research article valuing catastrophe bonds involving...
TRANSCRIPT
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
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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 Mathematical Problems in Engineering
[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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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
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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 Mathematical Problems in Engineering
[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
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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 Mathematical Problems in Engineering
[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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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 Mathematical Problems in Engineering
[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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
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 Mathematical Problems in Engineering
[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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
[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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of