MULTI-PERIOD OPTIMIZATION MODEL FOR A HOUSEHOLD, AND OPTIMAL INSURANCE DESIGN

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1 Journal of he Operaions Research Sociey of Japan 27, Vol. 5, No. 4, MULTI-PERIOD OPTIMIZATION MODEL FOR A HOUSEHOLD, AND OPTIMAL INSURANCE DESIGN Norio Hibiki Keio Universiy (Received Ocober 17, 26; Revised May 7, 27) Absrac We discuss an opimizaion model o obain an opimal invesmen and insurance sraegy for a household. In his paper, we exend he sudies in Hibiki and Komoribayashi (26). We inroduce he following poins, and examine he model wih numerical examples. 1ç We consider cash çow due o a serious disease and involve medical insurance. 2ç An opimizaion model is formulaed wih erm life insurance which variable insurance money is received. 3ç We propose a model o decide opimal life and medical insurance money received a each ime. 4ç Sampling error is examined wih kinds of 5, sample pahs. Keywords: Finance, muli-period opimizaion, ånancial planning, invesmen and insurance sraegy, insurance design 1. Inroducion We discuss an opimizaion model o obain an opimal invesmen and insurance sraegy for a household. Recenly, ånancial insiuions have promoed giving a ånancial advice for individual invesors. 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 insiuions need o recommend appropriae ånancial producs in order o hedge risk agains hese accidens. We clarify how a se of asse mix, and life, åre and medical insurance aãec asse and liabiliy managemen for a household. We develop a muli-period opimizaion model which involves deermining a se of ånancial producs, hedging risk associaed wih a life cycle of a household and saving for he old age. The simulaed pah approach [3, 4] can be used o solve his problem. There are some sudies in he lieraure for individual opimal invesmen sraegy; Bodie, Meron and Samuelson [1], Meron [7, 8], Samuelson [1]. Chen, Ibboson, Milevsky and Zhu [2] advocae an opimizaion model wih he inclusion of wage income, consumpion expendiure, and boh opimal asse allocaion and life insurance. Yoshida, Yamada and Hibiki [12] solve an opimal asse allocaion problem for a household using a muli-period opimizaion approach. Hibiki, Komoribayashi andtoyoda[6]describeamuli-periodopimizaion model o deermine an opimal se of asse mix, life insurance and åre insurance in conjuncion wih heir life cycle and characerisics. The model is examined wih numerical examples. In addiion, some ånancial advices for hree households are illusraed for 463

2 464 N. Hibiki pracical use, and resuls which coincide wih a pracical feeling are obained. Hibiki and Komoribayashi [5] exend he sudies in Hibiki, Komoribayashi and Toyoda [6] for pracical use. Risk associaed wih he householder's deah is hedged by life insurance. The model is proposed involving he associaed hree facors: receip of a survivor's pension, exempion from morgage loan paymens, and change in he consumpion level. Addiional eãecs by hree facors are examined wih numerical examples. Moreover, he sensiiviy of parameers associaed wih home buying is analyzed in order o examine he home buying sraegy. We obain he following pracical and ineresing resuls in Hibiki, Komoribayashi and Toyoda [6], and Hibiki and Komoribayashi [5]. (1) The older a householder is, he less opimal life insurance money is. (2) Opimal åre insurance money is nearly equal o he maximum loss of non-ånancial wealh. (3) Expeced erminal ånancial wealh does no aãec opimal life and åre insurance money. (4) If a household receives a survivor's pension and keeps he consumpion level lower afer a householder died, opimal life insurance money and invesmen unis of a risky asse are reduced. (5) If loan paymens are forgiven due o he householder's deah, opimal invesmen unis of a risky asse are reduced, bu opimal life and åre insurance money are no inçuenced. (6) Home buying sraegy aãecs an opimal asse mix and life insurance money. In his paper, we exend he sudies in Hibiki and Komoribayashi [5]. We inroduce he following poins in a muli-period opimizaion model, and examine he model wih numerical examples. 1ç We consider cash çow due o a serious disease and involve medical insurance o cover he expensive medical cos. 2ç An opimizaion model is formulaed wih erm life insurance which variable insurance money is received, and i is compared wih he consan receip of life insurance money by using numerical examples. 3ç We propose a model o decide opimal life and medical insurance money received a each ime. 4ç Sampling error is examined wih kinds of 5, sample pahs. This paper is organized as follows. We describe a household, income, consumpion expendiure, and four kinds of ånancial producs, or securiies, life insurance, åre insurance, and medical insurance o develop a model in Secion 2. Secion 3 shows he formulaion of a muli-period ALM opimizaion model for a household. We analyze he sensiiviy of parameers associaed wih a serious disease, and examine he eãec of medical insurance. We solve he problem wih decreasing life insurance money over ime, and compare i wih consan life insurance money over ime by using numerical examples. In Secion 4, we propose a model o decide opimal life and medical insurance money a each ime, and numerical examples are shown. Sampling error is examined wih kinds of 5, sample pahs in Secion 5. Secion 6 provides our concluding remarks. 2. Model Srucure We deåne a household, and describe income and consumpion expendiure. We clarify he characerisics of ånancial insrumens such as securiies, life insurance, åre insurance, and

3 Muli-Period Model and Insurance Design 465 medical insurance. We aach a superscrip (i) o a random and pah dependen parameer in order o formulae a model in he simulaed pah approach Household We deåne a household as a group composed of a householder and members of family as in he previous papers [5, 6]. Wealh a ime held by a household can be divided ino wo kinds of wealh: ånancial wealh W (i) 1; and non-ånancial wealh W (i) 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. I is assumed ha a deah of a householder makes wage earnings sop, a serious disease of a householder decreases wage income and makes large paymen, and a åre of a house damages a fracion ã of non-ånancial wealh. A householder can purchase a life insurance policy, a åre insurance policy and a medical insurance policy o hedge risk in addiion o he invesmen in securiies such as socks and bonds. 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. ú (i) 1; : one if a householder dies on pah i a ime and zero oherwise. ú (i) 2; :oneifaåre of a house occurs on pah i a ime and zero oherwise. ú (i) 3; : one if a householder is alive on pah i a ime and zero oherwise. ú (i) 4; : one if a householder has a serious disease on pah i a ime and zero oherwise. 1 ï 1; : moraliy rae a ime, or he probabiliy ha a person who is alive a ime will die a ime, ï 1; =Pr(ú 1; =1)= 1 IX ú (i) 1; where I is he number of simulaed I i=1 pahs. ï 2 : rae of a åre (which is assumed o be ime independen), or he probabiliy ha aåre occurs, ï 2 =Pr(ú 2; =1)= 1 IX ú (i) 2;. I i=1 ï 4; : disease rae a ime, or he probabiliy ha a person who is alive a ime will have a serious disease a ime, ï 4; =Pr(ú 4; =1)= 1 IX ú (i) 4;. I i= Income Income a ime is a householder's wage m if a householder is alive and invesmen reurn from ånancial wealh W (i) 1;. If a householder dies, a household canno ge wages, bu receive severance pay, and draw a survivor's pension. Amouns of severance pay and a survivor's 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. An amoun of severance pay e (i) is also dependen on years of coninuous employmen(age). When a householder has a seriousdisease,iisassumed ha a fracion ó 3 (ó 3 < 1) of wage income decreases because a householder has o ake a res from work. Amouns of wage income, severance pay and a survivor's pension, or cash inçow excep invesmen reurn, borrowing, and insurance money, M (i) can be shown as follows : M (i) = ú (i) 3; m (i) + ê 1 Ä ú (i) ë (i) 3; a m + ú (i) 1; e (i) + 1 f=t g ú (i) 3;T e (i) T Ä ó 3 ú (i) 4; m (i) ( =1;:::;T) (1) where 1 fag is an indicaor funcion which shows one if he condiion A is saisåed, and zero oherwise. 1 If ú (i) 3; =,henú(i) 4; =.

4 466 N. Hibiki 2.3. Consumpion expenses There assumes o be wo kinds of expenses: living expenses C (i) 1; and paymens associaed wih non-ånancial wealh C 2;, (i) 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. (1) Expenses for purchasing a house We assume ha a household purchases a house by making a down paymen and a deb loan a a bank (H ). Le e be he ime when a house is purchased. The deb loan H e is he diãerence beween he price of he house and he down paymen. The expendiure for he house C 2;e is he price of he house, and herefore non-ånancial wealh W 2;e increases by he expendiures C 2;e a ime e. However, 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 ineresraeandheloanperiodaferhe ime e + 1. We include periodic paymens C1; 2 in he living expense for life (C 1;) (i) inhis paper. (2) Resoraion cos due o a åre I is assumed ha a fracion ãof 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) = ú (i) 2; ã(1 Ä ç )W (i) 2;Ä1 (2) where ç is a depreciaion raio of non-ånancial wealh a ime. A (i) does no aãec non-ånancial wealh. 2 Insead, i aãecs cash çow sreams as shown in Equaion (12) in Secion 3.2. (3) Medical cos I is assumed ha a household pays ó 2 if a householder has a serious disease such as cancer, cardiac infarcion, apoplexy. Paymen is ú (i) 4; ó 2, and i is included in he living expense C 1;. (i) (4) Living expenses C (i) 1; The following four kinds of parameers are used o describe living expenses. C 1(i) 1; : cos independen of he householder's deah, such as educaion cos and ren. : annual paymen for a morgage loan (when a householder is alive). C1; 2 ó 2 : medical cos due o a serious disease a householder has. C 3(i) 1; : oher living expense excep C 1(i) 1;, C1;, 2 andó 2 (when a householder is alive). Nex, we explain how o compue he annual paymen for he deb loan and oher living coss dependen on he householder's deah. 1ç Morgage loan If a household purchases a group credi insurance policy, he loan paymens are forgiven afer a householder died. This shows ha an amoun of annual paymen can be ú (i) 3; C1;. 2 However, he loan paymen is no forgiven if a household purchases a house afer a householder died. By using he condiion ha ú (i) 3; =for> e if ú (i) 3; e =, he amoun of annual paymen for morgage loan can be ê 1 Ä ú (i) 3; e + ú (i) ë 3; C 2 1;. 2ç Change of he consumpion level I is assumed ha a household can keep a normal consumpion level if a householder is alive, however a consumpion level mus be îimes a normal level if a householder is dead, 2 Non-ånancial wealh decreases by A (i) non-ånancial wealh increases by A (i). due o a åre, bu he same money is spen o recover he loss, and

5 Muli-Period Model and Insurance Design 467 where î is a parameer associaed wih a consumpion level. For example, we se î =1 when a household keeps a normal level, and we se î= :7 when i has o allow for he 7% consumpion level. Therefore, oher living cos becomes The oal living cos is n (i) ú 3; + ê 1 Ä ú (i) ë o 3(i) 3; î C 1; = n î+(1ä î)ú (i) o 3(i) 3; C 1; : (3) C (i) 1; = C 1(i) 1; + ê 1 Ä ú (i) 3; e + ú (i) 3; ë C 2 1; + n î+(1ä î)ú (i) 3; o C 3(i) 1; + ú (i) 4; ó 2 : (4) 2.4. Securiies Invesmen in risky asses conribues o a hedge agains inçaion. We inves in n risky asses and cash. Using a price ö j, a rae of reurn of a risky asse j a ime is R j = ö j ö j;ä1 Ä 1(j =1;:::;n; =1;:::;T): (5) A risk-free rae r a ime (= ; 1;:::;T Ä 1) is åxedinheperiodfromime o +1. We can assume any probabiliy disribuions of R j and r in he simulaed pah approach if we can sample random pahs for R j and r. However, i is assumed ha R is normally disribued wih he mean vecor ñ, and he covariance marix Ü (R ò N(ñ; Ü)), and r is consan for all in his paper. We calculae a price ö j by using R j Life insurance We use erm life insurance wih mauriy T agains he householder's deah. If a householder purchases a erm life insurance policy 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. We assume ha a household makes level paymen. Because only insured person who is alive pays a premium, a premium of level paymen per uni is y l = T Ä1 X = 1 Ä P! Ä1 k= ï 1;k (6) (1 + g 1 ) where g 1 is a guaraneed ineres rae of life insurance wih mauriy T. Using he principle of equalizaion of income and expendiure, insurance money is calculaed for he corresponding presen value of premium income. We explain how o compue variable life insurance money wih various kinds of paymen çow a each ime. 3 Le í 1; be variable life insurance money per uni of presen value of premium income a ime. We have 1= TX =1 í 1; ï 1; (1 + g 1 ) = T X =1 ë 1; í 1 ï 1; (1 + g 1 ) (7) where í 1; = ë 1; í 1. Equaion (7) is ransformed, and life insurance money per uni is í 1; = ë 1; ( T X k=1 ) Ä1 ë 1;k ï 1;k : (8) (1 + g 1 ) k 3 Insurance money of well-known ype of life insurance is consan over ime, and he problem is solved wih consan life insurance in Hibiki, Komoribayashi and Toyoda [6], and Hibiki and Komoribayashi [5].

6 468 N. Hibiki If ë 1; is consan, life insurance money is consan over ime regardless of he ime of he householder's deah Fire insurance A household purchases one year åre insurance o hedge loss of non-ånancial wealh due o aåre. 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, he relaionship beween one uni of presen value of premium income and he corresponding insurance money í 2 is shown as: 1= í 2ï 2 1+g 2 ; or í 2 = 1+g 2 ï 2 (9) where g 2 is a guaraneed ineres rae of one year åre insurance. I is independen of ime. We can only selec a single paymen because of one year åre insurance. A premium of single paymen per uni y F is equal o a uni of he presen value of fuure premium income (y F =1) Medical insurance We use erm medical insurance wih mauriy T agains a householder's serious disease. If a householder purchases a erm medical insurance policy and has a serious disease by ime T, i can receive medical insurance money. 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. We assume ha a household makes level paymens, and is premium per uni is calculaed as 5 y b = T Ä1 X = 1 Ä P! Ä1 k= ï 1;k ; (1) (1 + g 1 ) as well as life insurance. The same guaraneed ineres rae g 1 as life insurance is used. We explain how o compue variable medical insurance money as well as life insurance. Le í 4; be variable medical insurance money per uni of he presen value of premium income a ime. Wehave í 4; = ë 4; ( T X k=1 ) Ä1 ë 4;k ï 4;k : (11) (1 + g 1 ) k We deåne he funcion of medical insurance money wih he ë 4; values as well as he ë 1; values for life insurance. If ë 4; is consan, medical insurance money is consan over ime. 4 We show he following wo kinds of funcions of life insurance money besides a consan funcion, which have higher values as a householder dies earlier. Decreasing linear funcion : ë 1; = T Ä +1 Reciprocal of moraliy rae : ë 1; = 1 ï 1; 5 A premium of level paymen per uni of medical insurance is he same as ha of life insurance (Equaion (6)) wih he same mauriy because only insured person who is alive pays a premium. A disease rae inçuences insurance money.

7 Muli-Period Model and Insurance Design Muli-period ALM Opimizaion Model for a Household We formulae a muli-period opimizaion modelinhesimulaedpahapproach. Condiional value a risk (CVaR) is used as a risk measure [9]. We assume ha he curren ime is ( = ), and a householder reires a ime T, which is a planning horizon. As menioned in Secion 2.4, a household invess in n risky asses and cash, and i can rebalance posiions a each ime. I purchases a T -years life insurance and medical insurance policies a ime, and makes level paymens. I also purchases an one-year åre insurance policy which is updaed every year in he planning period Noaions (1) Subscrip/Superscrip j : asse (j =1;:::;n). :ime( =1;:::;T). i :pah(i =1;:::;I). (2) Parameers 6 ö j : price of risky asse j a ime (j =1;:::;n). : price of risky asse j on pah i a ime (j =1;:::;n; =1;:::;T; i =1;:::;I), ö (i) j ö (i) j1 = ê 1+R (i) ë j1 öj (j =1;:::;n; i =1;:::;I); ö (i) j = ê 1+R (i) ë (i) j ö j;ä1 (j =1;:::;n; =2;:::;T; i =1;:::;I) where R (i) j isaraeofreurnofriskyassej on pah i a ime. r : ineres rae in period 1 or a ime. r (i) Ä1 g 1 : ineres rae on pah i in period or a ime Ä 1( =2;:::;T; i =1;:::;I). : guaraneed ineres rae on life insurance policies. y l : premium of level paymen life insurance per uni, calculaed in Equaion (6). y (i) L; : premium of level paymen life insurance per uni on pah i a ime, calculaed as y (i) L; = ú (i) 3; y l. í 1; : life insurance money per uni a ime, calculaed in Equaion (8). L (i) : life insurance money per uni on pah i a ime, calculaedasl (i) = ú (i) 1; í 1;. y b : premium of level paymen medical insurance per uni, calculaed in Equaion (1). y (i) B; : premium of level paymen medical insurance per uni on pah i a ime, calculaed as y (i) B; = ú (i) 3; y b. í 4; : medical insurance money per uni a ime, calculaed in Equaion (11). B (i) g 2 : medical insurance money per uni on pah i a ime, calculaedasb (i) : guaraneed ineres rae on åre insurance policies. = ú (i) 4; í 4;. y F : premium of one year åre insurance per uni, se as y F =1. í 2 : one year åre insurance money per uni, calculaed in Equaion (9). F (i) :oneyearåre insurance money per uni on pah i a ime, calculaedasf (i) ú (i) 2; í 2. ã : loss raio of non-ånancial wealh due o a åre of a house. ç : depreciaion raio of non-ånancial wealh a ime. 6 The oher parameers, ú (i) 1; ;ú(i) 2; ;ú(i) 3; ;ú(i) 4; ;ï 1;;ï 2 ; and ï 4;,canbereferredinSecion2.1. =

8 47 N. Hibiki A (i) M (i) : loss of non-ånancial wealh on pah i a ime, calculaedinequaion(2). : cash income associaed wih wage, severance pay, and survivor's pension on pah i a ime, calculaedinequaion(1). : deb loan on pah i a ime. : oal consumpion expendiures on pah i a ime,calculaedasc (i) H (i) C (i) = C 1;+C (i) 2;. (i) W (i) 1; :ånancial wealh on pah i a ime. W 1; is an iniial ånancial wealh a ime. W (i) 2; :non-ånancial wealh on pah i a ime, calculaedasw (i) 2; =(1Äç )W (i) W 2; is an iniial non-ånancial wealh a ime. W E : lower bound of expeced erminal ånancial wealh. å : probabiliy level used in he CVaR calculaion. L v; : lower bound of cash a ime. WhenL v; <, he borrowing can be allowed. (3) Decision variables z j : invesmen uni of risky asse j a ime (j =1;:::;n; =;:::;T Ä 1). :cashaime. v v (i) :cashonpahi a ime ( =1;:::;T Ä 1; i =1;:::;I). u L : number of life insurance bough a ime. u F; : number of one year åre insurance bough a ime ( =;:::;T Ä 1). u B :numberofmedicalinsuranceboughaime. : å-var used in he CVaR calculaion. V å q (i) : shorfall below å-var (V å )oferminalånancial wealh(w (i) 1;T )onpahi, q (i) ë max ê V å Ä W (i) 1;T ; ë (i =1;:::;I). 2;Ä1+C 2;. (i) 3.2. Formulaion Cash çow consrains are imporan in a muli-period opimizaion approach. Cash çow excep rading ånancial asses D (i) is associaed wih income, expendiures, and insurance. Premium paymen is no required a ime T. I is formulaed as: D (i) = M (i) + H (i) Ä C (i) ê (i) Ä 1 f6=t g y L;u L + y F u F; + y B;u (i) ë (i) B + L u L + F (i) u F;Ä1 +B (i) u B Ä A (i) ( =1;:::;T Ä 1; i =1;:::;I): (12) The objecive is he maximizaion of he CVaR associaed wih erminal ånancial wealh subjec o he minimum reurn requiremen. 7 Namely, CVaR å = Max ( V å Ä 1 (1 Ä å)i IX i=1 å ) åååå q (i) W (i) 1;T Ä V å + q (i) ï (i =1;:::;I) Expeced erminal ånancial wealh E[W 1;T ] is deåned as a reurn measure, and herefore he minimum reurn requiremen is formulaed as 1 I IX i=1 W (i) 1;T ï W E : 7 Even if he CVaR of W ÄW (i) 1;T is used o minimize he objecive, we have he same soluion as he soluion derived from he maximizaion of he CVaR of W (i) 1;T.

9 Muli-Period Model and Insurance Design 471 The model is formulaed as follows: Maximize 1 IX V å Ä q (i) ; (1 Ä å)i i=1 (13) subjec o nx ö j z j + v + y L; u L + y F u F; + y B; u B = W 1; ; (14) (W (i) 1;1 =) (W (i) 1; =) j=1 nx j=1 nx j=1 ö (i) j1 z j +(1+r )v + D (i) 1 = nx j=1 ö (i) j z j;ä1 + ê 1+r (i) ë (i) Ä1 v Ä1 + D (i) = 8 W (i) < 1;T = : 1 I IX i=1 nx j=1 ö (i) jtz j;t Ä1 + ê 1+r (i) ë (i) T Ä1 v ö (i) j1 z j1 + v (i) 1 (i =1;:::;I); (15) nx j=1 T Ä1 ö (i) j z j + v (i) ( =2;:::;T Ä 1; i =1;:::;I); (16) 9 = ; + D(i) T (i =1;:::;I); (17) W (i) 1;T ï W E ; (18) W (i) 1;T Ä V å + q (i) ï (i =1;:::;I); (19) z j ï (j =1;:::;n; =;:::;T Ä 1); v ï ; v (i) ï L v; ( =1;:::;T Ä 1; i =1;:::;I); u L ï ; u F; ï ( =;:::;T Ä 1); u B ï ; q (i) ï (i =1;:::;I); : free: V å 3.3. Numerical analysis Seing We es numerical examples using he same seing of he family and he parameers as in Hibiki and Komoribayashi [5]. We show he sensiiviy analysis associaed wih a serious disease because he model involves medical insurance. All of he problems are solved using NUOPT (Ver ) { mahemaical programming sofware package developed by Mahemaical Sysem, Inc. { on Windows XP personal compuer which has 2.13 GHz CPU and 2GB memory. The householder is hiry years old and he spouse is weny-eigh years old. The års child is an infan aged, and he second child will be born in hree years. 8 The householder works a a ånancial insiuion, and he household plans ha i will prepare weny million yen as a down paymen en years laer and buy an aparmen in he cener of Tokyo which coss åfy million yen. Tweny million yen is paid a he ime ( e = 1) when he house is bough. Thiry million yen is borrowed, and he morgage loan is equally paid over weny years. Equal yearly paymen is calculaed a a morgage invesmen rae of 6%. The parens make an educaional plan ha he children will go o a privae elemenary school, a privae 8 I is assumed ha he second child is no born if he householder dies in wo years.

10 472 N. Hibiki junior high school, a privae high school, and a privae universiy. The parameer values used in he examples are shown in Table 1. Table 1: Parameer values Parameers Values number of risky asses n =1 lengh of one period one year reiremen age of a householder 6 years old number of periods T = 3 expeced rae of reurn of a risky asse ñ= :1 sandard deviaion of rae of reurn of a risky asse õ= :2 risk-free rae r =:4 moraliy rae ï 1; (*1) rae of a åre ï 2 =:5 disease rae (*2) ï 4; = ó 1 ï 1; life insurance money per uni í 1; = í 1 (consan) medical insurance money per uni í 4; = í 4 (consan) guaraneed rae on life and medical insurance g 1 =:5 guaraneed rae on åre insurance g 2 =:5 maximum coeécien of severance pay (*3) é U =2 ime of reaching maximum coeécien of severance pay T é =2 iniial ånancial wealh W 1; = 1 (million yen) iniial non-ånancial wealh W 2; = 1 (million yen) depreciaion rae of non-ånancial wealh ç =:3 loss of non-ånancial wealh due o a åre ã= 1 lower bound of cash (million yen) L v; =,L v; = Ä1( 6= ) lower bound of expeced erminal ånancial wealh W E = 7 (million yen) probabiliy level å= :8 number of pahs I =5; *1 The raes are esimaed by he life insurance sandard life able 1996 for men [11]. *2 There mus be a serious disease rae able for medical insurance, and a premium mus be calculaed by using he able. However, he able is no published ouside. In his paper, we assume ha he disease rae ï 4; is ó 1 (> 1) imes he moraliy rae ï 1;. The reason is ha a serious disease causes deah, and he disease rae becomes higher wih age. *3 An amoun of severance pay e (i) is calculaed by muliplying an amoun of wage when a householder reires or dies wih he provision coeécien é in Equaion (2). I is assumed ha he provision coeécien is a piecewise linear funcion wih an upper bound é U a ime T é in Equaion (21). Explicily, e (i) = é m (i) ( =1;:::;T); (2) îí ì ï é = min ; 1 é U ( =1;:::;T; i =1;:::;I; T é î T ): (21) T é Wage income depends on householder's age and occupaion. We calculae wage income of a householder over ime based on he Census of wage in 23 by Minisry of Healh, Labor and Welfare [15]. Consumpion expendiure depends on wage income, family srucure and

11 Muli-Period Model and Insurance Design 473 school (educaion) plan. We calculae average consumpion expendiures wih respec o each number of family and each income level of family based on he naional survey of family income and expendiure in 1999 by Saisic Bureau, Minisry of Inernal Aãairs and Communicaions [16]. We calculae average educaional expenses based on he survey of household expendiure on educaion per suden in 21 [13], he survey of suden life by Minisry of Educaion, Culure, Spors, Science and Technology [14]. We clarify he eãecs of four parameers (facors) associaed wih a serious disease : 1ç a serious disease rae, 2ç medical cos, 3ç he decrease in wage income, 4ç he possibiliy of deah afer having a serious disease. We solve åve kinds of problems for each parameer as in Table 2 o examine he sensiiviy of four parameers. When one of he parameers is examined, he oher parameer values are åxed a `P3' values. We es 17 combinaions in oal. 9 Table 2: Parameers associaed wih a serious disease Parameers Noaions P1 P2 P3 P4 P5 Coeécien of a disease rae ó Medical cos ó Decreasing rae of wage income ó Deah probabiliy afer a serious disease(*4) ï *4 A serious disease causes deah, and herefore i is assumed ha ï 3 (< 1:consan)is he probabiliy ha a householder having a serious disease dies afer one year. For example, he probabiliy is 6% if we se ï 3 =: Resul CVaR Life insurance money CVaR(million yen) ν1 ó 1 ν2 ó 2 ν3 ó 3 λ3 ï 3 Insurance money(million yen) ν1 ó 1 ν2 ó 2 ν3 ó 3 λ3 ï P1 P2 P3 P4 P5 P1 P2 P3 P4 P5 Parameer Parameer Figure 1: CVaR and opimal life insurance money Figure 1 shows he CVaR on he lef-hand side and life insurance money (í 1 u É L)onhe righ-hand side for each combinaion of parameers. For example, a broken line of ó 1 shows 9 For example, when we examine he sensiiviy of ó 1 value, we solve he problems wih one of åve kinds of ó 1 and `P3' values of ó 2, ó 3 and ï 3 (i.e. ó 2 = 15, ó 3 =:2 and ï 3 =:5). Four combinaions are overlapped, and herefore 17(= 5 Ç 4 Ä 3) combinaions are esed.

12 474 N. Hibiki values of he CVaR for åve kinds of ó 1,i.e.P1=2: hrough P5= 4: intable2. 1 When a ó 1 value becomes larger, he CVaR value is smaller because he probabiliies of he decrease in wage income and he increase in medical cos are higher due o a serious disease. When ó 2 and ó 3 values become larger, wage income decreases and medical cos increases, and herefore he CVaR value is smaller. However, he CVaR value is no inçuenced by he ï 3 value, or he probabiliy of he householder's deah afer a householder has a serious disease. Life insurance money is no aãeced by four facors associaed wih a serious disease. 6 Unis of medical insurance 5. Medical insurance money 4 Premium of level paymen unis ν1 ó 1 ν2 ó 2 ν3 ó 3 λ3 ï 3 P1 P2 P3 P4 P5 Parameer Insurance money(million yen) ν1 ó 1 ν2 ó 2 ν3 ó 3 λ3 ï 3 P1 P2 P3 P4 P5 Parameer Premium (housand yen) ν1 ó 1 ν2 ó 2 ν3 ó 3 λ3 ï 3 P1 P2 P3 P4 P5 Parameer Figure 2: Opimal unis, insurance money, and premium for medical insurance Figure 2 shows unis of medical insurance (u É B) on he lef-hand side, medical insurance money (í 4 u É B) on he middle, and premium paymens (y b u É B) on he righ-hand side. When ó 2 and ó 3 values become larger, he householder purchases more unis of medical insurance policy o hedge agains he decrease in wage incomeandheamounofmedicalcos. I means ha he number of unis of medical insurance and premium paymens become larger. Medical insurance money is no inçuenced by he increase in a disease rae ó 1 because wage income does no decrease and medical cos does no increase. However, medical insurance money per uni (í 4 ) becomes small, and he householder needs o purchase more unis of medical insurance policy(u É B) in order o receive medical insurance money which can cover cash ouçow due o a serious disease. Therefore, premium paymens (y b u É B) become large. A ï 3 value does no inçuence he number of unis of medical insurance, medical insurance money, and premium paymens as well as he CVaR and life insurance money. When a householder has a serious disease a ime,ånancial wealh decreases by paymen for he expensive medical cos and he decrease in wage income a ime. Weexaminehe relaionship beween he annual average or maximum decrease in wealh during hiry years and opimal medical insurance money in Figure 3. The decrease in wealh consiss of he decrease in wage income and he increase in medical cos. The average and maximum decreases are almos equal o medical insurance money. I shows ha medical insurance money is used o hedge agains he decrease in cash inçow. Figure 4 shows opimal invesmen unis of a risky asse for each parameer. Invesmen unis decrease gradually over ime. When ó 1, ó 2,andó 3 values become large, invesmen unis increase. The reason is ha a household needs o inves in more amouns of a risky asseocoverhedecreaseinwageincomeandheincreaseinmedicalcos,andoincrease expeced erminal wealh. A ï 3 value does no inçuence opimal invesmen unis. 1 When we show values of he CVaR for åve kinds of ó 1,wecanwrieåveó 1 insead of he expression of P1 hrough P5 on he verical axis so ha readers undersand he meaning of graphs easily. However, we employ he expression as in Figure 1 for lack of space. Readers can know parameer values of P1 hrough P5 by checking Table 2.

13 Muli-Period Model and Insurance Design average decrease 5. average decrease decrease (million yen) maximum decrease insurance money decrease (million yen) maximum decrease insurance money ν2 ó 2 νó 3 Figure 3: Relaionship beween he decrease in wealh and opimal medical insurance money νó 1 1 νó2 2 Invesmen unis of a risky asse ó1 ν 1=2. = 2: ó1 ν 1=2.5 = 2:5 ó1 ν 1=3. = 3: ó1 ν 1=3.5 = 3:5 ó1 ν 1=4. = 4: Invesmen unis of a risky asse νó2 2=.5 = :5 νó2 2=1. = 1: νó2 2=1.5 = 1:5 νó2 2=2. = 2: νó2 2=2.5 = 2: im e im e νó 3 λ3 3 ï 3 Invesmen unis of a risky asse νó3 3=.1 = :1 νó3 3=.15 = :15 νó3 3=.2 = :2 νó3 3=.25 = :25 νó3 3=.3 = :3 Invesmen unis of a risky asse λï3 3=.3 = :3 ï3 λ 3=.4 = :4 ï3 λ 3=.5 = :5 ï3 λ 3=.6 = :6 ï3 λ 3=.7 = : im e im e Figure 4: Opimal invesmen unis of a risky asse 3.4. Life insurance wih a decreasing linear funcion Term life insurance of receiving consan insurance money is a very popular produc. We solve he problem o examine he characerisics of he produc. Four cases are he combinaionsofwokindsofpf and wo kinds of np inroduced in Hibiki and Komoribayashi [5], i.e. pf =andnp =,pf =andnp =1,pf =1andnp =,pf =1andnp =1. pf is one if he problem is solved wih receip of a survivor's pension and zero wihou receip, and np is one if he problem is solved wih exempion from a morgage loan and zero wihou exempion.

14 476 N. Hibiki Figure 5 shows condiional expeced erminal ånancial wealh a he ime of he householder's deah. 11 A value a ime shows an expeced value under he condiion ha a householder does no die in he planning period. Expeced erminal ånancial wealh is increasing as he ime of he householder's deah becomes lae. 12 The reason is ha a household ges a wage in he longer period before a householder dies, and receives life insurance money when a householder dies. If a householder dies earlier, expeced erminal ånancial wealh ends o be lower because a household receives a lower survivor's pension relaive o an amoun of wage. Expeced erminal financial wealh (million yen) price pf=, np= pf=, np=1 pf=1, np= pf=1, np= Expeced erminal price of a risky asse ime of he householder's deah (: a householder does no die.) Figure 5: Condiional expeced erminal ånancial wealh a each ime I is imporan for a household no o have a ånancial problem and o lead a sable life by receiving life insurance money even if a householder dies earlier. Therefore, we should design life insurance ha a household can receive more insurance money as a householder dies earlier. We solve he problem wih a decreasing life insurance policy ha insurance money decreases in proporion o a householder's age. We calculae í 1; wih ë 1; = T Ä+1 in Equaion (8). We solve he problem, and compare a decreasing ype of life insurance wih a consan ype. 11 Condiional expeced erminal ånancial wealh a each ime W ú1 in Figure 5 can be calculaed as follows: where jú 1; j = IX i=1 W ú 1 = 1 jú 3;T j IX i=1 ú (i) 1;,andjú 3;T j = ú (i) 3;T W (i) T ; W ú 1 = 1 jú 1; j IX ú 3;T. i=1 IX i=1 ú (i) 1; W (i) T ( =1;:::;T) 12 If a householder dies afer ime 12, he loan paymen is forgiven. Therefore, expeced erminal ånancial wealh afer ime 12 for np = 1 are larger han hose for np = because he reducion in loan paymen conribues o he increase in erminal ånancial wealh. Broken lines are no smooh, especially a ime 1, 14, 15, and 21. The reason is ha erminal ånancial wealh (W (i) T ) is inçuenced by a erminal price of a risky asse (ö (i) jt ).

15 Muli-Period Model and Insurance Design Consan Decreasing CVaR premium of level paymen for life insurance (housand yen) Consan Decreasing premium of level paymen for medical insurance (housand yen) Consan Decreasing 3 pf=, np= pf=, np=1 pf=1, np= pf=1, np=1 pf=, np= pf=, np=1 pf=1, np= pf=1, np=1 2 pf=, np= pf=, np=1 pf=1, np= pf=1, np=1 Figure 6: CVaR and premium paymens for life insurance and medical insurance Figure 6 shows he CVaR values, and premium paymens for life insurance and medical insurance. The CVaR values of he decreasing ype increases abou 7 million yen or 18%, compared wih he consan ype for pf =1andnp = 1. Premiums of he decreasing ype decreases abou 35% for np =, and abou 45% for np = 1, compared wih he consan ype. We obain dramaic eãecs by inroducing he decreasing ype of life insurance. The change in design of life insurance does no inçuence medical insurance money. pf= pf= life insurance money (million yen) np=(decreasing) np=1(decreasing) np=(consan) np=1(consan) life insurance money (million yen) np=(decreasing) np=1(decreasing) np=(consan) np=1(consan) ime ime Figure 7: Opimal life insurance money Figure 7 shows life insurance money a each ime wihou receiving a survivor's pension (pf = ) onhelef-handside, andwihreceiving(pf = 1) on he righ-hand side. Life insurance money wihou receiving a survivor's pension is larger han insurance money wih receiving for boh ypes. Life insurance money of he decreasing ype is larger unil ime 1, bu smaller afer 1 han ha of he consan ype. The reason premiums of consan ype are larger han hose of he decreasing ype is ha a moraliy rae becomes higher as a householder ges older, and he period more life insurance money can be received for he consan ype is longer han he period for he decreasing ype. Figure 8 shows condiional expeced erminal ånancial wealh a each ime of he householder's deah. When a householder dies earlier, condiional expeced erminal wealh becomes larger because a household can receive larger life insurance money. 13 Expeced erminal ånancial wealh enirely becomes çaer han hose in Figure 5. A decreasing ype of life insurance reduces risk, and has similar expeced erminal ånancial wealh regardless of he ime of he householder's deah, while a household saves premium paymens. 13 We migh se up a decreasing linear funcion wih higher life insurance money when a householder dies earlier.

16 478 N. Hibiki Expeced erminal financial wealh (million yen) price pf=, np= pf=, np=1 pf=1, np= pf=1, np= Expeced erminal price of a risky asse ime of he householder's deah (: a householder does no die.) Figure 8: Condiional expeced erminal ånancial wealh a each ime 4. Opimal Insurance Design The resuls derived in Secion 3.4 show ha we had beer design life insurance so ha a household can receive insurance money o å cashçow needs. We propose a model o decide opimal life and medical insurance money received a each ime, insead of he consan or decreasing insurance money Modiåcaion for opimal design (1) Life insurance Life insurance money a ime (x ) is calculaed by muliplying life insurance money per uni (í 1; ) by he number of unis (u L ), i.e. x = í 1; u L. Life insurance money per uni is no given as inpu parameers, and herefore we need o add a consrain which shows he principle of equalizaion of income and expendiure insead of Equaion (7). We muliply he number of unis of life insurance (u L ) by boh sides in Equaion (7), and we obain u L = TX =1 L (i) u L in Equaion (12) is ransformed as û x where û = ï 1; (1 + g 1 ) : (22) L (i) u L = ú (i) 1; í 1; u L = ú (i) 1; x ( =1;:::;T; i =1;:::;I): (2) Medical insurance Medical insurance money a ime (w )iscalculaedasw = í 4; u B. The principle of equalizaion of income and expendiure is u B = TX =1 B (i) u B in Equaion (12) is ransformed as w where = ï 4; (1 + g 1 ) : (23) B (i) u B = ú (i) 4; í 4; u B = ú (i) 4; w ( =1;:::;T; i =1;:::;I):

17 Muli-Period Model and Insurance Design 479 (3) Oher consrains In addiion o he modiåcaion of he principle of equalizaion of income and expendiure, we need o add and modify he consrains in he formulaion. 1ç Cash çow excep rading asses (D (i) ) :modiåcaion of Equaion (12) D (i) = M (i) + H (i) Ä C (i) ê (i) Ä 1 f6=t g y L;u L + y F u F; + y B;u (i) ë (i) B + ú 1; x + ú (i) 2; í 2 u F;Ä1 +ú (i) 4; w Ä ú (i) 2; ã(1 Ä ç )W (i) 2;Ä1 ( =1;:::;T; i =1;:::;I): (24) 2ç Addiional non-negaiviy consrains x ï ( =1;:::;T); (25) w ï ( =1;:::;T): (26) Excep for he above-menioned consrains, we do no have o change he formulaion in Secion Numerical analysis We compare he combinaion (a) of a decreasing linear funcion for life insurance and a consan funcion for medical insurance iled `Decreasing LI'(Life Insurance) wih he combinaion (b) of opimal funcions for life and medical insurance iled `opimal' in Table 3. We solve he problems for four cases generaed by he combinaion of pf and np. Table 3: Combinaion of funcions for life and medical insurance life insurance money medical insurance money (a) Decreasing LI decreasing funcion (í 1; u L ) consan funcion (í 4 u B ) (b) opimal opimal funcion (x ) opimal funcion (w ) pf=, np= CVaR Decreasing LI Opimal pf=, np=1 pf=1, np= pf=1, np= premium of level paymen for life insurance (housand yen) pf=, np= pf=, np=1 Decreasing LI Opimal pf=1, np= pf=1, np= premium of level paymen for medical insurance (housand yen) pf=, np= Figure 9: CVaR and premiums of level paymen pf=, np=1 Decreasing LI Opimal pf=1, np= pf=1, np=1 Figure 9 shows he CVaR values on he lef-hand side, premium paymens for life insurance on he middle, and premium paymens for medical insurance on he righ-hand side. The CVaR values wih opimal funcions are larger han he CVaR values wih a decreasing linear funcion for life insurance by 1.2 million yen or abou 3% for pf =1andnp =1. Premium paymens for life insurance wih opimal funcions can be reduced by abou 1%. Premium paymens for medical insurance can be reduced by 13% for pf =andnp =, and 1% for pf =1andnp =. We show opimal funcions for life insurance in Figure 1, and describe he characerisics as follow.

18 48 N. Hibiki pf= pf=1 life insurance money (million yen) np=(opimal) np=1(opimal) np=(decreasing) np=1(decreasing) life insurance money (million yen) 15 5 np=(opimal) np=1(opimal) np=(decreasing) np=1(decreasing) ime ime Figure 1: Opimal life insurance money 1ç The opimal funcions also become decreasing funcions enirely as we expec and examine he eãec in Secion ç Amouns of opimal life insurance money rise sharply from ime 2 o 3. The reason is ha he second child will be born a ime 3, and i coss for he household o raise he second child (e.g. educaional cos 14 ) if he householder dies afer ime 3. 3ç Opimal life insurance money drops sharply from ime 1 o ime 11 for np = 1. The reason is ha a house is bough a ime 1, and loan paymens are forgiven if a householder dies afer ime When we solve he problem wih a decreasing ype of life insurance, he CVaR value becomes much larger, and premium paymens become much lower drasically, compared wih a consan ype of life insurance. On he oher hand, when we solve he problem wih an opimal funcion of life insurance, he CVaR value becomes larger by 3%, and premium paymens become lower by 1%, compared wih a decreasing ype of life insurance. The eãec using he opimal funcion is no dramaic. The reason is ha he decreasing linear funcion is similar o he opimal funcion as shown in Figure 1. Figure 11 shows condiional expeced erminal ånancial wealh a each ime of he householder's deah. Theyare çaer han hose in Figure 8. This shows ha condiional expeced erminal ånancial wealh a each ime has similar values, regardless of he ime of he householder's deah. 16 They have almos he same values, regardless of he values of pf and np. The reason is ha opimal life insurance money can be adjused, depending on he values of pf and np as shown from he resuls in Figure 1. The model involving he opimal insurance design is solved usefully so ha he above-menioned feaure can be reçeced, and erminal ånancial wealh can ge less aãeced by he ime of he householder's deah. Figure 12 shows medical insurance money on he lef-hand side, and åre insurance money on he righ-hand side. The broken line wih he legend `decrease' on he lef-hand side shows he decrease in wealh due o a serious disease, i.e. he decrease in wage income and he amoun of medical cos. The broken line wih he legend `loss' on he righ-hand side shows 14 The increases in life insurance money from ime 2 o 3 are abou 6.7 million yen for pf =, and abou 8.8 million yen for pf = 1. The value a ime 3 of he educaional cos is abou 8.95 million yen wih 4% discoun rae. 15 The decrease in life insurance money from ime 1 o 11 is abou 36.4 million yen for np = 1. Annual paymen is million yen for a morgage loan of 3 million yen wih 6% morgage ineres rae, and herefore he value a ime 11 of he morgage loan paymen is million yen wih 4% discoun rae. 16 The reason he values a ime 1 and 21 are higher is ha average prices of a risky asse are higher, and here exiss sampling errors.

19 Muli-Period Model and Insurance Design 481 he loss due o a åre. Thelef-handsideofFigure12showshesameresulsonaverage ha a medical insurance policy is purchased o cover he decrease in wage income and he amoun of medical cos in Secion 3.3. However, he amouns of medical insurance money a each ime are unsable, and he opimal funcion of medical insurance is easily aãeced by sampling error. Opimal åre insurance money is nearly equal o he maximum loss of non-ånancial wealh as well as he resuls derived by he previous models. Expeced erminal financial wealh (million yen) price pf=, np= pf=, np=1 pf=1, np= pf=1, np= Expeced erminal price of a risky asse ime of he householder's deah (: a householder does no die.) Figure 11: Condiional expeced erminal ånancial wealh a each ime medical insurance money (million yen) pf=, np= pf=, np=1 pf=1, np= pf=1, np=1 decrease fire insurance money (million yen) pf=, np= pf=, np=1 pf=1, np= pf=1, np=1 loss ime ime Figure 12: Opimal medical and åre insurance money 5. Examining sampling error We ånd he characerisics of he model wih numerical examples. However, sampling error occurs because a hiry-periods model is solved wih 5, simulaed pahs. We solve kinds of problems wih diãeren random seeds, and we examine sample disribuions of opimal soluions. We need o provide kinds of daase associaed wih prices of a risky asse (ö (i) j;), -1 parameers for he householder's deah (ú (i) 1;, ú (i) 3; ), -1 parameer for a house

20 482 N. Hibiki of a åre (ú (i) 2; ), and -1 parameer for he householder's serious disease (ú (i) 4; ). We call he model wih consan life insurance money in Secion 3.2 `model A', and he model wih opimal life insurance money in Secion 4 `model B' Numerical Analysis : Model A Figure 13 shows he change in he average of he CVaR on he lef-hand side, life insurance money on he middle, and medical insurance money on he righ-hand side as we increase he problems solved wih diãeren random seeds. There are åve kinds of perceniles derived wih diãeren random seeds in Figure 13. The average values converge beween fory percenile and sixy percenile by solving abou hiry problems wih diãeren random seeds CVaR (million yen) average 3% 4% 5% 6% 7% number of random seeds Life insurance money (million yen) average 3% 4% 5% 6% 7% number of random seeds Medical insurance money (million yen) average 3% 4% 5% % 7% number of random seeds Figure 13: Convergence of he CVaR and life and medical insurance money for model A 275 random seeds 275 Percenile % 25% 5% 75% % invesmen unis invesmen unis ime ime Figure 14: Opimal invesmen unis of a risky asse We show he number of invesmen unis of a risky asse for kinds of random seeds on he lef-hand side, and for åve kinds of perceniles on he righ-hand side in Figure 14. We ånd ha he number of invesmen unis of a risky asse decreases hrough ime as in Figure 4. Perceniles change smoohly hrough ime because perceniles are calculaed separaely a each ime. The values over ime are volaile as in Figure 4 when each problem is solved. However, he values are expeced o çucuae around he average values over ime. Le z1 ké be he opimal number of invesmen unis for a risky asse when he problem is solved wih he k-h random seed. The average of z1 ké a ime for kinds of problems is as z É 1 = 1 X k=1 z ké 1 ( =;:::;T Ä 1):

21 Muli-Period Model and Insurance Design 483 Le Dz ké be he average of deviaion from he average z É 1 for each random seed as Dz ké = 1 T T Ä1 X = ê z ké 1 Ä z É 1ë (k =1;:::;): A sandard deviaion of Dz ké is 1.69, a maximum value is 4.35, and a minimum value is Ä3:72. These values are much smaller han he number of invesmen unis, and herefore i can be said ha he values z1 ké çucuae around he average z É 1. We show condiional expeced erminal ånancial wealh a each ime of he householder's deah for kinds of random seeds on he lef-hand side, and for seven kinds of perceniles on he righ-hand side in Figure 15. We ånd he same characerisics as in Figure 5 even if we use diãeren random seeds. Expeced erminal financial wealh (million yen) random seeds Time of he householder's deah (: a householder does no die.) Expeced erminal financial wealh (million yen) Percenile % 25% 5% 75% 9% 95% % Time of he householder's deah (: a householder does no die.) Figure 15: Condiional expeced erminal ånancial wealh a each ime for model A 5.2. Numerical analysis : Model B We show life insurance money for kinds of random seeds on he lef-hand side, and for åve kinds of perceniles on he righ-hand side of Figure 16. We obain opimal soluions sably even if we use diãeren random seeds. The amouns of life insurance money rise unil ime 3, and decline aferward. We show medical insurance money for kinds of random seeds on he lef-hand side, and for åve kinds of percenile on he righ-hand side of Figure 17. The åfy percenile (median)onherigh-handsideoffigure17is almos equal o he sum of he decrease in wage income and he amoun of medical cos, and herefore we ånd a medical insurance policy is purchased o cover he loss due o a serious disease. However, he amouns of opimal medical insurance money çucuae around he average as shown on he lef-hand side of Figure 17, and we canno ignore he inçuence of sampling error. We show condiional expeced erminal ånancial wealh a each ime of he householder's deah for kinds of random seeds on he lef-hand side, and for seven kinds of perceniles on he righ-hand side of Figure 18. We can ånd he characerisics shown in Figure 11 ha erminal ånancial wealh can ge less aãeced by he ime of he householder's deah. The åfy percenile (median) on he righ-hand side of Figure 18 is almos ça. Some values

22 484 N. Hibiki 14 random seeds 14 Percenile Life insurance money (million yen) Life insurance money (million yen) % 25% 5% 75% % ime ime Figure 16: Opimal life insurance money for model B Medical insurance money (million yen) random seeds Medical insurance money(million yen) Percenile % 25% 5% 75% % decrease ime ime Figure 17: Opimal medical insurance money for model B Expeced erminal financial wealh (million yen) random seeds Expeced erminal financial wealh (million yen) Percenile % 25% 5% 75% 9% 95% % Time of he householder's deah (: a householder does no die.) Time of he householder's deah (: a householder does no die.) Figure 18: Condiional expeced erminal ånancial wealh a each ime for model B