Multi-Period Optimization Model for ånancial planning

Transcription

1 Muli-Period Opimizaion Model for ånancial planning Norio Hibiki É July 2, 28 1 Inroducion We discuss an opimizaion model o obain an opimal invesmen and insurance sraegy for a household. A household is exposed o risk associaed wih he decrease in real ånancial wealh due o inçaion, loss of wage income due o he householder's deah, loss of a house or non-ånancial wealh due o he åre, and he increase in medical cos due o a serious disease. Financial planners develop long-erm invesmen and insurance sraegies in consideraion of household's family srucure, income, consumpion, asse, liabiliy, he fuure hope, goal of home buying, children's educaion and pos-reiremen plan, and help he household implemen he sraegies. We develop a muli-period opimizaion model which involves deermining a se of ånancial producs and hedging risk agains he above-menioned accidens. The simulaed pah approach [4, 5] can be used o solve his problem. Campbell [2] describes he imporance of household ånance. There are some sudies in he lieraure for individual opimal invesmen sraegy; Bodie, Meron and Samuelson [1], Meron [11], Samuelson [16]. Chen, Ibboson, Milevsky and Zhu [3] advocae an opimizaion model wih he inclusion of wage income, consumpion expendiure, and boh opimal asse allocaion and life insurance. Yoshida, Yamada and Hibiki [18] solve an opimal asse allocaion problem for a household using a muli-period opimizaion approach. Hibiki, Komoribayashi and Toyoda [9] describe a muli-period opimizaion model o deermine an opimal se of asse mix, life insurance and åre insurance. The model is examined wih numerical examples, and some ånancial advices for hree households are illusraed for pracical use. We obain he resuls which coincide wih pracical feeling. Hibiki and Komoribayashi [8] propose he model involving he associaed hree facors wih he householder's deah for pracical use: receip of a survivor's pension, exempion from morgage loan paymens, and change in he consumpion level. Moreover, he sensiiviy of parameers associaed wih home buying is analyzed in order o examine he home buying sraegy. Hibiki [7] reånes cash çow sreams associaed wih severance pay, fuure cos for children, medical cos and insurance. A model is formulaed wih decreasing erm life insurance, which is compared wih erm life insurance by using numerical examples. A model is proposed o decide ime-dependen opimal life and medical insurance money during he period for insurance design. Sampling error is examined wih 1 kinds of 5, sample pahs. É Deparmen of Adminisraion Engineering, Keio Universiy, hibiki@ae.keio.ac.jp 1

2 We obain he following pracical and ineresing resuls. Hibiki, Komoribayashi and Toyoda [9] (1) The older a householder is, he less opimal amouns of life insurance money is. (2) Opimal amouns of åre insurance money is nearly equal o he maximum loss of nonånancial wealh. (3) Expeced erminal ånancial wealh does no aãec opimal amouns of life and åre insurance money. Hibiki and Komoribayashi [8] (4) If a household receives a survivor's pension and keeps he consumpion level lower afer a householder died, opimal amouns of life insurance money and invesmen unis of a risky asse decrease. (5) If loan paymens are forgiven due o he householder's deah, opimal invesmen unis of a risky asse decrease, bu opimal amouns of life and åre insurance money are no inçuenced. (6) Home buying sraegy aãecs opimal asse mix and opimal amouns of life insurance money. Hibiki [7] (7) The opimal unis of medical insurance becomes larger wih he increase in a disease rae, medical cos and a decreasing rae of wage income. Risk also becomes worse. (8) The opimal amoun of medical insurance money is almos equal o he decrease in ånancial wealh due o a serious disease, or he sum of medical cos and he decrease in wage income. (9) Premium of decreasing erm life insurance reduces more drasically han premium of erm life insurance. Moreover, a household can receive a larger amoun of life insurance money and obain large erminal wealh even if a householder dies earlier. (1) We can obain he resul ha he condiional expeced erminal ånancial wealh do no depend on he ime of he householder's deah by using he model which decides imedependen opimal life insurance money during he period. (11) Sampling error occurs because a hiry-periods model is solved wih 5, pahs, bu he average value and he values beween 25 percenile and 75 percenile clearly show he feaure of he opimal soluions. In his paper, we inroduce he following poins based on our previous sudies so ha ånancial planners can use he model in real life. 1ç We include a survivor's pension, exempion from morgage loan, and severance pay in our previous models. Moreover, we reåne he cash çow sreams for a household in a real world. We include paymen and deducion of naional and local income axes and social securiy premium, paymen of åxed asse ax, morgage loan ax credi, premium of group credi life insurance, axes imposed on a house, and so on. 2ç We obain he numerical resuls of opimal insurance sraegy wihou loading of premium in our previous sudies. We examine he eãecs for various loading of premium in his paper. We esimae loading of premium for various life insurance by using life insurance sandard life able 27, and refer o he esimaes for he numerical analysis. 2

3 3ç Hibiki [7] proposes he model o design opimal insurance for a household, however, he implemenaion was no shown. The model is formulaed wih decision variables for surrendering a life insurance policy a he ime when he house is bough in addiion o purchasing i a ime. The reason is ha he opimal design shows ha he amoun of insurance money should be reduced when he house is bough and purchasing group credi life insurance is required of he household. We hedge income risk by using combinaion of hree kinds of erm life insurance policies; erm life insurance, decreasing erm life insurance, and family income insurance. We also compare he resuls wih he cases ha one of life insurance is used. 4ç Decreasing erm life insurance and family income insurance are similar life insurance producs. Decreasing erm life insurance is axable inheriance, bu mos households do no have o pay inheriance ax, and can receive he full amoun of insurance money. On he oher hand, insurance proceeds of family income insurance are reaed as oher income, and we have o pay income ax in Japanese ax sysem. The amoun of ax is an increasing piecewise linear funcion of income, and we can formulae he model as a linear programming model. We compare he resuls for hem in aking ino accoun he ax diãerence. 5ç We use hedge raios agains loss as decision variables for åre and medical insurance o avoid sampling error. 6ç Chen e al. [3] examine he eãecs of he correlaion beween he volailiy of wage and rae of reurn of a risky asse in he one-period model. We also examine he sensiiviy in he muli-period model. 7ç We solve he problem in he simulaed pah model wih åxed invesmen raio sraegy, insead of åxed invesmen uni sraegy used in our previous sudies [7, 8, 9]. 8ç The simple mehod of invesing `1 - age'(%) of ånancial wealh in a risky asse is wellknown. In addiion, we need o solve he problem wih consan rebalance sraegy in one period model such as mean-variance model. We show he model including only insurance sraegy under he given asse mix, and compare i wih he base model. This paper is organized as follows. We deåne a household, and describe income, consumpion expendiure, and four kinds of ånancial producs, or securiies, life insurance, åre insurance, and medical insurance in Secion 2. Secion 3 shows he cash çow and formulaion of a muliperiod opimizaion model for a household. We se he parameers and examine he model wih numerical examples in Secion 4. Secion 5 provides our concluding remarks. 2 Model Srucure We assume ha he curren ime is ( = ), and a householder reires a ime T,whichisa planning horizon. We deåne a household, and describe income and consumpion expendiure. We clarify he characerisics of ånancial insrumens such as asses, life insurance, åre insurance, and medical insurance. We aach a superscrip (i) o random and pah dependen parameers in order o formulae a model in he simulaed pah approach. 3

4 2.1 Household We deåne a household as a group composed of a householder and oher members of family such as spouse and children as in he previous papers [7, 8, 9]. I is assumed ha members of family excep a householder do no die and have serious diseases. Wealh a ime held by a household can be divided ino wo kinds of wealh: ånancial wealh W (i) (i) 1; and non-ånancial wealh W 2;. A household is exposed o risk associaed wih hree kinds of accidens: a deah and a serious disease of a householder, and a åre of a house. Cash çow sreams are inçuenced by risk exposure associaed wih income and expendiure. We se he following parameers associaed wih he accidens o describe cash çow sreams. 1 ú (i) D; : one if a householder is dead on pah i a ime and zero oherwise. ú (i) L; : one if a householder dies on pah i a ime and zero oherwise. ú (i) F; : one if a åre of a house occurs on pah i a ime and zero oherwise. ú (i) M; : one if a householder has a serious disease on pah i a ime and zero oherwise. ï L; : moraliy rae a ime, or he probabiliy ha a person who is alive a ime will die a ime, ï L; = 1 IX ú (i) L; I. i=1 ï F : rae of a åre (which is assumed o be ime independen), or he probabiliy ha a åre occurs, ï F = 1 IX ú (i) F; I. i=1 ï M; : disease rae a ime, or he probabiliy ha a person who is alive a ime will have a serious disease a ime, ï M; = 1 IX ú (i) M; I. 2.2 Income i=1 If a householder is alive and has a job, i can ge wages. Wage is divided ino monhly salary and bonus, i is assumed ha monhly salary is deerminisic, bu bonus may vary according o he corporaion's performance. Wage ~! m + a ime of he householder who is m year old a ime is calculaed as 2 ~! m + = 12! 1;m + +! 2;m +(1 + ~ñ W; )( =1;...;T); (1) ~ñ W; = ñ W; + õ W; ~" W; ; ~" W; ò N(; 1); (2) ~ñ S; = ñ S; + õ S; ~" S; ; ~" S; ò N(; 1); (3) cov(~" W; ; ~" S; )=c W;S ; (4) where! 1;m + is monhly saraly,! 2;m + is he average bonus, ~ñ W; is he rae of change of bonus which is normally disribued wih he mean ñ W; and he sandard deviaion õ W,and ~ñ S; is he rae of reurn of sock which is normally disribued wih he mean ñ S; and he sandard deviaion õ S. I is assumed ha he sock index and he corporaion's performance are correlaed, and he correlaion coeécien is c W;S. 1 If ú (i) L; =1,henú(i) D;k =1for î k î T.Ifú(i) D; =1,henú(i) M; =. 2 If he calculaed wages are adjused by he inçaion rae f, hen hey are ~! m +Ä 1(1 + f) Ä 1 for =1;...;T. 4

5 We calculae income and inhabian axes wih axable income which is he amoun afer deducion for employmen income, social insurance premiums, life insurance premiums, and reiremen income which a household receives a reiremen ime, basic exempion, and exempion for spouses and dependens. If a household ges a housing loan, special credi for loans relaing o a dwelling is deduced from income ax. We calculae disposable income w (i) by deducing income and inhabian axes, social insurance premiums, and he decrease in wage due o a serious disease 3 from wage income and reiremen income. If a householder dies, a household canno ge wages, bu receive severance pay, and draw a survivors' pension(basic and employees' pension). Amouns of severance pay and a survivors' employees' pension are calculaed based on he wage level. Le a (i) m be he amoun of a survivor's pension. The amoun of a survivors' pension is dependen on he ime of he householder's deah m. The amoun of severance pay e (i) is also dependen on years of coninuous employmen(age). The sum of disposable income, a survivors' pension and severance pay a he ime of deah or E (i) can be shown as E (i) ê = 1 Ä ú (i) ë D; w (i) + ú (i) D; a(i) m + ú (i) L; e(i) ( =1;...;T): (5) 2.3 Consumpion expendiure There assumes o be wo kinds of expenses: living expenses C (i) 1; and paymens associaed wih non-ånancial wealh C (i) 2;, such as a house, goods, and repair coss. Besides hese coss, we need o pay he resoraion cos if a åre of a house occurs Living expenses Living expenses consis of four iems; 1ç expenses for purchasing a house, 2ç expenses for children such as educaion and living coss, 3ç medical cos, 4ç oher living expenses. They are calculaed as he sum of four iems C k(i) 1; (k =1; 2; 3; 4) or 4X C (i) 1; = k=1 C k(i) 1; ( =1;...;T): (6) (1) Expenses for purchasing a house Making decision wih respec o he house aãecs a household ånance. The household resides in a rened house a ime, and we assume he following hree plans a ime e in he fuure. 1ç A household purchases a house a ime e. 2ç A household purchases a house if a householder is alive a ime e. Tha is, a household does no purchase a house and i keeps residing in a rened house if a householder is no alive a ime e. 3ç A household keeps residing in a rened house. We deåne he following parameers as he household's hope in he fuure. : one if a household plans on buying a house a ime e and zero oherwise. ü 1 3 If a householder has a serious disease, i is assumed ha a fracion ó 3(ó 3 < 1) of wage income decreases because a householder akes a res from work. 5

6 ü 2 : one if a household does no plan on buying a house in he case ha a household is no alive a ime e for ü 1 = 1 and zero oherwise. We assign ü 1 = 1 and ü 2 =ocase 1ç, ü 1 = 1 and ü 2 = 1 o case 2ç,andü 1 = o case 3ç. We assume ha a household purchases a house by making a down paymen H and a deb loan a a bank H 1. The deb loan H 1 is he diãerence beween he price of he house and he down paymen. A household needs o pay real propery acquisiion ax, regisraion license ax, samp ax and consumpion ax when i purchases a house(a ime e ), åxed propery ax and ciy planning ax when i keeps a house(afer ime e + 1). Ne cash ouçow of purchasing he house a ime e is no he price of he house, bu he down paymen. The household pays he deb loan periodically under he deermined morgage ineres rae and he loan period afer ime e + 1. If a household purchases a group credi insurance policy, he loan paymens are forgiven afer a householder died. Expenses for purchasing a house C 1(i) 1; is calculaed as C 1(i) 1; = C 11(i) 1; + 1 fïe gc 12(i) 1; ( =1;...;T); (7) C 11(i) ê 1; = n1 Ä 1 fïe+1gü 1 1 Ä ü 2 ú (i) ëo D; e c 11 ; (8) C 12(i) 1; = ü 1 ê 1 Ä ü 2 ú (i) D;ë n 1f=egH + 1 fïe+1g ê 1+ú (i) D; e Ä ú (i) D; ë c 12(i) o ; (9) where c 11 is ren a ime, c 12(i) is oal paymen of morgage loan and axes associaed wih he house a ime, and1 fag is an indicaor funcion which shows one if he condiion A is saisåed, and zero oherwise. (2) Expenses for children Expenses for children such as educaion and living coss C 2(i) 1; are calculaed as C 2(i) 1; = N b X n=1 c 2n(i) N b X+N a + n=n b +1 ê 1 Ä ú (i) ë D; c;n Ä1 c 2n(i) ( =1;...;T); (1) where c 2n(i) is cos for he nh child, c;n isheimewhenhenh child is born, 4 N b is he number of children a ime, and N a is he number of children who will be born afer ime 1. (3) Medical cos I is assumed ha a household pays C F if a householder has a serious disease such as cancer, cardiac infarcion, apoplexy. Therefore, paymen C 3(i) 1; is shown as C 3(i) 1; = ú (i) M; C F ( =1;...;T): (11) (4) Oher living expenses Le c 41(i) be oher living expenses. I is assumed ha a household can keep an usual consumpion level c 41(i) if a householder is alive, however a consumpion level mus be îimes he level c 41(i) if a householder is dead, where îis a parameer associaed wih a consumpion level. For example, we se î= 1 when a household keeps an usual level, and we se î= :7 wheni has o allow for he 7% consumpion level. If cash amouns afer deducing all of he expenses from disposable income are posiive, a household can use a par of he posiive cash amouns in addiion. Therefore, oher living coss become c 42(i) C 4(i) 1; = n 1 Ä ú (i) o D; (1 Ä î) c 41(i) + c 42(i) ( =1;...;T): (12) 4 We assume ha he child will be born if a householder is alive a ime c;n Ä 1. 6

7 2.3.2 Expenses for non-ånancial wealh (1) Expenses for non-ånancial asse and non-ånancial wealh We assume ha expenses for non-ånancial asse C (i) 2; are esimaed from he oher living coss. Moreover, he paymen of a house price is included in he expenses a ime e if he house is bough. Non-ånancial asse is added o non-ånancial wealh, which is calculaed as W (i) 2;1 =(1Äç 1)W 2; + C (i) 2;1 ; W(i) 2; =(1Äç 1)W (i) 2;Ä1 + C(i) 2; ( =2;...;T); (13) where W 2; is an iniial value of non-ånancial wealh, and ç 1 is a depreciaion rae of non-ånancial wealh. (2) Resoraion cos due o a åre I is assumed ha a fracion ç of non-ånancial wealh W (i) 2;Ä1 is damaged and he resoraion cos A (i) is paid if a åre of a house occurs. Explicily, A (i) 1 = ú (i) F;1 ç (1 Ä ç 1 )W 2; ; A (i) = ú (i) F; ç (1 Ä ç 1 )W (i) 2;Ä1 ( =2;...;T): (14) Non-ånancial wealh decreases by A (i) due o a åre, bu he same money is spen o recover he loss, and non-ånancial wealh increases by A (i). Therefore, A (i) does no aãec non-ånancial wealh, bu cash çow sreams. 2.4 Invesmen asses We inves in n risky asses and cash. A risk-free rae r a ime (= ; 1;...;T Ä 1) is åxed in he period from ime o +1. Le ö j denoe a price of risky asse j a ime. A rae of reurn of a risky asse j a ime is calculaed using a price ö j as ñ j = ö j Ä 1(j =1;...;J; =1;...;T): (15) ö j;ä1 We can assume any probabiliy disribuions of ñ j and r in he simulaed pah approach. We generae sample rae of reurns using Mone Carlo simulaion echnique, and calculae sample prices from sample rae of reurns sequenially. 2.5 Life insurance If a householder purchases a erm life insurance policy wih mauriy T and dies by ime T,a household can receive insurance money. In his model, we look upon life insurance as a ånancial produc which can hedge risk associaed wih wage income earned by a householder. Hibiki[7] develops he model in which he ime-dependen opimal amouns of insurance money are derived. However, i is diécul for a householder o purchase he life insurance policy because life insurance companies need o design i so ha he household can receive life insurance money o mee he cash çow. Hibiki[7] shows ha he ime-dependen opimal amouns of life insurance money decrease hrough ime, and drop a he ime of home buying due o exempion from morgage loan paymens in he case ha a householder dies afer purchasing a house wih group credi life insurance. Life insurance companies need o design he policy conrac which includes he condiion ha life insurance money can be decreased a he ime of home buying. On he oher hand, i is diécul for a household o decide he ime of home buying a ime. However, we develop he model in consideraion of surrendering insurance policies so ha 7

8 a household can derive he life insurance porfolio o mee cash çow sreams for a household. We use hree kinds of life insurance wihou cancellaion reurn: erm life insurance(b = ), decreasing erm life insurance(b = 1), family income insurance(b =2). 5 We expec o derive he opimal soluion so ha ånancial planners can give a pracical advice. The model allows he household o surrender he policies a he ime of home buying because he householder migh ake advanage of surrendering he life insurance conrac when he group credi life insurance is bough. 6 We calculae life insurance premiums and money wih moraliy raes in life insurance sandard life able 27 [17] for men. 7 Le d m denoe a moraliy rae of m years old person. I is he rae a which m years old person dies wihin one year. When a householder is m years old a ime, he condiional survival rae l m of a householder who is alive a ime is l m =(1Äd m +) l m Ä1 where lm =1: Therefore, he condiional moraliy rae of a householder who is alive a ime is calculaed as ï L; = l m Ä1 Ä lm = d m +l m Ä1 : Premiums A householder purchases life insurance wih mauriy T a ime, and makes level paymen. I has o pay premiums from ime o ime T Ä 1. Because only insured person who is alive pays premiums, a level premium of life insurance b per uni y L;b is calculaed as ( T X Ä1 l m ) Ä1 y L;b = (1 + g = L ) (1 + ê L;b) (b 2 B); (16) where g L is a guaraneed ineres rae of life insurance, and ê L;b is he raio of loading of premium o a ne premium. A premium is 1 + ê L;b o an uni of expeced amoun of insurance Insurance money Using he principle of equalizaion of income and expendiure, hree kinds of life insurance money are calculaed for he corresponding presen value of premiums. (1) erm life insurance and decreasing erm life insurance We explain how o compue ime-dependen life insurance money in he period. Le í L;b; be ime-dependen amouns of insurance money of life insurance b a ime per uni of presen valueofpremium.wehave 5 We exclude a whole life insurance policy which is one of he major life insurance because we examine he inçuence of life insurance in he planning period. 6 The model includes he decision variables which are he numbers of unis of life insurance wih mauriy T purchased a ime and he numbers of unis of life insurance surrendered a ime e orheimeofhomebuying. The reader migh hink ha he model should include he decision variables which are he numbers of unis of life insurance wih mauriy e purchased a ime and he numbers of unis of life insurance wih mauriy T Ä e purchased a ime e because he opimizaion problem is solved under he condiion ha he ime of home buying is deerminisic. However, he household is exposed o mauriy risk if i purchases a life insurance policy wih mauriy e because he ime of home buying is no deermined in pracice. Therefore, we formulae he model wih he decision variables which are he number of unis of life insurance surrendered a ime e o avoid mauriy risk. 7 Life insurance sandard life able 1996 for men is used in our previous papers [7, 8, 9]. 8

9 ( T ) Ä1 X ë L;b;k ï L;k í L;b; = ë L;b; (1 + g k=1 L ) k ; (17) where ë L;b; is a proporionaliy coeécien o insurance money over ime. We se ë L;; =1for erm life insurance(b = ). We also se ë L;1; = T Ä + 1 for decreasing erm life insurance(b = 1), for example. Cash çows associaed wih life insurance b are Äy L;b a ime, ú (i) L; í L;b; Ä ê 1 Ä ú (i) ë D; y L;b from ime 1 o T Ä 1, and ú (i) L;T í L;b;T a ime T per uni of insurance. (2) family income insurance I is assumed ha we use family income insurance wihou minimum guaraneed period and he period when insurer can receive income afer erm of insurance. The amoun of family income insurance money per uni in he års year í L;2 is calculaed as ( X T 1 Ä l m ) Ä1 í L;2 = =1 (1 + g L ) : (18) Le r g denoe he increasing rae of he amoun. The amoun a ime is í L;2; = í L;2 (1+r g ) Ä Fire insurance A household purchases one year åre insurance o hedge loss of non-ånancial wealh due o a åre. I can updae he insurance conrac every year, and purchase he åre insurance policy corresponding o he fuure non-ånancial wealh. Using he principle of equalizaion of income and expendiure, åre insurance money per uni í F is shown as í F = 1+g F : ï F A premium of åre insurance wih one year mauriy y F is (19) y F =1+ê F ; (2) where ê F is he raio of loading of premium o a ne premium. The numbers of unis of åre insurance every year are decision variables in our previous papers [7, 8, 9], and we show ha opimal amouns of åre insurance money are nearly equal o he maximum loss of non-ånancial wealh wihou loading of premium. Hibiki [7] examines sampling error associaed wih opimal soluions of åre insurance, and shows ha he decision variables are aãeced in some degree. We deåne he hedge raio h F of one year åre insurance agains loss of non-ånancial wealh afer depreciaion ç (1 Ä ç 1 )W 2; and ç (1 Ä ç 1 )W (i) 2; ( =1;...;T Ä 1) as he decision variable o avoid sampling error. ç is he loss raio. Le F; and (i) F; denoe he numbers of unis of åre insurance o hedge loss of non-ånancial wealh afer depreciaion perfecly as F; = ç (1 Ä ç 1 )W 2; ; (i) F; í = ç (1 Ä ç 1 )W (i) 2; ( =1;...;T Ä 1): (21) F í F Cash çows associaed wih åre insurance are Äy F F; h F a ime, ê a ime 1, ú (i) F; í F (i) F;Ä1 Ä y F (i) ë F; ê ú (i) F;1 í F F; Ä y F (i) F;1 h F from ime 2 o T Ä 1, and ú (i) F;T í F (i) F;TÄ1 h F a ime T. ë h F 9

10 2.7 Medical insurance If a householder purchases a erm medical insurance policy wih mauriy T, i can receive medical insurance money each ime i has a serious disease by ime T. In his model, we look upon medical insurance as a ånancial produc which can hedge loss associaed wih he expensive medical cos and he decrease in wage income. Because only insured person who is alive pays premiums as well as life insurance, a premium of level paymen per uni y M is calculaed as TX Ä1 l m! Ä1 y M = (1 + g = L ) (1 + ê M ) ; (22) where ê M is he raio of loading of premium o a ne premium. We use he same guaraneed ineres rae g L as life insurance. 8 Hibiki[7] shows he opimal amouns of medical insurance money wihou loading of premium are almos expeced values of he sum of medical cos and he decrease in wage income. Therefore, we use he hedge raio for he presen value of he sum of medical cos and he decrease in wage income insead of he number of unis of medical insurance a ime. 9 Using he principle of equalizaion of income and expendiure, insurance money is calculaed for he corresponding presen value of premium income as ( T ) Ä1 X ï M; í M = (1 + g =1 L ) : (23) The hedge raio h M is decided so ha he presen value of unis of medical insurance money can be equal o he presen value of he sum of medical cos and he decrease in wage income or C M; muliplied by he hedge raio as ( TX í M u T ) M (1 + g =1 L ) = X C M; (1 + g =1 L ) h M : (24) Therefore, we have ( ) g L TX u M = (1 + g L ) T C M; (1 + g L ) T Ä h M = M h M ; (25) Ä 1 =1 where M is he number of unis of medical insurance o hedge expeced presen value of he sum of medical cos and he decrease in wage income perfecly. Cash çows associaed wih ê medical insurance are Äy M M h M a ime, ú (i) ë M; í M Ä y M M h M from ime 1 o ime T Ä 1, and ú (i) M;T í M M h M a ime T. 3 Muli-period Opimizaion Model for a Household We formulae a muli-period opimizaion model in he simulaed pah approach. Condiional value a risk (CVaR) is used as a risk measure. As menioned in Secion 2.4, a household invess in n risky asses and cash a ime, and i can rebalance posiions from ime 1 o T Ä 1. One period is one year. I purchases T years life insurance and medical insurance policies a ime, 8 A level premium of medical insurance per uni is he same as ha of life insurance (Equaion (16)) wih he same mauriy because only insured person who is alive pays a premium. A disease rae inçuences insurance money. 9 I is expeced ha sampling error is improved using he decision variable for he hedge raio. 1

11 and pay level premiums. Boh policies can be surrendered a he ime when he house is bough. I also purchases an one-year åre insurance policy which is updaed every year in he planning period. 3.1 Noaions We describe noaions used in he formulaion including noaions shown in Secion 2. (1) Subscrip/Superscrip j :asse(j =1;...;J).J is he number of risky asses. :ime( =;...;T).T is he number of planning periods. i :pah(i =1;...;I).I is he number of pahs. b : life insurance (b 2 B = f; 1; 2g). B is he se of life insurance. (2) Parameers 1 ö j : price of risky asse j a ime (j =1;...;J). ö (i) j : price of risky asse j on pah i a ime (j =1;...;n; =1;...;T; i =1;...;I), r : ineres rae in period 1 or a ime. r (i) Ä1 : ineres rae on pah i in period or a ime Ä 1( =2;...;T; i =1;...;I). y L;b : level premium of life insurance b per uni, calculaed in Equaion (16). í L; : erm life insurance money per uni, calculaed in Equaion (17). í L;1; : decreasing erm life insurance money per uni a ime, calculaed in Equaion (17). í L;2; : family income insurance money per uni a ime, calculaed in Equaion (18). y M : level premium of medical insurance per uni, calculaed in Equaion (22). í M : medical insurance money per uni, calculaed in Equaion (23). M : number of unis of medical insurance o hedge expeced presen value of he sum of medical cos and he decrease in wage income perfecly, calculaed in Equaion (25). í F : one year åre insurance money per uni, calculaed in Equaion (19). y F : premium of one year åre insurance per uni, calculaed in Equaion (2). F; : number of unis of åre insurance o hedge loss of non-ånancial wealh afer depreciaion a ime perfecly, calculaed in Equaion (21). (i) F; : number of unis of åre insurance o hedge loss of non-ånancial wealh afer depreciaion on pah i a ime perfecly, calculaed in Equaion (21). E (i) : cash income associaed wih disposable income or survivor's pension, and severance pay on pah i a ime, calculaed in Equaion (5). H 1 : deb loan a ime e. C (i) : oal consumpion expendiures on pah i a ime, calculaed as C (i) = C (i) 1; + C(i) 2;. W (i) 1; : ånancial wealh on pah i a ime. W 1; is an iniial amoun of ånancial wealh a ime. L C : lower bound of CVaR of erminal ånancial wealh. 1 The oher parameers, ú (i) D; ;ú(i) L; ;ú(i) F; ;ú(i) M; ;ï L;;ï F ; and ï M;, can be referred in Secion

12 L v : lower average raio of cash. (3) Decision variables z j : base invesmen variable of risky asse j and ime (j =1;...;J; =;...;T Ä 1;). v : cash a ime. v (i) : cash on pah i a ime ( =1;...;T Ä 1; i =1;...;I). u + L;b : number of unis of life insurance b bough a ime. u Ä L;b : number of unis of life insurance b surrendered a ime e when he house is bough. h F : hedge raio of one year åre insurance agains loss of non-ånancial wealh afer depreciaion. h M : hedge raio of medical insurance agains expeced presen value of he sum of medical cos and he decrease in wage income. V å : å-var q (i) :shorfallbelowå-var (V å ) of erminal ånancial wealh(w (i) 1;T ) on pah i, ê q (i) ë max V å Ä W (i) ë 1;T ; (i =1;...;I). 3.2 Objecive funcion and cash çow Objecive funcion We formulae he objecive funcion and he consrains wih respec o reurn and risk measures as follows. (1) Reurn measure : expeced erminal ånancial wealh a ime T W T = 1 I IX i=1 W (i) 1;T (2) Risk measure : Condiional value a risk(cvar) The CVaR associaed wih erminal ånancial wealh is deåned as a risk measure. Namely, ( å ) 1 IX åååå CVaR å = Max V å Ä q (i) W (i) 1;T (1 Ä å)i Ä V å + q (i) ï (i =1;...;I) : (27) i=1 TheCVaRisusedinhemaximizedobjecive funcion, or lower bound consrain. 11 (26) Cash çow associaed wih life insurance money A household has o pay inheriance ax for life insurance money(e.g. erm life insurance money) received as a whole a he ime when a householder dies. However, life insurance money is ax-exemp subsanially wihin 16 million yen on he ax sysem in Japan if he recipien is a spouse. We formulae he model wihou inheriance ax o avoid modeling complexiy because i is rare ha axable income is beyond 16 million yen. If a household receives family income insurance money in every period, i is reaed as oher income, and a household has o pay income ax. We assume ha a spouse is unwaged. If a 11 Even if he CVaR of W Ä W (i) 1;T is used o minimize he objecive, we havehesamesoluionashesoluion derived from he maximizaion of he CVaR of W (i) 1;T. 12

13 householder dies a ime m, oher income a ime or Z m is shown as 12 ( Z m =max í L;2 (1 + r g ) Ä1 Ä ) my L;2 r g (1 + r g ) Äm ê (1 + r g ) T Äm+1 Ä 1 ; u + L;2 Ä 1 ë f mï e+1gu Ä L;2 for r g > ; (28) ö í ì õ or Z ê m m =max í L;2 Ä y L;2 ; u + L;2 T Ä m +1 Ä 1 ë f mï e+1gu Ä L;2 for r g =: (29) We calculae axable income by excluding exempions for a spouse and dependens from oher income. If children are born in he fuure, exempion for dependens from income depends on he ime of householder's deah because he number of children depends on he exisence of a householder. Taxable income is max(z m ÄEX m ; ) where exempions from income are EX m. are calculaed as Equaion (3) wih consrains (31) and Income and inhabian axes TAX m (32) because income ax raes are variable on he six levels of income. 13 TAX m = 6X k=1 TR a kz m ;k ( m =1;...;T; = m ;...;T); (3) Z m Ä EX m î 6X k=1 Z m ;k ( m =1;...;T; = m ;...;T); (31) î Z m ;k î TR b k (k =1;...; 5); Z m ;6 ï ( m =1;...;T; = m ;...;T): (32) The inequaliy consrain (31) 14 is used in order o describe he axable income or max(z m Ä EX m ; ). Toal sum of premiums a ime or X Ä is calculaed as in Equaion (33), and oal income from insurance money on pah i a ime or X +(i) is calculaed as in Equaion(34). X Ä = X y L;b êu + L;b Ä 1 ë fï e gu Ä L;b ( =;...;T Ä 1); (33) b2b X +(i) = ú (i) ê L; ní L; +ú (i) D; where 1 fm ï e +1g = u + L; Ä 1 ë ê fï e+1gu Ä L; + í L;1; u + L;1 Ä 1 ëo fï e+1gu Ä L;1 ê ní L;2; u + L;2 Ä 1 ë o f mï e+1gu Ä L;2 Ä TAX m ( =1;...;T); (34) TX k= e +1 ú (i) L;k. 12 If family income insurance is surrendered, i.e. u Ä L;2 >, Equaions (28) and (29) are approximae. Toal sum of premiums is m y L;2 u + L;2 Ä max( m Ä e ; )y L;2 u Ä L;2, and oal sum of income is P T k= m í L;2 (1 + r g ) Ä kä 1 u + L;2 Ä 1 f mï e+1gul;2å Ä. Oher income is calculaed largely for m ï e +1 if Equaions (28) and (29) are used for u Ä L;2 >. Therefore, disposable income becomes small. However, we do no have an above-menioned inçuence on he resuls because oal premium is much smaller han oal income, and we use approximae equaions o be linear consrains. I is exac for m î e because equaions do no include he erm u Ä L;2. 13 We se TR1 a = :15, TR2 a = :2, TR3 a = :3, TR4 a = :33, TR5 a = :43, TR6 a = :5, TR1 b = 1:95, TR2 b =1:35, TR3 b =3:65, TR4 b =2:5, TR5 b = 9 on he ax sysem in 27. The uni of TRk b is million yen. 14 If Z m Ä EX m <, hen Z m ;k =. IfZm Ä EX m >, hen axes are assigned o he decision variables Z m ;k in order which have smaller coeéciens of TRk a o pay ax as few as possible under consrains (31) and (32). When we solve he problem in Secion 4, we shrink he problem size by using only hree decision variables of Z m ;k (k =1; 2; 3) for each combinaion of and m, because i is hough ha he opimal amouns of insurance money received every year are no larger han wage income. 13

14 3.2.3 Cash çow equaions Cash çow equaions excep rading ånancial asses are shown as D = X Ä + y M M h M + y F F; h F ; (35) = E (i) + 1 f=e gh 1 Ä C (i) nê Ä 1 f6=t g 1 Ä ú (i) ë D; ëêx Ä + y M M h M + y F (i) o F; h F +X +(i) + ú (i) M; í M M h M Ä ú (i) ê F; 1f=1g F; + 1 f6=1g (i) ë F;Ä1 í F (1 Ä h F ) ( =1;...;T; i =1;...;I): (36) D (i) The componens of cash inçow are wage, loan, life, åre and medical insurance money. The componens of cash ouçow are consumpion, premiums of life, åre and medical insurance, and loss due o a åre. Premium paymen is excluded a ime T. 3.3 Modeling Formulaion The objecive is he maximizaion of he expeced erminal ånancial wealh subjec o he minimum CVaR requiremen or L C. The model is formulaed as follows: 1 IX ; Maximize W (i) 1;T (37) I (W (i) i=1 subjec o nx ö j g(z j ;ö j ;W )+v + D = W 1; ; (38) j=1 1;1 =) n X (W (i) 1; =) j=1 nx j=1 ö (i) j1 g(z j;ö j ;W )+(1+r )v + D (i) 1 = nx j=1 ö (i) j g(z j;ä1;p (i) ê j;ä1 ;w(i) Ä1 )+ 1+r (i) ë Ä1 v (i) Ä1 + D(i) = nx j=1 8 W (i) < 1;T = : V å Ä ö (i) j g(z j;p (i) nx j=1 1 (1 Ä å)i j ;w(i) )+v (i) ö (i) j1 g(z j1;p (i) j1 ;w(i) 1 )+v(i) 1 (i =1;...;I); (39) ( =2;...;T Ä 1; i =1;...;I); (4) ö (i) jt g(z j;t Ä1;p (i) ê j;t Ä1 ;w(i) T Ä1 )+ 1+r (i) ë T Ä1 v (i) T Ä1 9 = ; + D(i) T (i =1;...;I); (41) IX q (i) ï L C ; (42) i=1 W (i) 1;T Ä V å + q (i) ï (i =1;...;I); (43) W (i) 1;T ï (i =1;...;I); (44) z j ï (j =1;...;n; =;...;T Ä 1); (45) 14

15 1 nx v ï L ö j g(z j ;ö j ;W )+v A ; (46) j=1 IX IX! v (i) ï L v W (i) 1; ( =1;...;T Ä 1); (47) i=1 i=1 v (i) ï ; (48) u + L;b ï uä L;b ï (b 2 B); (49) h F ï ; h M ï ; (5) (51) q (i) ï ; (i =1;...;I) (52) We need o add Equaions (3){(36) wih he above-menioned formulaion. The invesmen ê ë uni funcion is shown as g(z; p;w) = w p z Ieraive algorihm for he åxed-proporion sraegy Opimal soluions depend on he invesmen sraegies in he simulaed pah model. We can describe he various invesmen sraegies by subsiuing he associaed values for he parameers p and w. Hibikie al.[7, 8, 9] use he model wih åxed-uni sraegy in which we have he same values of invesmen unis for risky asses and he diãeren values for cash a ime on all pahs. Bu, we solve he problem by using he model wih åxed-proporion sraegy in which we have he same values of invesmen proporions for risky asses and cash on all pahs a ime. We describe he åxed-proporion sraegy by subsiuing ö (i) j for p (i) (i) j,andw 1; for w(i) in he formulaion, bu he problem is a large-scale and non-convex program. I is diécul o solve he problem and derive he global opimal soluion even if we use he sophisicaed commercial mahemaical programming sofware package. Therefore, we derive he approximae soluion wih ieraive algorihm proposed by Hibiki[6] as follows. 15 Sep 1: We derive he opimal soluion wih he invesmen uni funcion of p (i) j w (i) p (i) j = w (i) = 1, and calculae he value of wealh on pah i a ime or W (i)é 1;(). Le Obj denoe he objecive funcion value, and se k =1. Sep 2: We se up p (i) j = ö(i) j and w (i) = W (i)é (kä1) a he k-h ieraion, and solve he problem. We calculae wealh of pah i a ime or W (i)é 1;(k), and he objecive funcion value Obj k. Sep 3: Sop if a value of Obj k Ä Obj kä1 or he diãerences in cash raios v(i) among all pahs W (i) 1; are lower han a olerance. Oherwise, se k k + 1, and reurn o Sep 2. The algorihm does no guaranee o derive he global opimal soluions for he åxed-proporion sraegy. This algorihm is a heurisic one, andanysoluionderivedmaybelocallyopimal. 15 Suppose ha we derive he opimal soluions of he model wih he åxed-proporion sraegy, and calculae he value of wealh on pah i a ime or W (i)é 1;. If we solve he problem wih he value of wealh in Sep 2, he same soluions are supposed o be obained. This would only be rue if he global soluion could be compued. However, using he characerisics we use he ieraive algorihm o solve i approximaely. See Hibiki[6] in deail. or 15

16 However, i is more useful in pracical use ha we have he abiliy o derive a soluion by solving linear programming problems successively raher han he solver does no deliver any soluion o a large-scale problem. As a resul of ieraion, he invesmen raio a ime is assigned o he decision variables z j,andcashraioor v(i) (i =1;...;I) are almos he same values. W (i) 3.4 Formulaion of he model including only insurance sraegy under he given asse mix Financial planners someimes give some advice ha he household would be beer oãinvesing `1 - age'(%) of ånancial wealh in a risky asse. On he oher hand, we implemen consan rebalance sraegy when we use he one-period model such as mean-variance model. We can use he model including only insurance sraegy under he given asse mix in order o evaluae hose sraegies. We can describe he formulaion simply as a linear programming model, unlike he model in Secion 3.3. The rae of reurn of porfolio or ñ (i) p is calculaed as ñ (i) n p1 = X ñ (i) j1 x j; + r x ; (i =1;...;I); (53) ñ (i) p = j=1 nx j=1 ñ (i) j x j;ä1 + r (i) Ä1 x ;Ä1 ( =2;...;T; i =1;...;I); (54) where x j is an invesmen raio a ime for risky asse j, andx is a cash raio a ime. The value of ånancial wealh on pah i a ime or W (i) 1; from ime 1 o T are calculaed as W (i) ê 1;1 = 1+ñ (i) ë p1 (W Ä D )+D (i) 1 (i =1;...;I); (55) ê ë W (i) 1; = W (i) 1;T = ê where CR (i) u = 1+ñ (i) p W (i) 1;Ä1 + D(i) ( =2;...;T Ä 1; i =1;...;I); (56) 1+ñ (i) ë pt W (i) T 1;T Ä1 + D(i) T = CR(i) 1 (W XÄ1 Ä D )+ k=u u=1 CR (i) u+1 D(i) u + D (i) T (i =1;...;I); (57) TY ê 1+ñ (i) ë pk. We can solve he problem wih Equaion (57) insead of Equaions (38){(41), (45){(48) in he model in Secion 3.3. The formulaion is described as Maximize 1 I IX i=1 W (i) 1;T ; (58) subjec o Equaions (42){(44); (49){(52); (57): 4 Numerical Analysis We es numerical examples for a hypoheical household. All of he problems are solved using NUOPT (Ver. 9.2) { mahemaical programming sofware package developed by Mahemaical Sysem, Inc. { on Windows XP personal compuer which has 2.13 GHz CPU and 2GB memory. 16

17 4.1 Seing Household The householder(husband) is hiry years old and he spouse(wife) is weny-eigh years old. They go married hree years ago. The års child is one year old, and he second child will be born in wo years. 16 The householder graduaed from universiy, and began o work a a ånancial insiuion a he age of 22. He will work unil his reiremen a he age of 6. The household lives in he aparmen which ren is 125 housand yen a monh, however i plans ha i will prepare eigh million yen as a down paymen en years laer and buy an aparmen which coss fory million yen. Eigh million yen is paid a he ime ( e = 1) when he house is bough. Thiry wo million yen is borrowed, and he morgage loan is equally paid over weny years. Equal yearly paymen is calculaed a a morgage ineres rae of 3%. The parens make an educaional plan ha he children will go o public elemenary schools, privae junior high schools, and privae high schools, he års child will go o humaniies course in a privae universiy, and he second child will go o science course in a privae universiy. A risky asse is a sock only, and we inves ånancial wealh in sock and cash Parameers We show he parameers which are esimaed by using he associaed daa. (1) Income è wage Wage may depend on he age, he lengh of service, and he ype of job. However, we simply esimae wage in he form of a fourh-order polynomial of he age because we can esimae he funcion wih large coeécien of deerminaion. The funcion is esimaed using he able of monhly salary and bonus for universiy-educaed men's employee working in `over 1 employees' company of ånance and insurance indusry in he foureen kinds of large classiåcaion of indusries in wage income of he household over ime based on he Census of wage in 26 surveyed by Minisry of Healh, Labor and Welfare[14]. An inçaion rae is no considered in he numerical examples. è severance pay An amoun of severance pay e (i) is calculaed by muliplying an amoun of wage! (i) m + when a householder reires or dies wih he provision coeécien é m + in Equaion (59). I is assumed ha he provision coeécien is a piecewise linear funcion wih zero a he age of m d and an upper bound é U a he age of m d + T é in Equaion (6). Explicily, e (i) é m + = é m +! (i) m + ( =1;...;T); (59) îí ì ï m Ä m d + = min ; 1 é U ( =1;...;T): (6) T é We se m = 3, m d = 25, é U =2,andT é = 3 in he analysis. (2) Consumpion expendiure 16 I is assumed ha he second child will no be born if he householder dies in one year. 17 For simpliciy, we assume ha a risk free rae is consan for all, and he rae of reurn of sock is normally disribued. 17

18 è educaion coss Educaion coss from kindergaren o high school are calculaed by oal learning expendiure of households for children of `Survey of Household Expendiure on Educaion per Suden in 26' by Minisry of Educaion, Culure, Spors, Science and Technology[13]. We assume ha educaion coss unil enering kindergaren are 2 housand yen which is he same cos as a public kindergaren. Coss of a privae universiy are calculaed by he daa published in `Survey of average års year paymen of mariculan o he privae universiy in 24' by Minisry of Educaion, Culure, Spors, Science and Technology[12]. Enrance fees, uiion and equipmen expenses are included a he års year, and uiion and equipmen expenses are included each year from he second year. Coss of a naional universiy are he enrance fee and he uiion a he års year, and uiion each year from he second year. The enrance fee of a naional universiy is 282 housand yen, and he uiion is housand yen. Suden living coss from he household excep he cos of universiy are calculaed by `Survey on Suden Life in 24' by Japan Suden Services Organizaion[1]. è oher living expenses Two kinds of oher living expenses excep housing coss and educaion coss are esimaed wih disposal income w (i) as c 41(i) 1; = :4w (i) ; (61) c 42(i) h 1; = :2 w (i) n Ä C 1(i) ê 1; Ä 1 f=e gü 1 1 Ä ü 2 ú (i) ë D; H + C 2(i) 1; + C 3(i) 1; + C (i) oi 2; : (62) The sum of oher living expenses or C 4(i) 1; is calculaed as in Equaion (12) 18. è Coss associaed wih non-ånancial wealh Coss associaed wih non-ånancial wealh or C (i) 2; are esimaed wih oher living expenses excep housing coss and educaion coss or c 41(i) 1; as 19 C (i) 2; =:2c41(i) 1; We show average income and expendiure calculaed wih he above-menioned seing as in Figure 1. Average amouns of income excep capial gain and insurance money are locaed on he op-lef side, average amouns of expendiure excep insurance premiums and recovery cos of a house due o a åre are locaed on he op-righ side, he diãerences beween hem are locaed on he boom-lef side, and he cumulaive diãerences calculaed wih a risk-free rae of.5% are locaed on he boom-righ side. A household needs o hedge income risk wih life insurance because he cumulaive diãerence of income and expendiure becomes larger when a householder dies earlier. (63) 18 We esimae oher living expenses wih disposal income by using 'Naional survey of family income and expendiure in 24' by Saisic Bureau, Minisry of Inernal Aãairs and Communicaions[15] as follows. However, we use Equaions (61) and (62) because he problems are someimes infeasible o solve. C 4(i) 1; =55:9 + :5836w (i) 19 We esimae coss associaed wih non-ånancial wealh by using 'Naional survey of family income and expendiure in 24' by Saisic Bureau, Minisry of Inernal Aãairs and Communicaions[15] as follows. However, we use Equaion(62) because he problems are someimes infeasible o solve. C (i) 2; = Ä13:45 + :1935c41(i) 1; 18

19 average income(mil.yen) Deah a ime 1 Deah a ime 5 Deah a ime 1 Deah a ime 15 Deah a ime 2 Deah a ime 25 living a ime 3 average expendiure(mil.yen) Deah a ime 1 Deah a ime 5 Deah a ime 1 Deah a ime 15 Deah a ime 2 Deah a ime 25 living a ime ime ime difference beween income and expendiure (mil. yen) Deah a ime 1 Deah a ime 5 Deah a ime 1 Deah a ime 15 Deah a ime 2 Deah a ime 25 living a ime cumulaive difference beween income and expendiure(mil. yen) Deah a ime 1 Deah a ime 5 Deah a ime 1 Deah a ime 15 Deah a ime 2 Deah a ime 25 living a ime ime ime Figure 1: Average income and expendiure (3) Insurance è moraliy rae We compue premiums and insurance money of life insurance wih he moraliy raes ï L; derived from `life insurance sandard life able (27) for men' by The Insiue of Acuaries of Japan[17]. è disease rae There mus be a serious disease rae able for medical insurance, and premiums mus be calculaed by using he able. However, he able is no published ouside. In his paper, we assume ha he disease raes ï M; are ó L (> 1) imes he moraliy raes ï L;. The reason is ha a serious disease causes deah, and he disease raes become higher wih age. We se ó L = 3 in he numerical examples. è raioofloadingofpremiumoanepremium Premiums of life insurance depend on insurance age, lengh of he conracs, and insurance money. We calculae raios of loading of premium o a ne premium wih he guaraneed rae of 1.5% for 45 erm life insurance for six companies, 8 decreasing erm life insurance for wo companies, and 21 family income life insurance for seven companies as in Figure 2. Raios of loading of premium o a ne premium depend on he ype of life insurance. However, we se ê L =:5 in he numerical examples. 19

20 Raio of loading of premiu o a ne premium Term Insurance Decresing Term Insurance Family Income Insurance Insurance age Figure 2: Raio of loading of premium o a ne premium (4) Iniial wealh Iniial wealh are se under he following assumpions. è iniial ånancial wealh The householder had one million yen in savings a he age of 22, and go he deermined wages from 22 o 3 in he same way. He had lived wih his parens, paid 3 housand yen o his parens unil he go married, and spen his disposal income of 24% every year. They spen ne cash ouçow of 1.49 million yen for wedding ceremony and he preparaion of new life. 2 Afer he go married, he household spen he same money as afer he curren ime or ime. Invesmen yield is.9% before ime. We calculae iniial ånancial wealh, which is W 1; = 865:4 in he numerical examples. è iniial non-ånancial wealh No iniial non-ånancial wealh exiss a he householder's age of 22, and i is calculaed from he age of 22 o 3 in he same way as afer ime. We se W 2; =24:4 inhenumerical examples. (5) Oher parameers Oher parameer values used in he numerical examples are shown in Table 1. 2 Wedding expense is 4.14 million yen, ånancial assisance from parens is 1.82 million yen, and wedding gif of money is 2.23 million yen on avearge in `Zexy repor of wedding rend 27' by Recrui Co., Ld. Average cos of new life is 1.4 million yen in `Zexy repor of preparaion for new life 27' by Recrui Co., Ld. Ne cos of 1.49 million yen is calculaed wih hem. 2

21 Table 1: Oher parameers Parameers Values expeced rae of reurn of sock ñ S =2:5% sandard deviaion of rae of reurn of sock õ S = 1% sandard deviaion of he rae of change of wage õ W =2% correlaion coeécien beween wage and sock c WS =:2 risk-free rae r = :5% rae of a åre ï F =:5% guaraneed rae of life and medical insurance g L =1:5% guaraneed rae of åre insurance g F =1:5% raio of loading of premium o a ne premium ê L =:5, ê F =:3, ê M =:3 medical cos due o a serious disease C F =1:5(million yen) loss raio of non-ånancial wealh due o a åre ç =1 depreciaion rae of non-ånancial wealh ç 1 =:3 lower bound of cash L v = 1% probabiliy level of CVaR å = :8 number of pahs I = Seing cases for numerical analyses We call he model wih åxed-proporion sraegy `model 1' in Secion 3.3. We call he model including only insurance sraegy under he given asse mix `model 2' in Secion 3.4. We examine some cases wih hese models as follows. We also examine he sensiiviy of he associaed parameers. Case #1 : Basic analysis Case #2 : Comparison of he resuls for respecive life insurance among hree. Case #3 : Comparison of he resuls for various loading of premiums (1) Comparison of he resuls for various raios of loading of premiums for respecive insurance : ê k =:; :3; :5; 1:; 1:5 (k 2fL; F; Mg) (2) Three kinds of raios of.5,.75, and 1. of loading of premiums for decreasing erm life insurance are examined under he raio of.25 for family income insurance. 21 Case #4 : Comparison of he resuls for various correlaion coeéciens beween wage and sock : c WS = Ä:4; Ä:2; :; :2; :4 We examine he above-menioned analyses for he wo models as follows. We show he resuls of Case #4 only for he model 1 because we examine how invesmen raios are aãeced by he correlaion beween wage and sock. Model 1 : We solve and compare six kinds of problems which are he maximizaion of CVaR 21 The raio for family income insurance is se o be smaller han he raio for decreasing erm life insurance as in Figure 2. 21