1、 Andreas Wrth Hans Schumacher CVaR Pricing and Hedging in Unit- Linked Insurance Products Discussion Paper 10/2008 - 048 October 17, 2008 CVaR pricing and hedging in Unit-Linked insurance products Andreas W urth Hans Schumacher y October 17, 2008 Abstract This paper describes a way how to nd the min
2、imal sellers price for a unit-linked insurance product in order to make the claim acceptable, un- der the assumption that information about the insurance process is only available at the time of maturity. For the general case, the price calcu- lated here provides still an upper bound. Acceptability
3、is dened through the CVaR criterion. Furthermore, the paper shows how to nd the cor- responding hedging strategy. We show how CVaR pricing is connected to earlier results of F ollmer/Leukert about minimization of Expected Short- fall, and apply and extend those results for the case of discrete insur
4、ance probabilities. Wearriveatanalgorithmwhichisstraightforwardanddoes not involve any optimization problem. For an example of a unit-linked survival insurance, we provide analytical formulas for the corresponding hedge, as well as an explicit numerical solutions for the CVaR price. Keywords: Risk m
5、easure pricing, unit-linked insurance products, incomplete markets, CVaR, utility maximization 1 Introduction Unitlinkedinsuranceproductsbecomemoreandmorepopularininsurancein- dustry, because they combine the classical coverage against risks such as death, A.M. W urth, CentER, Department of Economet
6、rics and Operations Research, Tilburg University, Tilburg, the Netherlands, Tel.: +41-76-3814579, Email: a.m.wurthuvt.nl. y J.M. Schumacher, CentER, Department of Econometrics and Operations Re- search, Tilburg University, Tilburg, the Netherlands, Tel.: +31-13-4662050, Email: j.m.schumacheruvt.nl.
7、Research supported in part by Netspar. 1longevityanddisabilityinlifeinsurancewiththepossiblechanceoflargecapital earnings that traditionally banks o er. The pricing of such insurance products needs a combination of classical ac- tuarial principles as well as principles from nancial mathematics. Such
8、 com- binations have been treated in Mller (2002), where the focus was mainly on pricing using a standard deviation principle, but also some hints for a general utility function were given. In general, nancial valuation principles are based on a replication of a claim, whereas in insurance, a risk-l
9、oading is charged, because the claims can- not be hedged. For unit-linked insurance products, one can assume that a full hedge is not possible, because there is a nonhedgeable component. The insur- ance company willtherefore still askfora riskloading. However, becauseof thenancial component of those
10、 products, at least a partial hedge should be pos- sible, which gives the ability to reduce the minimal necessary risk loading. The general aim of this paperis to nda minimal price as well as the corresponding hedging strategy which make a unit-linked insurance claim acceptable for the insurer, wher
11、e acceptability is de ned in terms of coherent risk measures, as in Artzner et al. (1999). We will take CVaR as risk measure. From a practical point of view, this pricing method has the advantage that it is more closely related to the cost of capital method than other pricing rules are, a method whi
12、ch is typically applied in insurance industry. Thetopicofriskmeasurepricingandminimizationhasbeentreatedrecently in several papers. One of the rst papers which presented this idea was Carr et al. (2001). In the sequel, general principles of risk measure pricing have been developed in Kl oppel& Schwe
13、izer (2007), Xu (2006), Cheridito &Kupper (2006) as well as Jobert & Rogers (2008), see also Cherny (2008). The pricing principles are similar to the one in this paper. An abstract framework for pricingon thebasisof coherent risk measureshasbeen rmlyestablished in the cited papers. However, to calcu
14、late prices in speci c situations one typically still needs to solve an optimization problem. In this paper, we address the optimization problem for the case of unit-linked insurance products, and for CVaR as speci c risk measure. A concrete result for the problem of minimization of Worst Conditiona
15、l Expectation has been developed in Sekine (2004). In this paper, the author obtained a formula for the solution of the minimization problem. However, the problem stated there is not the same as the one in this paper. We will talk about this issue again later in the paper. In Ilhan et al. (2008), th
16、e authors solved, apart from theoretical considerations, the problem of numerical risk measure pricing in the example of expected shortfall in the sense of F ollmer 2& Leukert (2000). For their solution, a Hamilton-Jacobi-Bellman method has been applied, which leads to a nonlinear partial di erentia
17、l equation with two variables. The authors pointed out there the computational challenge for doing this. Risk-minimizing strategies for unit-linked insurance products have already been treated in Mller (2001), where risk-minimizing is understoodin the sense of local risk minimization, see for exampl
18、e Schweizer (1991). Other papers of the same author consider the variance as de nition of risk. Minimization of value at risk in unit-linked insurance products as well as corresponding pricing principles have been looked at in Melnikov & Skornyakova (2004), where the authors have mainly focused on t
19、he case with only one insured person, and in thiswayobtainedanalyticformulas. Forthecaseofmanyinsuredpersons,they obtain boundswhichare derived byconsideringthe nancialand insurancerisk separately. As already stated, in our paper, we consider CVaR as risk measure. We are interestedinariskminimizings
20、trategywhenthe nancialandinsuranceriskare considered in an integrated manner. Assuming, as in Melnikov & Skornyakova (2004), that all information about the insurance process is arriving only at the endofthetimeperiod,weobtaintheminimalpricemakingtheclaimacceptable aswellasthecorrespondinghedge. Ittu
21、rnsoutthatwiththissimpli cation,the problembecomeseasyto calculate, andforthespeci cmodelwe use,weobtain analytic formulas. And even in situations where this assumption is unrealistic, the method presented here still leads to an upper bound for the CVaR price, including the corresponding hedge. Actu
22、ally, the results presented in this paper can be applied not only for unit-linked insurance products, but also for other situations of CVaR pricing in incomplete markets. Thekey issueisonlythepricingofapayo which depends on a complete nancial market, as well as on another source of uncertainty, inde
23、pendentofthe nancialmarket,whichcannotbereplicated. Onemaythink about the option of a company to buy a speci c commodity at a speci c time in a speci c currency, where the decision whether or not to buy dependson the foreign exchange rate, but also on other circumstances which are independent of the
24、 nancial market. Our approach is based on a result of Rockafeller & Uryasev (2002), with which we can connect the problem of CVaR pricing to the earlier results of F ollmer & Leukert (1999) and F ollmer & Leukert (2000). These papers con- nect the problem of minimization of expected shortfall to the
25、 Neyman-Pearson theory, see Witting (1985), and for some speci c cases they develop analytic formulas. We further develop those results for the speci c case of unit-linked 3insurance products. In particular, we show in general how those papers can be connected for obtaining a CVaR price for unit-lin
26、ked insurance products, as well as in a speci c example. Furthermore, we extend a theorem presented in F ollmer &Leukert(2000) forthecase wheretheinsuranceprobabilities aredis- crete. Finally, we apply the results obtained to a speci c unit-linked insurance model, and show explicitly the formulas an
27、d the numerical results. The structure of the paper is as follows: In section 2, we formulate the general model, as well as the CVaR pricing principle. In section 3, we present an algorithm for calculating the CVaR price under the additional assumption of continuous distribution. We give an example
28、in which we approximate the discrete insurance probabilities by a normal approximation. In section 4, we prove an extension of the theorem mentioned, in order to be able to apply the resultfordiscreteprobabilities. Withthisresult,weareabletoobtainanalytical formulas for some speci c models, or to so
29、lve the problem numerically. In section 5, we present again an algorithm for the calculation of the CVaR price, without assumption of continuous distributions. We present a speci c example of a unit linked survival insurance, where we obtain analytical formulas. We give an explicit numerical example
30、 for the CVaR price, as well as an analytical formula for the corresponding hedge. Section 6 concludes. 2 Problem speci cation and general statements 2.1 Insurance model and problem speci cation We are dealing with a probability space ( ,F,P). On this probability space, a vector-valued stochastic pr
31、ocess Z t is de ned representing the insurance state process. The states of the nancial market are represented by another vector- valued processX t on ( ,F,P). The ltrationF t is given by the natural ltra- tion generated byX t andZ t . ItisassumedthatZ t andX t areMarkovian,X t is continuous, andtha
32、tthismarket iscomplete, inthesensethat everycontingent claim F(X T ) can be replicated by a suitable trading strategy F(X T ) =S 0 +T 0 t dS t whereS t is the vector-valued process representing the available nancial assets. It is assumed that this process is a vector-valued function of time and then
33、ancial state variables, that is S t = S(t,X t ) with S a measurable function, such thatS t is a vector-valued continuous semi- martingale. A sucient condition for this would be, by the It o formula, that 4X t is a continuous semimartingale and S is twice di erentiable. It is assumed that only the as
34、sets S t can be used for trading. Throughout the whole paper, it is assumed that there exists an equivalent measure Q such that S t is a local martingale. It is assumed furthermore that X t and Z t are independent under P. The option payo due to a unit-linked insurance product at the terminal time T
35、 is given by a nonnegative product-measurable function g(X T ,Z T ), de- pending on the nancial market as well as on the insurance process. Remark2.1. Byextendingthestatespace,itisalwayspossiblethatthepayo depends on some states at t T. Therefore, to restrict to payo s depending only on states at ti
36、me T is not really a restriction. Remark 2.2. A genuine restriction is that payo s can only take place at time T, even if they may depend on earlier times. This has to be assumed because we aim to calculate the CVaR at the terminal timeT. It is in general not clear how to de ne the CVaR if there are
37、 di erent payo times. In some speci c examples, it may besensible to dividethe payments at all times bya num eraire (a reasonable choice may be a zero bond with expiry at time T), and take the CVaR at the xed time T. This situation is also covered by our model. We repeat at this point the CVaR risk
38、measure from Rockafeller & Uryasev (2002): De nition 2.3. LetX be a random variable on a probability space ( ,F,P). ThenCVaR (X) at a certain level is the mean of the distribution function X () := 0 if ( X ()/(1) if where X isthecumulativedistributionfunctionofX,and theValue-at-Risk at level , that
39、is =min| X () Thecredit-constrainedCVaRpricingrule,whichwewilldenoteinthesequel for simplicity CVaR price, is now the following: De nition 2.4. Let the CVaR level be given, as well as a self- nancing portfolio B t . Then the CVaR price is the minimal capital V 0 such that there exists a self- nancin
40、g predictable strategy with respect to the ltration F t such that (2.1) CVaR (g(X T ,Z T )Y T )0 5and such that Y t B t for all t, where the wealth processY t is de ned as Y t =V 0 +t 0 u dS u The problem is now to nd the CVaR price due to de nition 2.4, as well as the corresponding strategy . Remar
41、k 2.5. In the context of the general theory of coherent risk measures, see Artzner et al. (1999), equation (2.1) means that the risk is acceptable for the insurance company. Remark 2.6. At this stage, it is not clear that such a CVaR price exists. This is the issue of the Proposition 2.15. Remark 2.
42、7. In contrast to Kl oppel & Schweizer (2007) or Xu (2006), we consideronlystrategies with wealth processesuniformly(thatisindependentof thespeci cprocess)boundedfrombelowbyaself- nancingportfolio. Typically, one may think about a zero-bond. Economically, this makes sense, because no insurance compa
43、ny has an unlimited credit line. A lower bound of cB t , c a constant and B t a zero-bond, means then the limit until which the insurance company can use credit. Mathematically, one needs this uniform bound, because Assumption 5.4 from Kl oppel & Schweizer (2007) (or Assumption 2.3 from Xu (2006) is
44、 not satis ed by the CVaR in (for instance) the Black-Scholes model. Assumption 5.4 says, in our terminology, that inf gC (g), where is the coherent risk measure andC is the set of superreplicable claims at 0 wealth. Essentially the same problem leads also to CVaR prices which are not necessarily ma
45、rket consistent, an issue which will be discussed in the subsequent section. Asaconsequenceofthelimitedcreditcondition,theCVaRpriceisnottrans- lation invariant, in contrast to the risk measure prices in Kl oppel & Schweizer (2007). Remark2.8. Mathematically, auniformlowerboundB t ,whereB t isagenera
46、l self- nancing strategy, leads to the same problem as the assumption that the wealthprocessY t remainsnonnegative. Indeed,addingcapitalB 0 totheinitial available capital, andB T totheterminalpayo , therestrictionofanonnegative wealth process Y t of this modi ed problem is the same as the restrictio
47、n that Y t B t of the original problem. In the sequel, we will therefore always assume that B t =0, that is that the wealth cannot be negative. Remark 2.9. Conversely, for a xed initial capital V 0 , one can also ask the question what is the minimal possible CVaR and the corresponding hedging 6strat
48、egy. Thisquestion issensibleifthemarketiscompetitive andan insurance company is not able to price independently its products. Remark 2.10. The pricing method described here di ers completely from the one by using an equivalent martingale measure. Actually, there does typically not exist any absolute
49、ly continuous probability measure such that the CVaR price of any insurance claim is given as an expectation under this measure. To see this, consider a simple economy which consists only of two insurance states which have both 50% probability, and only one risk-free nancial asset with return 0. For an insurance option which pays 1 in the rst state and 0 in the second, one can see that the CVaR price must be 1. As a consequence, the only probability measureunderwhich the expected payo
