Optimal consumption and investment problem incorporate. housing and life insurance decision: The continuous time case

Transcription

1 Optimal consumption and investment problem incorporate housing and life insurance decision: The continuous time case Shang-Yin Yang 1 Ko-Lun Kung 2 Abstract This study considers the optimal consumption-investment-insurance problem incorporating the housing decision for a household when interest rates and labor income are stochastic. Due to the insurance industry has an important role in the world economy (the premium penetration is roughly 6.5% to 7.5%), we investigate the household who has life insurance demand for his/her posterity with future lifetime uncertainty. We extended Kraft and Munk (2011) to allow the household s insurance purchase, examining the impact of insurance demand to the household s consumption and investment strategy. In complete market, the closed-form solution for the optimal strategy allows us analysis the household s investment behavior analytically. Using the data from Taiwan financial market, the household invest more in bond and housing portfolio and reduce his stock holding to against the mortality risk. Keyword: Portfolio management, Life insurance, Housing, Labor income risk, Interest rate risk. 1 Assistant Professor in the Department of Finance, Tunghai University, Taichung, Taiwan. R.O.C., [email protected]. 2 Ph.D. Candidate in the Department of Risk Management and Insurance, National Cheng Chi University, Taipei, Taiwan. The presenter. 1

2 1. Introduction In this study, we solve the optimal consumption-investment-insurance strategy incorporating housing decision for a household when interest rates and labor income are stochastic. To point out the economic significance of life insurance demand in the household s financial planning, premium penetration 1 gives considerable information. Table 1 presents the premium penetration of the main region in world during Premium penetration is an index that indicates the relative importance of the insurance business in the domestic economy. Before the subprime crisis in 2008, the average premium penetration in the world is 7.49% in 2007, and the highest level of premium penetration is 10.38% in Japan and newly industrialized Asian region. Due to the financial credit crunch in 2008, the average premium penetration in the world drop 0.42% in After 2009, the Eurozone crisis had affected the economy in many European countries and the premium penetration had dropped in the Eurozone since During , the premium penetration is roughly 7% of world s GDP, meaning the insurance demand of the household plays an important role to the global economy. Zietz (2003) organizes the comprehensive literature review of the insurance demand on different point view, Li, Moshirian, Nguyen and Wee (2007) Table 1 Premium Penetration in the World Region Premiums in % of GDP America North America Latin America and Caribbean Europe Western Europe Central and Eastern Europe Asia Japan, South Korea, Hong Kong, 11.8 Singapore, and Taiwan South and East Asia Middle East and Central Asia Africa Oceania World Data source: Swiss Re (Sigma) 1 Insurance penetration is the ratio of direct gross premiums to GDP. 2 In 2008, American and Europe are the main source of the drop of the average premium penetration. Because the original source of subprime crisis is American and Europe. 1

3 has point out the determinates of insurance demand in the OECD countries and Liebenberg, Carson and Dumm (2012) focus on the dynamic behavior for these determinates to the insurance demand. Our study focuses on the long-term financial planning of a household. We would like to know how much life insurance a household should purchase to protect against the loss of its breadwinner s human capital. During its lifetime, the household should utilize the labor income to optimize its consumption, housing, investment, and insurance strategy. Life insurance protects the most important asset for a household: Human capital. The breadwinner provides for the family by earning labor income. As long as breadwinner lives, the human capital generates a stochastic labor income to provide for rest of the family. Unfortunately, the breadwinner can die and the family loses a major source of income. Life insurance has offered a way to hedge against the risk of losing the main income of the household. The second question we are trying to answer is how the family should allocate its financial resources between risk-free, risky assets and real estate before and after a formal retirement date. Real estate is another major asset that a family may own. The housing decision for the household is affected by the house price. If the house price is not affordable, the household might choose to rent the house instead of owning it. Lin and Grace (2007) suggest that life insurance is a complement to the real estate for a young household. From Merton (1969, 1971), the seminal research that discusses the household s optimal consumption and portfolio selection in a continuous time and stochastic environment. Richard (1975) extends Merton (1971) by incorporating the life insurance for a household with an arbitrary but known distribution of death time. Campbell (1980) considers the impact of labor income uncertainty to the optimal insurance purchase for the household. Lewis (1989) investigates the insurance demand from the beneficiary s viewpoint. Pliska and Ye (2007) studied the optimal saving, consumption and investment problem when household s lifetime is random. Huang and Milevsky (2008) and Huang, Milevsky and Wang (2008) focus on difference risk preference: HARA case and CRRA case. Pirvu and Zhang (2012) extended considering the mean-reverting return of the investing asset. The literature of the dynamic portfolio selection with the interest rate uncertain include Sorensen (1999), Brennan and Xia (2000), Deelstra, Grasselli and Koehl (2000), Campbell and Viceira (2001), Munk and Sorensen (2004), Sangvinatsos and Wachter (2005), and Liu (2007). In the labor income issue, Bodie, Merton and Samuelson (1992) considered fully hedgeable income risk, the optimal unconstrained strategies can be deduced from the optimal strategies without labor income. Other studies with stochastic income include He and Pages (1993), Heaton and Lucas (1997), Duffie, 2

4 Fleming, Soner and Zariphopoulou (1997), Koo (1998), Munk (2000), Viceira (2001), Constantinides, Donaldson and Mehra (2002), and Cocco, Gomes and Maenhout (2005). Munk and Sorensen (2010) solve the optimal portfolios when interest rate and labor income are stochastic. Several papers consider housing in portfolio selection. Grossman and Laroque (1990) and Damgaard, Fuglsbjerg and Munk (2003) discuss the housing implication without labor income. Yao and Zhang (2005) consider the household s renting/owning decisions under borrowing constraints. Van Hemert (2010) incorporates the stochastic interest rate and focuses on the interest rate exposure. Kraft and Munk (2011) discuss the optimal housing, consumption, and investment decisions. Because of the importance of the insurance decision in the economy and housing decisions implicate the household s portfolio selection. This study investigates the household who has a random life and life insurance demand for his/her posterity. We extended Kraft and Munk (2011) to allow the household s insurance purchase, examining the impact of insurance demand to the household s consumption and investment strategy. Our main findings are as follows. First, we derive the closed-form solution to the optimal consumption, housing, investment and insurance problem. The optimal insurance premium for the household is related to the expected future labor income remaining and the currently mortality to the household. Second, the loading factor of the insurance premium causes the household losses 1% of the total wealth. Therefore, the household should invest more housing units and risky bonds to against the loading in the premium. Furthermore, we calibrate the parameters by using the data from the Taiwan financial market to analyze the decision for a household. which optimize its investment strategy for fifty years that includes thirty years working period and twenty years retirement period. Finally, the numerical results suggest the household should gradually increase housing portfolio and reduce holding stocks and risk bonds. Our empirical work is based on Taiwan life insurance and housing market. Taiwanese has the greatest interest in life insurance and housing. According to Swiss Re s Sigma report, Taiwan has the highest life insurance penetration (around 15%) in the world during the period and still growing. Taiwanese housing market is also one of the hottest housing market in Asia. Despite suffering from the high price-to-rent ratio 3, the homeownership rate for household only decreased 3%, from 87% to 84% in past 15 years. 4 3 The price-to-rent ratio data can be found in GlobalPropertyGuide.com. Currently the price-to-rent ratio of Taiwan is 64 years. For comparison, Hong Kong and Singapore is 35 years, India is 45 years, Thailand is 20 years, and Japan is 27 years. 4 According to Report on the Survey of Family Income and Expenditure 2013, published by National Statistic of Republic of China. It is the Taiwanese version of Panel Study of Income Dynamics. 3

5 The roadmap of our paper is as follow. We illustrate the setup of the problem and HJB equation in the next section. Solution methodology and the closed-form solution are discussed in the Optimal solution section. In the Numerical Analysis section, the data description, model calibration, and some results are presented. Finally, the Conclusion section summarizes our findings. 2. Market Framework This study considers a consumption, housing, investment, and life insurance purchase problem of a household in the economic environment investing his wealth in stocks, bonds, houses, and money market account, buying life insurance product, and receiving a stochastic labor income. 2.1 Financial Assets Following Kraft and Munk (2011), we assume that the instantaneous short-term interest rate r t satisfies the Vasicek (1977) model, dd t = κ(r r t )dd σ r dz rt, (1) where Z r = Z rt t 0 is a standard Brownian motion, and r, the long run mean, σ r, the volatility, and κ, the mean reversion coefficient, are positive constants. The money market account M(t) with initial value of M(0) = 1 is given by The bond price B t = B(r t, t) is given by t 0 M(t) = exp r s dd. (2) db t = B t r t + λ B σ B (r t, t) dd + σ B (r t, t)dz rt, (3) where σ B (r t, t) = σ r B r (r, t)/b(r, t) > 0 is the bond price volatility, λ 5 B is the Sharpe ratio of the bond, and ρ 6 Br is the correlation between the interest rate and the bond price. The price of a zero-coupon bond maturing at some time T is given by where B t T B T (r t, t) = e a(t t) b κ (T t)r t, (4) a(τ) = r + λ Bσ r κ b κ (τ) = 1 κ (1 e κκ ), σ 2 r 2κ 2 [τ b κ(τ)] + σ 2 r 4κ b(τ)2. 5 λ B is identical to the market price of interest rate risk. 6 The interest rate and the bond price have perfectly negative correlation, i.e. ρ BB = 1. 4

6 Consistent with Vasicek s model, the volatility of a zero-coupon bond with a maturity T is σ r b κ (T t), which only relies on time until expiration T t, but not depends on the level of interest rates. The stock price S t is assumed to have dynamics ds t = S t (r t + φ)dd + σ S ρ SS dz rt + 1 ρ 2 SS dz St, (5) where Z S = Z St t 0 is a standard Brownian motion independent of Z r, σ S is the constant volatility, φ is the risk premium of the stock, and ρ SS = ρ SS is the constant correlation between the stock and the bond. Following Kraft and Munk (2011), we suppose the household can invest in the real estate at a unit 7 price. The unit house price H t satisfies dh t = H t (r t + λ H σ H ζ)dd + σ H ρ HH dz rt + ρ HH dz St + ρ H dz Ht, (6) where Z H = Z Ht t 0 is a standard Brownian motion independent of Z r and Z S, σ H is the constant volatility, λ H is Sharpe ratio of the houses, ρ HH is the constant correlation between the house and the bond, ρ HH = (ρ HH ρ SS ρ HH )/ 1 ρ 2 SS, 2 ρ H = 1 ρ HH ρ 2 HH, where ρ HH is the constant correlation between house and stock, and ζ is the cost of holding per unit house. This study considers the household may renting the house instead of buying it. The cost of renting a unit house νh t 8 is assumed to the proportion of the house price. Therefore, the return of the household investing and renting out a unit house satisfies dd t + νh t dd = H t r t + λ Hσ H dd + σ H ρ HH dz rt + ρ HH dz St + ρ H dz Ht, (7) where λ H = λ H + (ν ζ)/σ H. 2.2 The life insurance In the insurance market, the household buys life insurance to hedge against mortality risk. Following Pliska and Ye (2007) and Pirvu and Zhang (2012), we assume that the household is alive the initial age z and the future lifetime is a non-negative random variable T x, defined on the probability space (Ω, F, P) that is independent of Z r, Z S, and Z H. Let μ x+t be the force of mortality and t p x be the 7 The unit is one square meter. 8 ν is a constant. 5

7 probability of the household who survives from age x to age x + t, under the force of mortality μ x+t. Then tp x satisfies t 0 tp x = P(T x > t) = exp μ x+s dd. (8) The probability of the household who dies before age x + t, under the force of mortality μ x+t is given by tq x = P(T x t) = 1 exp μ x+s dd. (9) The household who buy the life insurance will pay a premium rate I x+t9. θ x+t is the t 0 insurance premium rate with loading, and θ x+t μ x+t is called loading factor. If we ignore the operating cost of the life insurance company, the insurance premium rate is equal to the force of mortality i.e. θ x+t = μ x+t. If we consider the operating cost of the life insurance company, the insurance premium rate will large than the force of mortality i.e. θ x+t > μ x+t. In this study, we also evaluate the effect of the operating cost and insurance premium rate to the household s insurance demand. 2.3 Labor income The household receives a labor income from non-financial sources at a rate of y t which given by where dy t = y t μ y (r t, t)dd + σ y (t) ρ yy dz rt + ρ yy dz St + ρ yy dz Ht, (10) μ y (r t, t) = η 0(t) + η 1 r t 0 σ y (t) = σ y 0 0 t < T T t T, 0 t < T T t T, ρ yy = ρ yy ρ SS ρ yy, ρ yy is the constant correlation between the income grows and 1 ρ2 SS the stock and ρ yy = ρ yy is the constant correlation between the income grows and bond returns, ρ yy = 1 ρ 2 yy ρ 2 yy, and ρ yy = ρ HH ρ yy + ρ HH ρ yy + ρ H ρ yy is the constant correlation between the income grows and the house price. We assume that the drift term of labor income is η 0 (t) + η 1 r 10 t is the deterministic function of the labor income volatility. 9 μ x+t, x p t, and x q t are standard actuarial notations. Without loss of generality, this study use I x+t to determine the premium rate of the household who buy insurance at age x. 10 The drift term of labor income has linear relation to the interest rate. 6

8 2.4 Optimal strategies In the economy, the household has to deal with an optimize consumption, housing, investment and insurance problem to maximizes the household s expected utility. Assume that the insured s investment horizon T is shorter than the maturing dates of the bond, i.e. T T. This ensures that the bond is a long-lived asset from the household s viewpoint. Let π Bt and π St denote the proportion that household invested in the bond and the stock at time t. We suppose the household has two consumption goods: housing and perishable good 11. Let the household owned φ oo house units and rented φ rr house units. The household can investment φ RR house units via house price-linked financial assets such as real estate investment trusts (REITs). Let φ CC = φ oo + φ rr and φ II = φ oo + φ RR, φ CC is the total house units that used by the household and φ II is the total house units that invested by the household. W t denotes the household s financial wealth and c t denotes the consumption rate of the perishable good at time t. Therefore, the dynamics of the financial wealth satisfies dw t = π St W t ds t S t + π Bt W t db t B t + W t 1 π St π Bt φ II H t dd +φ oo dh t + φ RR (dh t + νh t dd) φ rt νh t dd c t dd + y t I z+t dd = W t r t + π St φ + π Bt σ B (r t, t)λ B + φ II λ Hσ H H t φ CC νh t c t + y t I z+t dd + W t π St σ S ρ SS + π Bt σ B (r t, t) + φ II H t σ H ρ HH dz rt + W t π St σ S 1 ρ 2 SS + φ II H t σ H ρ HH dz St +φ II H t σ H ρ H dz Ht (11) where t [0, mmm(t x, T)] and W t 1 π St π Bt φ II H t is the position that invested in the money market account. If the household dies at time t, 0 < t T, then his/her beneficiary will receives the insurance benefit I x+t θ x+t and financial wealth W t. Therefore, the total legacy of the household leaves to posterity is denoted by N t = W t + I x+t θ x+t. (12) 11 Perishable good is the numeraire good. 7

9 σ B 0 0 For simplify, let λ = (λ B, λ S, λ H ), Σ = 2 σ S ρ SS σ S 1 ρ SS 0, σ H ρ HH σ H ρ HH σ H ρ H Z = (Z r, Z S, Z H ), and π t = W t π St, W t π Bt, φ II H t, where λ S = 1 1 ρ2 SS φ σ S ρ SS λ B, λ H = λ H ρ HH λ B ρ HH λ S. We revise the financial wealth as the following process: ρ H dw t = [r t W t + π t Σ(r t, t)λ φ CC νh t c t + y t I z+t ]dd + π t Σ(r t, t)dz t (13) We assume that the household uses an expected utility to evaluate the consumption-investment-insurance problem incorporating the housing decisions in a stochastic interest rate and stochastic labor income environment. The value function for this optimization problem is denoted by V(W, r, y, h, t) = sup E t e δ(s t) U(c s, φ CC )dd, c t,π St,π Bt,φ II,φ CC,I x+t A t where the conditional expectation is measured given the values of W, r, y, h, at time t and given the strategy c t, π St, π Bt, φ II, φ CC, I z+t. A t is the set of all such strategy c t, π St, π Bt, φ II, φ CC, I z+t and δ is the time preference rate. We supposed the household s preference is a time-additive Cobb-Douglas style utility so that value function satisfies V(W, r, y, h, t) = sup E t e (c δ(s t) s α φ 1 α CC ) 1 γ dd, c t,π St,π Bt,φ II,φ CC,I x+t A t 1 γ where γ > 1 is the relative risk aversion. The Hamilton-Jacobi-Bellman (HJB) equation associated with this dynamic optimization problem is (δ + μ)v = sup U(c, φ C ) + μμ + V t + V W (rr + π Σλ φ C νν (c,π S,π B,φ I,φ C,I x ) c I x + y) V WWπ ΣΣ π + V r κ(r r) V rrσ r 2 + t t T T V y y(η 0 (t) + η 1 r) V yyy 2 σ y 2 + V h h r + λ Hσ H v V hhh 2 σ H 2 V WW π Σe 1 σ r + V WW yσ y π Σρ yy + V Wh hσ H π Σρ h V rr yρ yy σ y σ r V rh hρ HH σ H σ r + V yh yh ρ yy ρ h σ H σ y, (14) 8

10 where ρ yy = ρ yy, ρ yy, ρ yy, λ h = λ H, e 1 = (1,0,0), and ρ h = (ρ HH, ρ HH, ρ H ). The subscripts on V denote partial derivatives. Ψ is the value function that the household with no labor income and no life insurance and Ψ satisfies the following HJB equation. δδ = sup U(c, φ C ) + Ψ t + Ψ W (rr + π Σλ φ C νν c) + (c,π S,π B,φ I,φ C ) 1 2 Ψ WWπ ΣΣ π + Ψ r κ(r r) Ψ rrσ r 2 + Ψ h h r + λ Hσ H v Ψ hhh 2 σ H 2 Ψ WW π Σe 1 σ r + Ψ Wh hσ H π Σρ h Ψ rh hρ HH σ H σ r. (15) According to Kraft and Munk (2011) theorem 3.1, Ψ is given as Ψ(W, r, h, t) = 1 1 γ g(r, h, t)γ W 1 γ, where g(r, h, t) satisfies the following partial differential equation 0 = 1 2 σ r 2 g rr h2 σ 2 H g hh ρ HH σ r σ H hg rh + κ(r r) + γ 1 σ γ r λ B g r + r + 1 γ σ Hλ H v hg h + g t + ηη 1 α hk δ γ + γ 1 γ r + γ 1 2γ 2 λ λ g, with the zero utility at time T, the terminal condition g(r, h, T) = 0, and we can obtain g(r, h, t) as the following equation. where and g(r, h, t) = ηη 1 α hk T e d 1 (s t) α t γ 1 γ b κ (s t)r dd d 1 (τ) = δ γ + γ 1 2γ 2 λ λ k 1 γ σ Hλ H v (k 1)σ H 2 (τ) α r + γ 1 σ B λ B γ κ kσ rσ H ρ HH κ + α2 σ B 2 2κ 2 γ 1 γ 2 (τ b κ (τ) b κ (τ) 2 ), η = α 1/γ αα 1 α k 1, k = (1 α) 1 1 γ. γ 1 τ b γ κ (τ) (16) 9

11 The terminal condition for the HJB equation V is V(W, r, y, h, T) = Ψ(W, r, h, T). The first-older condition for consumption satisfies and the housing consumption satisfies c t = η αα 1 α hk [V W (W t, r t, y t, h t, t)] 1 γ, (17) φ CC = ηh k [V W (W t, r t, y t, h t, t)] 1 γ. (18) The first-order condition for the portfolio allocation π t satisfies π t = V W V WW (Σ(r t, t) ) 1 λ V WW V WW y t σ y (t)(σ(r t, t) ) 1 ρ yy V Wh V WW h t σ H (Σ(r t, t) ) 1 ρ h + V WW V WW (Σ(r t, t) ) 1 e 1 σ r. (19) The first term of the right hand side (RHS) is the myopic portfolio, the second term of the RHS is the income rate hedging portfolio, the third term of the RHS is the house price hedging portfolio, and the fourth term of the RHS is the interest rate hedging portfolio. The first-order condition for the insurance premium satisfies I x+t = θ γ x+t γ 1 1 γ μ x+t g(r, h, t)v W (W t, r t, y t, h t, t) 1 γ θ x+t W t (20) 3. Optimal solution Following Kraft and Munk (2011), in a complete market, the labor income L t can be replicated by financial assets, and the labor income is the present value of the household s future income flow. At time t, the expected value of the labor income from time t to T is T t L t L(y t, r t, t) = E Q t s t p x+t y s e t r vdd dd, (21) where Q is the equivalent martingale measure. The explicit solution of the labor income is in the proposition 1. Proposition 1. The expected value of labor income is given by and L t = y tf(r t, t), y T F(r t, t), t < T T t T T s t p x+t e A(t,s) (1 η 1 )b κ (s t)r dd t t < T T F(r, t) = + T tp x+t Υ s T p x+t e A (t,s) b κ (s t) η 1 b κ (T t) r dd, T T Υ s t p x+t e a(s t) b κ (s t)r dd, T t T t s (22) (23) 10

12 where s t p x+t is the probability of the household who survives from age x + t to age x + s, s s A(t, s) = η 0 (u)dd (1 η 1 )σ r ρ yy σ y (u)b κ (s u)dd t t + (1 η 1 )r s t b κ (s t) 1 2 (1 η 1) 2 σ r 2 κ 2 s t b k(s t) κ 2 b k(s t) 2, A (t, s) = A t, T + a s T b κ s T σ r 2 (1 η 1 ) 1 κ b k T t b 2k T t r κb k T t σ r 2 b 2k T t b k s T T + σ r ρ yy σ y (u)e κ(t u) dd, t η 0 (t) = η 0 (t) σ y (t)λ y, r = r + σ rλ B, and λ κ y = ρ yy Σ 1 λ. Proof: According to Kraft and Munk (2011), we consider the mortatility in to the labor income, it is not diffcult to obtain expected value of the labor income. In proposition 1, the expected value of the labor income is depended on the income rate y t, the interest rate r t, and the survivor probability s t p x+t. At retirement, the household s income is based on replacement ratio Υ and the income at retirement day y T. Following Bodie, Merton and Samuelson (1992), the household sells his future income flow for the amount L t. Therefore, we can think the household s total wealth equal to W t + L t. The optimal problem become to the case without labor income but the financial wealth equal to W t + L t. Based on the complete market with Vasicek (1977) interest rate model and power utility function, the indirect utility function J is the following equation: V(W, r, y, h, t) = J(W + L, r, h, t) = 1 g(r, h, 1 γ t)γ (W + L) 1 γ. (24) We substitute the above equation in to the HJB equation and summarize the solution in the following proposition. Proposition 2. The value function is given by V(W, r, y, h, t) = 1 1 γ g(r, h, t)γ (W + L) 1 γ where L is referred in proposition 1 and g(r, h, t) = ηη 1 α hk T e K(s)+d 1 (s t) α t γ 1 γ b κ (s t)r dd (25) 11

13 with and d 1 (τ) = δ γ + γ 1 2γ 2 λ λ k 1 γ σ Hλ H v (k 1)σ H 2 (τ) α r + γ 1 σ B λ B γ κ kσ rσ H ρ HH κ + α2 σ B 2 2κ 2 γ 1 γ 2 τ b κ (τ) κ 2 b κ(τ) 2, γ 1 τ b γ κ (τ) K(s) = (θ x+s ) γ 1 γ (μ x+s ) 1 γ + 1 γ θ γ x+s 1 γ μ x+s. 12 The optimal consumption rate is the optimal housing consumption rate is c t = ηαv h 1 α t k W t+l t, g(r t,h t,t) (26) φ CC = ηh t k 1 while the optimal investment in bond, stock, and housing are π Bt X t = ξ B (W γσ t + L t ) σ y (t)ζ B β and the insurance demand σ B W t+l t g(r t,h t,t) (27) L t + σ r σ B y t F r (r t, t) σ r σ B (W t + L t ) g r (r t,h t,t) g(r t,h t,t), (28) π St X t = ξ S γσ S (W t + L t ) σ y (t)ζ S σ S L t, (29) φ It H t = ξ I (W γσ t + L t ) σ y (t)ζ I H σ I L t + (W t + L t ) H tg h (r t,h t,t), (30) g(r t,h t,t) I x+t = θ x+t γ 1 γ (μ x+t ) 1 γ(w t + L t ) θ x+t W t. (31) (ξ B, ξ S, ξ I ) are the mean return of the risk asset and (ζ B, ζ S, ζ I ) are the coefficients of the income Sharpe ratio λ Y, where ξ B = λ B (1 ρ 2 SS ) φ σ (ρ SS ρ SS ρ BB ) λ H (ρ BB ρ BB ρ HH ) S 1 + 2ρ SS ρ HH ρ SS ρ 2 2 2, SS ρ HH ρ SS 12 (θ x+s ) γ 1 γ (μ x+s ) 1 γ + 1 γ γ θ x+s 1 γ μ x+s < 0 12

14 ξ S = ξ I = φ σ (1 ρ 2 BB ) λ B (ρ SS ρ SS ρ BB ) λ H (ρ SS ρ SS ρ HH ) S 1 + 2ρ SS ρ HH ρ SS ρ 2 2 2, SS ρ HH ρ SS λ H (1 ρ 2 SS ) φ σ (ρ SS ρ SS ρ HH ) λ B (ρ BB ρ BB ρ HH ) S 1 + 2ρ SS ρ HH ρ SS ρ 2 2 2, SS ρ HH ρ SS ρ H ρ yy ρ yy ρ HH ζ B = ρ yy ρ SS 2 ρ H 1 ρ SS The proof of proposition 2 is in Appendix. ζ S = ρ Hρ yy ρ yy ρ HH, 2 ρ H 1 ρ SS ζ I = ρ yy ρ H. ρ HH ρ yy ρ H, In proposition 2, the function K(s) determines the utility loss that the household needs to reduce the mortality risk and pays the loading premium for the life insurance. According to equation (19), the optimal investment amount in equation (28) to (30) can be decomposed to four funds which conclude in the following table. Table 2 The four funds Myopic ξ B ξ S ξ I,, γσ β γσ S γσ H Income hedging Interest rate hedging Housing hedging σ y(t)ζ B σ B L t + σ r y t F r (r t, t) W t + L t σ B W t + L t, σ y(t)ζ S σ S σ r g r (r t, h t, t) σ B g(r t, h t, t) H t g h (r t, h t, t) g(r t, h t, t) L t W t + L t, σ y(t)ζ I σ I L t W t + L t 4. Numerical Analysis At the first of this section, we describe the data in Taiwan financial market. Second, we illustrate the method and result in model calibration. Finally, we demonstrate an example to illustrate the household s behavior on investment and insurance demand. 4.1 Data We use the quarterly data of socio-economic variables of Taiwan in the period 1999Q1 2014Q4, which consist of 64 observations. The average labor income and Consumer Price Index (CPI) are obtained from the National Statistics of Republic of China. The average labor income is the total labor income includes regular salary, bonus, and other compensations divided by the total amount of employed. 13

15 The cum-dividend stock index we used is the MSCI Taiwan Gross Index obtained from the Morgan Stanley Capital International (MSCI). The MSCI Taiwan Gross Index starts from 1999Q1 and therefore is the limiting factor of our empirical data. We construct the quarterly cum-dividend stock return from the MSCI Taiwan Gross Index in the last month of each quarter. We use Sinyi housing index published by Sinyi Real Estate Planning and Research monthly. The Sinyi housing index is the first housing price index in Taiwan starting from 1991Q1 and covers previously occupied house transactions of 6 most populated metro areas in Taiwan Island. The nominal labor income, stock index, and housing index are all deflated to real term using CPI, with 1999Q1 as base period. The nominal interest rate is the average 1-year saving rate provided by five major Taiwanese commercial banks, which served as the main reference rate of personal loan and mortgage in the Taiwanese market. 13 The average interest rate is published by the Central Bank of Taiwan. Real interest rate is obtained by subtracting the inflation rate from nominal interest rate. The inflation rate subtracted is the realized inflation rate of the last 4 quarters. Figure 1 plots the labor income, stock index, housing price index, and real interest rate. We normalize the labor income, stock index, and the housing price index with the beginning period (1999Q1) and then take logarithm. 13 These 5 banks are government-owned before the deregulation of the banking industry. They are generally perceived as credible and dominates the market in saving deposits and mortgage in early 90s. 14

16 Figure 1: Labor income, stock index, house index, and real interest rate time series. The data we used to calibrate the mortality model is the Second Experience Annuity Life Table in Taiwan, which is based on the annuitant s actual mortality experience in the past. The underlying population covers all annuitant that had purchased commercial annuity sold in Taiwan. The table was recently published in Model calibration To calibrate the benchmark parameters, we estimate a restricted VAR(1) model. The log-income is first decomposed into the common component and idiosyncratic income following Cocco, Gomes, and Maenhout (2005), that is, llly t = P(t) + u t + v t. The common component includes a deterministic component P(t) and a stochastic common factor u t. The deterministic component is described by a third degree polynomial of remaining lifetime to capture the life-cycle feature in the income, i.e., P(t) = a + bb + ct 2 + dt 3, where a, b, c, and d are coefficients. To extract the common stochastic factor and the idiosyncratic income, we subtracted the fitted third degree polynomial from the log income data. We assume the stochastic common factor of log-income and idiosyncratic income are both geometric Brownian motion. The stochastic component of log-income is 15

17 l t = u t + v t. Since our calibration is based on the aggregate labor income, the idiosyncratic income becomes averaged out and can only extract common factor u t from the aggregate data. We follow Cocco, Gomes, and Maenhout (2005) on the assumption of the idiosyncratic income is random-walk component with no drift, i.e., dv t = σ v dz vv. This implies that the common stochastic component dominates in the aggregate stochastic labor income by governing the drift term and idiosyncratic income only contributes to the variation. Let the aggregate stochastic component be l t = u t. In accordance with the labor income process, the dynamic of l t can be written as dl t = (η 0 + η 1 r t 1 2 σ u 2 )dd + σ u dz uu, where η 0 and η 1 are the parameters of drift term, σ u is the volatility of stochastic component, and Z ut is a standard Brownian motion VAR(1) model The parameter of labor income, stock index, housing index, and interest rate dynamic can be estimated via a restricted VAR(1) model. To do this, we first discretize the joint dynamic of labor income, stock index, housing index, and interest rate. Let Y = (l t, llls t, r t, lllh t ). We can write down the VAR(1) model for the stochastic differential equation dy t = (dl t, dddds t, dr t, ddooh t ) : Y t+δ = A + BY t + ε t+δ, ε t+δ N(0, Ω), where and η σ u 2 A = φ η σ S , B = κ 0 κr φ ζ σ H 2 2 σ u ρ uu σ u σ S ρ uu σ u σ r ρ uu σ u σ H 2 Ω = ρ uu σ u σ S σ S ρ SS σ S σ r ρ SS σ S σ H 2. ρ uu σ u σ r ρ SS σ S σ r σ r ρ rr σ r σ H 2 ρ u Hσ u σ H ρ S Hσ S σ H ρ rh σ r σ H σ H The parameter we are interested in is the set {η 0, η 1, φ, κ, r, φ, σ u, σ S, σ H, σ r, ρ uu, ρ uu, ρ uu, ρ SS, ρ SS, ρ rr }. 14 We can estimate the model with maximum likelihood estimation (MLE) or Feasible Generalized Least 14 We fixed the rental yield ζ at 1.5% in the estimation to reflect the high price/rent ratio of the current housing market in Taiwan and φ = λ H σ H 16

18 Square (FGLS) method. Assuming that Y t follows a multivariate Gaussian distribution, it is fairly simple to estimate the VAR(1) model with MLE. We follow the algorithm in Lutkepohl (2005) and implement the MLE in MATLAB Mortality rate model We calibrate the Gompertz mortality model to the second annuity experience table in Taiwan. We use the male mortality rate from age 30 to 100 to cover the hypothetical person s lifetime. We assume the constant force of mortality between fractional years. The probability of death q x of an individual of age x is first converted to force of mortality μ x using the relation q x = 1 e μ x. Our mortality model estimate is the following: 1 μ x = ex The parameters Table 3 reports the parameters and the corresponding standard errors of relevant economic variable. The relatively low level of income growth parameter η 0 and η 1 reflects the real income growth in Taiwan has been staggering since the last decade. The volatility of stochastic common factor in labor income is about 2.91%. The risk premium of stock is about 1.72%, and the volatility is about 27% per annum. We set the benchmark value of risk premium to a slightly higher 4.5% to reflect more optimistic market history as generally expected. The long-term real interest rate is 0.8%, which is the consequence of both rising level of inflation (average of 1.03% during the period) and low interest rate (average of 2.16% during the period). The estimated risk premium of housing market is about 6.6%. The housing market performed relatively well as the house price in Taiwan increases steadily during The estimated house price volatility is about 7.2%. We adjust the benchmark value of housing price volatility to 7.5% to compensate for the overly optimistic and monotonic increasing housing market over the past decade. The risk premium for housing is 0.75% in real term. We find the yield of housing price is largely offset by the rising price level, for that the average spread between rate of return on housing and real interest rate is only 0.35%. We set the rental yield ζ to 1.5% and the cost of renting a house v to 3%. 17

19 Table 3: The parameter estimate and benchmark value of the income, stock, house price, and interest rate dynamic. Parameter Value Standard error Benchmark η η σ u ρ uu ρ uu φ κ r σ S σ r ρ SS φ σ H ρ uu ρ SH ρ rr ζ v An example for Taiwan market In this study, we consider the complete market case that income risk is spanned perfectly by the stock, bond, and house price. According to the optimal solution in section 3, Figure 2 illustrates the pattern of human wealth, financial wealth, and total wealth. 18

20 Fig. 2. The expected wealth of the household The variation of the financial wealth, human wealth, and total wealth at different time shows in Figure 2. The difference between the solid curve and the dashed curve describes the impact of the mortality and loading. The effect of the mortality consideration will roughly lead 5% decreasing of the total wealth and the loading will probably result 1% loss of the total wealth. In Figure 2, the human capital will decrease gradually that due to the low labor income in Taiwan in the past decade. The financial wealth will increase to the highest level at retirement time than decreasing gradually at the retirement period. Fig. 3. The expected consumption 19

21 Fig. 4. The expected housing consumption Figure 3 and 4 are the expected consumption and housing consumption units (square feet) for the household. Because of the total wealth decreased by the mortality, the household reduced the consumption and housing consumption units. In Figure 3 and 4, loading factor has an important effect to the household consumption. At beginning, the household has large human capital, he reduces his consumption and pay loading premium for insurance to protect his human capital. In the last few years, protecting the remaining human capital for the household is not reasonable. Therefore, the household has more consumption in the last few years. Fig. 5. Optimal insurance and investment strategy According to Proposition 2, the optimal insurance and investment strategy illustrate in Figure 5. Figure 5 shows the portfolio weight of financial assets at difference times 20

22 and the insurance premium (the green line) has only a little effect to the household s optimal strategy. The result in Figure 5 illustrates the household should gradually transfer his investment form stocks to bank account, risky bond and housing investment at the first twenty year 15. In the last ten years of working period, the household should gradually transfer his investment form risk bond and stocks to bank account and housing investment. At retirement, the investment strategy will dramatically change from bond and stock to housing. Because the decline income, housing provides more attractive risk-return trade off and the demand of the hedging against the future housing expenditures. Fig. 6. The investment difference in income hedging portfolio Fig. 7. The investment difference in interest rate hedging portfolio 15 The total wealth includes the human capital. Therefore, the sum of the investment portfolio weight is not equal to one. 21

23 Figure 6 and 7 present the effect of the mortality risk to the optimal investment strategy. Figure 6 describes the investment difference in the income hedging portfolio and Figure 7 illustrates the investment difference in the interest rate hedging portfolio. To hedge against the mortality risk, the household will increase the investment in bond and housing position and reduce the stock holding in the income hedging portfolio. The loading factor enlarges the influence for the investment strategy. In Figure 7, the mortality risk has no effect to the interest rate hedging portfolio but the loading factor reduces the bond investment in the interest rate hedging portfolio. The result is obvious due to the utility loss is caused by the loading factor Premium/Wage Percentage Time in years Fig. 8. Premium-to-wage ratio. Figure 8 reports the spending on premium as percentage of wage throughout the lifespan of breadwinner. In early stage of life, the premium-to-wage ratio is low because the insurance is relatively cheap. As breadwinner ages, the death probability increases and premium becomes more expensive. The death probability increases exponentially while the human capital decreases in an approximately linear fashion (see Figure 2) explains the increase in premium-to-wage ratio before retirement. The jump in premium-to-wage ratio reflects that the wage (now retirement income) is halved by the replacement ratio immediately after retirement. The premium-to-wage ratio reaches its maximum at 11% when the breadwinner is about 68 years old. At the end of the lifespan, both human capital decreases and insurance premium increases at a faster speed because the death probability increases exponentially. If we convert the natural premium to level premium, the level premium-to-wage ratio is around 5.8%, an extra 0.5% if loading is considered. 22

24 5. Conclusion In this study, we derive the closed form solution for optimal consumption, housing, investment, and insurance purchasing. The overall investment strategy is affected by the variation of the risky assets and the changing of the labor income. The changing in risky assets investment illustrates the effect of the household s decline income at retirement period. The effect of mortality will reduce roughly 5% total wealth to the household which create the demand of insurance and the loading factor will increase another 1% loss to total wealth. In complete market assumption, the closed-form solution for the optimal strategy allows us analyze the household s investment behavior analytically. We find that the household should hold more bond and housing portfolio and reduce stock position to against the mortality risk. The loading factor enlarges the demand of this investment change for the household. The cases of unspanned income risk with constrained investment and stochastic parameters need further analysis in the future. Using the data from Taiwan financial market, we calibration the parameters to demonstrate the condition of the Taiwan market. To our knowledge, we are the first to estimate a full-scale life-cycle model in insurance literature using Taiwanese data. We hope that this exercise not only provide a demonstration for our theoretical improvement on modeling, but also provide a reference point for estimating the behavior in the economically or ethically similar Asian country. Reference Bodie, Z., R. C. Merton, and W. F. Samuelson, 1992, Labor supply flexibility and portfolio choice in a life-cycle model, Journal of Economic Dynamics & Control 16, Brennan, Michael J, and Yihong Xia, 2000, Stochastic interest rates and the bond-stock mix, European Finance Review 4, Campbell, J. Y., and L. M. Viceira, 2001, Who should buy long-term bonds?, American Economic Review 91, Campbell, R. A., 1980, The demand for life-insurance - an application of the economics of uncertainty, Journal of Finance 35, Cocco, J. F., F. J. Gomes, and P. J. Maenhout, 2005, Consumption and portfolio choice over the life cycle, Review of Financial Studies 18, Constantinides, G. M., J. B. Donaldson, and R. Mehra, 2002, Junior can't borrow: A new perspective on the equity premium puzzle, Quarterly Journal of Economics 117, Damgaard, A., B. Fuglsbjerg, and C. Munk, 2003, Optimal consumption and investment strategies with a perishable and an indivisible durable consumption 23

25 good, Journal of Economic Dynamics & Control 28, Deelstra, G., M. Grasselli, and P. F. Koehl, 2000, Optimal investment strategies in a cir framework, Journal of Applied Probability 37, Duffie, D., W. Fleming, H. M. Soner, and T. Zariphopoulou, 1997, Hedging in incomplete markers with hara utility, Journal of Economic Dynamics & Control 21, Grossman, SJ, and G Laroque, 1990, Asset pricing and optimal portfolio choice in the presence of illiquid durable consumption goods, Econometrica 58, He, H., and H.F. Pages, 1993, Labor income, borrowing constraints, and equilibrium asset prices, Economic Theory 3, Heaton, J., and D. Lucas, 1997, Market frictions, savings behavior, and portfolio choice, Macroeconomic Dynamics 1, Huang, H. X., and M. A. Milevsky, 2008, Portfolio choice and mortality-contingent claims: The general hara case, Journal of Banking & Finance 32, Huang, H. X., M. A. Milevsky, and J. Wang, 2008, Portfolio choice and life insurance: The crra case, Journal of Risk and Insurance 75, Koo, H. K., 1998, Consumption and portfolio selection with labor income: A continuous time approach, Mathematical Finance 8, Kraft, H., and C. Munk, 2011, Optimal housing, consumption, and investment decisions over the life cycle, Management Science 57, Lewis, F. D., 1989, Dependents and the demand for life-insurance, American Economic Review 79, Lutkepohl, H., 2010, New Introduction to Multiple Time Series Analysis. Springer. Li, D. H., F. Moshirian, P. Nguyen, and T. Wee, 2007, The demand for life insurance in oecd countries, Journal of Risk and Insurance 74, Liebenberg, Andre P., James M. Carson, and Randy E. Dumm, 2012, A dynamic analysis of the demand for life insurance, Journal of Risk and Insurance 79, Lin, Y. J., and M. F. Grace, 2007, Household life cycle protection: Life insurance holdings, financial vulnerability, and portfolio implications, Journal of Risk and Insurance 74, Liu, J., 2007, Portfolio selection in stochastic environments, Review of Financial Studies 20, Merton, R. C., 1969, Lifetime portfolio selection under uncertainty - continuous-time case, Review of Economics and Statistics 51, Merton, R. C., 1971, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory 3, Munk, C., 2000, Optimal consumption/investment policies with undiversifiable 24

26 income risk and liquidity constraints, Journal of Economic Dynamics & Control 24, Munk, C., and C. Sorensen, 2004, Optimal consumption and investment strategies with stochastic interest rates, Journal of Banking & Finance 28, Munk, C., and C. Sorensen, 2010, Dynamic asset allocation with stochastic income and interest rates, Journal of Financial Economics 96, Pirvu, T. A., and H. Y. Zhang, 2012, Optimal investment, consumption and life insurance under mean-reverting returns: The complete market solution, Insurance Mathematics & Economics 51, Pliska, S. R., and J. C. Ye, 2007, Optimal life insurance purchase and consumption/investment under uncertain lifetime, Journal of Banking & Finance 31, Richard, S.F., 1975, Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model, Journal of Financial Economics 2, Sangvinatsos, A., and J. A. Wachter, 2005, Does the failure of the expectations hypothesis matter for long-term investors?, Journal of Finance 60, Sorensen, C., 1999, Dynamic asset allocation and fixed income management, Journal of Financial and Quantitative Analysis 34, Van Hemert, O., 2010, Household interest rate risk management, Real Estate Economics 38, Vasicek, O., 1977, Equilibrium characterization of term structure, Journal of Financial Economics 5, Viceira, L. M., 2001, Optimal portfolio choice for long-horizon investors with nontradable labor income, Journal of Finance 56, Yao, R., and H. H. Zhang, 2005, Optimal consumption and portfolio choices with risky housing and borrowing constraints, Review of Financial Studies 18, Zietz, Emily Norman, 2003, An examination of the demand for life insurance, Risk Management and Insurance Review 6,

27 Appendix This appendix illustrates the detailed proof of proposition 2. We substitute equation (17), (18), (19), and (24) into the HJB equation (14). Therefore, the HJB equation (14) can be written as 0 = γ V g vv 1 α hk σ r 2 g rr σ H 2 h 2 g hh ρ HH σ r σ H hg rh + κ(r r) + γ 1 γ δ γ 1 γ γ 1 σ rλ B g r + h r + 1 γ λ Hσ H v g h + g t + θ γ 1 γ μ 1 1 γ γ + γ θ 1 γ μ γ 1 r 2γ 2 λ λ g + g(1 γ)y F γ(w+l) t + F r (κ(r r) ρ YY σ r σ Y (t)) + 2 F rrσ r 2 F (1 η 1 )r + θ η 0 (t) + 1. (A.1) Reducing γ V g in all terms of equation (A.1), we get 0 = vv 1 α hk σ r 2 g rr σ H 2 h 2 g hh ρ HH σ r σ H hg rh + κ(r r) + γ 1 γ δ γ 1 γ γ 1 σ rλ B g r + h r + 1 γ λ Hσ H v g h + g t + θ γ 1 γ μ 1 1 γ γ + γ θ 1 γ μ γ 1 r 2γ 2 λ λ g + g(1 γ)y F γ(w+l) t + F r (κ(r r) ρ YY σ r σ Y (t)) + 2 F rrσ r 2 F (1 η 1 )r + θ η 0 (t) + 1 (A.2) For the solution to exist in equation (A.2), we have to satisfy the following two PDE. 0 = F t + F r (κ(r r) ρ YY σ r σ Y (t)) F rrσ r 2 F (1 η 1 )r + θ η 0 (t) + 1. (A.3) 0 = vv 1 α hk σ r 2 g rr σ H 2 h 2 g hh ρ HH σ r σ H hg rh + κ(r r) + γ 1 γ σ rλ B g r + h r + 1 γ λ Hσ H v g h + g t + θ γ 1 γ μ 1 1 γ γ + γ θ 1 γ μ δ γ γ 1 γ γ 1 r 2γ 2 λ λ g. (A.4) In equation (22), the function F(r, t) satisfied the PDE in equation (A.3) under the real-world measure. In the risk-neutral measure, the equation (A.3) can be rewrite by replaced η 0 (t) to η (t), 0 r to r, and θ to μ. In equation (25), the function g(r, h, t) satisfied the PDE in equation (A.4). 26