An Optimal Strategy of Natural Hedging for. a General Portfolio of Insurance Companies

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1 An Opimal Sraegy of Naural Hedging for a General Porfolio of Insurance Companies Hong-Chih Huang 1 Chou-Wen Wang 2 De-Chuan Hong 3 ABSTRACT Wih he improvemen of medical and hygienic echniques, life insurers may gain a profi whils annuiy insurers may suffer losses because of longeviy ris. In his paper, we invesigae he naural hedging sraegy and aemp o find an opimal collocaion of insurance producs o deal wih longeviy riss for life insurance companies. Differen from previous lieraures, we use he eperienced moraliy raes from life insurance companies raher han populaion moraliy raes. This eperienced moraliy daa se includes more han 50,000,000 policies which are colleced from he incidence daa of he whole Taiwan life insurance companies. On he basis of he eperienced moraliy raes, we demonsrae ha he proposed model can lead o an opimal collocaion of insurance producs and effecively apply he naural hedging sraegy o a more general porfolio for life insurance companies. Keyword: Longeviy ris, Naural hedging sraegy, Eperienced moraliy raes 1 Professor, Deparmen of Ris Managemen and Insurance, Naional Chengchi Universiy, Taipei, Taiwan. *Corresponding auhor. 2 Associae Professor, Deparmen of Ris Managemen and Insurance, Naional Kaohsiung Firs Universiy of Science and Technology, Kaohsiung, Taiwan. 3 Deparmen of Ris Managemen and Insurance, Naional Chengchi Universiy, Taipei, Taiwan. 1

2 1.Inroducion According o he publicaion-sigma of Swiss Re, he life epecancy of human around he world will increase 0.2 year per year. Because he moraliy rae has improved rapidly for he pas decades, longeviy ris has become an imporan opic. In he pas wo decades, a wide range of moraliy models have been proposed and discussed (Lee-Carer, 1992; Brouhns e al., 2002; Renshaw and Haberman, 2003; Koissi e al., 2006; Melniov and Romaniu, 2006; Cairn, Blae and Dowd, 2006). Among hem, he Lee-Carer (LC) (1992) model is probably he mos popular choice, because i is easy o implemen and provides accepable predicion errors. Consrucing delicae moraliy model for he use of pricing is one soluion o hedge longeviy ris for boh life insurance and annuiy producs. However, his soluion is ofen difficul o apply ino pracice because of mare compeiion. Even hough insurance companies have abiliy o build a delicae moraliy model o cach he acual fuure moraliy improvemen, hey may no be able o price and sell annuiy producs using he moraliy rae derived from his moraliy model since i migh be oo epensive o sell hese annuiy producs wih mare compeiion. Anoher possible soluion o hedging longeviy ris is o use he moraliy derivaives, such as Survival Bonds and Survival Swaps. Blae and Burrows (2001) propose he concep of Survival Bond and insurance companies can hedge longeviy ris based on i. Cairn, Blae and Dowd (2006) propose Survival Swaps, which is a conrac o echange cash flows in he fuure based on he survivor indices. Alhough moraliy derivaives are easy and convenien o use bu here are sill many obsacles in moraliy derivaive. The special purpose vehicles mus pay more aenion on heir cusomers and counerpary, and ha means insurance companies have o pay a huge amoun of ransacion cos on moraliy derivaives. Anoher soluion is naural hedging. 2

3 Insurance companies can opimize he collocaion of is producs, annuiies and life insurances, o hedge longeviy ris. This approach can be done inernally in an insurance company. Therefore i is more convenien and pracical for insurance company o hedge longeviy ris by using his mehod. Naural hedging is a relaively new opic in acuarial field, so few papers have sudied his issue. Wang, Yang and Pan (2003) invesigae he influence of he changes of moraliy facors and propose an immunizaion model o hedge moraliy riss. Co and Lin (2007) indicae ha naural hedging uilizes he ineracion of life insurance and annuiies o a change in moraliy o sabilize aggregae cash ouflows. And hey drew a conclusion ha naural hedging is feasible and moraliy swaps mae i available widely. Wang, Huang, Yang and Tsai (2010) analyze he immunizaion model menioned above and use effecive duraion and conveiy o find he opimal produc mi for hedging longeviy ris. However, heir paper uses he same moraliy rae for boh life insurance and annuiy producs due o lac of eperience moraliy daa. This is definiely no rue in pracice. Differen from hose previous lieraures, we consider he variance effec relaed o boh uncerainies of moraliy rae and ineres rae and he miss-pricing effec induced by he moraliy improvemen in consrucing naural hedging porfolio. In addiion, we use he eperienced moraliy raes from life insurance companies raher han populaion moraliy raes. This eperienced moraliy daa se includes more han 50,000,000 policies which are colleced from he incidence daa of he whole Taiwan life insurance companies. Because we don have real annuiy moraliy daa, we regard he eperience moraliy rae of life insurance policies wih heavy principal repaymen as annuiy moraliy rae. Boh eperience moraliy raes beween wih and wihou principal repaymen are correlaed-bu no perfecly negaive correlaed. 3

4 Therefore, i is no possible o perfecly hedge longeviy ris as Wang, Huang, Yang and Tsai (2010) when we consider boh life and annuiy moraliy raes. Besides, we consider he pricing differences in hose insurance producs beween period-moraliy basis and cohor-moraliy basis. Our objecive is o minimize he variaion of he change of oal porfolio s value and he differences beween differen pricing bases. On he basis of hese eperienced moraliy raes, he proposed model in his paper appropriaely provides an opimal collocaion of insurance producs and effecively applies he naural hedging sraegy o a more general porfolio for life insurance companies. The reminder of his paper is organized as follows. In Secion 2, we review he moraliy model and ineres rae model, proposing our porfolio model wih variance effec and miss-pricing effec. In Secion 3, we calibrae he Lee-Carer model by using he eperienced moraliy raes from Taiwan Insurance Insiue (TII). Secion 4 conains he numerical analysis of our model. Secion 5 summarizes he paper and gives conclusions and suggesions. 2.Model Seing 2.1. Moraliy Model: Lee-Carer Model Lee and Carer proposed his moraliy model in They found ou he fiing and forecasing abiliy of his model is superior and simple bu powerful one-parameer model. Since hen, Lee-Carer model has been one of he mos popular sochasic moraliy models. Their model is as follows: ln( m ), (1),, where 4

5 m, : The cenral deah rae for a person aged a ime. : The average age-specific moraliy facor. : The age-specific improving facor. : The ime-varying effec inde. In heir sudy, he ime effec inde can be esimaed by using an ARIMA (0,1,0) process. Noe ha he coninuous-ime limi of he ARIMA (0,1,0) process of can be epressed as d u d dz (), where Z () T 0 is a sandard Brownian moion. There are los of papers invesigaing he fiing of Lee-Carer parameers and wo mos popular mehods are SVD (Singular Value Decomposiion) and Approimaion. In heir paper, Lee and Carer use SVD o find he parameers. Wilmoh (1993) propose a modified Approimaion mehod, Weighed Leas Squares, o avoid he zero-cell problem. We use Approimaion mehod o esimae he parameers of Lee-Carer model because missing values problem did eis in our daa. The compuaional seps can be wrien as follows: 1.The fied values of equal he average of, ln( m ) a age over ime. 2.Employ he sandard normalizing consrains on boh and, such ha b 1 and is equal o he sum of ln( m, ) over each age. 4.Using he regression model wihou inercep bu wih dependen variables and, ln( m ), we can obain he coefficien. 5

6 Following he above seps, we can obain he esimaed parameers, and o forecas he fuure moraliy raes Ineres Rae Model: CIR Model Co, Ingersoll and Ross (1985) specifies ha he insananeous ineres rae follows a square roo process, also named he CIR process as follows: dr a( b r ) d r dz ( ), (2) r r where a denoes he speed of mean-revering adjusmen; b represens he long erm mean of ineres raes; r is ineres rae volailiy; and Z () T r 0 is sandard Brownian moion modeling he random mare ris facor. In CIR model, he ineres rae owards he long run level b wih speed of adjusmen governed by he sricly posiive parameer a. in his paper, we assume ha 2 2ab r ; herefore, an ineres rae of zero is also precluded. We can ransform he coninuous-ime process ino a discree-ime version as follows: r r a( br ) r, (3) 1 1 r 1 where is a sandard normal random variable. We can use Equaion (3) o simulae he fuure ineres raes wih he seing origin r 0. The long-erm ineres rae can be esimaed by erm srucure of ineres rae. According o CIR model, we can calculae he price of one uni zero-coupon bond a ime wih mauriy dae +T as follows: Pr T AT e, (4) B(, T) r (,, ) (, ) where 6

7 [( a) T]/2 2 e AT (, ) T ( a)( e 1) ab/ r, (5) T 2( e 1) BT (, ) T, (6) ( a )( e 1) 2 ( ) 2 r 2 2 a, (7) 2.3. General Porfolio Model In our model, we adop Lee-Carer moraliy model and CIR Ineres rae model. The insurance company s porfolio conains zero-coupon bonds, annuiies and life insurances wih differen ages and genders. The facors affecing he oal value of his porfolio are moraliy rae and ineres rae. The model is as follows:,, () sg sg (),, B B, sa, L g1,2 i V N V m r NV r, (8) B where V represens he value of he insurance porfolio; V ( r, ) is he value of one uni of zero-coupon bond wih fac value equal o one; A,1 A,2 m () ; V( m ( ), r, ) ( Vm ( ( ), r, ) ) denoes he value of one uni of m, annuiy policy issued o he cohor ha consiss of males (females) aged a ime 0; L,1 V m r ( V( m ( ), r, )) denoes he value of one uni of life insurance policy L,1 ( ( ),, ) issued o he males (females) ha consiss of females aged a ime 0; N B represens he number of unis invesed in zero-coupon bonds; and, N ( N L g ) Ag, denoes he number of unis allocaed in he annuiy (life insurance) policies. From Lee-Carer model, we can ge he following informaion, and by using approimaion mehod and he esimaed force of moraliy is as follows: 7

8 ln( m ). (9), Because he coninuous-ime limi of he ARIMA (0,1,0) process of can be epressed as d u d dz (), differencing Equaion (9) yields he coninuous -ime represenaion of log moraliy rae as follows: or equivalenly, dln( m ) u d dz ( ), (10) ln( m ) ln( m ) u Z ( ), s L or A, g 1 or 2. (11) sg, sg, sg, sg, sg, sg, sg,,,0 By using he Io s lemma, we can ransform he dynamic of logarihm of moraliy ino he dynamic of moraliy rae as follows: or equivalenly, ln( m 2, ) 1 d m, d e m, dln m, m, dln m, 2 1 = m, ud dz( ) d m, d = m, u m, dm, dz( ), (12) sg, sg, sg, sg, 1 sg, sg, sg, sg, sg, sg, sg, dm () = m () + u m () + dm () + dz () 2 for s L or A, g 1 or 2, (13) In our model, o capure he covariance srucure of moraliy raes, we assume ha he four moraliy ris facors, A,2 Z, A,1 Z, L,2 Z and L,1 Z are dependen sandard Brownian moions. For he racabiliy of our analysis, we decompose hem ino he linear combinaion of four independen sandard Brownian moions () T 0, i =1,2,3,4 Z Z i i, by using Chelosy decomposiion as follows: 8

9 dz () dz ( ) A,2 1 A,1 () a21 a dz 2( ) dz L,2 dz () a31 a32 a33 0 dz 3( ) L,1 dz () a41 a42 a43 a44 dz 4(). (14) Uilizing he Io s lemma, we invesigae he change of oal insurance porfolio value wih respec o he change of moraliy rae and ineres rae as follows: V dv () () V 1 V 1 V sg, 2 sg, 2 sg, sg, sg, sg, 2 2 N dm ( dr ) ( ( )) ( ) sg, N dm dr sg, 2 2 sl, A g1,2 i m r s L, A g 1,2 i2 m 2 r 4 Q d Q dz () Q dz (), (15) 0 i i 5 r i1 Q, Q, Q, Q, Q and Q 5 are defined as follows: where sg, sg, V sg, sg, sg, 1 sg, sg, 2 sg, 2 Q0 N ( m ( ), + m ( ) ) sg sl, A g1,2 i m 2 2, 2 sg 1 sg, V sg, sg, sg, 2 V 1 V 2 + N ( m ( ) ) ( ) sg, 2 a br 2 r r, (16) 2 m r 2 r V V Q N m a N m A,2 A,1 A,2 A,2 A,2 A,2 A,1 A,1 A,1 A,1 1 (), () A A,1 + i m i m L,2 L,1 L,2 V L,2 L,2 L,2 L,1 V L,1 L,1 L,1 a31 N m (),2 + a41 N m () L L,1 + i m i m, (17) Q a N V m a N V m A,1 L,2 A,1 A,1 A,1 A,1 L,2 L,2 L,2 L, (), () A L,2 i m i m L,1 L,1 V L,1 L,1 L,1 a 42 N m () L,1 + i m, (18) Q a N V m a N V m L,2 L,1 L,2 L,2 L,2 L,2 L,1 L,1 L,1 L, (),2 43 () L L,1 i m i m, (19) L,1 L,1 V L,1 L,1 L,1 Q4 a 43 N m () L,1 i m, (20) 9

10 V r ; (21) Q5 r r he V r and 2 V r 2 are of he form: V V V r r r sg, B sg, N NB, (22) sl, A g1,2 i V V V. (23) 2 2 sg, 2 B sg, N 2 N 2 B 2 r sl, A g1,2 i r r Under he assumpion ha moraliy raes and financial ris are independen, he erms dz dz, i 1,2,3,4 are zero in our model. We apply he conceps of effecive i r duraions and conveiies o esimae he firs-order and second-order derivaives in our model as follows: V V( m, r) V( m, r), (24) m 2 V( m, r) m V V( m, r ) V( m, r ), (25) r 2 V( m, r) r 2 V V( m, r) V( m, r) 2 V( m, r) 2 2, (26) m V( m, r)( m) 2 V V( mr, ) V( mr, ) 2 V( mr, ) 2 2, (27) r V( m, r)( r) where m m(1 ); m m(1 ); r r(1 ); r r(1 ); represens a sricly posiive change rae of moraliy rae and ineres rae; and m and r saisfy he following epression: m m m 1, (28) r r r 1. (29) 10

11 By virue of Equaion (15), he variance of he change of oal insurance porfolio value is as follows: Var( dv ( )) ( ) 5 Q j 2, (30) j =1 which is deermined by he parameers of Lee-Carer model and CIR model as well as he firs-order and second-order derivaives defined in Equaions (24)-(27). Insurers minimize he variance of porfolio reurns wih respec o he change in moraliy rae and ineres rae because heir ris profiles are mainly consis of he longeviy ris and ineres rae ris. Therefore, we incorporae he variance in Equaion (30), named as variance effec, in he objecive funcion for opimally allocaing he insurance porfolio wih annuiy policies and life insurance policies. Besides, we consider differen pricing basis beween period-moraliy basis and cohor-moraliy basis. The moraliy rae wihou he improvemen effec is called period moraliy rae which los of insurance companies apply o price insurance policies. The moraliy rae wih improvemen effec, called cohor moraliy rae, consiss wih he endency of fuure moraliy rae. Consequenly, he difference of porfolio value evaluaed beween hese wo bases can be regarded as a miss-pricing error. The pricing difference of his porfolio is of he form: D N V m r V m r sg, sg, sg, period ( ( ),, ) cohor ( ( ),, ), (31) sa, L g1,2 i where V m r denoes he policy value wih period-moraliy basis whils, ( sg period ( ),, ) V m r denoes he policy value wih cohor-moraliy basis. As a resul,, ( sg cohor ( ),, ) we also incorporae he pricing difference, named as miss-pricing effec, in he objecive funcion. 11

12 The purpose of our paper is o find a feasible policy collocaion in he ris profile of insurance companies o minimize he variance effec wih a weigh (1-θ) and he miss-pricing effec wih a weigh θ. If he insurance companies pu more emphasis on he variance effec, hey end o conrol he change in he insurance porfolio due o he unepeced shocs in moraliy rae and ineres rae. If hey pu more emphasis on he miss-pricing effec, hey aim o minimize he miss-pricing error by increasing he weigh θ. Our objecive funcion can be epressed as follows: 5 sg, 2 2 B = - q sg, j + q N, NB j=1 f ( N, N ) min (1 ) å ( Q ) D. (32) 3.Moraliy Daa The moraliy daa were colleced via Taiwan Insurance Insiue (TII) and include more han 50,000,000 policies of life insurance companies in Taiwan. The original daa are caegorized by ages, gender and sors. We use he original daa o consruc four Lee-Carer moraliy ables, including female annuiy (fa), male annuiy (ma), female life insurance (fl) and male life insurance (ml). The maimal age of he original daa is abou 85. The parameers of LC model are calibraed by using approimaion mehod and are dipiced in Figure 1. [Inser Figure 1 here] In order o capure longeviy ris, we uilize erapolaion o eend maimal age from 85 o 100. For he purpose of forecasing he fuure moraliy rae, in Table 1 we fa ma fl calculae he sandard deviaion ( ) of,, and ml. [Inser Table 1 here 12

13 We hen use Cholesy decomposiion mehod o ransform he dependen random variables ino a linear combinaion of independen random variables. Firs, we compue he correlaion mari (M) of fa,, and ma fl ml as follows: M (33) Based on he Cholesy decomposiion, he lower riangular mari (R), which saisfies M T R R, can be epressed as follows: R. (34) Therefore, based on he lower riangular mari (R), Equaion (14) can be rewrien as follows: dz () dz () A,2 1 A,1 () dz 2() dz L,2 dz () dz 3() L,1 dz () dz 4(). (35) 4.Numerical Analysis 4.1Scenario 1 : θ = 0 (variance effec) We firs assume ha he weigh θ is zero, and hen we can observe he effec caused by he variance effec only. Whih loss of generiy, in he squeal we assume ha a=0.1663, b= and r =4.733% for he parameers of CIR model. We begin 13

14 wih a simple case, in which here are only wo policies in he porfolio: an annuiy and a life insurance policy. We can capure he corresponding hedging relaion from hose simple cases. Given one uni value of annuiy due, we can find he opimal value (uni) of life insurance. Assuming ha (he unepeced change rae of fuure moraliy rae and ineres rae) are 5%, 10%, and 15%, Table 2 represens he opimal unis of life insurance o hedge one uni value of annuiy due for differen unepeced change raes. We see from Table 2 ha he unepeced change rae of moraliy and ineres rae doesn have obvious impac on he opimal unis held in life insurance. The reason is ha he effecive duraions and conveiies of each produc are almos he same. In Table 3 we ae he life insurance policy of female wih age 30 (fl30) as an eample. The variance increases only slighly wih larger unepeced change rae because of he slighly change of duraion and conveiy. We use 10% unepeced change rae of boh moraliy and ineres rae for he following scenario analyses. [Inser Table 2 here] [Inser Table 3 here] We calculae he corresponding opimal unis of differen life insurances o hedge one uni of female annuiy due. The resuls are shown in Table 4. From Table 4, we find ha we are no able o reduce he oal variance by holding female life insurances o hedge annuiy policy, bu we can mae i hrough male life insurances. Thus, Table 4 shows ha he variance of he porfolio of female annuiy and male life insurance are smaller han ha of he porfolio of female annuiy and female life insurance. [Inser Table 4 here] 14

15 We furher invesigae hold one uni value of male annuiy, and he following he opimal unis of life insurance wih differen genders and ages. Table 5 shows he corresponding opimal unis of differen life insurances o hedge one uni of male annuiy due. We find ha resuls are similar o hose observed from Table 4. In addiion, comparing he righ-hand side of Table 4 wih ha of Table 5, we observe ha he opimal unis of male life insurance o hedge one uni of male annuiy60 (aged 60, male annuiy) is larger han hose of female annuiy60. [Inser Table 5 here] We epec o hedge longeviy ris of annuiy produc by holding some unis of life insurances because we now ha he sign of effecive duraion of annuiy and life insurance are opposie. We can obain he hedging effec from holding life insurance because he coefficien of Cholesy decomposiion of female life insurance is negaive. However, he coefficien of male life insurance is posiive, so i is possible o minimize he variance effec by holding some male life insurance policies. The reason ha we mus hold more male life insurance o hedge male annuiy60 han female annuiy60 is ha he magniude of male annuiy s duraion and moraliy rae are larger han hose of female, as shown in Table 6. Consequenly, we mus hold more unis of male life insurances o hedge 1 uni of male annuiy60. [Inser Table 6 here] We furher wan o compare he differences of opimal hedging sraegies beween holding one uni of annuiy due and one uni of deferred annuiy. The comparison resuls are as presened in Tables 7 and 8. From Table 7-1 & 7-2 and Table 8-1 & 8-2, for deferred annuiies, we need o hold less life insurance policy o hedge he moraliy uncerainy. In addiion, we observe ha, wih he increase of insured s ages, 15

16 we need less life insurance unis o hedge he corresponding annuiy. Alhough he duraion of younger life insurances is larger han ha of elder, he elder s moraliy rae is much higher han he younger s moraliy rae. Therefore, i is more possible for he insurance company o suffer an insan claim by he elder insured. So he insurance company needs less unis of he elder life insurance o offse he longeviy ris of annuiies. [Inser Table 7 here] [Inser Table 8 here] 4.2Scenario 2: θ=1 (miss-pricing effec ) In his secion we ignore he variance effec and discuss he miss-pricing effec. Wihou consideraion of he variance effec in he objecive funcion, we can find a perfec-hedge uni of life insurance produc o hedge one uni of annuiy, and vice versa. Therefore, he value of objecive funcion is always zero in he opimal siuaion. In he following analysis, we jus show he opimal unis of life insurance polices, corresponding o fiedly holding one uni of annuiy policy. In Figure 2, we depic he levels of under-pricing for annuiy policies and over-pricing for life insurance policies under differen ages. We find ha, for annuiy producs, under-pricing problem of male is more serious han ha of female. For life insurance producs, under-pricing problem of female is more serious han ha of male. In addiion, we observe ha he magniude of miss-pricing problem decreases as he issued age increases. [Inser Figure 2 here] 16

17 According o Table 9 and Table 10, unlie he resul in Scenario 1, we are able o hedge longeviy ris of annuiy producs hrough female life insurance producs in Scenario 2. However, wih he issued age increases, he holding endency goes up. This oucome is oally differen from he resuls in Scenario 1. This implies ha insurance companies need o ae very differen hedging sraegies o reduce variance effec and miss-pricing effec respecively. [Inser Table 9 here] [Inser Table 10 here] 4.3Scenario 3: 0 < θ <1 In pracice, insurance companies should ae accoun of boh variance and miss-pricing effecs simulaneously o hedge longeviy ris. This means he value of θ should beween 0 and 1 bu no equal o 0 or 1. In his secion, we discuss opimal hedging sraegies by varying he level of θ. We firs consider holding one uni of annuiy due and find he opimal unis of life insurance. The resuls of opimal unis of female life insurance are epressed in Table 11. [Inser Table 11 here] From Table 11, we observe ha he opimal unis of female life insurance decrease as he weigh θ decreases. Tha is, when we pu less emphasis on miss-pricing effec, we end o minimize he variaion of he change of oal porfolio value. As a resul, he less uni of life insurance we hold, he smaller he objecive funcion is. According o Table 11, he paerns of opimal unis of female life insurance o hedge one uni of annuiy60 for boh male and female under differen value of θ are similar, bu he magniude of ma60 is larger han ha of female annuiy60, which is consisen wih as he resuls in Scenarios 1 & 2 menioned above. In addiion, we find ha he female life insurances are unable o reduce he variance of oal porfolio value in Scenario 1, 17

18 which eplains he reason why we need less unis of life insurance as he weigh θ decreases. In Table 11, insurance companies shall hold smalles opimal unis as θ = 0 and hold he larges opimal unis as θ =1. When θ is equal o 0.1, 0.01 and respecively, he paerns of opimal unis are similar o he case wih θ =1. In he case of θ = 0.001, we find he paern of opimal unis is similar o he case of θ =1 in younger age, bu in older ages he paern of opimal unis will decrease lie he case of θ = 0. In Scenario 1, we observe ha he variance effec is sronger in older age because he variance effec is posiively relaed o he moraliy rae and effecive duraion. So we will observe he ineracion of boh variance and difference effec in he case of θ = in he following analysis. In Table 12, we invesigae opimal unis of male life insurance o hedge one uni of annuiy60 for boh male and female under differen value of θ, and he resuls are shown as follows. Compared he resuls in Table 12 wih hose in Table 11, he opimal unis of male life insurance do no always decrease as he weigh θ decreases. In Scenario 1, we can uilize male life insurances o reduce oal variance erm, however, we also ae miss-pricing effec ino consideraion a he same ime. Therefore, he oucomes in Table 11 & 12 are he ineracion beween hese wo effecs. [Inser Table 12 here] According o he resuls of Table 11 and 12, we choose he weigh θ=0.001 as an eample because here eiss an ineracion beween wo effecs in average. The resuls of opimal unis of male and female life insurance o hedge one uni of annuiy for differen ages and genders are epressed in Figure 3. Figure 3 shows ha he paerns of opimal unis of male and female life insurance o hedge one uni of 18

19 annuiy are similar for differen ages and genders. We see from Figure 3 ha he rend of opimal unis among differen ages and genders are similar. The main differences among hem are he magniude of opimal unis. In general, opimal unis of male life insurance o hedge one uni of annuiy is slighly larger han hose of female life insurance. Opimal unis of male or female life insurance o hedge a younger age of annuiy are larger han hose o hedge an older age of annuiy. In addiion, opimal unis of male or female life insurance o hedge one uni of male annuiy are larger han hose o hedge a female annuiy. [Inser Figure 3 here] 5.Conclusion and Suggesion In his paper, we propose a naural hedging model, which can ae accoun of wo imporan effecs of longeviy ris a he same ime. The firs one is he variance of he change of oal porfolio value, and he second one is he miss-pricing effec. We can hedge variaions of he fuure moraliy rae by he firs effec, and hedge he presen miss-pricing by he second one. Previous researches on naural hedging only ae accoun of variance effec, bu we incorporae boh variance and miss-pricing effecs ino our model in his paper. In pracice, los of insurance companies indeed have a miss-pricing problem due o moraliy improvemen over ime. Therefore, we conribue o he eising lieraure by showing ha i is no suiable o ignore miss-pricing effec o hedge longeviy ris. Unlie he previous lieraures, we use he eperienced moraliy raes from life insurance companies insead of using populaion moraliy raes. These differences mae our model more general and easier o apply ino pracice. 19

20 Using he eperienced moraliy raes, we separae he moraliy rae from gender and he ype of insurance producs, life insurance policies and annuiy policies. We use he correlaions beween hese four ypes of moraliy raes o hedge he variaions of he fuure moraliy rae. Furhermore, we use Lee-Carer model o forecas he fuure moraliy rae and calculae he level of miss-pricing in pracice. Then, insurance companies can decide he relaive significance of he variance and miss-pricing effecs by a weigh θ. So we can effecively apply he naural hedging sraegy o a more general porfolio of life insurance companies. When we consider he variance effec only, he opimal unis of life insurance is affeced by he effecive duraion and moraliy rae mosly. As a resul, he opimal unis of life insurance decrease as he age of life insurer increases. However, when we consider he miss-pricing effec only, he opimal unis are oally decided by he period-cohor difference of each produc. The opimal collocaion sraegy decided by considering he variance effec comes o an opposie conclusion by considering he variance effec case. In his paper, we obain opimal collocaion soluions for boh effecs o hedge longeviy ris. Furhermore, he model proposed in his paper is easy o eend o a general porfolio wih realisic combinaions of annuiy and life insurance policies. 20

21 6.References Blae, D., and Burrows, W. (2001), Survivor Bonds: Helping o Hedge Moraliy Ris, The Journal of Ris and Insurance 68: Brouhns, N., Denui, M., and Vermun, J. K. (2002), A Poisson Log-Bilinear Formaed: Paern: Clear (Whie) Regression Approach o he Consrucion of Projeced Life-Tables, Mahemaics and Economics, 31: Cairns, A. J. G., Blae, D., and Dowd, K. (2006), A Two-Facor Model for Sochasic Moraliy wih Parameer Uncerainy: Theory and Calibraion, The Journal of Ris and Insurance, 73: Co, J. C., Ingersoll, Jr., J. E., Ross, S. A. (1985), A Theory of he Term Srucure of Ineres Raes, Economerica, 53: Co, S. H., and Lin, Y. (2007), Naural Hedging of Life and Annuiy Moraliy Riss, Norh American Acuarial Journa, 11(3): Dowd, K., Blae, D., Cairns, A. J. G., and Dawson, P. (2006), Survivor Swaps, The Journal of Ris & Insurance, 73: Koissi, M. C., Shapiro, A. F., and Högnäs, G. (2006), Evaluaing and Eending he Formaed: Paern: Clear (Whie) Lee Carer Model for Moraliy Forecasing: Boosrap Confidence Inerval, Mahemaics and Economics, 38: Lee, R. D., and Carer, L. R. (1992), Modeling and Forecasing U.S. Moraliy, Journal of he American Saisical Associaion, 87(419): Melniov, A., and Romaniu, Y. (2006), Evaluaing he Performance of Gomperz, Formaed: Paern: Clear (Whie) Maeham and Lee Carer Moraliy Models for Ris Managemen wih Uni-Lined Conracs, Mahemaics and Economics, 39:

22 Renshaw, A. E., and Haberman, S. (2003) Lee-Carer Moraliy Forecasing wih Age Specific Enhancemen, Mahemaics and Economics, 33: Wang, J. L., Huang, H.C., Yang, S. S., and Tsai, J. T. (2010), An Opimal Produc Mi For Hedging Longeviy Ris in Life Insurance Companies: The Immunizaion Theory Approach, The Journal of Ris and Insurance, 77: Wang, J. L., Yang, L. Y., and Pan, Y. C. (2003), Hedging Longeviy Ris in Life Insurance Companies, In Asia-Pacific Ris and Insurance Associaion, 2003 Annual Meeing. Wilmoh, J. R. (1993), Compuaional Mehods for Fiing and Erapolaing he Lee-Carer Mehod of Moraliy Change, Technical Repor, Deparmen of Demography, Universiy of California, Bereley. 22

23 TABLE 1 Sandard deviaion of in four differen moraliy groups fa ma fl ml Sd TABLE 2 Opimal unis of life insurances wih differen (θ = 0) 1 uni fa60 1 uni ma60 fl30 ml30 fl30 ml30 N N =5% f(n) =5% f(n) N N =10% f(n) =10% f(n) N N =15% f(n) =15% f(n) TABLE 3 Effecive duraion (dur) and conveiy (conv) wih differen (5%, 10%, 15%) fl30 =5% =10% =15% dur(m) dur(r) conv(mm) conv(rr) 2, , , conv(mr)

24 TABLE 4 Opimal unis of life insurances, given 1 uni of female annuiy (θ = 0) 1 uni fa60 1 uni fa60 N f(n) N f(n) fl ml fl ml fl ml fl ml TABLE 5 Opimal unis of life insurances, given 1 uni of male annuiy (θ = 0) 1 uni ma60 1 uni ma60 N f(n) N f(n) fl ml fl ml fl ml fl ml TABLE 6 Effecive duraion and moraliy rae of each produc (insured) effecive duraion fa ma fl ml moraliy rae fa ma fl ml

25 TABLE 7-1 female annuiy due TABLE 7-2 female deferred annuiy Comparison of deferred effec of female annuiy (θ = 0) 1 uni fa60 1 uni fa30 N f(n) N f(n) ml ml ml ml ml ml ml ml TABLE 8-1 male annuiy due TABLE 8-2 male deferred annuiy Comparison of deferred effec of male annuiy (θ = 0) 1 uni ma60 1 uni ma30 N f(n) N f(n) ml ml ml ml ml ml ml ml TABLE 9 Opimal unis of life insurances, given 1 uni of annuiy (θ = 1) 1 uni fa60 1 uni ma60 fl N ml N fl N ml N fl ml fl ml fl ml fl ml fl ml fl ml fl ml fl ml TABLE 10 Comparison of gender effec of annuiy (θ = 1) 1 uni fa30 1 uni ma30 fl N ml N fl N ml N fl ml fl ml fl ml fl ml

26 fl ml fl ml fl ml fl ml TABLE 11- Opimal unis of female life insurance o hedge one uni of annuiy60 for boh male and female under differen value of θ 1 uni fa60 1 uni ma60 θ fl30 fl40 fl50 fl60 θ fl30 fl40 fl50 fl TABLE 12 Opimal unis of male life insurance o hedge one uni of annuiy60 for boh male and female under differen value of θ 1 uni fa60 1 uni ma60 θ ml30 ml40 ml50 ml60 θ ml30 ml40 ml50 ml

27 FIGURE 1 Parameer Esimaes of,, in LC model Panel A. Parameer Esimaes of Panel B. Parameer Esimaes of 27

28 Panel C. Parameer Esimaes of FIGURE 2 Pricing differences of each produc beween period and cohor bases FIGURE 3 Opimal unis of male and female life insurance o hedge one uni of annuiy for differen ages and genders 28

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