Optimal Longevity Hedging Strategy for Insurance. Companies Considering Basis Risk. Draft Submission to Longevity 10 Conference

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1 Opimal Longeviy Hedging Sraegy for Insurance Companies Considering Basis Risk Draf Submission o Longeviy 10 Conference Sharon S. Yang Professor, Deparmen of Finance, Naional Cenral Universiy, Taiwan. Hong-Chih Huang* Professor and Chairman, Deparmen of Risk Managemen & Insurance, Naional Cheng-Chi Universiy, Taiwan. *Corresponding auhor Jin-Kuo Jung Acuarial Specialis, PCA Life, Taiwan. 1

2 Opimal longeviy hedging sraegy for Insurance Company Considering Basis Risk Absrac This paper proposes he hedging sraegy o deal wih longeviy risk for life insurer considering basis risk. Differen o he exising lieraure, we build up a hedging sraegy for longeviy risk considering no only he inernal naural hedging bu also exernal hedging by using longeviy-linked securiies. Exending from he immunizaion heory, we find he opimal hedging sraegy based on he insurer's profi funcion. We ake ino accoun basis risk in he hedging sraegy and employ Yang and Wang (2013) s muli-populaion moraliy model o capure he moraliy dynamics for annuiy and life insurance business. Insead of using populaion moraliy, we adop a unique daa se of annuiy and life insurance policies ha enable us o calibrae he muli-populaion moraliy dynamics for differen lines of insurance policies and calculae heir liabiliies in he profi funcion. In addiion, we derive he opimal hedging sraegy analyically by applying Taylor expansion on he profi funcion. As a resul, he basis risk is examined empirically and numerical analysis on he opimal hedging sraegy. Keywords: Naural Hedging; Longeviy Risk; Basis Risk; Longeviy-linked securiies 2

3 1. Inroducion Recen increase in longeviy has increased pressures on defined benefi (DB) pension plan providers and annuiy provider. Longeviy risk has become non-negligible and is influence is increasing gradually and globally. As a resul, hedging longeviy risks has aken on an increasingly imporan role for life insurance companies. The sudy of hedging sraegies for managing longeviy risk has been explored in recen years. In general, he hedging sraegy can be caegorized as an inernal or exernal mehod. Naural hedging is regarded as he inernal hedging sraegy ha he insurer can hedge longeviy risks wih heir own business producs beween life insurance and annuiy because hese wo ypes of producs are sensiive in opposing ways o he changes in moraliy raes. If he fuure moraliy of a cohor improves relaive o curren expecaions, life insurers gain a profi because hey can pay he deah benefi laer han iniially expeced, whereas annuiy insurers suffer losses because hey mus pay annuiy benefis for longer han hey iniially expeced. Cox and Lin (2007) find he empirical evidence ha annuiy wriing insurers who have more balanced business in life and annuiy risks end o charge lower premiums han oherwise similar insurers and indicaes ha insurers who have a naural hedge have a compeiive advanage. Wang e al.(2010) invesigae he naural hedging sraegy o deal wih longeviy risks for life insurance companies and propose an immunizaion model o invesigae he naural hedging sraegy by calculaing he opimal life insurance annuiy produc mix raio o hedge agains longeviy risks. Wang e al. (2013) propose a naural hedging model ha can accoun for boh he variance and 3

4 mispricing effecs of longeviy risk a he same ime. Therefore, life insurance can serve as a dynamic hedge vehicle agains unexpeced moraliy risk. Alernaively, he life insurer and pension provider can seek o hedge longeviy risk exernally using capial marke soluions. Blake and Burrows (2001) firs proposed ha issuing survivor bonds could help a pension fund insure agains he longeviy risk. To uilize he capial marke for ransferring longeviy risk, more recen sudies focus on he issue of securiizaion of longeviy risk and a variey of survivor securiies and survivor derivaives have been developed in boh academic and pracice ((e.g., Lin and Cox 2005; Cox e al. 2006; Dowd e al. 2006; Blake e al. 2006; Denui e al. 2007; Biffis and Blake, 2009; Blake e al. 2010; Dawson e al., 2010). For example, he EIB/BNP longeviy bond aimed o ransfer longeviy risk, hough i was never ulimaely issued. The world s firs capial marke derivaive ransacion, a q-forward conrac beween JPMorgan and he U.K. company Lucida, ook place in January 2008; he firs capial marke longeviy swap, execued in July 2008, enabled Canada Life o hedge is U.K.-based annuiy policies. In December 2010, Swiss Re launched a series of eigh-year longeviy-based insurance-linked securiy noes, which i called Koris noes. Blake e al.(2012) noe ha he emergence of a raded marke in longeviy-linked capial marke insrumens would ac a caalys o help faciliae he developmen of annuiy markes. There are some discussions regarding exernal and inernal hedging sraegies for he life insurer. Cox and Lin (2007) sugges ha naural hedging is good bu may be oo expensive o be effecive in he conex of inernal life insurance and annuiy producs. They show ha insurers ha exploi naural hedging by using a moraliy swap can charge a lower risk premium han ohers. In addiion, he resricion of using he naural hedging sraegy for he insurers is ha hey mus adjus he sales volume of life insurance and annuiy producs o remain an opimal liabiliy proporion which 4

5 is someime no feasible in pracice. To overcome such resricion, we propose a longeviy hedging framework ha considers no only he naural hedging bu also he exernal hedging by using longeviy-linked securiies. Assume a life insurer wih boh annuiy and life insurance business. Greaer longeviy risk implies ha he insurer can earn profis from selling life insurance policies bu suffer losses for selling annuiy insurers. The annuiy providers usually has more annuiy policies han life insurance policies in heir liabiliy, i.e. he longeviy risk canno be fully naural hedged and he insurer can consider he exernal hedging by using he longeviy-linked securiies o deal wih he remaining longeviy risk. To deal wih he proposed hedging sraegy ha combines boh naural hedging and exernal hedging, we define he profi funcion according he enire business of life insurance and annuiy business and he opimal hedging sraegy is obained based on he insurer s profi funcion. Longeviy risk exposure may be differen in differen lines of business and populaion groups. The grea concern in hedging longeviy risk is longeviy basis risk. Coughlan e al. (2011) poin ou he imporance of basis risk in longeviy hedging because he moraliy experience may differ from ha of life insurance and annuiy porfolio, he hedge will be imperfec and leaving a residual amoun of risk, known as basis risk. The use of populaion-based moraliy indices for managing he longeviy risk inheren in specific blocks of annuian liabiliies may resul in basis risk. However, he exising lieraure on longeviy hedging mainly demonsrae wih populaion moraliy experience or reaing life insurance and annuiy business on he same moraliy basis. For example, Wang e al.(2010) employ he same moraliy rae measure (populaion moraliy raes) for boh life insurance and annuiy producs in finding he opimal produc mix. I may happen he mismach in moraliy raes beween life insurance and annuiy producs. Therefore, o hedge longeviy risk, i is imporan o consider basis risk. To fill he gap, we consider basis risk on finding he 5

6 opimal hedging sraegy. Paricularly, we deal wih basis risk in he following hree aspecs. Firs, insead of populaion moraliy daa, we employ a unique experience moraliy daa 1 from life insurance indusry ha includes he moraliy experience for boh life insurance and endowmen insurance for men and women separaely. Wih his unique moraliy experience, i enables us o capure he longeviy risk for differen produc lines o avoid mismach in moraliy raes. Second, we measure he basis risk for he longeviy exposed business based on Yang and Wang (2013) s muli-populaion moraliy framework. This model is calibraed o capure he basis risk beween differen lines of insurance policies. The empirical analysis of basis risk for life insurance and annuiy policy are sudied. Third, we ake ino he muli-populaion moraliy model in he longeviy hedging framework and calculae he profi funcion for differen lines of life insurance business. The opimal hedging sraegy is obained according o he profi funcion. Therefore, we can benefi from he unique experience moraliy daa o measure longeviy risk and model moraliy dynamics in he presence of basis risk. In addiion, we consider no only he naural hedging bu also he exernal hedging in he proposed longeviy hedging framework. The life insurer can uilize he longeviy-linked securiies o hedge longeviy risk. Thus, he profi funcion is calculaed based on boh he cash flow of he insurer s liabiliy and he cash flow of he longeviy-linked securiies. The sochasic moraliy dynamic is considered o capure he longeviy risk and he valuaion framework for pricing various longeviy-linked securiies is also derived. Exending from he immunizaion heory, he insurer aemps o sabilize is profi in response o he change of moraliy improvemen. We apply Taylor expansion on he profi funcion o measure he impac 1 These incidence daa include more han 50,000,000 policies, colleced from all Taiwanese life insurance companies. 6

7 of moraliy improvemen and derive he closed-form soluion for he opimal hedging sraegy under he firs order approximaion. The conribuions of his research are fourfold. Firs, his paper builds a longeviy hedging framework for he insurer ha uilizes boh naural and exernal hedging sraegy. Second, his paper considers he basis risk in finding he opimal hedging sraegy. We employ Yang and Wang (2013) s muli-populaion moraliy framework and obain he opimal hedging sraegy analyically according he insurer s profi funcion. Third, we deal wih basis risk using a real moraliy daa from life insurance indusry. We can model he moraliy dynamics for differen lines of business. Fourh, he opimal hedging sraegy is examined numerically. The remainder of his paper is organized as follows. In Secion 2, we presen a muli-populaion moraliy framework o model he moraliy dynamics for differen produc lines. To avoid basis risk, his muli-populaion moraliy framework is calibraed o he real moraliy experience from insurance indusry. The moraliy daa is inroduced and he corresponding parameers in he moraliy model are obained. Secion 3, we build he proposed hedging framework for he insurance company. The profi funcion and objecive funcion are inroduced. The opimal hedging sraegy is derived analyically. In Secion 4, he opimal hedging is analyzed numerically. Afer we provide a numerical analysis of he opimal hedging sraegy, we draw some key conclusions and implicaions. 2. Modeling Moraliy dynamics for he Life Insurer Considering Basis Risk The muli-populaion moraliy dynamic The insurer has differen moraliy paern for life insurance and annuiy business. 7

8 We consider he basis risk o model longeviy risk for differen lines of business. Thus, we need a moraliy model o projec he fuure moraliy raes for differen groups of populaion simulaneously. We employ Yang ad Wang (2013) s muli-populaion model which is buil on he Lee-Carer framework (Lee and Carer, 1992). We analyze changes in moraliy as a funcion of boh age x and ime on a filered probabiliy space (,,,( ) + P T s s= 0 ) ΩI I, where P is he physical probabiliy measure and I is he informaion available a ime. If we consider N produc lines for life j insurance,, hen m x,, j = 1,..., N, is he moraliy force a age x for in he j h insurance produc during calendar year. Tha is, ln m = a + bk + e, j 1,..., N. (1) j j j j j x, x x x, where he parameers b j x and j k are subjec o b j x = 1 and k j = 0, o x ensure he model idenificaion. The Lee-Carer model can capure he age period effec for he j h populaion wih j a x coefficiens. Furhermore, he acual force of moraliy for he j h populaion changes according o an overall moraliy index j k, which is modulaed by an age response j j b x. The error erm x, e reflecs a paricular, age-specific hisorical influence ha is no capured by he model. To beer presen he muli-populaion moraliy forecas clearly, we express he fuure moraliy raes for a person of age x a ime in marix form: ln m x, ax bx 0 0 k e x, ln m x, ax 0 bx 0 k e x, = + +. (2) 0 0 N N N N N ln mx, ax 0 0 b e x k x, Lee and Carer (1992) assume ha he parameers of a x and b x remain consan over ime and forecas fuure moraliy by projecing he moraliy ime index 8

9 using a sandard ARIMA ime-series model for a single populaion. However, o deal wih longeviy exposure for a pool of policies across differen produc lines, we need o forecas fuure moraliy for differen group of populaions. To do so, we adop a mulivariae ime-series approach o analyze he moraliy ime index. Tha is, we use a co-inegraion analysis o invesigae wheher a common sochasic rend appears in he fuure moraliy ime rends K for populaions across counries. We model K wih a VECM. If each series in process wih p uni roos, hen K is an I( p ) process, ha is, a nonsaionary K is co-inegraed. According o he Granger represenaion heorem (Engle and Granger 1987), he VECM of order p (or VECM(p)) for K follows he form: p 1 K = ω +Π K + Γ K + ε, (3) 1 d d d = 1 where is he firs-order difference operaor; ω, a N-by-1 vecor, conains deerminisic erms (e.g., consan, rend, seasonal dummies); Π is a N-by-N long-run impac marix; Γ for d = 1,..., p 1, is a N-by-N shor-run impac marix; d and 1 N ε = ε,..., ε is a shock vecor, where he ε, ε+ 1, ε are muually independen, each following a N-variae normal disribuion wih 0 mean and a N-by-N covariance marix of Σ. Calibraion of moraliy model To calibrae he moraliy model for capuring he moraliy dynamics of annuiy and insurance business, we employ a unique experience moraliy daa from he life insurance indusry in Taiwan. This daa se includes he acual moraliy experience for boh life insurance and endowmen insurance and for men and women separaely. Thus, we can capure he acual moraliy paern for differen lines of business raher 9

10 han demonsraing wih populaion moraliy raes. These daa was colleced from all Taiwanese life insurance companies. The daa covers more han 50,000,000 policies issued by he life insurance companies in Taiwan from he period of 1972 o 2008 (27 years of daa) 2. To capure he moraliy paern for life insurance and annuiy policies, we divide all policies ino wo groups: wih or wihou endowmens (single endowmen or serial periodic endowmens). The insurance company in Taiwan has a special produc sraegy ha selling endowmen insurance insead of annuiy business. Therefore, we use he policies wih endowmens as he proxy for annuiy policies. Table 1: Summary of policy numbers derived from Taiwan insurance daa Table 1: Policy Summary Insurance Type Female Male Wih Endowmens 7,175,200 7,509,730 No Endowmens 18,254,681 18,072,776 In addiion, we demonsrae ha he insurer uses q-forward conracs as he exernal hedging insrumen. Assume he underlying moraliy index for he q-forward conrac is based on he UK male populaion moraliy. To deal wih basis risk, we calibrae he muli-populaion model o each line of business and he underlying moraliy index. In oher words, we have five populaions: Taiwan female annuiy(tw-fa), Taiwan male annuiy(tw-ma), Taiwan female life(tw-fl), Taiwan male life(tw-ml) and UK male populaion moraliy(uk-male). In order o model he basis risk among differen populaions, we apply Co-inegraion analysis o capure he inerrelaed moraliy improvemen among populaions according o he moraliy experience. 2 The experience daa were sored by he Insurance Agency Associaion of he Republic of China (CIAA) before 2007, and TII has been in charge of he daa since TII now mainains all of he experience daa available from 1972 unil he presen. 10

11 Figure 1: Moraliy Time Trend( k ) for Differen Populaion Groups j Figure 1 exhibis he relaionships of moraliy ime rend(k ) among he 5 differen group of populaions. We can observe ha he moraliy improvemen for annuiy populaions seems more faser han for life insurance populaions and UK male populaion. However here is an ineresing behavior ha he process of all k 's move in a similar paern for differen groups of populaions. Based on he co-inegraion analysis, i indicaes ha here is a long-erm equilibrium relaionships among all k 's and he k 's have coinegraed effec. Then we can wrie down he esimaed parameers for he error correcion model as follows. where Δk 1, Δk 2, Δk 3, Δk 4, ε 1, = βi k i, 1 + ε 1, i=1 ε 1, Δk 5, ε 1, ε 1, and β 1 β 2 β 3 β 4 β =

12 Σ = Cov ε 1, ε 2, ε 3, ε 4, ε 5, = We can rea he firs difference of k as a ime-varying mulivariae normal disribuion. Tha is Δk 1, Δk 2, Δk 3, Δk 4, Δk 5, ~N(μ(k 1 ), Σ) where he mean μ(k 1 ) is adjused by he error correcion erm as follows μ(k 1 )= i=1 βi k i, 1 3. Opimal Hedging Sraegy for longeviy risk Profi Funcion of he Insurer The purpose of his research is o sudy he opimal hedging sraegy of he insurer o hedge longeviy risk using boh naural hedging and exernal hedging insrumen. We consider a life insurance company wih boh life insurance and annuiy policies. The insurance company is assumed o have more annuiy policies han life insurance policies in heir liabiliy, i.e. he longeviy risk canno be fully naural hedged. Thus, he insurer also considers he exernal hedge insrumen and aemps o find he opimal level of longeviy securiies o effecively reduce longeviy risk. In oher words, he insurer can use he longeviy securiies o sabilize he profi funcion in response o a change moraliy improvemen. Thus, we exend he immunizaion heory o find he opimal hedging sraegy based on he insurer s 12

13 profi funcion. The profi funcion is calculaed according o he cash flow of he insurer s liabiliy and he cash flow of he hedging insrumen. Le U() denoe he insurer s profi a ime. The profi funcion of he insurance company including all life and annuiy policies a ime is defined as follows: U() = M V S () c L,j V L x j, j c A,j V A x j, j, (4) where V S () is he value of he exernal hedge insrumen a ime and M is he amoun of he hedge insrumen, V L x j, is he value of jh life insurance policy for he insured aged xj wih face amoun c L,j ; V A x j, is he value of jh annuiy policy for he insured aged xj wih face amoun c A,j. Equaion (4) is a general form of he profi funcion for he life insurance company. If he insurer only provides annuiy business or he pension provider, we can exclude he erm for life insurance policies in Equaion (4). Here we consider q-forward as he hedging insrumen agains longeviy risk. The presen value of payoff funcion of q-forward can be represen as V S (0) = 1000(1 + r) 10 f (q UK male65 real q UK male65 ) f where q UK male65 is he q forward rae deermined a he beginning of he conrac and he mauriy is 10 years. The ineres rae is assumed o be Objec Funcion of finding opimal hedging sraegy According o he profi funcion, he insurer s objecive is o find an effecive hedging sraegy for avoiding unexpeced changed on he profi caused by he moraliy shifing. Our objecive is o deermine he opimal value of M, i.e. how many unis of longeviy-linked securiies he insurer should ake o reduce he variaion for he insurer s profi. To esimae he effec of he moraliy change on he profi, we apply Taylor expansion of U() wih respec o moraliy rend (k ). Tha is, he change of profi is measured by ΔU() U() Δk. 13

14 We invesigae hree differen hedging sraegies based on he mean-variance framework for he insurer's profi process. The objecive funcions under differen hedging sraegies are shown below: (1) Maximizing mean-variance approach: max E[ΔU()] θvar ΔU() M (2) Maximizing mean- value a risk approach: max E[ΔU()] z p Var ΔU() M where z p is he inverse cumulaed disribuion funcion of sandard normal disribuion. (3) Maximizing mean- condiional ail expecaion approach: max M E[ΔU()] Var ΔU() p 2 e z p 2 2π Under he assumpion ha he change of profi is approximaely normal disribued, we can obain he mean and variance analyically. Therefore, under he firs order approximaion of ΔU(), we can ake he properies of normal disribuion o solve he opimal hedging sraegy analyically. Derivaion of Opimal Hedging Sraegy We demonsrae how he insurer hedge he longeviy risk using q-forward conracs when he insurance company has more annuiy produc han life conrac in heir liabiliy. The hedging sraegy is for single-period, ha is o conrol he unexpeced profi changes form his year o nex year under he assumpion ha moraliy dynamics follow Yang and Wang(2013). We define he profi funcion o analyze and measure he risk of each produc, hen find he opimal soluion o hedge moraliy risk for differen hedging sraegies. 14

15 Define where We firs esimae he change of U()wih respec o k s using Taylor expansion. U() U() k 1, k 1, U() U() U() = k 2, = k 2, = D 1 MD 2 U() M VS k 5, k 5, k 5, U() k 1, D 1 = U() k 2, U() k 4, and D 2 = VS k 5, k 5, We also define Δk 's Δk = Δk 1, Δk 2, Δk 3, Δk 4, Δk 5, where Δk (1) = (1) Δk μ(1) = (2) Δk ~N (k 1 ) μ (2) (k 1 ), Σ 11 Σ 12 Σ 21 Σ 22 Δk 1, Δk 2, Δk 3, and (2) Δk = Δk 5, Δk 4, Wih hese noaions, we can derive he disribuion of he change of profi. I is approximaely normal disribued. Tha is ΔU() U() Δk ~N( U() μ(k 1 ), U() Σ U()) We can furher express he mean and variance of he change of profi as follows: 15

16 E[ΔU()] = A 1 + MA 2 Var[ΔU()] = A 3 + MA 4 + M 2 A 5 where A 1 = D 1 μ (1) (k 1 ) A 2 = D 2 μ (2) (k 1 ) A 3 = D 1 Σ 11 D 1 A 4 = D 2 Σ 21 D 1 + D 1 Σ 12 D 2 A 5 = D 2 Σ 22 D 2 Proposiion 1: Under he maximizing mean-variance approach, he opimal soluion is M = A 2 θa 4 2θA 5. Proposiion 2: Under he maximizing mean- value a risk approach, :he opimal soluion is M = K 1± K 2 1 4K 2 K 0, where K 2K 0 = 4A 2 2 A 3 z 2 p A 2 4, K 1 = 4A 2 2 A 4 2 4A 4 A 5 z p 2, K 2 = 4A 2 2 A 5 4z p 2 A 5 2 and M 0. Proposiion 3: Under he m aximizing mean- condiional ail expecaion approach, he opimal soluion is M = K 1 ± K 2 1 4K 2 K 0 2K 2, where K 0 = 4A 2 2 A 3 K 2 A 4 2, zp 2 K 1 = 4A 2 2 A 4 4A 4 A 5 K 2, K 2 = 4A 2 2 A 5 4K 2 A 2 5, K = e 2 and p 2π M 0, 4. Numerical examples Assume boh he life insurance and annuiy conain wo policies, one is of male insured aged 55 and he oher is of female insured aged 55 respecively. For calculaing he value of life insurance conracs, annuiies and q-forward, we generae 50,000 sample pahs of moraliy rae and compue he averaged value of acuarial 16

17 presen value of each simulaed moraliy. Table 2 displays he opimal hedging sraegies for differen objecive funcions. We can see under mean variance assumpion, as θ increases, he opimal hedging unis decreases. The opimal hedging sraegies of VaR and CTE need fewer unis of q-forward o hedge he longeviy risk. Table 2: Opimal Hedging Sraegies Opimal M Sraegy 1: Dela mv (θ = 0.8), (θ = 1), (θ = 1.2) Sraegy 2: Dela VaR Sraegy 3: Dela CTE (To be coninued ) 17

18 Reference: Biffis, E. and D. Blake, 2009, Moraliy-Linked Securiies and Derivaives, Discussion paper PI-0901, The Pensions Insiue. Blake, D., A. J. G. Cairns and K. Dowd, 2006, Living wih Moraliy: Longeviy Bonds and Oher Moraliy-linked Securiies, Briish Acuarial Journal, 12, Blake, D., P. Dawson, K. Dowd and A. J. G. Cairns, 2010, Survivor Derivaives: A Consisen Pricing Framework, Journal of Risk and Insurance, 77, Blake, D., A. J. G. Cairns, G. D. Coughlan, K. Dowd and R. MacMinn, 2012, The New Life Marke, Working Paper. Blake, D. and W. Burrows, 2001, Survivor Bonds: Helping o Hedge Moraliy Risk, Journal of Risk and Insurance, 68, Cairns, A. J. G., D. Blake and K. Dowd, 2006, A Two-Facor Model for Sochasic Moraliy wih Parameer Uncerainy: Theory and Calibraion. Journal of Risk and Insurance, 73, Coughlan, G. D., K. A. Marwa, Y. Ye, S. Kumar, A. J. F. Cairns, D. Blake, and K. Dowd, 2011, A Framework for Longeviy Basis Risk Analysis and Hedge Effeciveness, Norh American Acuarial Journal on Longeviy, 15, Cox, S.H. and Y. Lin, 2007, Naural Hedging of Life and Annuiy Moraliy Risk, Norh American Acuarial Journal, 11, 3, Cox, S. H., Y. Lin and S. Wang, 2006, Mulivariae Exponenial Tiling and Pricing Implicaions for Moraliy Securiizaion, Journal of Risk and Insurance, 73, Denui, M., P. Devolder and A. C. Goderniaux, 2007, Securiizaion of Longeviy 18

19 Risk: Pricing Survivor Bonds Wih Wang Transform in he Lee-Carer Framework, Journal of Risk and Insurance, 74, Dowd, K., D. Blake, A. J. G. Cairns and P. Dawson, 2006, Survivor Swaps, Journal of Risk and Insurance, 73, Engle, R. F. and C. W. J. Granger, 1987, Coinegraion and Error-Correcion: Represenaion, Esimaion and Tesing, Economerica, 55, Larry Y. Tzeng, Jennifer L. Wang and Jeffrey T. Tsai (2011), Hedging Longeviy Risk When Ineres Raes Are Uncerain, Norh American Acuarial Journal, 15(2). Lee, R. D., and Carer, L. 1992, Modeling and Forecasing he Time Series of U.S. Moraliy, Journal of he American Saisical Associaion 87, Lin, Y., and S. H. Cox, 2005, Securiizaion of Moraliy Risks in Life Annuiies, Journal of Risk and Insurance, 72(2), Li, J. S. H. and M. R. Hardy, 2011, Measuring Basis Risk in Longeviy Hedges, Norh American Acuarial Journal, 15, Tsai, J.T., J.L. Wang, and L.Y. Tzeng (2010), On he opimal produc mix in life insurance companies using condiional Value a Risk, Insurance: Mahemaics and Economics, 46, Cox, S. H. and Y. Lin, 2007, Naural Hedging of Life and Annuiy Moraliy Risks, Norh American Acuarial Journal, 11(3): Wang, C. W., H. C. Huang and I. C. Liu, 2011, A Quaniaive Comparison of he Lee-Carer Model under Differen Types of Non-Gaussian Innovaions, Geneva Papers on Risk and Insurance Issues and Pracice, 36, Wang, C. W., H. C. Huang and De-Chuan Hong, 2013, A Feasible Naural Hedging Sraegy for Insurance Companies, Insurance : Mahemaics and Economics, 52(3),

20 Wang, J.L., H.C. Huang, S.S. Yang and J.T. Tsai, 2010, An Opimal Produc Mix for Hedging Longeviy Risk in Life Insurance Companies: The Immunizaion Theory Approach, The Journal of Risk and Insurance, Vol. 77, No. 2, Yang, S. S. and Wang, C. W. 2012, Pricing Survivor Derivaives wih Cohor Moraliy Dependence under he Lee-Carer Framework, Journal of Risk and Insurance, Forhcoming. 20

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