Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies

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1 Insurance: Mathematcs and Economcs Loss analyss of a lfe nsurance company applyng dscrete-tme rsk-mnmzng hedgng strateges An Chen Netspar, he Netherlands Department of Quanttatve Economcs, Unversty of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, he Netherlands Receved December 2005; receved n revsed form October 2007; accepted 23 December 2007 Abstract he present paper nvestgates the net loss of a lfe nsurance company ssung equty-lnked pure endowments n the case of perodc premums. Due to the untradablty of the nsurance rsk whch affects both the n- and outflow sde of the company, the ssued nsurance clams cannot be hedged perfectly. Furthermore, we consder an addtonal source of ncompleteness caused by tradng restrctons, because n realty the hedgng of the contngent clams s more lkely to occur at dscrete tmes. Based on Møller Møller,., Rsk-mnmzng hedgng strateges for untlnked lfe nsurance contracts. Astn Bull. 28, 17 47, we partcularly examne the stuaton, where the company apples a tme-dscretzed rskmnmzng hedgng strategy. hrough an llustratve example, we observe numercally that only a relatvely small reducton n run probabltes s acheved wth the use of the dscretzed orgnally rsk-mnmzng strategy because of the accumulated extra duplcaton errors caused by dscretzng. However, the smulated results are hghly mproved f the hedgng model nstead of the hedgng strategy s dscretzed. For ths purpose, Møller s Møller,., Hedgng equty-lnked lfe nsurance contracts. North Amer. Actuaral J. 5 2, dscrete-tme bnomal rsk-mnmzng strategy s adopted. c 2008 Elsever B.V. All rghts reserved. JEL classfcaton: G10; G13; G22 Subject and Insurance Branch Codes: IM10; IE10; IE50; IB10 Keywords: Net loss; Dscrete-tme rsk-mnmzng hedgng strateges; Pure endowment equty-lnked lfe nsurance; Run probablty 1. Introducton he topc of nsolvency rsk of lfe nsurance companes has attracted a great deal of attenton. Snce the 1980s, a long lst of defaultng lfe nsurance companes n Europe, Japan and USA has been reported. Here are two notceable examples from the Unted States: Frst Executve Lfe Insurance Co. n 1991, the 12th largest bankruptcy n the Unted States n the perod , and Conseco Inc. n 2002, the 3rd largest bankruptcy n the Unted States n the same perod. 1 In Japan, the followng lfe nsurance carrers defaulted: Nssan Mutual Lfe n 1997, Chyoda Mutual Lfe Insurance Co. and Kyoe Lfe Insurance Co. n 2000 and okyo Mutual Lfe Insurance n In Europe, there were the followng most notceable nsolvency cases: Garante Mutuelle des Fonctonnares n France n 1993, the world s oldest lfe nsurer Equtable Lfe n the Unted Kngdom n 2000 and Mannhemer Leben n Germany n herefore, the task of how to reduce nsolvency rsk becomes a more and more mportant topc. he nsolvency rsk of an nsurance company can usually be reduced n two dfferent ways: externally or nternally. Concernng external rsk management, a regulator may be ntroduced who mposes an nterventon rule n order to prevent the nsurance company from nsolvency. hs s the approach taken e.g. by Grosen and Jørgensen In el.: ; fax: E-mal address: A.Chen@uva.nl. 1 Data were taken from 2 Bernard et al extend Grosen and Jørgensen by ncorporatng the stochastc nterest rate. Chen and Suchaneck 2007 extend Grosen and Jørgensen by allowng for more general bankruptcy procedures /$ - see front matter c 2008 Elsever B.V. All rghts reserved. do: /j.nsmatheco

2 1036 A. Chen / Insurance: Mathematcs and Economcs ther model, the frm defaults and s lqudated f up to the maturty tme the value of the total assets has not been suffcently hgh to cover the nomnal lablty multpled by some pre-specfed constant parameter. he regulator controls the strctness of the nterventon by settng the sze of ths parameter. Concernng nternal rsk management, the nsurance company actvely manages ts exposure to nsolvency by approprately hedgng the rsks of the ssued contracts. hs approach has already been used e.g. n Mahayn and Schlögl hey manly nvestgate how to determne the contract parameters conservatvely and mplement robust rsk management strateges. It s worth mentonng that dfferent contract desgns and dfferent hedgng crtera would lead to very dfferent results f usng ths approach. In the present paper, we manly study the case when the nsurance company apples a rsk-mnmzng hedgng strategy to equty-lnked pure endowments. Moreover, we go one step further and nvestgate the net loss of the contract-ssung company. Equty-lnked lfe nsurance contracts are an example of the nterplay between nsurance and fnance. A rapdly ncreasng volume of equty-lnked lfe nsurance contracts s observed. Neglectng some detaled subtletes, many lfe nsurance products provded outsde Europe have smlar features to equty-lnked lfe nsurance, although they have completely dfferent names, lke segregated fund contracts n Canada whch have become a popular alternatve to mutual fund nvestment, unt-lnked nsurance products n Unted Kngdom and varable annutes n Unted States whch are smlar to segregated funds. A detaled descrpton of the hstory of equty-lnked lfe nsurance and dverse contract forms can be found n e.g. Hardy In contrast to the fnancal rsks, the nsurance rsk s not tradable n the fnancal market. here are dfferent methods to deal wth ths untradable rsk. Followng Brennan and Schwartz 1979, most authors e.g. Bacnello and Ortu 1993, Bacnello and Persson 2002, and Mltersen and Persson 2003 replace the uncertanty of the nsured ndvduals death/survval by the expected values accordng to the law of large numbers. So, the actual nsurance clams, ncludng mortalty rsk as well as fnancal rsk, are replaced by modfed clams, whch only contan fnancal uncertanty. hs allows the use of standard fnancal valuaton and hedgng technques for complete markets. Although some other authors add mortalty rsk to ther model, they neglect the hedgng perspectve and manly deal wth the far valuaton of the equty-lnked lfe nsurance contracts, see e.g. Aase and Persson 1994, Ekern and Persson 1996 and Nelsen and Sandmann 1995, 1996, In contrast to all the authors mentoned above, Møller 1998 attempts to hedge the combned actuaral and fnancal rsk. In hs work, contnuously adjustable rsk-mnmzng n the sense of varance-mnmzng hedgng strateges are determned for equty-lnked lfe nsurance contracts. In ths paper, we use Møller s rsk-mnmzng strategy wth a modfcaton, namely a tradng restrcton s mposed on ths contnuous strategy,.e., the hedgng of the contngent clam occurs at dscrete tmes only. herefore, the consdered model s ncomplete n two aspects where the ncompleteness results not only from the mortalty rsk but also from the tradng restrcton. hrough an llustratve smulaton example, t s observed numercally that a substantal reducton n the run probablty 3 s acheved by usng the tme-dscretzed rsk-mnmzng strategy, n comparson wth the scenaro where the nsurer nvests the premums n a rsk free asset wth a rate of return correspondng to the market nterest rate. However, the extent of the reducton becomes less apparent and the advantage of usng ths strategy almost dsappears when the tradng frequency s ncreased. hs s due to the fact that extra duplcaton errors are caused when the orgnal mean-self-fnancng rskmnmzng hedgng strategy s dscretzed wth respect to tme, and because these errors ncrease wth the frequency. In order to mprove the numercal results, another type of dscretetme rsk-mnmzng strategy s taken nto consderaton. It s obtaned by dscretzng the hedgng model nstead of the hedgng strategy. For ths purpose, we consder the Cox-Ross- Rubnsten 1979 model CRR, whch converges n the lmt to the Black and Scholes 1973 model. In ths dscrete-tme framework, the bnomal rsk-mnmzng strategy n Møller 2001 s adopted. When comparng the smulaton results wth the scenaro where the strategy s dscretzed, we observe consderably smaller run probabltes, n partcular, when the frequency s ncreased. hs paper s organzed as follows: In Secton 2, the net loss of an nsurance company s defned, and for two smple scenaros the loss s computed. Secton 3 focuses on the net loss caused by usng the tme-dscretzed orgnally contnuous rskmnmzng hedgng strateges. Secton 4 demonstrates how to calculate the relevant dscretzed rsk-mnmzng strategy wth the help of an example. In Secton 5, by smulatng the run probablty caused by dscretzng the hedgng strategy, we show that some of the numercal results are not very satsfactory. In Secton 6, the hedgng model s dscretzed nstead of the hedgng strategy, and the numercal results are mproved substantally. Secton 7 concludes the paper. 2. Net loss and two extreme cases hs secton ams at ntroducng the model setup and specfyng the underlyng fnancal market and the loss functon. We consder an nsurance company operatng on the tme horzon 0,. Suppose, at tme 0, n dentcal customers of age x close the same pure endowment wth the nsurance company, whch promses each of them a payment of f, S at f they survve the maturty date. he functon f, S descrbes the dependence of the fnal payment on the evoluton of the stock prce. It can be a functon of the termnal stock prce S only or of the whole path of the stock, and possbly t contans embedded optons. In return, each customer pays a premum of K perodcally, whch s determned at the begnnng of the contract and whch wll be kept constant through the duraton of the contract. In other words, as long as the customer s alve, the customer pays K at each t 3 Due to the specfc modellng of the contract pure endowments, the run probablty equals the relatve frequency the smulated net loss of the nsurer at the maturty of the contract s larger than zero.

3 A. Chen / Insurance: Mathematcs and Economcs {t 1,..., t } wth t M :. Let Y t n denote the number of customers who survve through tme t. As most authors do, we also assume that the resdual lfetmes of each customer are ndependent. hs leads to a bnomal dstrbuton of Y t n wth parameters n, t p x, where t p x gves the probablty that an x- aged nsured survves tme t. Further, the notaton t p x+t s used to descrbe the probablty that an x-age lfe survves tme, gven that he has survved tme t. Furthermore, t s assumed that mortalty and fnancal rsks are ndependent Fnancal market model For formalty, we assume that the stock s defned under some probablty space Ω, F, F t 0 t<, P, where P denotes the unque equvalent martngale measure and F t 0 t< the natural fltraton generated by the stock. More specfcally, the underlyng stock s assumed to evolve accordng to a geometrc Brownan moton under P as follows: ds t S t δ dt + σ dwt, 1 where δ s the rsk free nterest rate, σ > 0 the volatlty of the stock and Wt t 0, a martngale under P. he bank account correspondng to an accumulaton factor s defned by B t exp{δt} Net loss In realty, t s mpossble or sometmes unreasonable to make contnuous adjustments to a hedgng portfolo, because the securty markets do not trade at nghts, weekends or on holdays, and because frequent tradng could lead to very hgh transacton costs. In ths context, ths paper ams to nvestgate the net loss of a lfe nsurer when he trades n more realstc dscrete-tme hedgng strateges. Frst, the set of tradng dates s characterzed by a sequence of equdstant refnements τ of the nterval 0,, namely τ : {τ 0 0 < τ 1 < τ 2 < < τ Q } wth τ k+1 τ k 0 for Q. In the followng, we set τ Q ; hence τ j j τ. For smplfcaton reasons, Q s assumed to be a multple of M. herefore, τ Q can be consdered as a refnement of the premum payment dates. For example, Q s assumed to be equal to 2M n Fg. 1,.e., tradng occurs twce as frequently as premum payments,.e. τ 1 2 t, f t : t +1 t. Second, let φ τ ξ τ, η τ descrbe a dscrete-tme tradng strategy n the stock and bank account under the set of tradng dates τ. It s assumed that the tradng strategy φ τ satsfes the usual ntegrablty condton, but t s not necessarly selffnancng. 4 hroughout the paper, we consder dscounted 4 For example, f we denote the value process assocated wth φ by V t φ ξ t S t + η t B t. hs strategy s self-fnancng f V t φ V 0 φ + t 0 ξ u ds u + t 0 η u db u for all 0 t,.e., after tme t 0, no further nflows or outflows are needed. processes,.e. the bank account B s used as numerare and a star s put n the superscrpt to denote the dscounted value, e.g. St : Bt 1 S t denotng the dscounted asset value. he dscounted tradng gans G assocated wth the dscrete-tme strategy φ τ s descrbed by G φ τ : Q1 ξ τ j S τ j, 5 wth S τ j S τ j+1 S τ j denotng the dscounted asset ncrements. herefore, nspred by Møller 2001, the dscounted net loss of the nsurer gven that he uses the dscretetme hedgng strategy φ τ s descrbed by Q1 L n n φ Y f, Se δ ξ τ j Sτ j Y n t +1 Y n Y n t. 3 he dscounted net loss s gven as the dfference of dscounted payments the nsurer has to provde at tme and premum ncomes he obtans, mnus the tradng gans he acheves from the rsk management strateges. hose, who de durng t, t +1 only pay the premums untl t, and those who survve the end of the contract t M pay all the premums. Naturally the tradng gans losses depend on hedgng/nvestment strateges the nsurer chooses. In the remanng of ths secton, we consder two extreme, smple cases to analyze the dstrbuton of the dscounted net loss and carry out an nvestgaton under the market measure wo extreme cases Investment n a rsk free asset As a startng pont, we consder the net loss of the company when the nsurance company nvests ts premums n a rsk free asset wth a rate of return δ. Hence, the dscounted net loss of the nsurer at tme s smplfed to: L r f eδ Y n f, S Y n Y n t Y n t +1 he expected loss can be derved as follows: EL r f n p x eδ E f S, n t p x t+1 p x n p x he number of the bank account ητ j and the value B τ j do not appear n ths equaton because dscounted assets are consdered, and hence the value of B τ j s dentcal to 1. It leads to η τ j B τ j 0.

4 1038 A. Chen / Insurance: Mathematcs and Economcs Fg. 1. Partton of accordng to premum payment dates {t 0, t 1,..., t } and tradng dates {τ 0, τ 1,..., τ Q }. radng occurs e.g. twce as frequently as premum payments. he ndependence assumpton between the fnancal and mortalty rsks and the equalty EYt x n t p x are needed for the above dervaton. It s observed that the expected loss s equal to 0 f and only f K e δ p x E f, S e δt j t p x t+1 p x + p x e δt j. 6 It s observed that the optmal K does not depend on the number of the contracts n that the nsurer ssues. Only wth ths premum, EL r f /n 0 holds,.e., the expected loss per contract s equal to zero. If the charged premum s larger than K, then EL r f /n < 0,.e., lm n EL r f. hs means that n the expectaton, the company makes an nfntely large proft as the number of contract holders s ncreased to nfnty. On the contrary, f the charged premum s smaller than K, ths wll result n an nfntely large expected loss for the company as the number of contract-holders goes to nfnty. Due to the mpact of the mortalty rsk on both the payment of the nsurer and the payment of the nsured, the varance of the loss s much more complcated than n the sngle premum case. In order to calculate the varance of the dscounted net loss L r f, the relaton between varance and condtonal varance s appled,.e., VarX VarEX Z + EVarX Z, 7 where X, Z are two arbtrary random varables. Sophstcated choces of Z can smplfy the calculaton of VarX to a great extent. In our context, f the stock prce S s chosen as the random varable whch L r f s condtoned on, the ndependence between the fnancal market rsk and nsurance rsk can be exploted. Proposton 2.1. Gven the ndependence assumpton between the fnancal market and mortalty rsk, the varance of the loss s gven by L r f VarL r f VarELr f wth S + EVarLr f S VarEL r f S n p x eδ 2 Var f, S EVarL r f S n p x 1 p x e2δ E f, S 2 { 2 n e δ p x 1 p x K e δ t j δ + 2 e K e δ t j n p x t+1 p x t p x } E f, S + n p x 1 p x + + n t+1 p x t p x t+1 p x n t p x 1 t p x 1 t p x+t n p x t+1 p x t p x Proof. A detaled proof s found n Appendx A. It s observed that the varance of the dscounted net loss n the case of a perodcal premum s much more complcated than that of the sngle premum case, where the varance corresponds to the frst two terms of the above varance. hs s due to the fact that the mortalty rsk not only decdes the occurrence of the clam payment, but also nfluences the payment of the perodc premums. Asymptotcally, that s, as n, t s notced that 1 Var n Lr f p 2 x e2δ Var f, S. 8 As expected, by ncreasng the number of the nsured, the nsurer can elmnate all the mortalty rsk,.e., the rsk related to the uncertanty concernng the number of the polcyholders who wll survve to tme and the uncertanty concernng the number of the polcyholders that wll de between t, t +1, M 1. hs s the so-called dversfcaton effect over sub-populaton. Meanwhle the fnancal uncertanty concernng the future development of the stock remans wth the nsurer, snce all contracts are lnked to the same stock. In ths case, the nsurer does not really hedge aganst the rsk, and the resultng varance gves an upper bound of the rsk that the nsurer can reach Net loss n the case of a statc hedge Snce there s an upper bound for the varance, naturally, the queston wll be asked of whether there exsts a lower bound. If there are some statc buy-and-hold hedgng strateges whch duplcate the fnal payment f, S perfectly, the nsurer can elmnate all the fnancal rsk from these strateges. Suppose that the company apples the statc strategy,.e., t purchases n p x fnancal contracts at the contract-ssung tme and holds them untl the maturty date. Each of these fnancal contracts pays the amount f, S at tme. Let V 0 be today s prce of such a fnancal contract. Hence, the net loss for ths case s descrbed as the dfference of the loss n the frst case and the.

5 A. Chen / Insurance: Mathematcs and Economcs proft/loss from tradng: L s Lr f e δ n p x f, S n p x V 0 n p x V 0 + e δ f, SY n n p x Y t n Y t n +1 Y n he expected loss s descrbed by. 9 EL s ELr f n p x eδ E f, S + n p x V 0 n p x V 0 n t p x t+1 p x n p x. In ths stuaton, the expected loss s equal to 0 f and only f K p x V 0 e δt j t p x t+1 p x + p x e δ t j Smlarly, just wth ths premum K, lm n EL s /n 0 holds. If the charged premum s larger than K, then lm n EL s /n < 0,.e., ELs as n. It leads to nfntely large expected profts for the company as the sze of the contract-holders s ncreased to. On the contrary, f the charged premum s smaller than K, t wll lead to nfntely large expected losses for the company as the sze of the contract-holders goes up to. Besdes, K corresponds to K f V 0 e δ E f, S. he varance of the net loss for ths statc hedge s determned by VarL s E VarL r f S. A detaled expresson for ths varance s gven n Proposton 2.1. Opposte to the frst case, t s notced that L s Var 0 n as n,.e., n ths case, the total rsk mortalty rsk + fnancal rsk can be elmnated by ncreasng the number of polces n the portfolo and buyng the optons on the stock. However, ths statc strategy s not realstc because the usual term nsurance contracts s qute long, e.g. t takes years n Germany, whle standard optons are typcally short-term transactons, say, less than one year. Due to ths unrealstc restrcton, ths scenaro wll not be consdered later. In the followng sectons, we examne the net loss of dscretetme rsk-mnmzng hedgng strateges. 3. me-dscretzed rsk-mnmzaton hedgng strategy here are dfferent ways of generatng a dscrete-tme strategy. In ths secton, we nvestgate a dscrete-tme hedgng strategy whch s generated by dscretzng a contnuous-tme. hedgng strategy wth respect to tme. In a later secton, we consder a dscrete-tme hedgng strategy n whch the underlyng asset s drven by a bnomal model. In the lterature on the hedgng of equty-lnked lfe nsurance n the above fnancal market specfcaton, dfferent optmzaton crtera have been used for the purposes of hedgng, e.g. Melnkov 2004a,b apply quantle and effcent hedgng crtera. Here, we rely on Møller 1998, who apples a rsk-mnmzng concept ntroduced by Föllmer and Sondermann 1986 n equtylnked lfe nsurance. hanks to the ndependence assumpton between the fnancal market and mortalty rsk, the resultng rsk-mnmzng hedgng strategy for a pure endowment can be expressed as a product of two parts: the expected number of customers who survve and the hedge rato when there s fnancal market rsk only. In the followng, we denote by φ ξ τ, η τ,...,q1 ths tme-dscretzed orgnally rskmnmzng hedgng strategy. Accordng to Møller 1998, the hedge rato number of stocks the nsurer should hold ξ τ takes the form of ξ τ Y n τ τ p x+τ F s τ,, S τ, 0,..., Q 1, 10 where Ft,, S t gves the tme t value of contngent clam f, S and F s t,, S t s the dervatve of Ft,, S t wth respect to the stock prce S t. It s noted that ξ τ0 n p x F s 0,, S 0. Further, the hedge rato at tme τ s descrbed as the product of the hedge rato n the case of fnancal rsk only and the average number of customers who survve the contract s maturty tme gven that they have survved tme τ. We do not wrte down the number of the bond because t s rrelevant n the loss analyss. Accordng to Eq. 3, the net loss of ths tme-dscretzed rsk-mnmzng hedgng strategy φ s gven by L n φ Y n f, Se δ Q1 f s τ j, S τ j Sτ j Y t n Y n Y τ n j τ j Y n t Proposton 3.1 Varance of L n φ. he varance of usng the tme-dscretzed rsk-mnmzng hedgng strategy φ specfed n Eq. 10 s gven by VarL φ Var f Se δ n p x Q1 + Var n p x f s τ, S τ Sτ Q1 2 n 2 p x 2 e δ Cov f S, f s τ, S τ Sτ + n p x 1 p x e 2δ E f S 2 Q1 Q1 + E f s τ j, S τ j Sτ j τ j

6 1040 A. Chen / Insurance: Mathematcs and Economcs f s τ, S τ Sτ of such a contract the payoff τ p x+τ n τ j p x 1 τ j p x Q1 f t M, S K e gt +1 2 E f S f s τ, S τ Sτ τ p x+τ e δ n τ p x St α + 1K e g t, 13 1 τ p x + K e δ t St j n t p x 1 t p x 1 t p x+t + n t+1 p x t p x t+1 p x + 2 +e n p x q x +2 δ e δ n p x t+1 p x t p x n p x t+1 p x t p x n p x 1 p x E f, S Q E f s τ, S τ Sτ τ j n 1 τ j t +1 p x + τ j t p x 1 τ j t p x τ j t +1 p x j n p x 1 τ j p x Proof. A detaled proof s gven n Appendx B. It s observed that asymptotcally 1 lm Var n n L n φ e 2δ p x 2 Q1 Var f, S f s τ, S τ S τ. 12 By ncreasng the number of contracts to nfnty, all the uncertanty about the customers survval can be dversfed away the number of survvors can be substtuted by the correspondng expected number and only some fnancal rsk s left to the nsurer. 4. An llustratve example Due to ts popularty, an equty-lnked lfe nsurance contract wth guarantee s appled as an llustratve example. Our goal s not only to prce the ssued contract, but to derve the dscretzed orgnally contnuous rsk-mnmzng strategy, to study the cost process, and further to nvestgate the hedger s net loss. We consder a specfc guaranteed equty-lnked pure endowment lfe nsurance contract, whch provdes the buyer. f he survves the maturty of the contract. he fnal payment depends on the mnmum guaranteed nterest rate g, the partcpaton rate n the surpluses α, the duraton of the contract M, and more mportantly the stock prces. Specfed at the begnnng of the contract, the premum K s pad perodcally by the nsured untl the maturty of the contract or the death of the nsured, whchever comes frst. If the nsured survves the maturty of the contract, he obtans the guaranteed amount and the accumulated bonuses partcpaton n the surplus of the company, whch are represented by a sequence of European call optons wth strke e g t. Followng Eq. 10, at tme t τ Q, we need to calculate the prce Ft,, S of the clam at tme t and ts frst dervatve wth respect to the stock prce at that tme, n order to obtan the rsk-mnmzng strategy. It s well-known that the prce of a contngent clam at tme t equals the expected dscounted value of the termnal payoff condtonal on the nformaton structure tll tme t, t 0, under the equvalent martngale measure: Ft,, S E e δ t f t M, S F t δ t e K e +1g t + αk + 1 wth e δ t St+1 S t e δ t St +1 Nd t, 2 + e g t 1 {t>t+1} S t Nd t, 1 e g t e δt +1t 1 {t <t t +1 }e δ t e δt +1t N d 1 e gδ t N d 2 1 {t t } d t, 1/2 ln S t/s t gt +1 t + δ ± 1 2 σ 2 t +1 t σ t +1 t d 1/2 δ g ± 1 2 σ 2 t +1 t σ. t +1 t N s the cumulatve standard normal dstrbuton functon. From the derved prce of the contngent clam, we take the dervatve wth respect to S t and obtan for t t, t +1 F s t,, S αk + 1e δ t +1 1 S t Nd t,t In our context, only those dervatves at t τ Q are of mportance. At dfferent tradng dates, F s t,, S could take very smlar values as long as all of these tradng dates le n a

7 A. Chen / Insurance: Mathematcs and Economcs same tme nterval resultng from the partton of accordng to the premum payments. For nstance, n Fg. 1, both τ 1 and τ 2 le n the nterval t 0, t 1, the dervatves F s dstngush from each other only by Nd τ 1,t 0 1 and Nd τ 2,t 0 1. In the next secton, we use the Monte Carlo smulaton method to obtan some numercal results and to fgure out whch strategy s more benefcal to the nsurance company by comparng the smulated techncal run probabltes. An nsurance company ams at reachng a run probablty whch s as small as possble. Usually, run s defned as a frst passage event, but due to the contract specfcaton pure endowment, run s defned as the event where the dscounted net loss of the nsurance company s larger than zero. Hence, the run probablty s gven as the frequency that the net loss of the nsurer s larger than zero. 5. Numercal results Smulatons are performed n ths secton to examne the nsurer s net losses for dfferent cases: 1 he nsurer nvests the premums n the rsk free asset at a fxed rate of nterest δ. 2 he hedger uses a tme-dscretzed rsk-mnmzng hedgng strategy and the hedgng frequency s the same as the premum payments Q M t τ. If t s assumed that premums are pad yearly, then the adjustment of the hedgng strateges occurs yearly as well. 3 he hedger uses a tme-dscretzed rsk-mnmzng hedgng strategy and the hedger adjusts hs tradng strategy 12 tmes as frequently as the premum payments Q 12M t 12 τ. hat means the nsurer adjusts the tradng portfolo monthly. he dstngushng between the last two scenaros s done n order to fnd out whether the hedger s able to reach a smaller run probablty by ncreasng the tradng frequency. Due to the ndependence assumpton between the mortalty rsk and the fnancal rsk, n prncple the smulaton of the losses reduces to smulatng: a the survval process } t 0,, and b the payoff of the pure endowment nsurance contract f, S or the correspondng dervatve of f, S wth respect to the stock respectvely. In order to smulate the survval process, we just need to know the survval probablty { t p x } t 0,, whch can be calculated by a hazard rate functon. For the numercal calculaton, the Gompertz Makeham hazard rate functon from Møller 1998 s adopted,.e: {Y t n µ x+t x+t t 0. hs functon was used n the Dansh 1982 techncal bass for men. Consequently, the survval probablty of an x-aged lfe s gven by t } t p x { exp x+u du. 0 Another parameter whch should be consdered before startng a smulaton s the far premum K. Accordng able 1 Far partcpaton rates α s wth followng parameters: δ 0.05, x 35, σ 0.2 Duraton, M Mnmum guarantee, g Far partcpaton rate, α to the analyses n Secton 2, non-optmal K -values could cause nfnte losses or profts to the hedger asymptotcally. However, n ths specfc example, the far premum 6 cannot be determned explctly, because the fnal payment of the contract depends on the perodc premums. Substtutng ths fnal payment n the expresson of the optmal premum equaton Eq. 6, the K -terms would be left out n the calculaton. Hence the optmal K can only be determned mplctly through the far relatonshp between the partcpaton rate α and the mnmum guaranteed nterest rate g. hat s, for a gven g, we obtan a correspondng partcpaton α whch makes the contract far. Under the equvalent martngale measure, α as a functon of g s gven by the equaton n Box I. In able 1, some exemplary far values are lsted. Obvously, there exsts a negatve relatonshp between far α s and g s. Furthermore, the far α rses substantally as the duraton of the contract ncreases. hs s due to the fact that the perodc bonuses n the ssued contract are held by the nsurer untl the maturty date, wthout gvng any compensatons to the customer. A long duraton of the contract mples that the nsurer keeps more bonuses of the nsured for a longer tme, whch hampers the nsured to renvest the perodc bonuses to a large extent. Accordng to the prncple of equvalence, a larger α -value becomes necessary to make the contract far. hese values for the far partcpaton rate α, combned wth the correspondng g s and M s, are used n smulatng the run probabltes. Of course the far partcpaton rate also depends on some other parameters lke σ and the survval probabltes. However, these dependences are not of nterest here. Smulatng the loss dstrbuton of the frst case, where the company nvests the premum ncomes n a rsk free asset, s relatvely smple. Smulate the prce processes S t +1 St, 0,... M 1 under the market measure and substtute them nto the f, S expresson; then one sample of the clam f, S s obtaned. Combned wth the smulated Y n 1,..., Y n, one path of the loss s generated. If the whole smulaton s repeated m tmes, the run probablty of the nsurance company s approxmated as the rato: the number of the paths where the smulated loss s above 0. m he run probabltes for the rsk-mnmzng strateges are acheved smlarly accordng to Eq. 11. Followng the 6 akng mortalty rsk nto consderaton, a premum s called far f the expected dscounted accumulated premum ncome equals the expected dscounted accumulated payoff of the contract under the equvalent martngale measure.

8 1042 A. Chen / Insurance: Mathematcs and Economcs α g e δt j t p x t+1 p x + p x e δt j p x e + 1e δt N d 1 e gδt +1t N d 2 δ e gt +1 Box I. able 2 Run probabltes for dfferent µ s wth parameters: n 100, α , g , M 12, δ 0.05, x 35, σ 0.2 Case 1 Case 2: Q M Case 2: Q 12M µ Run prob. µ Run prob. % µ Run prob % able 3 Run probabltes for dfferent M wth parameters: n 100, α M 12, α M 20, α M 30, g , µ 0.04, δ 0.05, x 35, σ 0.2 Case 1 Case 2: Q M Case 2: Q 12M M Run prob. M Run prob. % M Run prob. % procedure we ntroduced above, the run probabltes for Cases 1, 2 and 3 are obtaned after smulatng the losses tmes. he study of the run probablty s carred out under the market measure. Frst, let us have a look how the run probablty depends on the market performance of the stock, whch s descrbed by the rate of return µ. able 2 exhbts the run probablty for three dfferent µ values, µ < δ, µ δ, and µ > δ. he percentage numbers n the last column of the table gve the rato of the run probablty n the case of Q M and Q 12M to the run probablty n Case 1 respectvely. Frst of all, t s observed that the run probablty n the case of dscretzed rsk-mnmzng hedgng s consderably smaller than n the frst case. In the stuaton Q M, the run probabltes are reduced by 69.28%, 62.47% and 77.95% respectvely for µ 0.04, µ 0.05 and µ he same phenomenon s observed for the stuaton of Q 12M wth the percentage numbers 76.02%, 74.45% and %. Second, a common observaton for the frst case and the case Q 12M s that the run probablty ncreases wth the value of µ. hs s due to the fact that a better performance of the stock leads to a hgher lablty of the nsurer. However, ths relatonshp between µ and the run probablty n the dscretzed rskmnmzng hedge Q 12M s not so notceable as n Case 1. And n case Q M ths relatonshp ceases to be vald,.e. the relatonshp between the run probablty and µ s qute ambguous see also ables 3 5. heoretcally, t s vald that the more frequently the nsurer updates hs rskmnmzng hedgng strateges, the more the fnancal rsks are reduced. Furthermore, the nsurer can elmnate all the fnancal rsks f he could hedge contnuously. However, the accumulated hedgng error caused by dscretzng the contnuous rskmnmzng hedgng strategy has destroyed ths argument. hs s why t s observed that not all the run probabltes n the case Q 12M are smaller than n the case M Q. he relaton between the run probablty and the duraton of the contract s llustrated n ables 3 5 for dfferent µ- values. Above all, M plays a very mportant role n determnng able 4 Run probabltes for dfferent M wth parameters: n 100, α M 12, α M 20, α M 30, g , µ 0.05, δ 0.05, x 35, σ 0.2 Case 1 Case 2: Q M Case 2: Q 12M M Run prob. M Run prob. % M Run prob. % able 5 Run probabltes for dfferent M wth parameters: n 100, α M 12, α M 20, α M 30, g , µ 0.06, δ 0.05, x 35, σ 0.2 Case 1 Case 2: Q M Case 2: Q 12M M Run prob. M Run prob. % M Run prob. % the far partcpaton rate α c.f. able 1. For dfferent g s and M s dfferent far α s are obtaned. Also n these cases the run probabltes are reduced substantally, wth the use of dscretzed rsk-mnmzng strateges. Almost throughout, a postve relatonshp between the run probablty and M s observed. In the frst case, obvously the effect of M on the nsurer s lablty domnates that of M on hs accumulated premum ncomes. Run appears more lkely as M ncreases. In the second case, on the one hand, t s known that some dscretzaton and duplcaton errors exst when the dscretzed rsk-mnmzng hedgng strategy s used and that they are an essental part of the hedger s loss. As tme goes by, the hedge errors accumulate negatve effect. On the other hand, a longer duraton of the contract leads to hgher premum nflows. Consequently, n the long run ths reduces the nsurer s loss

9 A. Chen / Insurance: Mathematcs and Economcs able 6 Run probabltes for dfferent combnatons of α and g wth parameters: n 100, µ 0.06, M 12, δ 0.05, x 35, σ 0.2 g, α Case 1 Q M % Q 12M % g , α g , α g , α to a certan extent postve effect. Here the negatve effect domnates the postve effect overall. hs negatve mpact s so dstnct that qute bg run probabltes have resulted for M 30 for the case of Q 12M. In ths subcategory, the nsurer adjusts hs portfolo much more frequently than the premum payment occurrence. he more often the hedger updates hs strategy, the more duplcaton and dscretzaton errors arse. Consequently, relatvely hgh run probabltes are caused as the duraton of the contract ncreases. able 6 demonstrates how the run probablty changes wth the far combnaton of α and g. Overall, the effect of the mnmum guarantee g domnates that of α. hs s due to the fact that the resultng α s are relatvely small, and consequently the bonus part of the payment does not play a role as mportant as the mnmum guarantee parameter g. Hence, a hgher mnmum nterest rate guarantee leads to a hgher run probablty. Conversely, t s expected that the effect of the α s wll domnate that of the g s for relatvely small mnmum nterest rate guarantees g, say near 0, and relatvely hgh partcpaton rates. 6. Rsk-mnmzaton n a bnomal model Naturally, the queston wll be asked of how dfferent the results wll be caused by usng some other dscretetme hedgng strateges, e.g. whether dscretzng the hedgng model nstead of dscretzng the strategy would mprove the results. 7 It s well-known that the bnomal model converges to the Black-Scholes model when the number of tme perods ncreases to nfnty and the length of each tme perod s nfntesmally short. hs argument s proven e.g. by Cox et al and Hsa Due to ths convergence reason, the bnomal model s used as our dscrete-tme model setup. More specfcally, the followng analyss reles on Møller 2001 who derves rsk-mnmzng strategy for equty-lnked lfe 7 Accordng to Mahayn and Schlögl 2003, dscretzng the hedgng model Cox-Ross-Rubnsten-based hedgng model yelds a more favorable result for the hedger than dscretzng the contnuous hedgng strategy, n the sense that the bnomal hedge wth a sutably adjusted drft component s meanself-fnancng, whle the dscretzed Gaussan hedge sub-replcates the convex payoff for both a postve or a negatve drft component. 8 he proof of Cox et al s elegant but long and specfc. It reles on Central Lmt heorem. Furthermore, they choose specfc up and down factors so that the dstrbuton of the stock return has the same parameters as the desred lognormal dstrbuton n the lmt. No restrctons are mposed on up and down parameters n Hsa Hs proof s shorter and requres few cases of takng lmts. nsurance contracts n the Cox-Ross-Rubnsten CRR model. In a pure endowment, the amount of the stock n the rskmnmzng hedgng strategy s descrbed by ξ B τ Y τ n τ p x+τ ατ f, 0,..., Q 1, 15 where α f t stands for the level of the stock at tme t n a bnomal hedge wthout mortalty rsk, and also we assume the bnomal model contans Q perods. In order to avod the lengthy calculaton of α f t for each tme perod, we proceed wth the followng lemma. Lemma 6.1. he dscounted tradng gans of a strategy φ whch s used to hedge a -contngent clam X can be expressed as the functon of total hedgng errors assocated wth ths strategy as follows: G φ X V 0 φ C,tot φ, wth V 0 φ denotng the ntal nvestment n the strategy and C,tot φ the dscounted accumulated total hedgng costs assocated wth φ. Proof. Snce the dscounted accumulated total hedgng costs consst of two parts: the dscounted duplcaton costs X V φ and the dscounted rebalancng costs V φ V 0φ + G φ,.e.: C,tot φ X V φ + V φ V 0φ + G φ. Lemma 6.1 together wth Eq. 3 leads to the net loss of the nsurer as follows: L n φ B V 0 φ B + C,tot φ B premum ncomes. Furthermore, accordng to Møller 2001, the dscounted accumulated hedgng error from usng the rsk-mnmzng strategy at tme τ Q has the form of C,tot φ B Q e δτ j f τ j, S τ j τ j j1 Y n τ j Y n τ j1 τ 1, 16 where f t, S t gves the value of the contngent clam at tme t. he last term of the above equaton Y τ n j Y τ n j1 τ 1 ndcates that ths unhedgeable rsk results exactly from the dfference between the actual number of survvors at tme τ j and the expected number of survvors at tme τ j condtonal on tme τ j1. In ths case, all the hedge errors are caused by mortalty rsk and the expected hedge errors are zero under both the subjectve and the martngale measure,.e., the bnomal rsk-mnmzng strategy s mean-self-fnancng. Consequently, the net loss of the nsurance company can be decomposed nto three parts: the ntal nvestment V 0 φ B plus the hedgng errors gven n Eq. 16 and mnus the premum ncomes. herefore, only the values f τ j, S τ j of the contngent clams at certan dscrete tradng tmes are relevant for the examnaton of the net loss of the hedger. Here, we follow

10 1044 A. Chen / Insurance: Mathematcs and Economcs able 7 up, down and w-values wth σ 0.2 µ Up Down w M Q Q 12M M Q Q 12M M Q Q 12M able 9 Run probabltes wth a bnomal hedge wth parameters: n 100, x 35, σ 0.2, µ 0.06 Left: g ; Rght: M 12 Bnomal hedge: Q M Bnomal hedge: Q M M Run prob. g, α Run prob g , α g , α g , α able 8 Run probabltes wth a bnomal hedge wth parameters: n 100, x 35, σ 0.2, M 12, r 0.05, g , α Bnomal hedge: Q M Bnomal hedge: Q 12M µ Run prob. µ Run prob the orgnal parameter constellaton as n Cox et al. 1979,.e., specfc up and down factors are chosen so that the dstrbuton of the stock returns to have the same parameters as the desred lognormal dstrbuton n the lmt. Frst, the market rate of return µ n the bnomal model can be expressed as a functon of the weghted sum of up and down values as follows: µm E ln S St 0 F t0 Qw ln up + 1 w ln down, 17 where w gves the probablty that the stock moves upwards under the market measure and E denotes the correspondng expected value under ths measure. Accordngly, the up, the down movement and the nterest rate per perod are set as follows: up exp { σ } Q { δq exp δ } 1. Q, down exp { σ }, Q 18 Agan s the duraton of the contract and Q denotes how many tmes the nsurer adjusts hs portfolo. Pluggng Eq. 18 nto 17, the market performance can also be characterzed consequently by w: w µ M 2σ Q. Although µ/w s rrelevant n determnng the hedgng strategy n the bnomal model, t does decde how the market performs and wth what probablty the underlyng asset reaches a certan knot under the market measure. able 7 demonstrates several values of up, down and w, whch are used later for the calculaton of the run probablty. In addton, the determnstc nterest rate δ used n the prevous work s the conform nterest rate. Wth the equaton r exp{δ} 1, we obtan the yearly nterest rate r. able 8 llustrates how the run probablty depends on the market performance of the stock for two subcases: Q M and Q 12M. Frst, an ncrease n the run probablty s observed as µ goes up for M Q, but ths effect s not as pronounced as n the frst case. Furthermore, t ceases to be vald as the tradng frequency ncreases to Q 12M. Second, wth a more frequent rebalancng of the portfolo Q M Q 12M, the run probablty becomes very small. Almost all the fnancal rsks are elmnated when the tradng occurs 12 tmes as often as the premum payment. Accordngly, qute small run probabltes result n the scenaro Q 12M n the bnomal hedge. hs advantage obtaned from the bnomal hedge can be explaned by the followng theory to some extent. In the bnomal hedge, there exsts no duplcaton errors, and all the hedgng errors are caused by the mortalty rsk. By contrast n the use of the tmedscretzed rsk-mnmzng hedgng strategy, duplcaton errors are encountered wth each adjustment of the portfolo. As the adjustment frequency rses, the advantages from ths rse can be largely dmnshed by these duplcaton errors and consequently hgher run probabltes are caused c.f. ables 2 5. able 9 s generated for the case M Q and shows the dependence of the run probabltes on the duraton of the contract M left table and on the dfferent α gcombnatons rght table. In contrast to Case 1 and the case of the orgnally contnuous rsk-mnmzng strategy, the run probablty does not go up wth the duraton of the contract M. It s known that only some ntrnsc hedgng errors wll result from the use of ths bnomal hedgng strategy, whch are completely caused by the mortalty rsk. he sze of these ntrnsc hedgng errors s small n comparson wth the premum nflows of the nsurer. herefore, a qute small run probablty s observed, e.g for M 30. It could easly be shown that almost no run probablty wll result f a long duraton of the contract s combned wth a hgh adjustment frequency. Hence, a bnomal hedge mproves the stablty of those nsurers who manly deal wth long-term contracts or/and adjust ther tradng portfolos very frequently. he effect of the combnaton of α and g on the run probablty remans unchanged the effect of g domnates α. Rather, larger values of the run probablty are observed compared to the orgnally contnuous rsk-mnmzng strategy. hs s due to the fact that both the duraton of the contract M 12 and the frequency of adjustng the tradng portfolo are chosen to be qute low Q 12. Consequently, the advantages from the bnomal hedge are not so pronounced.

11 7. Concluson hs paper represents a smulaton study to examne the value of applyng rsk-mnmzng hedgng strateges by nvestgatng the net loss of a lfe nsurance company ssung dentcal pure endowments to n dentcal customers. More specfcally, the probablty of run s used as the crteron. It s observed that a consderable decrease n the run probablty s acheved when the hedger uses a tme-dscretzed rsk-mnmzng strategy. Nevertheless, the magntude of the reducton becomes qute small and the advantage of usng ths tme-dscretzed strategy almost dsappears as the hedgng frequency s ncreased. hs s due to the fact that by dscretzaton, the orgnally mean-self-fnancng contnuous rsk-mnmzng hedgng strategy s not mean-self-fnancng any more. Furthermore, t causes some extra duplcaton errors, whch may ncrease the nsurer s net loss substantally. It s shown that the smulaton results are greatly mproved when the hedgng model nstead of the hedgng strategy s dscretzed. he effect s partcularly dstnct when long-term contracts are taken nto consderaton, or when the hedgng strategy s adjusted qute frequently. Acknowledgements Earler versons of ths paper have been presented at the 12. Annual DGF Meetng n Augsburg, at the AFFI 2006 Meetng n Poters and at the 4th World Congress of the Bacheler Fnance Socety 2006 n okyo. I would lke to thank the partcpants n those conferences, Klaus Sandmann, Mchael Suchaneck and an anonymous referee for helpful comments and suggestons. All errors are mne. I acknowledge fnancal support from Netspar. Appendx A. Dervaton of the varance of L r f In order to calculate the varance of the dscounted net loss L r f, the relaton between varance and condtonal varance s appled. VarL r f VarELr f f S + EVarLr S Var E Y n f, Se δ 1 Y t n Y t n +1 Y n 1 S + E Var Y n f, Se δ 1 Y t n Y t n +1 Y n 1 S Var f, Se δ EY n EY n t Y t n +1 EY n A. Chen / Insurance: Mathematcs and Economcs E + Var 2 Cov f, Se δ Y n t f, Se δ Y n Y n t +1, Y t n Y n t +1 2 VarY n, where VarY n n p x 1 p x because of the bnomal dstrbuton of Y n. In order to calculate the varance further, the followng lemma s needed. Lemma A.1. Cov Y n, Y t n n p x 1 t p x. Proof. Analogous relaton as n Eq. 7 holds for the covarance too,.e., CovX, Y ECovX, Y Z + CovEX Z, EY Z, where X, Y, Z are three arbtrary random varables. If Yt x s chosen as the random varable whch s condtonal, we obtan CovY n, Y t n E CovY n, Y t n Y t n + Cov EY n n Y t, EY t n Y t n 0 + Cov Y t n t p x+t, Y t n t p x+t VarY n t t p x+t n t p x 1 t p x n p x 1 t p x. CovY n, Y t n Y t n equals zero because Y t n can be consdered as a constant f t s condtonal on tself. Furthermore, the equalty p x t p x t p x+t s used. In the followng, every separate component n the above varance expresson s calculated. By usng Lemma A.1 and the fact that Y t n Y t n +1, 0,..., M 1 are ndependent, we obtan Var Y t n Y t n CovY t n, Y n 2 VarY n t t +1 Y n t +1 VarY t n + VarY t n +1 2 K e n δj t p x 1 t p x

12 1046 A. Chen / Insurance: Mathematcs and Economcs n t+1 p x 1 t+1 p x 2 n t+1 p x 1 t p x n p x t+1 p x t p x 2 K e δj n t p x 1 t p x n t+1 p x 2 n t+1 p x +2 n t+1 p x t p x 2 K e n δj t p x 1 t p x + n t+1 p x t p x t+1 p x + n t+1 p x t p x 1 2 K e n δj t p x 1 t p x + n t+1 p x t p x t+1 p x + n t p x t p x+t t p x 1 2 n t p x 1 t p x 1 t p x+t K e δ t j + n t+1 p x t p x t+1 p x. Due to a repeated use of the ndependence of the ncrements n the survvng customers, we obtan Cov Y n, Y t n K e δ t j Y n t +1 CovY n, Y n t n p x t+1 p x t p x. CovY n, Y n t +1 Fnally, we obtan the varance of the net loss when the nsurer nvests the premum ncomes n a rsk free asset: VarL r f Var +E + f, Se rδ n p x K e δt j n t p x t+1 p x n p x f, Se r 2 n p x 1 p x 2 K e δ t j n t p x 1 t p x 1 t p x+t + n t+1 p x t p x t+1 p x 2 f, Se δ n p x e δ 2 Var f, S + n p x 1 p x e 2δ { E f, S 2 K e δ t j n p x t+1 p x t p x n e δ p x 1 p x δ + 2 e } K e δ t j E f, S + n p x 1 p x K e δ t j 2 n t p x 1 t p x 1 t p x+t + n t+1 p x t p x t+1 p x + 2 n p x t+1 p x t p x Appendx B. Dervaton of the varance of the tmedscretzed rsk-mnmzng hedgng strategy From Lemma A.1, we obtan frst the followng corollary whch s used several tmes for the varance calculaton. Corollary B.1. Cov Y τ n j, Y τ n n τ j p x 1 τ j p x wth and denotng the maxmum and mnmum operator respectvely. he varance of the net loss gven n Eq. 11 conssts of three parts: Q1 Var f, Se δ Y τ n j τ j f s τ j, S τ j Sτ j Y n + Var 2Cov Q1 Y t n Y n Y n t Y n t +1 f, Se δ + Y n Y n τ j τ j f s τ j, S τ j S τ j, Y n t +1 + Y n..

13 A. Chen / Insurance: Mathematcs and Economcs In the followng, we calculate each term separately. Var Y n Var Var +E Var Q1 f, Se δ Y n E f, Se δ Q1 Y n τ τ p x+τ f s τ, S τ S τ Y n Q1 Y n Var Q1 Y n Q1 +E n p x f, Se δ Y n τ τ p x+τ f s τ, S τ S τ τ τ p x+τ f s τ, S τ Sτ S Y n f, Se δ τ τ p x+τ f s τ, S τ Sτ S f, Se δ n τ j p x τ p x+τ f s τ, S τ Sτ + Var f, Se δ 2 VarY n Q1 Y n τ τ p x+τ f s τ, S τ Sτ S Q1 2 f, Se δ τ p x+τ f s τ, S τ Sτ CovY n, Y τ n Var f, Se δ n p x Q1 + Var n p x f s τ, S τ Sτ Q1 2 n 2 p x 2 e δ Cov f, S, f s τ, S τ Sτ + n p x 1 p x e 2δ E f, S 2 Q1 Q1 +E f s τ j, S τ j Sτ j τ j f s τ, S τ Sτ τ p x+τ CovY τ n, Y τ n j Q1 2 E f, S f s τ, S τ Sτ τ p x+τ e δ CovY n, Y τ n Var f, Se δ n p x Q1 + Var n p x f s τ, S τ Sτ Q1 2 n 2 p x 2 e δ Cov f, S, f s τ, S τ Sτ + n p x 1 p x e 2δ E f, S 2 Q1 Q1 +E f s τ j, S τ j Sτ j τ j f s τ, S τ S τ τ p x+τ n τ j p x 1 τ j p x Q1 2 E f, S f s τ, S τ Sτ τ p x+τ e δ n τ p x 1 τ p x Var Var + Var +2Cov Y n t Y n t Y n Y n t +1 Y n t +1 Y n t + Y n Y n t +1, Y n 2 K e δ t j n t p x 1 t p x 1 t p x+t + n t+1 p x t p x t+1 p x t+1 p x t p x. 2 Cov Q1 Y t n 2 + e 2 n p x q x Y n n p x f, Se δ Y n τ j τ j f s τ j, S τ j S τ j, Y n t +1 δ e δ + Y n n p x t+1 p x t p x n p x 1 p x E f, S

14 1048 A. Chen / Insurance: Mathematcs and Economcs Q +2 E f s τ, S τ Sτ + St+1 τ j e g t 1 {t t } S t F t F t δ t n τ j t p x 1 τ j t p x e K e +1g t δ t + αk + 1 e + St+1 e τ j t +1 p x 1 τ j t +1 p x g t 1 {t>t+1}e δ t St +1 Nd t, S t S 1 t + K e δt e g t e δt+1t Nd t, j 2 1 {t <t t +1 } n p x 1 τ j p x. j e Put these three terms together, we obtan the varance of L φ. δ t e δt +1t N d 1 e gδ t N d 2 1 {t t } Appendx C. Dervaton of the t-value of f, S he far prce Ft,, S t of the contngent clam f, S at tme t s derved as follows: Ft, t M, S t E e δ t f S F t E e δ t K e +1g t e + α + 1K δ t St+1 S t K e +1g t + e g t F t + αk + 1E e δ t St+1 S t e e δ t E + E + E K e +1g t + αk + 1 δ t e δ t St+1 S t e δ t St+1 S t e δ t St+1 S t + e g t F t + e g t 1 {t>t+1} F t + e g t 1 {t <t t+1} F t + e g t 1 {t t } F t K e +1g t + αk e δ t St+1 e g t 1 {t>t+1} S t + e δ t e δt+1t E e δt +1t + St+1 e g t 1 {t <t t+1} S t F t + e δ t e δt +1t E E e δt +1t wth d t, 1/2 ln S t/s t gt +1 t + δ ± 1 2 σ 2 t +1 t σ t +1 t d 1/2 δ g ± 1 2 σ 2 t +1 t σ. t +1 t References Aase, K., Persson, S.A., Prcng of unt-lnked nsurance polces. Scandnavan Actuaral Journal Bacnello, A.R., Ortu, F., Prcng equty-lnked lfe nsurance wth endogenous mnmum guarantees. Insurance: Mathematcs and Economcs 12, Bacnello, A.R., Persson, S.A., Desgn and prcng of equty-lnked lfe nsurance under stochastc nterest Rates. he Journal of Rsk Fnance 3 2, Bernard, C., Le Courtos, O., Quttard-Pnon, F., Market value of lfe nsurance contracts under stochastc nterest rates and default rsk. Insurance: Mathematcs and Economcs 36, Black, F., Scholes, M., he prcng of optons and corporate labltes. Journal of Poltcal Economy 81, Brennan, M.J., Schwartz, E.S., Prcng and nvestment strateges for equty-lnked lfe nsurance. In: Huebner Foundaton Monograph, vol. 7. Wharton School, Unversty of Pennsylvana, Phladelpha. Chen, A., Suchaneck, M., Default rsk, bankruptcy procedures and the market value of lfe nsurance labltes. Insurance: Mathematcs and Economcs 40, Cox, J., Ross, S., Rubnsten, M., Opton prcng: A smplfed approach. Journal of Fnancal Economcs 7, Ekern, S., Persson, S.A., Exotc unt-lnked lfe nsurance contracts. he Geneva Papers on Rsk and Insurance heory 21, Föllmer, H., Sondermann, D., Hedgng of non-redundant contngent clams. In: Hldenbrand, W., Mas-Collel, A. Eds., Contrbutons to Mathematcal Economcs n Honor of Gerard Debreu. North Holland, Amsterdam, pp Grosen, A., Jørgensen, P.L., Lfe nsurance labltes at market value: An analyss of nsolvency rsk, bonus polcy and regulatory nterventon rules n a barrer opton framework. Journal of Rsk and Insurance 69 1, Hardy, M.R., Investment Guarantees: Modellng and Rsk Management for Equty-Lnked Lfe Insurance. Wley. Hsa, C.-C.A., On Bnomal opton prcng. he Journal of Fnancal Research 6, Mahayn, A., Schlögl, E., he rsk management of mnmum return guarantees. Bonn Econ Papers 18/2003.

15 A. Chen / Insurance: Mathematcs and Economcs Melnkov, A., 2004a. Quantle hedgng of equty-lnked lfe nsurance polces. Doklady Mathematcs 69, Melnkov, A., 2004b. Effcent Hedgng of equty-lnked Lfe nsurance polces. Doklady Mathematcs 69, Mltersen, K.R., Persson, S.A., Guaranteed nvestment contracts: Dstrbuted and undstrbuted excess returns. Scandnavan Actuaral Journal 4, Møller,., Rsk-mnmzng hedgng strateges for unt-lnked lfe nsurance contracts. Astn Bulletn 28, Møller,., Hedgng equty-lnked lfe nsurance contracts. North Amercan Actuaral Journal 5 2, Nelsen, J.A., Sandmann, K., Equty-lnked lfe nsurance: A model wth stochastc nterest rates. Insurance: Mathematcs and Economcs 16, Nelsen, J.A., Sandmann, K., Unqueness of the far premum for equtylnked lfe nsurance contracts. he Geneva Papers on Rsk and Insurance heory 21, Nelsen, J.A., Sandmann, K., he far premum of an equty-lnked lfe and penson nsurance. In: Schönbucher, P., Sandmann, K. Eds., Advances n Fnance and Stochastcs: Essays n Honor of Deter Sondermann. Sprnger-Verlag, Hedelberg.