On The Risk-neutral Valuation Of Life Insurance Contracts .

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On the risk-neutral valuation of life insurance contractswith numerical methods in viewDaniel Bauer*Daniela BergmannRüdiger KieselAbstract:In recent years, market-consistent valuation approaches have gained an increasing importance for insurance companies. This has triggered an increasing interestamong practitioners and academics, and a number of specific studies on such valuation approaches have been published.In this paper, we present a generic model for the valuation of life insurance contracts and embedded options. Furthermore, we describe various numerical valuationapproaches within our generic setup. We particularly focus on contracts containing early exercise features since these present (numerically) challenging valuationproblems.Based on an example of participating life insurance contracts, we illustrate thedifferent approaches and compare their efficiency in a simple and a generalizedBlack-Scholes setup, respectively. Moreover, we study the impact of the consideredearly exercise feature on our example contract and analyze the influence of modelrisk by additionally introducing an exponential Lévy model.Keywords: Life insurance · Risk-neutral valuation · Embedded options · Bermudan options · Nested simulations · PDE methods · Least-squares Monte Carlo* Corresponding author

11 IntroductionIn recent years, market-consistent valuation approaches for life insurance contracts have gainedan increasing practical importance.In 2001, the European Union initiated the “Solvency II” project to revise and extend currentsolvency requirements, the central intention being the incorporation of a risk-based frameworkfor adequate risk management and option pricing techniques for insurance valuation. Furthermore, in 2004 the International Accounting Standards Board issued the new International Financial Reporting Standard (IFRS) 4 (Phase I), which is also concerned with the valuation oflife insurance liabilities. Although Phase I just constitutes a temporary standard, experts agreethat fair valuation will play a major role in the future permanent standard (Phase II), which isexpected to be in place by 2010 (see International Accounting Standards Board (2007)).However, so far, most insurance companies only have little knowledge about risk-neutralvaluation techniques and, hence, mostly rely on simple models and brute force Monte Carlosimulations. This is mainly due to the fact that predominant software solutions (e.g. Moses,Prophet, or ALM.IT) were initially designed for deterministic forecasts of an insurer’s tradeaccounts and only subsequently extended to perform Monte Carlo simulations. In academicliterature, on the other hand, there exists a variety of different articles on the valuation of lifeinsurance contracts. However, there are hardly any detailed comparisons of different numericalvaluation approaches in a general setup. Moreover, some studies do not apply methods fromfinancial mathematics appropriately to the valuation of life insurance products (e.g. questionableworst-case scenarios in Gatzert and Schmeiser (2008) and Kling, Ruß and Schmeiser (2006);see Sec. 3.1 below for details).The objective of this article is to formalize the valuation problem for life insurance contractsin a general way and to provide a survey on concrete valuation methodologies. We particularlyfocus on the valuation of insurance contracts containing early-exercise features or interventionoptions (cf. Steffensen (2002)), such as surrender options, withdrawal guarantees, or optionsto change the premium payment method. While almost all insurance contracts contain suchfeatures, insurers usually do not include these in their price and risk management computationseven though they may add considerably to the value of the contract.The remainder of the text is organized as follows: In Section 2, we present our genericmodel for life insurance contracts. Subsequently, in Section 3, we describe different numericalvaluation approaches. Based on an example of participating life insurance contracts, we carryout numerical experiments in Section 4. Similarly to most prior literature on the valuation of lifecontingencies from a mathematical finance perspective, we initially assume a general BlackScholes framework. We compare the obtained results as well as the efficiency of the differentapproaches and analyze the influence of a surrender option on our example contract. However,as is well-known from various empirical studies, several statistical properties of financial marketdata are not described adequately by Brownian motion and, in general, guarantees and optionswill increase in value under more suitable models. Therefore, we analyze the model risk forour valuation problem by introducing an exponential Lévy model and comparing the obtainedresults for our example to those from the Black-Scholes setup. We find that the qualitativeimpact of the model choice depends on the particular model parameters, i.e. that there exist(realistic) parameter choices for which either model yields higher values. Finally, the last sectionsummarizes our main results.2 Generic contractsWe assume that financial agents can trade continuouslyin a frictionless and arbitrage-freefinan!"cial market with finite time horizon T .1 Let ! F , F F , Q F , FF (FtF )t [0,T ] be a complete,1 In actuarial modeling, it is common to assume a so-called limiting age meaning that a finite time horizonnaturally suffices in view of our objective.

2filtered probability space, where QF is a pricing measure and FF is assumed to satisfy theusual conditions. In this probability space, we introduce the q1 -dimensional, locally bounded,F,(1)F,(q ), . . .,Yt 1 )t [0,T ] , and call it the state process ofadapted Markov process (YtF )t [0,T ] (Ytthe financial market.Within this market, we assume the existence of a locally risk-free asset (Bt )t [0,T ] with# %!"Bt exp 0t ru du , where rt r t,YtF is the short rate. Moreover, we allow for n N other(i)risky assets (At )t [0,T ] , 1 i n, traded in the market with2(i)At A(i) (t,YtF ),1 i n.!"In order to include the mortality component, we fix another probability space ! M , G M , P Mand a homogenous population of x-year old individuals at inception. Similar to Biffis (2005) andDahl (2004), we assume that a q2 -dimensional, locally bounded Markov process (YtM )t [0,T ] !"M,(q1 1)M,(q), . . .,Yt)t [0,T ] , q q1 q2 , on ! M , G M , P M is given. Now let µ (·, ·) : R (Ytthe time"of death Tx of an individual asRq2 R be a positive continuous function and define!the first jump time of a Cox process with intensity µ (x t,YtM ) t [0,T ] , i.e.& '( t)'M'Tx inf t ' µ (x s,Ys ) ds E ,0where E is a unit-exponentially distributed random variable independent of (YtM )t [0,T ] and mu!"tually independent for different individuals. Also, define subfiltrations FM FtM t [0,T ] andH (Ht )t [0,T ] as the augmented subfiltrations generated by (YtM )t [0,T ] and (1{Tx t} )t [0,T ] ,! "respectively. We set GtM FtM Ht and GM GtM t [0,T ] .Insurance contracts can now be considered on the combined filtered probability space!"! , G , Q, G (Gt )t [0,T ] ,where ! ! M ! F , G F F G M , Gt FtF GtM , and Q Q F P M is the productmeasure of independent financial and biometric events. We further let F (Ft )t [0,T ] , whereFt FtF FtM . A slight extension of the results by Lando (1998) (Prop. 3.1) now yields thatfor an Ft -measurable payment Ct , we have for u t 3' *Bu EQ Bt 1 Ct 1{Tx t} ' Gu& (t)' ,' 1{Tx u} Bu EQ Bt 1 Ct exp µ (x s,Ys ) ds '' Fu ,uwhich can be readily applied to the valuation of insurance contracts. For notational convenience,we introduce the realized survival probabilities)& (t'* (t)Q'1{Tx t} Ft H0 exp µ (x s,Ys ) ds ,t px : E0(t)t u px u(t): t px(u)u px& (t) exp µ (x s,Ys ) ds , 0 u t,uas well as the corresponding one-year realized death probability(t)(t)(t)qx t 1 : 1 px t 1 1 1 px t 1 .(i)2 We denote by At only assets which are not solely subject to interest rate risk, e.g. stocks or immovableproperty. The price processes of non-defaultable bonds traded in the market are implicitly given by the short rateprocess."!3 In what follows, we write µ (x t,Y ) : µ (x t,Y M ) and r (t,Y ) : r t,Y F , where (Y )ttt t [0,T ] : tt! F M"Yt ,Yt t [0,T ] is the state process.

3While Q F is specified as some given equivalent martingale measure, there is some flexibilityin the choice of P M . In a complete financial market, i.e. if QF is unique, with a deterministicevolution of mortality and under the assumption of risk-neutrality of an insurer with respectto mortality risk (cf. Aase and Persson (1994)), Møller (2001) points out that if PM denotesthe physical measure, Q as defined above is the so-called Minimal Martingale Measure (seeSchweizer (1995)). This result can be extended to incomplete financial market settings whenchoosing QF to be the Minimal Martingale Measure for the financial market (see e.g. Riesner(2006)). However, Delbaen and Schachermayer (1994) quote “the use of mortality tables in insurance” as “an example that this technique [change of measure] in fact has a long history”in actuarial sciences, indicating that the assumption of risk-neutrality with respect to mortality risk may not be adequate. Then, the measure choice depends on the availability of suitablemortality-linked securities traded in the market (see Dahl, Melchior and Møller (2008) for aparticular example and Blake, Cairns and Dowd (2006) for a survey on mortality-linked securities) and/or the insurer’s preferences (see Bayraktar and Ludkovski (2009), Becherer (2003)or Møller (2003)). In what follows, we assume that the insurer has chosen a measure PM forvaluation purposes, so that a particular choice for the valuation measure Q is given.To obtain a model for our generic life insurance contract, we analyze the way such contractsare administrated in an insurance company. An important observation is that cash flows, such aspremium payments, benefit payments, or withdrawals, are usually not generated continuouslybut only at discrete points in time. For the sake of simplicity, we assume that these discretepoints in time are the anniversaries " {0, . . . , T } of the contract. Therefore, the value V" ofsome life insurance contract at time " under the assumption that the insured in view is alive bythe risk-neutral valuation formula is:V" B"T# EQµ "*' 'B 1µ Cµ F" ,where Cµ is the cash flow at time µ , 0 µ T .Since the value at time t only depends on the evolution of mortality and the financial market,and as these again only depend on the evolution of (Ys )s [0,t] , we can write:Vt Ṽ (t,Ys , s [0,t]).But saving the entire history of the state process is cumbersome and, fortunately, unnecessary:Within the bookkeeping system of an insurance company, a life insurance contract is usuallymanaged (or represented) by several accounts saving relevant information about the history ofthe contract, such as the account value, the cash-surrender value, the current death benefit, etc.(1)(m)Therefore, we introduce m N “virtual” accounts (Dt )t [0,T ] (Dt , . . ., Dt )t [0,T ] , the socalled state variables, to store the relevant history. In this way, we obtain a Markovian structuresince the relevant information about the past at time t is contained in (Yt , Dt ). Furthermore,we observe that these virtual accounts are usually not updated continuously, but adjustments,such as crediting interest or guarantee updates, are often only made at certain key dates. Also,policyholders’ decisions, such as withdrawals, surrenders, or changes to the insured amount,often only become effective at predetermined dates. To simplify notation, we again assume thatthese dates are the anniversaries of the contract. Therefore, to determine the contract value attime t if the insured in view is alive, it is sufficient to know the current state of the stochasticdrivers and the values of the state variables at )t* max{n N n t}, i.e. the value of thegeneric life insurance contract can be described as follows:Vt V (t,Yt , Dt ) V (t,Yt , D)t* ),t [0, T ].We denote the set of all possible values of (Yt , Dt ) by t .This framework is “generic” in the sense that we do not regard a particular contract specification, but we model a “generic” life insurance contract allowing for payments that depend onthe insured’s survival. While more general contracts depending on the survival of a second life

4(multiple life functions) or payments depending e.g. on the health state of the insured (multipledecrements) are not explicitly considered in our setup, their inclusion would be straightforwardakin to the classical case.Similar frameworks in continuous time have been e.g. proposed by Aase and Persson (1994)and Steffensen (2000), where the value – or, more precisely, the market reserve – of a genericcontract is described by a generalized version of Thiele’s Differential Equation. In contrast, welimit our considerations to discrete payments since (a) this is coherent with actuarial practice aspointed out above and (b) the case of continuous payments may be approximated by choosingthe time intervals sufficiently small. Hence, we do not believe that these limitations restrict theapplicability of our setup.In particular, many models for the market-consistent valuation of life insurance contractspresented in literature fit into our framework. For example, Brennan and Schwartz (1976) priceequity-linked life insurance policies with an asset value guarantee. Here, the value of the contract at time t only depends on the value of the underlying asset which is modeled by a geometric Brownian motion, i.e. we have an insurance contract which can be described by a onedimensional state processes and no state variables.Participating life insurance contracts are characterized by an interest rate guarantee andsome bonus distribution rules, which provide the possibility for the policyholder to participatein the earnings of the insurance company. Furthermore, these contracts usually contain a surrender option, i.e. the policyholder is allowed to lapse the contract at time " {1, . . . , T }. Suchcontracts are, for instance, considered in Briys and de Varenne (1997), Grosen and Jørgensen(2000) and Miltersen and Persson (2003). All these models can be represented within our framework. Moreover, the setup is not restricted to the valuation of “entire” insurance contracts, butit can also be used to determine the value of parts of insurance contracts, such as embeddedoptions. Clearly, we can determine the value of an arbitrary option by computing the value ofthe same contract in- and excluding that option, ceteris paribus. The difference in value of thetwo contracts is the marginal value of the option. For example, the generic model can be used inthis way to analyze paid-up and resumption options within participating life insurance contractssuch as in Gatzert and Schmeiser (2008) or exchange options such as in Nordahl (2008). Alternatively, the value of a certain embedded option may be determined by isolating the cash-flowscorresponding to the considered guarantee (see Bauer, Kiesel, Kling and Ruß (2006)).Bauer, Kling and Ruß (2008) consider Variable Annuities including so-called GuaranteedMinimum Death Benefits (GMDBs) and/or Guaranteed Minimum Living Benefits (GMLBs).Again, their model structure fits into our framework; they use one stochastic driver to model theasset process and eight state variables to specify the contract.3 A survey of numerical methodsThe contracts under consideration are often relatively complex, path-dependent derivatives, andin most cases, analytical solutions to the valuation problems cannot be found. Hence, one hasto resort to numerical methods. In this section, we present different possibilities to num

literature, on the other hand, there exists a variety of different articles on the valuation of life insurance contracts. Ho we ver,there are hardly an y detailed comparisons of different numerical valuation approaches in a general setup. Moreo ver, some studies do not apply methods from

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