Dynamic ånancial planning for a household in a multi-period optimization approach

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1 Dynamic ånancial planning for a household in a multi-period optimization approach Norio Hibiki É, Katsuya Komoribayashi y May 30, 2006 Abstract We discuss an optimization model to obtain an optimal investment and insurance strategy for a household. In this paper, we extend the studies in Hibiki, Komoribayashi and Toyoda(2005) for the practical use, and propose the model with three factors we need to consider if the householder is dead. Three factors are receipt of survivor's pension, exemption from housing loan payments, and change of the consumption level. We examine the additional eãect by three factors with numerical examples. We analyze the sensitivity of parameters associated with home buying in order to examine the home buying strategy. Moreover, We aggregate the paths we may not use to calculate the downside risk, and we formulate the simpliåed model which can be solved faster than the original model. keywords: multi-period optimization, ånancial planning for a household,optimal investment strategy, life insurance, home buying strategy 1 Introduction We discuss an optimization model to obtain an optimal investment and insurance strategy for a household. Recently, ånancial institutions have promoted giving a ånancial advice for individual investors. How much will the household need to save when the householder retires? What kind of ånancial products should be purchased to hedge various risk such as market risk, inçation risk, and catastrophe insurance risk? Financial institutions need to recommend appropriate ånancial products to answer these questions in conjunction with a life cycle, current asset, and future income. We clarify how a set of asset mix, life insurance, and åre insurance aãect asset and liability management for a household. We develop a multi-period optimization model which involves determining a set of ånancial products, hedging risk associated with a life cycle of the household and saving for the old age, and the simulated path approach (Hibiki, 2001b) can be used to solve this problem. There are some studies in the literature for individual optimal investment strategy; Samuelson(1969), Merton(1969), Merton(1971), Bodie, Merton and Samuelson(1992). Ibbotson, Chen, Milevsky and Zhu(2005) advocate an optimal model with the inclusion of wage income, consumption expenditure, and life insurance in addition to asset allocation. Yoshida, Yamada and Hibiki(2002) solve an optimal asset allocation problem for a household using a multi-period É Keio University, hibiki@ae.keio.ac.jp y The Center for Financial Industry Information Systems, komoribayasi@åsc.or.jp 1

2 optimization approach. Hibiki, Komoribayashi and Toyoda(2005) describe a multi-period optimization model to determine an optimal set of asset mix, life insurance and åre insurance in conjunction with their life cycle and characteristics. The model is examined with numerical examples. In addition, some ånancial advices for three households are illustrated for practical use, and the results which coincide with the practical feeling are obtained. Risk associated with the householder's death is hedged by life insurance. In this paper, we extend the studies in Hibiki, Komoribayashi and Toyoda(2005) for the practical use, and propose a model with three factors we need to consider if the householder is dead. Three factors are receipt of survivor's pension, exemption from the home loan payments, and change of the consumption level. We examine additional eãects by three factors with numerical examples. Given four kinds of parameters associated with home buying, or, down payment, loan period, and mortgage interest rate, we solve the problems. We analyze the sensitivity of these parameters in order to examine the home buying strategy. It takes much to solve the large-scale problems with a long planning period. For example, it is assumed that the householder is thirty years old, and will retire thirty years after. When one period is one year and the number of paths is åve thousand, the number of the planning period is thirty periods, both numbers of constraints and decision variables are about 150,000, and it takes about six minutes to solve a problem 1. It is necessary to solve the problems faster in order to examine the model with a lot of numerical examples. If we calculate a downside risk measure, we do not use paths where a terminal wealth is larger than a target associated with the downside risk. We aggregate paths we may not use to calculate the downside risk, and we formulate a simpliåed model which can be solved faster than the original model. This paper is organized as follows. We introduce the concept of the simulated path approach in Section 2. We deåne a household and three kinds of ånancial products, or securities, life insurance, and åre insurance to describe the model structure. We consider three factors in addition to Hibiki, Komoribayashi, and Toyoda[2005] for practical use: 1ç survivor's pension, 2ç exemption from housing loan payments, 3ç change of the consumption level. Section 3 shows the formulation of the multi-period ALM optimization model for a household. We demonstrate some numerical tests in Section 4. Section 5 shows the sensitivity analysis of the parameters associated with home buying in order to examine the home buying strategy. Section 6 shows the formulation using a grouped path and some numerical tests in order to solve problems fast. Section 7 provides our concluding remarks. 2 Solution technique and model structure At årst, we explain a multi-period optimization in the simulated path approach to determine an optimal set of asset mix and insurance. Next, we deåne a household, and describe the income and the consumption expense. We clarify the characteristics of ånancial instruments such as securities, life insurance and åre insurance. 1 Problems are solved using NUOPT (Ver ) { mathematical programming software package developed by Mathematical System, Inc. { on Windows XP personal computer which has 2.13 GHz CPU and 2GB memory. All of the problems in this paper are also solved using the same computer and software. 2

3 2.1 Simulated path approach It is critical for stochastic modeling to handle uncertainties and investment decisions appropriately. The decisions have to be independent from knowledge of actual paths that will occur. Thus, we must deåne a set of decision variables and a set of constraints to prevent an optimization model from being solved by anticipating events in the future. In addition, we need a suécient number of paths to get a better accuracy with respect to the future possible events. The concept of scenarios is typically employed for modeling random parameters in the multiperiod stochastic programming models. Scenarios are constructed via a tree structure as in the left-hand side of Figure 1 (see Mulvey and Ziemba, 1995 and 1998 for a detailed discussion). The model is based on the expansion of the decision space, taking into account a conditional nature of the scenario tree. Conditional decisions are made at each node, subject to the modeling constraints. To ensure that the constructed representative set of scenarios covers the set of possibilities to a suécient degree, the numbers of decision variables and constraints in the scenario tree may grow exponentially. This model is called a scenario tree model. Scenario Tree Simulated Paths Figure 1: A scenario tree and simulated paths Meanwhile, simulated paths give another description of scenarios shown as in the right-hand side of Figure 1. Hibiki [2000, 2001b] developed a simulated path model in a multi-period optimization framework. If we formulate stochastic diãerential equations or series models associated with asset returns, discrete asset returns are generated by a standard Monte Carlo simulation technique to describe uncertainties more accurately than would the scenario tree as in the left-hand side of Figure 1. However, if a decision is made on the associated path, the model is solved anticipating the event in the future. Therefore, the rule that the same investment decision is made at each is deåned to satisfy the non-anticipativity condition in the simulated path model. We do not use the scenario tree model, but the simulated path model. This is because the model involves determining a set of optimal life and åre insurance money, and we need a lot of 3

4 paths at each to describe the mortality rate and the rate of the åre. According to the life insurance standard life table(1996), the mortality rate for 50 years old men is 0.379%. Even if the householder dies on a path to describe the appropriate mortality rate, we need 264 paths at each. If we construct a scenario tree over 30 years, we have to generate an enormous tree, and we cannot solve the problem in practice. Therefore, it is essential for this type of the problem to use the simulated path approach, and we use the simulated path model in this paper Setting We attach a superscript (i) to the random (path dependent) parameters in order to formulate the model in the simulated path approach Household We deåne a household as a group composed of a householder and members of family in this paper. Wealth at t held by a household can be divided into two kinds of wealth: ånancial wealth W (i) (i) 1;t and non-ånancial wealth W 2;t. The household is exposed to risk associated with two kinds of accidents: the death of the householder and the åre of the house. It is assumed that the death of the householder makes wage earnings stop, and the åre of the house damages a fraction ã of non-ånancial wealth. The household can purchase the life insurance and the åre insurance to hedge risk in addition to the investment in securities such as stocks and bonds Incomes Income at t is the householder's wage m t if the householder is alive and investment returns from ånancial wealth W (i) 1;t. If the householder dies, the household cannot get wages, but draw the survivor's pension. The amount of the survivor's pension is calculated based on the wage level. Let a (i) t m be the amount of the survivor's pension because the amount of the survivors' employees' pension is dependent on the of the householder's death t m.theincomebythe wage or the survivor's pension M (i) t can be shown as follows : M (i) t = ú (i) ê 3;t m(i) t + 1 Ä ú (i) ë 3;t where ú (i) 3;t a (i) t m are one if a householder is alive on the path i at t and zero if otherwise comsumption expenses There assumes to be two kinds of expenses: the living expenses C (i) 1;t and the purchase of nonånancial assets C (i) 2;t, such as a house, goods, and repair costs. We need to pay the restoration cost if the åre of the house occurs. (1) Expenses for purchasing a house 2 Hibiki(2001c, 2003) developed the hybrid model, which not only describes the uncertainties on the simulated path structure but also makes conditional decisions on the tree structure. The hybrid model is allowed to expand the decision space and to make conditional decisions as in the scenario tree model. The simulated path model is a special version of the hybrid model. The formulation and numerical tests with the hybrid model are our future research. 4

5 We assume that a household purchases a house with a down payment and a debt loan from banks (H t ). Let t e be the when the house is purchased. The debt loan H te is the diãerence between the price of the house and the down payment. We explain the relationship between a cash çow of purchasing a house and the change of non-ånancial wealth. The debt loan H te is a cash inçow, and the consumption expenditures for non-ånancial asset C 2;t is a cash outçow. However, the non-ånancial wealth, W 2;te is increased by expenditures C 2;te at t e. The household has to pay the debt loan periodically under the determined mortgage interest rate and the loan period after the t e + 1. In this paper, we include periodic payments C1;t 2 in the expenditures for life (C (i) 1;t ). (2) Restoration cost due to the åre It is assumed that the fraction ã of non-ånancial wealth W 2;t is damaged and the following restoration cost A (i) t is paid if the åre of the house occurs 3. A (i) t = ú (i) 2;t W (i) 2;tÄ1ã(1 Ä ç) where ú (i) 2;t are one if the åre of the house occurs and zero if it does not occur, and çis depreciation ratio of non-ånancial wealth. A (i) t does not aãect the non-ånancial wealth. Instead, it aãects the cash çow as shown in Equation (7) and (8) in Section 3. (3) Expenditures for living C (i) 1;t We show three kinds of parameters as follows. C 1(i) 1;t : costs independent of the householder's death, such as education costs and rent C1;t 2 : annual payments for mortgage loan (when the householder is alive). C 3(i) 1;t : other costs for living except C 1(i) 1;t and C1;t 2 which are supposed when householder is alive. Next, we explain how to compute the annual payments for the debt loan and other expenditures for living dependent on the householder's death. 1ç Mortgage loan As mentioned earlier, ú (i) 3;t are one if a householder is alive and zero if otherwise. If the household purchases the group credit insurance 4, the loan payment is forgiven after the householder was dead. This shows that the annual payments can be ú (i) 3;t C2 1;t. However, the loan payment is not forgiven if the household purchases the house after the householder was dead. By using the condition that ú (i) 3;t =0fort>t e if ú (i) 3;t e = 0, the annual payments for mortgage loan can be ê 1 Ä ú (i) 3;t e + ú (i) ë 3;t C1;t: 2 2ç Change of the consumption level The household will think it cannot keep the consumption level if the householder dies. It is assumed that the household can keep the normal consumption level if the householder is alive, however the consumption level must be î s the normal level if the householder is dead, where î is the parameter associated with the consumption level. For example, we set î = 1 when the household keeps the normal level, and we set î= 0:7 when it has to allow for the 7 3 Non-ånancial wealth is decreased by A (i) t due to the åre, but it is increased by A (i) t by spending the same money to recover the loss. 4 Insurance fee for the mortgage loan from ånancial institutions in the private sector is included in the mortgage interest rate. The group credit insurance needs to be purchased for the mortgage loan from Government Housing Loan Corporation 5

6 consumption level. Therefore, other costs for living except C 1(i) 1;t and, 5 n ú (i) ê 3;t + 1 Ä ú (i) ë o 3;t î C 3(i) n 1;t = î+(1äî)ú (i) o 3;t C 3(i) 1;t : The expenditures for living are calculated in total as follows. C (i) ê 1;t = C1(i) 1;t + 1 Ä ú (i) 3;t e + ú (i) ë n 3;t C1;t 2 + î+(1äî)ú (i) o 3;t Securities C 3(i) 1;t ê 1 Ä ú (i) 3;t e + ú (i) ë 3;t C1;t 2 are Investment in risky assets contributes to a hedge against inçation. We invest in n risky assets and cash. A rate of return R jt of risky asset j at t is calculated using the price ö jt as follows. R jt = ö jt Ä 1; (t =1;...;T) (1) ö j;tä1 Arisk-freerater t at t(= 0; 1;...;T Ä 1) is åxed in the period from t to t +1. We can assume any probability distributions of R jt and r t in the simulated path approach if we can sample random paths for R jt and r t. However, it is assumed that R is normally distributed with the mean vector ñ, and the covariance matrix Ü (R ò N(ñ; Ü), and r t is constant for all t. We calculate the price ö jt by using R jt Life insurance We use the term life insurance with maturity T against the householder s death. If a householder purchases the term life insurance and dies until T, the household can receive the insurance money. In this model, we look upon the life insurance as the ånancial product which can hedge risk associated with the wage income earned by the householder. When the insurance policy is designed, pure premium should be determined so that the present value of future premium income can be equal to the present value of future premium payments. It is called the principle of equalization of income and expenditure. The yield used in calculating the present value is called the guaranteed interest rate. Using the principle of equalization of income and expenditure, the relationship between a unit of present value of premium income and the corresponding insurance money í 1 is shown as : ( TX í 1 ï T ) Ä1 1;t 1= (1 + g t=1 1 ) t ; or í X ï 1;t 1 = (1 + g t=1 1 ) t (2) where g 1 is the guaranteed interest rate of the insurance against death with maturity T,and ï 1;t is the mortality rate at t, or the probability that the person who is alive at 0 will die at t. We can select single payment or level payment when we pay the life insurance premium. Premium of single payment per unit y f1 is equal to a unit of the present value of future premium income. 5 If the householder is alive, the consumption level is ú (i) 3;t s the normal level because ú (i) 3;t = 1 and ê ë 1 Ä ú (i) 3;t î =0. If the householder is dead, the consumption level is î s the normal level because ú (i) 3;t ê ë =0 and 1 Ä ú (i) 3;t î= î. 6

7 y f1 =1 (3) The premium of level payment per unit y f2 is calculated as follows, because only insured person who is alive pays the premium tx 1 Ä ï T Ä1 X >< 1;i >= i=0 y f2 = 6 (1 + g 4 t=0 1 ) t 7 5 >: >; Ä1 (4) Fire insurance The household purchases one year åre insurance to hedge the damage of non-ånancial wealth due to the åre. The household can update the insurance contract every year, and purchase the åre insurance policy corresponding to the future non-ånancial wealth. Using the principle of equalization of income and expenditure, the relationship between one unit of present value of premium income and the corresponding insurance money í 2 is shown as : 1= í 2ï 2 ; or í 2 = 1+g 2 (5) 1+g 2 ï 2 where g 2 is the guaranteed interest rate of the one year åre insurance, and ï 2 istherateofthe åre, or the probability that the åre occurs. It is independent on t. We can only select single payment because of the one year åre insurance. The premium of single payment per unit y F is equal to a unit of the present value of future premium income. y F =1 (6) 3 Multi-period ALM optimization model for a household We formulate the multi-period optimization model in the simulated path approach. We assume that the current is 0(t = 0), and a householder retires at T, which is a planning horizon. As mentioned in Section 2, we invest in n risky assets and cash, and we can rebalance the positions at each. We purchase T -years life insurance at 0, and one-year åre insurance which is updated every year in the planning period. We can select single payment life insurance or level payment life insurance. 3.1 Notations (1) Subscript/Superscript j :asset(j =1;...;n). t :(t =1;...;T). i :path(i =1;...;I). (2) Parameters I ö j0 : number of simulated paths. : price of risky asset j at 0, (j =1;...;n). 7

8 ö (i) jt r 0 r (i) tä1 : price of risky asset j of path i at t, (j =1;...;n; t =1;...;T; i =1;...;I). ö (i) ê j1 = 1+R (i) ë j1 ö j0 ; (j =1;...;n; i =1;...;I) ö (i) ê jt = 1+R (i) ë jt ö (i) j;tä1 ; (j =1;...;n; t =2;...;T; i =1;...;I) where R (i) jt is the rate of return of risky asset j on path i at t. :interestrateinperiod1,(therateat0). : interest rate in period t (the rate of path i at t Ä 1), (t =2;...;T; i =1;...;I). ú (i) 1;t : one if a householder dies on path i at t and zero if otherwise. ú (i) 2;t : one if the åre of the house occurs and zero if it does not occur. ú (i) 3;t : one if a householder is alive on path i at t and zero if otherwise. ï 1;t : mortality rate at t : ï 1;t =Pr(ú 1;t =1)= 1 I ï 2 : rate of the åre(which is independent) : ï 2 =Pr(ú 2;t =1)= 1 I g 1 f 1 : guaranteed interest rate on life insurance policies. : one if single payment life insurance is bought and zero if level payment life insurance is bought. y f1 : premium of single payment life insurance per unit: y f1 =1 (A unit of insurance policy corresponds to the present premium of 1 yen.) y f2 : premium of level payment life insurance per unit : ( T X Ä1 1 Ä P ) Ä1 t y f2 = i=0 ï 1;i (1 + g t=0 1 ) t y (i) L;t : premium of life insurance per unit at t: y (i) L;t = y f 1 Åf 1 ú 4;t + y f2 Å(1 Ä f 1 )ú (i) 3;t,whereú 4;0 =1,ú 4;t =0(t6= 0). ( X T ) Ä1 í 1 : life insurance money per unit : í 1 = t=1 IX i=1 ï 1;t (1 + g 1 ) t L (i) t : life insurance money per unit on path i at t: L (i) t = ú (i) 1;t í 1 g 2 : guaranteed interest rate on åre insurance policies. y F : premium of åre insurance per unit : y F =1 í 2 : one year åre insurance money per unit : í 2 = 1+g 2 ï 2 F (i) t : one year åre insurance money per unit on path i at t : F (i) t = ú (i) 2;t í 2 ã A (i) ú (i) 1;t : loss ratio of non-ånancial wealth due to the åre of the house. t : loss of non-ånancial wealth due to the åre of the house on path i at t: A (i) t = ú (i) 2;t W (i) 2;tÄ1ã(1 Ä ç)å where ç is depreciation ratio of non-ånancial asset. M (i) t : wage income a householder earns or survivor's pension a household receives on path i at t: H (i) t : debt loan on path i at t. IX i=1 ú (i) 2;t 8

9 C (i) t : total consumption expenditures on path i at t. W (i) 1;t : ånancial wealth on path i at t. (W 1;0 is an initial ånancial wealth at 0.) W (i) 2;t : non-ånancial wealth on path i at t : W (i) (i) 2;t =(1Äç)W 2;tÄ1 + C(i) 2;t (W 2;0 is an initial non-ånancial wealth at 0.) W E å L v;t : lower bound of expected terminal ånancial wealth : Probability level used in the CVaR calculation. : lower bound of cash at t. WhenL v;t < 0, the borrowing can be allowed. (3) Decision variables z jt : investment unit of risky asset j at t. (j =1;...;n; t =0;...;T Ä 1) v 0 :cashat0 v (i) t : cash of path i at t. (t =1;...;T Ä 1) u L : number of life insurance bought at 0. u F;t : number of one-year åre insurance bought at t. V å q (i) : å-var used in the CVaR calculation. :shortfallbelowå-var(ë V å ) of terminal ånancial wealth(ë W (i) ê on path i, q (i) ë max V å Ä W (i) 1;T ë,(i ; 0 =1;...;I) 3.2 Objective function, return requirement and cash çow except trading asset (1) Objective function The objective is the maximization of the CVaR associated with terminal ånancial wealth subject to the minimum return requirement, as follows. ( ) 1 IX CVaR å = Max V å Ä q(i) (1 Ä å)i å W (i) 1;T Ä V å + q (i) ï 0; (i =1;...;I) i=1 Even if CVaR of W 0 Ä W (i) 1;T is used to minimize the objective, we have the same solutions as the solutions derived from the maximization of CVaR of W (i) 1;T. (2) Return requirement We deåne the expected terminal ånancial wealth E[W 1;T ] as a return measure. The lower bound is W E, and therefore the minimum return requirement is formulated as : IX W (i) 1;T ï W E i=1 (3) Cash çow except trading assets Cash çow constraints are important in the multi-period optimization approach. Cash çow except trading assets D (i) t is associated with income, expenditures, and insurance. It is formulated as follows. 1ç t =1; ÅÅÅ;T Ä 1 D (i) t = M (i) t + H (i) t Ä C (i) t Ä y (i) L;t u L Ä y F u F;t + L (i) t u L + F (i) t u F;tÄ1 Ä A (i) t ; (t =1;...;T Ä 1) (7) 1;T ) 9

10 2ç t = T : Insurance payment is not required at T. D (i) T = M (i) T + H(i) T Ä C(i) T + L(i) T u L + F (i) T u F;TÄ1 Ä A (i) T (8) 3.3 Formulation (W (i) Maximize 1 IX V å Ä q (i) (1 Ä å)i i=1 (9) subject to ö j0 z j0 + v 0 + y L;0 u L + y F u F;0 = W 1;0 () 1;1 =) n X (W (i) 1;t =) ö (i) j1 z j0 +(1+r 0 )v 0 + D (i) 1 = ö (i) ê jt z j;tä1 + 1+r (i) ë tä1 v (i) tä1 + D(i) t = 8 W (i) < 1;T = : 1 I IX i=1 ö (i) ê jt z j;t Ä1 + 1+r (i) ë T Ä1 v (i) ö (i) j1 z j1 + v (i) 1 ; (i =1;...;I) (11) ö (i) jt z jt + v (i) t ; (t =2;...;T Ä 1; i =1;...;I) (12) 9 = T Ä1; + D(i) T ; (i =1;...;I) (13) W (i) 1;T ï W E (14) W (i) 1;T Ä V å + q (i) ï 0; (i =1;...;I) (15) z jt ï 0; (j =1;...;n; t =0;...;T Ä 1) v 0 ï 0 v (i) t ï L v;t ; (t =1;...;T Ä 1; i =1;...;I) u L ï 0 u F;t ï 0; (t =0;...;T Ä 1) q (i) ï 0; (i =1;...;I) V å : free 4 Numerical examples 4.1 Preparation We test numerical examples using the parameters (Household B in Section 6) in Hibiki, Komoribayashi, and Toyoda[2005]. A householder is 30 years old and a spouse is 28 years old. The årst child is 0 years old, and the second child will be born in three years. The householder works at a ånancial institution, and the household plans that it will prepare 20 million yen as a down payment eleven years later and buy an apartment in the center of Tokyo which costs 50

11 million yen. 20 million yen is paid at the (t e = 11) when the house is bought. We borrow 30 million yen and the mortgage loan is equally paid over 20 years. Equal yearly payment is calculated with the mortgage investment rate(6%) 6. The parents make an educational plan that the children will go to the private elementary school, junior high school, high school, and university. The parameter values used in the examples are shown in Table 1. Table 1: Parameter values Parameter Value number of risky assets n =1 length of one period one year retirement age of a householder 60 years old number of periods T = 30 expected rate of return of a risky asset ñ= 0:1 standard deviation of rate of return of a risky asset õ= 0:2 risk-free rate r = 0:04 mortality rate ï 1;t (y) rate of the åre ï 2 =0:005 guaranteed rate on life insurance g 1 =0:05 guaranteed rate on åre insurance g 2 =0:05 payment of life insurance level payment: f 1 =0 initial ånancial wealth(million yen) W 1;0 = initial non-ånancial wealth(million yen) W 2;0 = depreciation rate of non-ånancial wealth ç= 0:03 loss of non-ånancial wealth due to the åre ã= 1 lower bound of cash (million yen) L v;0 =0,L v;t = Ä(t 6= 0) lower bound of expected terminal ånancial asset(million yen) W E =52:616 probability level å = 0:8 number of paths I =5; 000 y The rates are estimated by the \life insurance standard life table 1996 for men. The wage income depends on the householder's age and his occupation. We calculate the wage income of the household over based on the Census of wage by Ministry of Health, Labor and Welfare(2003). The household whose householder is a ånancial institution employee has a highly increasing rate in salary until 50 years old, however the wage income declines afterwards. The consumption expenditure depends on the wage income, family structure and school(education) plan. We calculate average consumption expenditures with respect to each number of family and each income level of family based on the national survey of family income and expenditure(1999) by Statistic Bureau, Ministry of Internal Aãairs and Communications. We calculate average educational expenses based on the survey of household expenditure on education per student(2001), the survey of student life by Ministry of Education, Culture, Sports, Science and Technology. 6 The annuity is million yen. We have 19 payments from t =12tot = 30, and one payment after retirement. It is assumed that the necessary terminal ånancial wealth is 70 million yen and the severance pay is 20 million yen. Therefore the lower bound of the expected terminal ånancial wealth (W E ) is million yen. 11

12 We test the following 20 combinations to clarify the eãects of three factors : 1ç receiving the survivor's pension, 2ç exemption from the mortgage loan, 3ç change of the consumption level. Yes No receiving the survivor's pension pf = 1 pf = 0 2cases exemption from the mortgage loan np =1 np =0 2cases change of the consumption level î= 0:6~1:0(by 0.1) 5 cases The model in Hibiki, Komoribayashi, and Toyoda[2005] corresponds to the model with pf = 0, np =0,î= Result CVaR (million yen) Insurance money (million yen) kappa kappa Figure 2: CVaR and life insurance money Figure 2 shows the CVaR on the left-hand side and life insurance money on the right-hand side. When the household keeps the consumption level lower { we solve the problem with lower î, the CVaR is higher and the household needs the lower life insurance money. This is because the receipt of the life insurance money and the reduction in consumption can make up for the loss of wage income caused by the householder's death. If the household receives the survivor's pension (pf = 1), the life insurance money can be bought lower by about 30 million yen for the same reason. Exemption from mortgage payment slightly improves the objective function value. However it does not aãect the objective function value and life insurance money. The reason is that we purchase the life insurance at 0, but the house is bought at 11, and therefore the loan payment is not forgiven if the house is bought after the householder was dead. However, this does not show that exemption from mortgage payment is not eãective. Figure 3 shows the conditional expected terminal ånancial wealth at the of the householder's death. The value at 0 shows the expected value under the condition that the householder does not die in the 12

13 planning period 7. If the householder dies after 12, the loan payment is forgiven. Therefore, the expected terminal ånancial wealth after 12 for np = 1 are larger than those for np =0 because the reduction in loan payment contributes to the increase in the terminal ånancial wealth. The eãect of the reduction in loan payment fades as the of the householder's death becomes late, and the diãerence between the value for np = 1 and the value for np =0 diminishes. Expected terminal financial wealth (million yen) risky asset price Expected terminal price of the risky asset of the householder's death (0: the householder does not die.) Figure 3: Conditional expected terminal ånancial wealth at the of the householder's death(î= 1) The expected terminal ånancial wealth is increasing as the of the householder's death becomes late. The reason is that the household gets the wage income in the longer period before the householder dies, and the life insurance money when the householder dies. The expected wealth when the householder is alive is smaller than the expected wealth when the householder dies after 14. If the householder dies earlier, especially before buying a house, the householder receives the life insurance money. However, the expected terminal wealth tends to be lower because of the lower survivor's pension 8. The left-hand side of Figure 4 shows investment units of the risky asset for åve kinds of îvalue, pf =1,andnp = 1. The trend of the optimal investment units at each is not dependent on î value, however the lower î value is, the smaller the investment unit is. The reason is that terminal ånancial wealth increases by reducing the consumption level, and therefore we do not have to take risk by investing in a risky asset. We have the same characteristics for the combination of both pf and np. The right-hand side of Figure 4 shows investment units 7 The reader should pay attention to looking at Figure 3. 8 The conditional expected terminal wealth at 1 is large because the conditional expected terminal price of a risky asset is high. This is a sampling error because there are four paths at 1 when the householder dies. 13

14 of a risky asset for four combinations of both pf and np, andî = 1. When we consider the receipt of survivor's pension and exemption from the loan payment, we can hedge risk against the householder's death and expect the increase in the terminal ånancial wealth. Therefore the investment units of a risky asset are reduced kappa=1 Investment unit of the risky asset kappa=0.6 kappa=0.7 kappa=0.8 kappa=0.9 kappa=1.0 Investment unit of the risky asset Figure 4: Investment unit of the risky asset Average cash (million yen) kappa=0.6 kappa=0.7 kappa=0.8 kappa=0.9 kappa=1.0 Average cash (million yen) kappa= Figure 5: Average cash Figure 5 shows the average cash for åve kinds of î value, pf =1,andnp = 1 on the lefthand side, and for four combinations of both pf and np, andî = 1 on the right-hand side. The average cash increases gradually until, however it decreases drastically at 11 because of purchasing a house. After 12, it increases slightly over years, however it increases drastically after 25 because the second child graduates from university and we do nothavetopaytheeducationcost. Whenîvalue is small, the household receives the survivor's pension, or the loan payment is forgiven, average cash increases because it expects to increase the terminal ånancial wealth, and it reduces the investment in a risky asset. Figure 6 shows the investment ratio of a risky asset for åve kinds of î value, pf =1,and np = 1 on the left-hand side, and for four combinations of both pf and np, andî = 1 on the right-hand side. The average cash increases much more than the investment unit of a risky asset 14

15 until the the house is bought. On the other hand, the average investment ratio of a risky asset is reduced slightly because the average price of a risky asset rises with the % expected rate of return. 7 9 kappa=1 Average ratio of the risky asset % kappa=0.6 kappa=0.7 kappa=0.8 kappa=0.9 kappa= Average ratio of the risky asset % Figure 6: Average ratio of the risky asset Though the investment unit of a risky asset does not change at 11, the average investment ratio of a risky asset increases drastically because the house is bought and a big cash outçow occurs to make a down payment. The average ratio of a risky asset increases gradually after the. However, it goes down as the average cash goes up. The result is summarized as follows. When the survivor's pension is received, the loan payment is forgiven, and the consumption level îis small, the cash increases and the average investment ratio of a risky asset decreases. Figure 7 shows the sampled distributions of the terminal ånancial wealth derived by the optimal solutions. The right-hand side is a magniåed view below the VaR. The tail of the distribution shifts to the right and the downside risk can be decreased by considering the survivor's pension. This is the result we expect from the CVaR in the left-hand side of Figure % 8 16% Cumulative probability Cumulative probability 14% 12% % 8% 6% 4% % 2% Terminal financial wealth (million yen) Terminal financial wealth (million yen) Figure 7: Sampled distribution of the terminal ånancial wealth 15

16 2 9 18% Optimal Cumulative probability Optimal Log-normal Cumulative probability 16% 14% 12% % 8% 6% Log-normal Normal 2 Normal 4% % 2% Terminal financial wealth Terminal fnancial wealth Figure 8: Comparison between the sampled distribution, normal and log-normal distributions We compare the sampled distribution with the associated normal and log-normal distributions in Figure 8. We derive the sampled distribution in the case where pf =1,np =1,î= 1. The normal and log-normal distributions have the same expected value and standard deviation as the sampled distribution. The sampled distribution has the thinner tail than the normal distribution, and it is similar in shape to the log-normal distribution.thereasonisasfollows.itisassumed that the returns of the risky asset have no -series correlation, and therefore the terminal price distribution of the risky asset is similar in shape to the log-normal distribution. Figure 9 shows the optimal åre insurance money for åve kinds of îvalue, pf =1,andnp =1 on the left-hand side, and for four combinations of both pf and np, andî = 1 on the righthand side. We ånd that it is not aãected by three factors, or receiving the survivor's pension, exemption from the mortgage loan, and change of the consumption level kappa=1 Insurance money (million yen) kappa=0.6 kappa=0.7 kappa=0.8 kappa=0.9 kappa=1.0 Insurance money (million yen) Figure 9: Optimal åre insurance money 16

17 5 Sensitivity analysis on home buying We have four kinds of parameters associated with home buying;, down payment, loan period, and mortgage interest rate. We solve the problem with the åxed parameters in the examples of Section 4. We analyze the sensitivity of these parameters. We set W E = 70 million yen because the lower bound constraint of the expected terminal ånancial wealth may not be active for the lower mortgage interest rate. All 36 combinations of three kinds of (t e =5; ; 15 (years)), three kinds of down payments (, 20, 30 million yen), and four kinds of mortgage interest rates (3%, 4%, 5%, 6%) are solved for pf =1,np =1,andî= 1. The loan period depends on the when the house is bought, and is set from the home buying to the retirement (T Ä t e =25; 20; 15 (years)). Figure consists of three ågures 9. Each ågure shows the CVaR(objective function value) for each home buying. If the mortgage interest rate is equal to the risk-free rate(4%), the CVaR is not dependent on the down payment. If the mortgage interest rate is lower than the risk-free rate, the smaller the down payment is, the larger the CVaR is. This reason is that it is better to take advantage of the investment in more cash with the higher risk-free rate instead of making more down payments, and the household can obtain the diãerence between two interests. When the home buying becomes later, the objective function values get less aãected by down payment and mortgage interest rate. CVaR (million yen) te = 5 down pay. = mil. yen down pay. = 20 mil. yen CVaR (million yen) te = down pay. = mil. yen down pay. = 20 mil. yen down pay. = 30 mil. yen CVaR (million yen) te = 15 down pay. = mil. yen down pay. = 20 mil. yen down pay. = 30 mil. yen % 4% 5% 6% mortgage interest rate 3% 4% 5% 6% mortgage interest rate 3% 4% 5% 6% mortgage interest rate Figure : Objective function values (1) Figure is transformed into Figure 11 to clarify the relationship between the CVaR and the combinations of the home buying and the mortgage interest rate. Figure 11 consists of three ågures, which show the CVaR for each down payment. When the mortgage interest rate islow,andwemakethesamedownpayment,theearlierthehomebuyingis,thehigher the CVaR is. The reason is that the household does not have to pay the rent instead of paying the interest rate if the home buying is earlier. If the mortgage interest rate is higher, the household has to make more loan payment, and the later home buying is superior to the earlier one. 9 We cannot obtain the feasible solution for the combination of 5 of home buying and 30 million yen of down payment. This reason is that we cannot make much down payment and purchase a house earlier. If the lower bound of cash L v;t goes down, we will obtain the feasible and optimal solution. 17

18 CVaR (million yen) Down payment = mil. yen 50 te = 5 45 te = 40 te = CVaR (million yen) Down payment = 20 mil. yen 50 te = 5 45 te = 40 te = CVaR (million yen) Down payment = 30 mil. yen te = 40 te = % 4% 5% 6% 3% 4% 5% 6% 3% 4% 5% 6% mortgage interest rate mortgage interest rate mortgage interest rate Figure 11: Objective function values (2) Figure 12 consists of three ågures, and shows the optimal life insurance money for each home buying. The higher the mortgage interest rate is, the more the optimal life insurance money is. The reason is that the higher loan payment due to the higher mortgage interest rate reduces more ånancial wealth, and the household has to cover the loss by the life insurance money if the householder is dead. On the other hand, the household needs to make low down payment for the low mortgage interest rate, and it needs to make high down payment for the high mortgage interest rate in order to make the optimal life insurance money low. If we make the lower down payment and the higher loan payment, and the householder dies earlier, it can keep more ånancial wealth in order to forgive the loan payment. However, the ånancial wealth is reduced due to the higher loan payment. Therefore, when the mortgage interest rate is lower, it can keep more ånancial wealth for the lower down payment. However, when the mortgage interest rate is higher, it keeps less ånancial wealth, and it has to purchase more life insurance to cover the loss. When the down payment is high, the ånancial wealth is not aãected by the mortgage interest rate. 1 te = 5 1 te = 1 te = 15 Insurance money (million yen) 5 0 down pay. = mil. yen down pay. = 20 mil. yen 95 3% 4% 5% 6% Insurance money (million yen) 5 0 down pay. = mil. yen down pay. = 20 mil. yen down pay. = 30 mil. yen 95 3% 4% 5% 6% Insurance money (million yen) 5 0 down pay. = mil. yen down pay. = 20 mil. yen down pay. = 30 mil. yen 95 3% 4% 5% 6% mortgage interest rate mortgage interest rate mortgage interest rate Figure 12: Optimal life insurance money (1) Figure 12 is transformed into Figure 13 to clarify the relationship between the life insurance 18

19 money and the combinations of the home buying and the mortgage interest rates. Figure 13 consists of three ågures, which show the life insurance money for each down payment. When the mortgage interest rate is low, and the household makes the same down payment, the earlier thehomebuyingis,thelowerthelifeinsurance money is. However, when the mortgage interest rate is high, the later the home buying is, the lower the life insurance money is. The reason is that the earlier the household purchases the house, the higher the probability the loan payment is forgiven due to the householder's death. However, when the interest rate is high, the interest payment is larger than the exemption from the loan payment, and we have the opposite result. When the home buying is late, the ånancial wealth is not aãected by the mortgage interest rate. Down payment= mil. yen Down payment= 20 mil. yen Down payment= 30 mil. yen Insurance money (million yen) 5 0 te = 5 te = te = 15 Insurance money (million yen) 5 0 te = 5 te = te = 15 Insurance money (million yen) 5 0 te = te = % 4% 5% 6% 3% 4% 5% 6% 3% 4% 5% 6% mortgage interest rate mortgage interest rate mortgage interest rate Figure 13: Optimal life insurance money (2) Figure 14 shows the optimal investment units of a risky asset when the household purchases the house at. When the mortgage interest rate is high, the household needs to invest in a risky asset and to increase the expected terminal ånancial wealth because the household has to make the high loan payment. The more the down payment is, the smaller the diãerence among the investment units because the result is not aãected by the mortgage interest rate. Due to space limitation, we omit the case for t e = 5 and t e = 15. Investment unit of the risky asset Down payment= mil. yen mr=3% mr=4% mr=5% mr=6% Investment unit of the risky asset Down payment= 20 mil. yen mr=3% mr=4% mr=5% mr=6% Investment unit of the risky asset Down payment= 30 mil. yen mr=3% mr=4% mr=5% mr=6% Figure 14: Optimal investment unit of the risky asset(t e = ) 19

20 6 Formulation using a grouped path The problem size depends on the number of the planning period (T ), the number of paths (I), and the number of risky assets (n). The number of decision variables and the number of constraints except upper and lower bound constraints and non-negativity constraints, as follows. Number of decision variables : T (n + I +1)+3 Number of constraints : TI + 2 Both numbers are almost TI. The number of paths (I) is larger than the number of period (T ) in the simulated path approach. We need a lot of paths to describe the mortality rate associated with life insurance and the rate of the åre associated with åre insurance. According to the life insurance standard life table(1996), the mortality rate for 40 years old men is 0.156%. Even if the householder dies on a path to describe the appropriate mortality rate, we need 641 paths. Therefore, we need to increase the number of paths to improve the accuracy of description of uncertainties, but the problem size gets large. On the other hand, the reason we need a lot of sample paths is that we describe the uncertainties accurately and calculate an appropriate value of risk measure. We do not use the paths where the terminal wealth are larger than the VaR if we use the CVaR as the risk measure. We aggregate the paths we may not use to calculate the CVaR, and we think of the aggregated paths as a virtual path with the probability jgj I where G is the set of the aggregated paths and jgj is the number of the paths. We call a virtual path `grouped path'and we formulate the model with the grouped path. Figure 15 shows a sketch of setting the grouped path Set G 2f1; 2; 3; 4; 5; 6; 7; 8; 9g Grouped Path Figure 15: Setting the grouped path The number of paths is twelve. Three paths we may use to calculate the CVaR are i = ; 11; 12, and the set G consists of nine paths. The more the paths included in the set G are, the smaller the problem size is. However, there is a possibility of the set G which has the necessary paths to calculate the CVaR. There is a tradeoã between the problem size(or computation ) and the accuracy, and determining the set G depends on the aspect of the We can say this idea for downside risk measure such as the LPM(lower partial moments). 20

21 problem. At årst, we formulate the model under a given set G. Second, we propose the method to determine the set G, and we test the method with numerical examples. 6.1 Formulation Constraints associatedwith the paths in the set G are aggregated, and the aggregated constraints are described such as Equations (19), (20), (22). If the paths are not in the set G, we describe the same constraints, such as Equations (18), (21), (23). 1 X Maximize V å Ä q (i) (16) (1 Ä å)i i62g subject to ö j0 z j0 + v 0 + y L;0 u L + y F u F;0 = W 1;0 (17) ö G j1z j0 +(1+r 0 )v 0 + D1 G = ö G j1z j1 + 1 X v (i) 1 (18) jgj i2g ö (i) j1 z j0 +(1+r 0 )v 0 + D (i) 1 = ö (i) j1 z j1 + v (i) 1 ; (i 62 G) (19) ö G jtz j;tä1 + 1 jgj X i2g ê 1+r (i) ë tä1 v (i) tä1 + DG t = ö (i) ê jt z j;tä1 + 1+r (i) ë tä1 v (i) tä1 + D(i) t = 8 < W1;T G = ö G : jtz j;t Ä1 + 1 jgj 8 W (i) < 1;T = : 1 jgjw1;t G + X W (i) 1;T I i2g X i2g ê 1+r (i) ë T Ä1 v (i) ö (i) ê jt z j;t Ä1 + 1+r (i) ë T Ä1 v (i)! T Ä1 ö G jtz jt + 1 jgj X i2g v (i) t ; (t =2;...;T Ä 1) (20) ö (i) jt z jt + v (i) t ; (t =2;...;T Ä 1; i 62 G) (21) T Ä1 9 = ; + D(i) T 9 = ; + DG T (22) ; (i 62 G) (23) ï W E (24) W (i) 1;T Ä V å + q (i) ï 0; (i 62 G) (25) z jt ï 0; (j =1;...;n; t =0;...;T Ä 1) v 0 ï 0 v (i) t ï L v;t ; (t =1;...;T Ä 1; i =1;...;I) (26) u L ï 0 u F;t ï 0; (t =0;...;T Ä 1) 21

22 q (i) ï 0; (i 62 G) V å : free where ö G jt = 1 X jgj i2g ö (i) jt D G t = M G t + H G t Ä C G t Ä y G L;tu L Ä y F u F;t + ú G 1;tí 1 u L + ú G 2;tí 2 u F;tÄ1 Ä 1 jgj X i2g ú (i) (i) 2;t (1 Ä ç)w 2;tÄ1ã; (t =1;...;T Ä 1) D G T = M G T + H G T Ä C G T + ú G 1;T í 1 u L + ú G 2;T í 2 u F;TÄ1 Ä 1 jgj M G t = 1 jgj ú G 1;t = 1 jgj X i2g X i2g M (i) t ; H G t = 1 jgj ú (i) 1;t ; úg 2;t = 1 jgj X i2g X i2g X i2g H (i) t ; C G t = 1 jgj ú (i) 2;t ; úg 3;t = 1 jgj X i2g ú (i) (i) 2;T (1 Ä ç)w 2;T Ä1 ã yl;t G = y f1 Åf 1 ú 4;t + y f2 Å(1 Ä f 1 )ú3;t G If the interest rate is constant(r t ) at each, and we replace Equation (26) for i 2 G with the following Equation (27), we can replace jgj decision variables v (i) t for i 2 G) with one decision variable vt G, and we reduce the problem size. vt G = 1 X v (i) t ï L t (27) jgj i2g In the case, we need to replace Equations (18), (20), (22), (26) with the following equations. ö G j1z j0 +(1+r 0 )v 0 + D1 G = ö G j1z j1 + v1 G (28) ö G jtz j;tä1 +(1+r tä1 ) vtä1 G + Dt G = ö G jtz jt + vt G ; (t =2;...;T Ä 1) (29) 8 9 < ê ë = W1;T G = ö G : jtz j;t Ä1 + 1+rT G Ä1 vt G Ä1; + DG T (30) v (i) t ï L v;t ; (t =1;...;T Ä 1; i 62 G) (31) The number of decision variables and the number of constraints except upper and lower bound constraints and non-negativity constraints, as follows. Number of decision variables : T (I ÄjGj + n +1)+3 Number of constraints : T (I ÄjGj +1)+1 In Section 6.3, we make the problems reduced under the condition that the interest rate is constant, and we test numerical examples. X i2g ú (i) 3;t C (i) t 22

23 6.2 Setting the set G We examine the optimal solutions derived in Section 4 in order to determine the set G. It is assumed that there is a risky asset, and the wage income and consumption expenditures are not random if the householder is not dead or the åre of the house does not occur because the household does not receive the life insurance money, survivor's pension or the åre insurance money. Therefore the return of a risky asset is a main factor that inçuences the terminal ånancial wealth if the householder is alive. The correlation coeécient between the terminal ånancial wealth and the terminal price of a risky asset for the paths where the householder is alive is The value is almost one, and we should include the paths where the terminal price of a risky asset(ö 1;T ) is high under the condition that the householder is alive in the set G. We need to examine what number of the paths is appropriate. We show the relationship between the rank of paths which is sorted by the terminal price of a risky asset and the rank of paths under the VaR in Figure 16. The right-hand side is a magniåed view of a part of the left-hand side. There is highly possible that the terminal ånancial wealth on the path with the lower price of a risky asset is smaller than the VaR. Paths under the VaR are reduced gradually in the hit exceeding about 1,000 lower ranks, and we do not have the paths under the VaR if the rank of path is over 1,871. This is just the ex-post analysis, but it shows that 2,622 paths with the higher terminal price of a risky asset become beyond the VaR, and they can be included in the set G. number of paths under VaR 1, number of paths under VaR ,000 2,000 3,000 4,000 5,000 number of paths 880 1,000 1,200 1,400 1,600 1,800 2,000 number of paths small ö (i) 1T large small ö (i) 1T large Figure 16: Number of paths under the VaR when paths are sorted by the terminal prices of risky asset We do not have the wage income instead of receiving the life insurance money and the survivor's pension when the householder is dead. The terminal ånancial wealth in the case where the younger householder dies is smaller than the case that the older householder dies because the lower wage income in total and the survivor's pension are lower. We show the relationship 23