905ge0 119875) where Ω is the state space F is the 120590-

algebra of measurable events and 119875 is the market probabilitymeasure The maturity date of a CAT bond is 119879 that isdivided into 119899 period with continuous trading interval Δ119905

An increasing filtration F119905sub F 119905 isin [0 119879] The investor

receives the predetermined interest at the end of each periodand the interest of the current period and the whole principalin time119879The amount of the interest or the principal dependson whether the loss of the catastrophic event exceeds thetrigger level In this paper when considering the credit riskof the CAT bond the investment income also depends on theprobability of default at each period Suppose the probabilityof default at period 119894 is 120582

119894and the recovery rate is a constant 120579

Let119867119894denote the payment amount of theCATbond at period

119894 which can be expressed as followsWhen 1 le 119894 le 119899 minus 1

119867119894=

119862119894 119868119894le 119872 if not default

120579119862119894 119868119894le 119872 if default

119896119862119894 119868119894gt 119872

(1)

When 119894 = 119899

119867119899=

119862119899+ 119865 119868

119899le 119872 if not default

120579 (119862119899+ 119865) 119868

119899le 119872 if default

119896 (119862119899+ 119865) 119868

119899gt 119872

(2)

where 119865 is the principal of the CAT bond 119862119894is the interest of

period 119894 119896 is the obtained ratio of interest and the principalwhen the CAT bond triggers 119868

119894is the loss amount of the

catastrophic event at period 119894 and 119872 is the amount of theloss trigger

Then the price of the CAT bond at time 0 is

119881 = 119864[

119899

sum119894=1

119890minus119903sdot119894Δ119905

119867(120582119894 119868119894)] (3)

where

119867(120582119894 119868119894) =

119896119862119894sdot 119875 (119868119894gt 119872)

+ [120582119894120579119862119894+ (1 minus 120582

119894) 119862119894]

sdot119875 (119868119894le 119872) 1 le 119894 le 119899 minus 1

119896 (119862119899+ 119865) sdot 119875 (119868

119899gt 119872)

+ [120582119899120579 (119862119899+ 119865)

+ (1 minus 120582119899) (119862119899+ 119865)]

sdot119875 (119868119899le 119872) 119894 = 119899

(4)

Let

119881(119897)=1

119897

119897

sum119895=1

119899

sum119894=1

119890minus119903sdot119894Δ119905

119867(120582(119895)

119894 119868(119895)

119894) (5)

Assuming expression (3) exists we can approximate the priceof the CAT bond as 119881 = lim

119897rarrinfin119881(119897) where the number of

simulation paths is 119897 So the price of the CAT bond at time 0can be calculated by Monte Carlo method We will give moredetail of the method in Section 4 Because 119875(119868

119894gt 119872) = 1 minus

119875(119868119894le 119872) so the key for the CAT bond pricing is to obtain

the probability of default 120582119894and the probability distribution

of loss process 119868119894





119894= minus(1119894) sum

119894

119905=1ln(1 minus 120582

119905) then we

getprod119894119905=1(1 minus 120582

119905) = 119890minus120582

lowast

119894sdot119894 If 120582


120582lowast


119905=1120582119905

119894 (6)




119894 For the


119894meet the




120582lowast

119894=1 minus 119890119903119894minus119903

lowast

119894

1 minus 120579 (7)





1119894 1198832119894 119883





119896 119886119896gt 0 forall119896 isin N


that

119875(119868119896minus 119887119896

119886119896



119866 (119911) = expminus[1 + 120585 (119911 minus 120583

120590)]minus1120585

(9)

Year2015201020052000199519901985

Adju

sted

loss

50000

40000

30000

20000

10000

000






1 1198682 119868



119871 (120590 120585 120583) = 119899 log120590 minus (1 + 1

120585)

119899

sum119894=1

log [1 + 120585 (119868119894minus 120583

120590)]

minus

119899

sum119894=1

[1 + 120585 (119868119894minus 120583

120590)]

minus1120585

(10)

where 1 + 120585((119868119894minus 120583)120590) gt 0 as 119894 = 1 2 119899






27 0168 47337 7046 10697 2539 0448 7193 0872



(120583 120590 120585) = (1622 2025 102) (11)


cov = [

[

2340 2599 minus053

2599 3836 014

minus053 014 011

]

]

(12)







Time period 0-1 (1205821) 1-2 (120582

2) 2-3 (120582

3) 3-4 (120582

4) 4-5 (120582

5)

Default rate 592 986 1377 872 823




5 Conclusions



08

06

04

02

00

00 02 04 06 08 10

Empirical

Empi

rical

Probability plot

Model

Mod

el

400

300

200

100

0

0 100 200 300 400 500 600

Quantile plot

0020

0010

0000

0 100 200 300 400 500

z

f(z)


01 1 10 100 1000

Return period

0

5000

15000

Retu

rn le

vel






Acknowledgments



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119894= minus(1119894) sum

119894

119905=1ln(1 minus 120582

119905) then we

getprod119894119905=1(1 minus 120582

119905) = 119890minus120582

lowast

119894sdot119894 If 120582


120582lowast


119905=1120582119905

119894 (6)




119894 For the


119894meet the




120582lowast

119894=1 minus 119890119903119894minus119903

lowast

119894

1 minus 120579 (7)





1119894 1198832119894 119883





119896 119886119896gt 0 forall119896 isin N


that

119875(119868119896minus 119887119896

119886119896



119866 (119911) = expminus[1 + 120585 (119911 minus 120583

120590)]minus1120585

(9)

Year2015201020052000199519901985

Adju

sted

loss

50000

40000

30000

20000

10000

000






1 1198682 119868



119871 (120590 120585 120583) = 119899 log120590 minus (1 + 1

120585)

119899

sum119894=1

log [1 + 120585 (119868119894minus 120583

120590)]

minus

119899

sum119894=1

[1 + 120585 (119868119894minus 120583

120590)]

minus1120585

(10)

where 1 + 120585((119868119894minus 120583)120590) gt 0 as 119894 = 1 2 119899






27 0168 47337 7046 10697 2539 0448 7193 0872



(120583 120590 120585) = (1622 2025 102) (11)


cov = [

[

2340 2599 minus053

2599 3836 014

minus053 014 011

]

]

(12)







Time period 0-1 (1205821) 1-2 (120582

2) 2-3 (120582

3) 3-4 (120582

4) 4-5 (120582

5)

Default rate 592 986 1377 872 823




5 Conclusions



08

06

04

02

00

00 02 04 06 08 10

Empirical

Empi

rical

Probability plot

Model

Mod

el

400

300

200

100

0

0 100 200 300 400 500 600

Quantile plot

0020

0010

0000

0 100 200 300 400 500

z

f(z)


01 1 10 100 1000

Return period

0

5000

15000

Retu

rn le

vel






Acknowledgments



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












27 0168 47337 7046 10697 2539 0448 7193 0872



(120583 120590 120585) = (1622 2025 102) (11)


cov = [

[

2340 2599 minus053

2599 3836 014

minus053 014 011

]

]

(12)







Time period 0-1 (1205821) 1-2 (120582

2) 2-3 (120582

3) 3-4 (120582

4) 4-5 (120582

5)

Default rate 592 986 1377 872 823




5 Conclusions



08

06

04

02

00

00 02 04 06 08 10

Empirical

Empi

rical

Probability plot

Model

Mod

el

400

300

200

100

0

0 100 200 300 400 500 600

Quantile plot

0020

0010

0000

0 100 200 300 400 500

z

f(z)


01 1 10 100 1000

Return period

0

5000

15000

Retu

rn le

vel






Acknowledgments



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










08

06

04

02

00

00 02 04 06 08 10

Empirical

Empi

rical

Probability plot

Model

Mod

el

400

300

200

100

0

0 100 200 300 400 500 600

Quantile plot

0020

0010

0000

0 100 200 300 400 500

z

f(z)


01 1 10 100 1000

Return period

0

5000

15000

Retu

rn le

vel






Acknowledgments



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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