Optimal consumption, portfolio, and life insurance policies under interest rate and inflation risks

Insurance: Mathematics and Economics 73 (2017) 54–67

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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime

Optimal consumption, portfolio, and life insurance policies under interest rate and inflation risks Nan-Wei Han a,∗ , Mao-Wei Hung b a

Department of Banking and Finance, College of Finance, Takming University of Science and Technology, No. 56, Section 1, Huanshan Road, Taipei, Taiwan

b

Department of International Business, College of Management, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei, Taiwan

article

info

Article history: Received July 2016 Received in revised form January 2017 Accepted 6 January 2017 Available online 17 January 2017 JEL classification: C61 G11 G22 Keywords: Life insurance Stochastic differential utility Interest rate risk Inflation Elasticity of intertemporal substitution Risk aversion

abstract This paper solves the optimal life insurance, consumption, and portfolio decisions of a wage earner before retirement under interest rate and inflation risks. The wage earner’s preferences are represented by the stochastic differential utility, which separates the coefficient of relative risk aversion from the elasticity of intertemporal substitution (EIS). The wage earner’s life insurance demand is affected by the volatile interest rates and inflation. The optimal life insurance demand decreases with the level of nominal interest rates. Under an assumption of deterministic nominal income, the demand for life insurance would not be affected by the level of inflation. However, if the wage earner’s income is indexed to inflation, the life insurance demand would increase with the level of inflation. Furthermore, under investment opportunities with greater volatilities, wage earners who optimally allocate their wealth to the financial market benefit more from financial investments and cut their demand for life insurance. An analysis of EIS and risk aversion on life insurance demand shows that the demand for life insurance over the planning horizon increases with the measure of relative risk aversion but decreases with EIS. Optimal consumption is affected by the insurance premium load and the direction depends on the size of EIS relative to unity. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Wage earners with bequest motives purchase life insurance to protect their dependents against the loss of their human capital, i.e., the present value of future labor income, in the event of premature death before retirement. Following Hakansson (1969) and Richard (1975), many researchers extend the intertemporal consumption–portfolio choice model to incorporate the optimal demand for life insurance under an uncertain lifetime (see, e.g. Babbel and Ohtsuka, 1989; Huang and Milevsky, 2008; Pirvu and Zhang, 2012; Pliska and Ye, 2007). However, in these intertemporal optimization models, the effect of intertemporal substitution on life insurance demand does not derive as much attention as it deserves. One of the possible reasons is that most existing models are established with the power utility function in which the elasticity of intertemporal substitution (EIS) is represented by the inverse of the coefficient of relative risk

∗

Corresponding author. Fax: +886 226571726. E-mail addresses: [email protected] (N.-W. Han), [email protected] (M.-W. Hung). http://dx.doi.org/10.1016/j.insmatheco.2017.01.004 0167-6687/© 2017 Elsevier B.V. All rights reserved.

aversion; as a result, the effects of EIS and risk aversion are entangled. In this study, a continuous-time model is proposed to analyze a wage earner’s optimal consumption, portfolio, and life insurance strategies before retirement. It is assumed that the wage earner’s lifetime utility includes the utility of consumption when alive and the utility of bequest when a premature death occurs before retirement. The legacy received by the wage earner’s dependents is composed of the payment of life insurance and the wage earner’s financials savings upon death. To disentangle the effects of risk aversion and EIS on optimal life insurance purchase, we employ the stochastic differential utility (SDU) proposed by Duffie and Epstein (1992) to represent the wage earner’s preferences. SDU is the continuous-time limit of the recursive utility studied by Kreps and Porteus (1978) and Epstein and Zin (1989) and nests the timeadditive power utility as a special case. Furthermore, since the problem of optimal consumption, portfolio, and life insurance is a long-term financial planning problem, it is more realistic to take the uncertainties of investment opportunities into consideration. In this paper, we consider the risks of interest rates and inflation in the environment. The exact solution of the optimal strategies under SDU with stochastic

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

investment opportunities is obtained in two special cases: the case of unit EIS and the case of time-additive power utility. In a more general case with EIS different from unity, we employ the loglinearization method introduced by Chacko and Viceira (2005) to find an approximate solution of the problem. Under the assumption of stochastic interest rates and inflation, we find that the optimal portfolio consists of a self-financing and a replicating portfolio. The self-financing portfolio is the portfolio as if there is no human capital. It includes a speculative demand determined by the risk-return trade-off of risky assets and a hedging demand which hedges against the uncertainties of interest rates and inflation. The replicating portfolio is referred to as the portfolio which replicates the returns on the wage earner’s human capital to hedge against the risk of human capital. The optimal portfolio of wage earners with low risk aversion is highly leveraged. For the purpose of speculation, it takes a short position in inflation-indexed bonds and invests heavily on nominal bonds and stocks. When the wage earners are highly risk averse, the portfolio is balanced. The demands for nominal bonds and stocks fall and the demand for indexed bonds rises to hedge against the inflation risk. As to the optimal life insurance demand, we investigate the effects of interest rates and inflation. We find that the optimal proportion of human capital being insured would decrease with the level of nominal interest rate. This is because that the wage earner’s human capital is the discounted present value of his or her future labor income and is equivalent to a nominal coupon bond. A high level of interest rate would reduce the value of human capital and thus the demand for life insurance. Under an assumption of deterministic nominal income, the wager earner’s human capital is equivalent to a nominal coupon bond and the optimal proportion of human capital being insured is not affected by the level of inflation. However, if the wage earner’s income is indexed to inflation, the value of human capital would be equivalent to an inflation-indexed coupon bond. In this case, a high inflation rate would increase the value of human capital and therefore the demand for life insurance. Taking the uncertainties of interest rates and inflation into consideration, we analyze the effects of interest rate volatility and inflation volatility on optimal life insurance demand. The results show that the wage earners who optimally allocate their wealth to the financial market would earn more risk premium when interest rates and inflation are more volatile. The risk premium would raise the growth rate of the wage earner’s financial savings and, as a result, would decrease the wage earner’s demand for life insurance. An analysis of EIS and risk aversion on life insurance demand shows that EIS has no cross-sectional effect on optimal life insurance demand, i.e., with certain amounts of human capital and financial savings, the wage earner’s demand for life insurance would increase with the extent of risk aversion while being independent of EIS. This result is consistent with the findings of previous studies based on the power utility. However, this crosssectional irrelevance of EIS with life insurance demand does not necessarily imply that the overall expenditure on life insurance would not be affected by EIS. In fact, at any specific time, the life insurance demand would be affected by the amount of the wage earner’s financial savings over the planning horizon since the life insurance payment and the financial savings act as substitutes for each other. For individuals endowed with the same initial wealth and wage income, their financial savings could be quite different depending on how they allocate their resources. In this study, we examine the intertemporal effects of EIS and risk aversion on life insurance demand. The wage earner with a higher EIS purchases less life insurance over the planning horizon. This is because with a higher EIS, the wage earner is more willing to substitute the future for current consumptions. This leads to

55

a higher level of financial savings over the planning horizon and, as a result, a lower life insurance demand. Next, the wage earner with a higher risk aversion purchases more life insurance over the planning horizon. As indicated by earlier research, the wage earner who is more risk averse would cross-sectionally purchase more life insurance to hedge against the loss of human capital in the event of premature death. Moreover, a more risk-averse wage earner would invest less in the risky asset and relinquish the opportunities to collect the risk premium. In the long run, this would result in a lower level of financial savings over the planning horizon and increase the demand for life insurance. The numerical examples in this study show that the effect of EIS is weaker with a higher risk aversion and the effect of risk aversion is stronger with a higher EIS. We also show that the optimal consumption–wealth ratio is affected by the load of insurance premium. The direction of this effect is decided by the level of EIS. When EIS is smaller than unity, the optimal consumption–wealth ratio would increase with the load of life insurance premium. When EIS is greater than unity, the optimal consumption–wealth ratio would decrease with life insurance premium load. 2. Related literature The Bulk of the published literature has been devoted to the research of the demand for life insurance from an individual’s perspective since the pioneering work of Yaari (1965), who shows that the intertemporal optimization problem with an uncertain lifetime could be transformed into an equivalent one with a certain horizon. Hakansson (1969) uses the discretetime dynamic programming approach to solve the optimal consumption, investment, and life insurance strategies in one model. Richard (1975) extends the seminal work of Merton (1971) to solve the optimal decisions of consumption, portfolio choice, as well as life insurance with a continuous-time model. Campbell (1980), who uses a single-period model, solves the demand function for life insurance explicitly in terms of the individual’s risk aversion, intensity for bequests, and the insurance company’s loading charge. Pliska and Ye (2007) and Zhu (2007) analyze the effects that a wide array of economic parameters, such as risk aversion, subjective time preferences, risk-free rate, and insurance premium load, have on optimal life insurance demand. Huang and Milevsky (2008) and Huang et al. (2008) solve the optimal portfolio and life insurance decisions by focusing on the correlation between the dynamics of human and financial capital. More recently, Duarte et al. (2014) extend the model of Pliska and Ye (2007) to solve the optimal insurance, consumption, and portfolio rules with multiple risky securities. Pirvu and Zhang (2012) and Kwak and Lim (2014) solve the optimal consumption, investment, and life insurance problem with stochastic investment opportunities. Pirvu and Zhang (2012) investigate the effect of the stochastic market price of financial risk on optimal insurance; and Kwak and Lim (2014) analyze the effect of inflation risk on life insurance demand. Most of these existing works are established under the widely used power utility function and consistently show that the optimal life insurance demand would increase with the measure of relative risk aversion. With the power utility function, EIS is represented by the reciprocal of the coefficient of relative risk aversion; as a result, the effects of EIS and risk aversion on optimal policies are entangled. To disentangle the effects of risk aversion and EIS, we use SDU proposed by Duffie and Epstein (1992) to represent the preferences of the wage earner. SDU has been used to investigate the optimal consumption–portfolio problem in a financial context (see, e.g. Bhamra and Uppal, 2006; Chacko and Viceira, 2005; Chou et al., 2011; Schroder and Skiadas, 1999; Svensson, 1989). As pointed out by Svensson (1989), EIS affects the consumption–saving decision but it has no effect on the optimal portfolio decision under constant investment opportunities. Under stochastic investment opportunities, Bhamra and Uppal (2006)

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N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

show that EIS affects the magnitude of the hedging demand of risky assets, while the sign of the hedging demand depends on the size of the coefficient of risk aversion relative to unity. Our research is related to a recent work of Jensen and Steffensen (2015) who solve the problem of optimal consumption, investment, and life insurance demand under the separation between risk aversion and EIS by forming consumption certainty equivalents in the utility function. They show that their solution is identical to the solution under SDU in the case without mortality risk. By separating risk aversion from EIS, Jensen and Steffensen (2015) show that under mortality risk, the consumption path of the decision maker could be hump-shaped. We note that the problem analyzed in Jensen and Steffensen (2015) is solved under the assumption of constant investment opportunities. To extend literature, we employ the SDU framework under a more realistic assumption by taking the stochastic investment opportunities into consideration. Nominal interest rates and inflation are assumed to be stochastic in the economy. Pirvu and Zhang (2012) and Kwak and Lim (2014) have analyzed the respective impacts of stochastic market price of financial risk and uncertain inflation on life insurance demand. However, the interest rate risk is not considered in both studies for tractability. In this study, we extend the work of Kwak and Lim (2014) to solve the problem under stochastic interest rates. The remaining parts of this study are organized as follows. In Section 3, we establish the optimization problem on consumption, portfolio, and life insurance under SDU. In Section 4, we solve the optimization problem and analyze the properties of the optimal rules. Some further discussions are given in Section 5 to compare our findings to the related work of Jensen and Steffensen (2015). Finally, a summary of our major findings is given in Section 6.

Xt ≡ Wt +

BR κ(R¯ − R) +

 λ(u)du ,

t

and lim

∆t →0

Prob(t < τ < t + ∆t |τ > t )

∆t

∂B 1 + BRR σR2 = BR − χR BR σR , ∂t 2

 t +∆ t = lim

∆t →0

t

s pt ds

∆t

= λ(t ), (2)

where λ(t ) is referred to as the mortality rate of the wage earner. The wage earner receives a deterministic income stream continuously at a rate Y (t ) for 0 ≤ t < T ∧ τ , where T ∧ τ ≡ min(τ , T ). To protect against the loss of the wage earner’s human capital, the wage earner attempts to buy life insurance before retirement. In line with previous research (Huang and Milevsky, 2008; Pliska and Ye, 2007; Richard, 1975), the life insurance offered is assumed to be an instantaneous term life insurance. For 0 ≤ t < T ∧ τ , the wage earner pays the premium at a rate Pt continuously. If the insured dies during the time interval from t to t +dt, his or her dependents will receive the face value of the life insurance Pt /η(t ), where η(t ) is called the premium-insurance ratio and η(t ) ≥ λ(t ) is assumed so that the insurance company’s expected profit would be non-negative. For simplicity, we assume that η(t ) = ξ λ(t ), where ξ ≥ 1 is defined as the life insurance premium load. Next, denoting Wt as the wage earner’s financial savings accumulated up to time t, the total legacy received by the dependents upon premature death at time t < T is given by

(5)

Bst = exp [b0 (t ; s) − b1 (t ; s)Rt ] ,

(6)

where b1 (t ; s) ≡ b0 (t ; s) ≡

1 k



1 − e−κ(s−t ) ,

R¯ +



σ2 χR σR − R2 κ 2κ

(7)



[b1 (t ; s) − s + t] −

σR2 2 b1 (t ; s). 4κ (8)

Applying Itô’s lemma, the returns on the bond could be described as the following diffusion process:

= (Rt + χR σR b1 (t ; s))dt + σR b1 (t ; s)dZtR .

(9)

Without loss of generality, we assume that the wage earner could invest in a bond market fund Bt which has a constant maturity m, and the instantaneous returns on Bt could be simplified as: dBt Bt

= (Rt + σB χR )dt + σB dZtR ,

(10)

where σB ≡ σR κ −1 1 − e−κ m . Next, we denote It as the time t consumer price index. The price index is normalized such that I0 = 1. The dynamics of It follow a diffusion process:



(1)

(4)

where χR is the market price of interest rate risk, and BR and BRR denote the first- and the second-order partial derivative of Bst with respect to R, respectively. The solution of Bst is given as follows:

Bst

s

(3)

where ZtR is a standard Brownian motion, R¯ is the long-run mean of Rt , and κ represents the speed of mean-reversion. Let Bst denote the time t value of a nominal zero-coupon bond which pays one dollar on its maturity date s. According to the arbitrage-free argument, Bst would satisfy the following partial differential equation with the boundary condition Bss = 1:

3.1. The basic assumptions

 

.

dRt = κ(R¯ − Rt )dt − σR dZtR ,

dBst

s pt = λ(s) exp −

η(t )

In the financial market, we assume that the instantaneous nominal interest rate Rt follows an Ornstein–Uhlenbeck process introduced by Vasicek (1977):

3. The model

In this study, a wage earner’s planning horizon is denoted by T , representing the prescribed retirement date for the wage earner. The wage earner’s lifetime is uncertain, and the time of death is denoted by a non-negative random variable τ with the probability density function for τ = s conditional on τ > t as follows:

Pt

dIt It



= π dt + σIR dZtR + σII dZtI ,

(11)

where π is the expected inflation rate, and ZtI is a standard Brownian motion independent of ZtR . In an environment with inflation risk, the inflation-indexed bond is introduced to hedge against the inflation risk. The time t price of an inflation-indexed zero-coupon bond Gst , which pays Is when it matures at time s would satisfy the following partial differential equation with the boundary condition Gss = Is :

∂G + GR κ(R¯ − R) + GI I π − GRI I σR σI1 ∂t 1

1

2

2

+ σR2 GRR + I 2 (σIR2 + σII2 )GII = RG − GR σR χR + GI I (χR σIR + χI σII ),

(12)

where χI represents the market price of risk with respect to ZtI . Solving Eq. (12), we obtain the price of the inflation-indexed bond: Gst = It exp [g0 (t ; s) − g1 (t ; s)Rt ] ,

(13)

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

 1 1 − e−κ(s−t ) , (14) κ  χR − σIR + σIR χR + σII χI − π (s − t ) g0 (t , s) = − R¯ + σR κ   χR − σIR σR2 σ2 ¯ − 2 g1 (t ; s) − R g12 (t ; s). + R + σR κ 2κ 4κ

g1 (t ; s) =

(15) For simplicity, we also assume that the indexed-bond to be traded has a constant maturity m such that dGt Gt

= (Rt + χR σGR + χI σGI ) dt + σGR dZtR + σGI dZtI ,

dSt St

In the aggregator, δ represents the subjective time preference, γ is the coefficient of relative risk aversion, and ϕ is the elasticity of intertemporal substitution of the wage earner. In contrast to the power utility representation, which forces EIS to be the reciprocal of relative risk aversion, the main advantage of SDU is that it represents EIS and risk aversion with two independent parameters. As indicated by Duffie and Epstein (1992), SDU nests the traditional power utility as a special case because when ϕ = 1/γ , SDU would be reduced to



(18)

where

σB 0 0 ≡ σGR σGI σSR σSI   χR 3 ≡ χI , χS  R

0 0



σSS

,

Jt (wt ) = Et

(19)

(25)

In this study, we assume that the wage earner’s utility is composed of three components: the interim real consumptions, ct ≡ Ct /It ; the real bequests for the wage earner’s dependents in the event of premature death, say, xτ ≡ Wτ /Iτ + Pτ /[Iτ η(τ )] = wτ + pτ /η(τ ); and the real wealth at retirement age, wT ≡ WT /IT . The utility of consumption is measured by the aggregator and the second and the third term on the right-hand side of Eq. (25) represent the utilities of bequest and terminal wealth, respectively. In our model, the utility of bequest is measured in terms of the wage earner’s lifetime utility as if the wage earner is endowed with the legacy xt when τ = t and the bequest motive is measured by β where 0 ≤ β ≤ 1. According to the following facts that E 1{τ >s} |τ > t = Prob(τ > s|τ > t )



  s  = exp − λ(u)du

(26)

t

and (21)

dZtS

E Jτ (xτ )1{τ ≤T } |τ > t =





T



(xs )ds,

s p t Js

(27)

t

the wage earner’s optimization problem with an uncertain lifetime can be transformed into the following one with a certain horizon T:

3.2. The objective The wage earner’s preferences are represented by SDU introduced by Duffie and Epstein (1992). The expected lifetime utility is represented by



f (cs , Js )ds

J t = Et

f (cs , Js )1{τ >s} ds

  1−γ  wT  + β Jτ (xτ )1{τ ≤T } + 1{τ >T }  τ > t ,  1−γ



dZt

T

(24)

T



(20)

  dZt ≡  dZtI  .



− δ Js ds ,

1−γ

which is ordinally equivalent to the recursive form of the power utility function. With an uncertain lifetime, the expected lifetime utility at time t is defined as follows:

(17)

dWt = Wt (Rt + α⊤ t 03)dt



cs

t

where ZtS is a standard Brownian motion independent of ZtR and ZtI and χS is the market price of risk with respect to ZtS . Denoting Ct as the instantaneous nominal consumption rate, and αt ≡ (αtB , αtG , αtS )⊤ as the portfolio weights of Bt , Gt and St , the intertemporal nominal budget constraint can be written as follows:

− Ct dt + Y (t )dt − Pt dt + Wt α⊤ t 0dZt ,

δ

t

= (Rt + σSR χR + σSI χI + σSS χS ) dt + σSR dZtR + σSI dZtI + σSS dZtS ,



1−γ

T

J t = Et (16)

where σGR ≡ σIR + σB and σGI ≡ σII . The third asset in the financial market is a stock market fund with market value St , and the instantaneous returns on the stock market fund satisfy the following stochastic differential equation:

57

(22)

max Jt (wt ) = Et

T



 

cs ,αs ,ps t ≤s≤T

t

f (c , J )

   −1 1−(1/ϕ) C 1   δ (1 − γ ) J −1 , 1−    ϕ ((1 − γ )J )1/(1−γ ) ≡ ϕ ̸= 1       δ(1 − γ )J log c − 1 log ((1 − γ )J ) , ϕ = 1. 1−γ

t

T

wT λ(s)ds 1−γ 

1−γ

 .

(28)

Optimal policies are identified by using stochastic dynamic programming. First, since the utility are measured in real terms, we rewrite the intertemporal budget constraint in terms of real variables: dwt = d(Wt /It ) =

(23)

 λ(u)du [f (cs , Js (ws ))

t

  + λ(s)β Js (xs )]ds + exp −

t

where Et [·] ≡ E[·|Ft ] is the conditional expectation operator and {Ft |0 ≤ t ≤ T } is the filtration generated by the Brownian motions Zti , i = R, I , S. The function f (c , J ) is called the normalized aggregator and is defined as follows:

s

exp −

+

Wt



dIt

dWt Wt Wt

2

It Wt

− 

Wt dIt It dWt

It



dIt



− It Wt It  ⊤  = wt rt +  αt 03 dt − ct dt + yt − pt dt + wt α⊤ t 0dZt , (29) It



It

58

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

with  α t ≡ α t − ( 0 −1 ) ⊤ σ I ,  3 ≡ 3 − σ I , and σ I ≡ (σIR , σII , 0)⊤ . The term rt ≡ Rt − π + σ ⊤ 3 denotes the real interest rate at time I t, which evolves as: drt = κ(¯r − rt )dt − σr dZtR ,

(30)

where σr ≡ σR and r¯ ≡ R¯ − π + σ ⊤ I 3. Let Ht denote the human capital of the wage earner, i.e., the nominal present value of the wage earner’s future wage income until τ ∧ T : Ht =

T

 

   s η(u)du Y (s)Bst ds, exp −

t ≤ T;

(31)

t

t



0,

t > T.

The dynamics of the wage earner’s real human capital ht ≡ Ht /It when t ≤ T are given by ⊤  dht = −yt dt + ht rt + η(t ) + σ ⊤ h (t )3 dt + ht σ h (t )dZt ,





(32)

where σ h (t ) ≡ σ H (t ) − σ I and

 ∂H 1  t −  ∂ R Ht   σH (t ) ≡    0 1

 Ht = 

t

   s  s exp − η(u)du Y (s)Bt σR b1 (t ; s)ds  , (33) t  0 0

is the duration of the human capital. Next, we define some new variables as follows:

w t ≡ wt + ht ,  pt ≡ pt − ht η(t ),

(34) (35) (36)

The variable w t is the sum of the wage earner’s real financial savings and real human capital and is referred to as the total real wealth of the wage earner. As to  pt defined in Eq. (35), it is the net life insurance demand without the inclusion of the life insurance needed to capitalize the wage earner’s future income. According to Eqs. (29) and (32), the dynamics of w t are given by:

 dw t = w t rt +  α⊤ pt dt + w t α⊤ t 03 dt − ct dt − t 0dZt , 

(37)

where

wt  αt + ht (0−1 )⊤ σ h (t ) . (38) w t By definition, wt + pt /η(t ) = w t + pt /η(t ) and w T = wT + hT = wT since hT = 0. Hence, after changing variables, Eq. (28) can be

 αt ≡

−V w Vw  r  α∗t = 6−1 0 3− 6 −1 0 σ r , Vw t Vw t w w w w   =x∗ = ηt Vw Vw .  w t t βλt

 Jt ( w t ) = Et

(42) (43)

By substituting these first-order conditions into Eq. (40), we obtain a partial differential equation of Vt . We could derive the optimal policies after we solve the partial differential equation of Vt .

The solution of the value function under SDU in an environment with stochastic investment opportunities involves a non-linear partial differential equation. It is hard to find the exact analytical solution in general cases. In a special case of unit EIS, the exact solution could be obtained and we give the optimal policies under ϕ = 1 in this section. When ϕ = 1, by an educated conjecture, the value function would have the following form: Vt = Qt

  s  T  exp − λ(u)du f (cs , Js ( ws )) t

T

   + λ(s)β Js ( xs ) ds + exp − t

1−γ

w T λ(u)du 1−γ 

∂ Q /∂ t Q

+

, (39)

∂V + f (ct , V ) + λ(t )(β V ( xt ) − V ) ∂t  + Vw t (rt +  α⊤ pt + Vr κ(¯r − rt ) w  ct − Vw  t 03) − Vw  1 1 ⊤ 2 ⊤ ⊤ + Vw t α 0σ r + Vw t  αt 6 αt + Vrr σ r σ r , r w w w 

0 = max

 αt ,ct , pt

2

− δ log Q +

1−γ 2γ

σr2



Qr



1−γ



γ

2

Q

 σ⊤ r − r) r 3 + κ(¯

1

Qrr

2

Q

+ σr2



Qr Q

+ (1 − γ ) [r + M (t )] = 0, (45)

and



1

⊤  3  3 + λ(t )   γ β 1/γ 1− γ1 1 × ξ+ ξ . − 1−γ 1−γ

M (t ) ≡ δ log δ − δ +

2γ

Qt = exp {(1 − γ ) [q0 (t ) + q1 (t )rt ]} ,



where  xs ≡ w s + ps /η(s). Denoting Vt as the value function of the optimization problem, the Hamilton–Jacobi–Bellman equation is given by

2

(44)

(46)

The solution of Qt is then given by:

t



wt 1−γ

where Qt is the solution of the following partial differential equation with the boundary condition QT = 1:

expressed in terms of the new variables:



(41)

ϕ = 1.

1−γ

 Jt ( wt ) ≡ J ( wt − ht ) = J (wt ).



Vw 

ϕ ̸= 1;

4.1. Optimal policies under unit EIS T



 −ϕ 1−1/ϕ V  −1  1 −γ  w [(1 − γ )V ] , δ ct∗ =   δ(1 − γ )V , 

4. Optimal policies

0



where σ r ≡ (−σR , 0, 0)⊤ and 6 ≡ 00⊤ . The optimal policies are determined by the first-order conditions with respect to ct , αt , and  xt :

(40)

q1 (t ) ≡ q0 (t ) =

1

κ +δ  T



1−e

e−δ(s−t )

 −(κ+δ)(T −t )



1−γ

(47)

,

(48)



 σ⊤ r 3 + κ r¯ q1 (s)    1 1 2 2 − 1− σr q1 (s) + M (s) ds. 2 γ t

γ

(49)

By substituting the value function into the first-order conditions in Eqs. (41)–(43), the optimal consumption, portfolio and life insurance are thus obtained and we demonstrate the optimal policies in the following proposition.

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

Proposition 1. The optimal consumption, portfolio, and life insurance coverage when ϕ = 1 are ct∗

= δ, (50) w t + ht     1/γ p∗t β β 1/γ = ht + − 1 wt , (51) ηt ξ ξ  wt + ht 1 −1 α∗t = 6 03 wt γ      1 (e−(κ+δ)(T −t ) − 1) −1 1 + 1− 6 0σ r + 1 − 6−1 0σ I γ κ +δ γ −

ht

wt

6−1 0σ H (t ).

(52)

With log utility of consumptions, i.e., ϕ = 1, the optimal consumption–wealth ratio is independent of the financial and mortality risks. The ratio is constant over time and equals to the subject discount factor, δ . According to Eq. (51), there is a negative relationship between the demand for life insurance and the load for insurance premium. The higher the insurance premium load, the more expensive is the life insurance to the insured and, as a result, less insurance coverage is purchased. In a special case of ξ = 1 and β = 1, the optimal life insurance coverage equals the wage earner’s human capital, i.e., p∗t /η(t ) = ht . This agrees with the findings of Campbell (1980), who shows that a full insurance coverage against ht is optimal to the wage earner when the utility of bequest is equally weighted as the wage earner’s lifetime utility (β = 1) and an actuarially fair insurance premium is charged. In this case, the optimal life insurance amount, pt /η(t ), is not affected by the wage earner’s risk aversion. This explains the finding in Huang and Milevsky (2008), which argues that the optimal life insurance is insensitive to the extent of risk aversion, since an unloaded insurance premium is assumed in their model. When the premium of life insurance is loaded, i.e., ξ > 1, the optimal life insurance coverage would be less than ht and the optimal life insurance would decrease with the amount of the wage earner’s savings at time t. To explain why the optimal insurance decreases with wt , we re-emphasize that the utility of bequest is measured by the total legacy received by the wage earner’s dependents in the event of premature death. Since the legacy (xt ) is the sum of the savings account balance (wt ) and the life insurance coverage (pt /η(t )), wt and pt /η(t ) are perfect substitutes; this implies that wealthier wage earners would buy less life insurance. This result agrees with the findings in previous research based on the power utility (see, e.g. Richard, 1975; Zhu, 2007). The optimal portfolio derived in Eq. (52) is composed of two portfolios weighted by (wt + ht )/wt and −ht /wt , respectively. The first portfolio is referred to as the self-financing portfolio, i.e., the portfolio when ht = 0. This portfolio is composed of three parts. The first part, 6−1 03, is the speculative demand for risky assets decided by the return–risk trade-off of risky assets. The second and the third part, e

−(κ+δ)(T −t )

 −1

κ +δ

6 −1 0 σ r = 



1−e

−(κ+δ)

κ +δ 0 0

σr σB  ,

(53)

6−1 0σ I =

1 0

6−1 0σ H (t )



1

1 H  t

=

    s s η(u)du Y (s)Bt σR b1 (t ; s)ds exp −  t  0

T

 t

σB 

(55)

0 is referred to as the replicating portfolio, which replicates the returns on the human capital, Ht . This portfolio is composed of nominal bonds and as a result would reduce the demand for nominal bonds according to Eq. (52). In our model, the human capital is composed of a stream of deterministic cash flows in the future. Being endowed with the human capital is equivalent to holding a long position in nominal coupon bonds and, as a result, would shrink the demand for nominal bonds. Furthermore, in contrast to the self-financing portfolio, we find that the replicating portfolio would be affected by the life insurance premium load. Remember that the term T



1

s

  exp −

Ht

t

t

 ∂ Ht 1 η(u)du Y (s)Bst σR b1 (t ; s)ds = − ∂ R Ht

(56)

is the duration of the human capital and the human capital could be viewed as a coupon  sbond with instantaneous coupon payment Y (s) weighted by − t η(u)du, where η(u) = ξ λ(u). When the insurance premium load ξ becomes higher, the weight on the coupon with longer maturity drops more sharply than the weight on the coupon with shorter maturity. Hence, the duration of the coupon bond would decrease. This implies that the demand for nominal bonds in the replicating portfolio would therefore decrease with the life insurance premium load.

4.2. Optimal policies when EIS ̸= 1 When ϕ ̸= 1, the value function would be solved as γ −1 1−ϕ

w1−γ , 1−γ

(57)

where Ft is the solution of the following non-linear partial differential equation with the boundary condition FT = 1:

    1 ∂F 1  + κ(¯r − r ) − 1 − σ⊤ 3 Fr + σr2 Frr r ∂t γ 2

and

  −1

are the hedging demands for risky assets which hedge against the interest rate risk and the inflation risk, respectively. The relative weights of the speculative and the hedging demand in the selffinancing portfolio are decided by the wage earner’s attitude toward risk aversion. Since 1 − e−(k+δ)(T −t ) > 0, the hedging demand against the interest rate risk is represented by a long position in nominal bonds. This reflects the fact that the returns on nominal bonds are perfectly negatively correlated to the interest rates. The hedging demand against the inflation risk is composed of a long position in indexed bonds and a short position in nominal bonds. This is because the returns on indexed bonds are positively correlated to both the inflation and the returns on nominal bonds. Hence, holding indexed bonds to hedge against the inflation risk would crowd out the demand for nominal bonds. We note that the components of the self-financing portfolio are not affected by the mortality risk and the life insurance premium load. The other portfolio

Vt = F t



59

,

1

(54)

−

γ

−ϕ

2(1 − ϕ)

σr2

Fr2 F

− [Φ (t ) + (1 − ϕ)r ]F + δ ϕ = 0,

(58)

60

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

where v¯ denotes the long-run unconditional mean of vt and

and



ζ1 = ev¯ , ζ0 = ζ1 (1 − log ζ1 + ϕ log δ).

1

⊤  3 + λ(t ) 3  2γ   γ β 1/γ 1− 1 1 × ξ+ ξ r − . 1−γ 1−γ

Φ (t ) = ϕδ + (1 − ϕ)

(59)

where

    1 ∂F  + κ(¯r − r ) − 1 − σ⊤ 3 Fr r ∂t γ     1 1 1 + σr2 Frr − Φ (t ) + 1 − r F + δ γ = 0, 2 γ

    1  φ(s) ≡ Φ (s) − ζ0 − κ r¯ − 1 − σ⊤ 3 ν1 (s) r γ  2 1 1 − 1/γ σr2 ν12 (s). − 2 1−ϕ

and it can be solved by employing the Feynman Kac theory as follows: Proposition 2. When the wage earner’s preferences are represented by the power utility function, the value function is solved as follows: γ

Vt = Ft

w1−γ , 1−γ

(61)

where Ft = δ

1

γ

−

e

s t

Φ (u)du

Uts ds

+e

−

T t

Φ (u)du

UtT

.

(62)

t

The function Φ (t ) is defined in Eq. (59) with ϕ replaced by γ −1 and the process Uts is defined as: Uts = exp

 1−

u1 (t ; s) =

1



γ

u0 (t ; s) − u1 (t ; s)rt



,

(63)

 1 1 − e−κ(s−t ) ,

κ 

(64)

σr χR 1 σr2 − 1− γ κ γ 2κ 2   2 1 σr 2 u (t ; s). × [u1 (t ; s) − s + t] − 1 − γ 4κ 1

u0 (t ; s) =



r¯ +

1−

Proof. See Appendix.

1









(65)



According to Proposition 2, even though the exact solution can be obtained when γ ϕ = 1, the solution is complicated and is hard to interpret. To obtain an analytical solution which is easy to interpret in a more general case of γ ϕ ̸= 1, we employ the loglinearization method introduced by Chacko and Viceira (2005) to find an approximate solution for Ft . According to Eqs. (41) and (57), the optimal consumption–wealth ratio is given by: ct∗

w t

= δϕ

1 Ft

.

(66)

Denoting vt ≡ log(ct∗ / wt ), the first-order Taylor’s expansion of ct∗ / wt around the unconditional mean of vt could be expressed as:

δϕ

1 Ft

Ft = exp (ν0 (t ) + ν1 (t )rt ) ,

 ϕ−1  1 − e−(κ+ζ1 )(T −t ) , κ + ζ1  T e−ζ1 (s−t ) φ(s)ds, ν0 (t ) = − ν1 (t ) =

= evt ≈ ev¯ + ev¯ (vt − v¯ ) = ζ0 − ζ1 log Ft

(67)

(70) (71) (72)

t

(73)

We can see that this approximate solution is exact when ϕ = 1 since in that case the consumption–wealth ratio is a constant, δ , over time and by definition ζ1 = δ . According to the value function, we can derive the optimal policies when γ ϕ ̸= 1. Proposition 3. The optimal policies under stochastic interest rates and inflation when γ ϕ ̸= 1 are given by ct∗

T



(69)

Substituting this approximate expression for δ ϕ /F in Eq. (58), it is straightforward to verify that the solution of Ft could be obtained as:

The non-linearity stems from the term Fr2 /F appeared in Eq. (58) and makes it difficult to find the closed-form solution of F . The nonlinearity can be eliminated when σr = 0, which is the case under constant investment opportunities; or when γ ϕ = 1, i.e., the wage earner’s preferences are represented by the time-additive power utility. When γ ϕ = 1, the partial differential equation in Eq. (58) reduces to a parabolic partial differential equation with the boundary condition FT = 1:

(60)

(68)

w t + ht

=

δϕ

(74)

Ft

    1/γ β 1/γ β = ht + − 1 wt , (75) η(t ) ξ ξ    1 e−(κ+ζ1 )(T −t ) − 1 −1 w t + ht 1 −1 6 03 + 1 − 6 0σ R α∗t = wt γ γ κ + ζ1    ht −1 1 6−1 0σ I − 6 0σ H (t ). (76) + 1− γ wt p∗t

We note that the optimal portfolio rule obtained in Proposition 3 is the same as the rule of ϕ = 1 given in Proposition 1 except that ζ1 substitutes for δ in the optimal portfolio. As a result, most properties of the optimal portfolio aforementioned remain unchanged. One distinguishing difference between Propositions 1 and 3 is that the optimal consumption–wealth ratio when ϕ ̸= 1 is no longer the subject discount factor δ over time. In this study, we emphasize that the consumption–wealth ratio would be affected by the insurance premium load and that the direction depends on the size of EIS relative to unity. Proposition 4. The relationship between optimal consumption– wealth ratio and the life insurance premium load depends on the value of EIS. The optimal consumption–wealth ratio ct∗ /(wt + ht ) would increase (decrease) with ξ when ϕ < 1(> 1). Proof. See Appendix.



To explain this, we note that there are two different effects when ξ varies. As we defined in Eq. (3), the legacy received by the dependents in the event of premature death is the sum of the wage earner’s savings and the insurance coverage. To maintain the

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67 Table 1 Baseline parameters. W0

1 0.0395 0.00002 0.1459 1 

κ σIR σSS β Y (t )

T R¯

40 exp 0.04t

σII χR ξ

40 0.0369 0.081 0.209 e0.1

δ π σSR χI m

λ(t )

0.03 0.0357 0.0124 −0.105 10  1 10.5

exp

long-run expected paths. This means that the nominal interest rate Rt = R¯ , It = eπ t , and

γ σR σSI χS t −63.18 10.5

2 0.0195 −0.01 0.343

dpt

η(t )

= 0.

dpt

= −η(t ) = −ξ λ(t ).

(77)

(78)

When ξ is higher, the wage earner benefits more by cutting his or her current consumption since it increases the wage earner’s savings and substitutes for more expenses on life insurance. This is the so-called substitution effect. Another effect when ξ changes is the income effect. According to Eqs. (35) and (75), the net life insurance demand without the inclusion of the life insurance needed to capitalize the wage earner’s future income is

   1 β γ  pt = pt − ht η(t ) = − 1 (wt + ht ) η(t ). ξ ∗

∗

t

r¯ − δ +  αs 0 3− ∗

0

    β 1/γ − 1 η(s)ds . ξ (80)

Thus, the marginal rate of substitution of savings for the premium paid for life insurance is given by dwt

w t = w 0 exp





legacy, or equivalently the utility of bequest, on a fixed level, we must have dxt = dwt +

61

(79)

The net insurance demand would be negative when ξ > 1 or β < 1. As pointed out by Huang and Milevsky (2008) and Pirvu and Zhang (2012), a negative life insurance policy could be viewed as the purchase of a pension annuity on the life of the wage earner. In our model, this means that the wage earner would annuitize part of his or her financial savings to enhance the returns on the savings account. When ξ increases, the return from the annuity purchased is higher. This higher return would make the wage earner consume more out of his or her total wealth and is referred to as the income effect. The net effect of ξ on the optimal consumption depends on the relative strength of the substitution and the income effect. When EIS is smaller than unity, the income effect would dominate, which implies that the optimal consumption would increase as ξ increases. On the other hand, when EIS is greater than unity, the substitution effect dominates and the optimal consumption would decrease with ξ . 4.3. Numerics In this section, several numerical examples are implemented to investigate the properties of the optimal portfolio and life insurance strategies. In this section, we assume ϕ = 1 since the optimal portfolio and life insurance policies of ϕ ̸= 1 are qualitatively quite similar to the solutions under ϕ = 1. The following set of parameters listed in Table 1 is used in the numerical experiments in this section. Most of the financial market parameters are chosen from Munk et al. (2004). We note that χI is negative because if the asset returns are positively related to the inflation, the asset provides compensation for the loss of purchasing power when the inflation is high and, as a result, deserves a negative risk premium. In the first example, we show the optimal portfolio rules under γ = 2 and 5. For simplicity, the numerical results are obtained such that the sample paths of the state variables follow their

Fig. 1 plots the optimal portfolio weights on the stock market fund, the nominal bond, and the inflation-indexed bond. Since the returns on indexed bonds are positively related to the inflation, returns on indexed bonds would be lower than the returns on stocks and nominal bonds because of the negative risk premium associated with the inflation risk. Fig. 1(a) shows that when the risk aversion is low (γ = 2), the portfolio is highly leveraged and concentrates on the stock market fund and the nominal bond. A short position in the indexed bond reflects the fact that the returns on indexed bonds are inferior to the returns on the other two assets. When the risk aversion is high (γ = 5), the incentive to speculate becomes weaker and the hedging demand would dominate the portfolio. According to Eq. (54), the hedging demand against the inflation risk is composed of a long position in indexed bonds and a short position in nominal bonds. Fig. 1(b) shows that the demand for nominal bonds drops when γ is high and, at the same time, the demand for indexed bonds rises to hedge against the inflation risk. The following example examines the effects of the inflation and the interest rates on life insurance demand. In Fig. 2(a), we show the optimal demand for life insurance under three different levels of R. We use R = 3.69% given in Table 1 as the benchmark case. The high and low interest rate scenarios are represented by R = 5.69% and R = 1.69%, respectively. The life insurance demand is represented by the proportion of human capital being insured, p i.e., the ratio of η(tt ) to ht : p∗t /η(t ) ht

  γ1  β w t =1− 1− . ξ ht 

(81)

We note that most existing works express the demand for life insurance in terms of the amount of life insurance premium. However, it is implausible to say that the wage earners who pay less life insurance premium than others do would have a weaker incentive to buy life insurance. According to Eq. (51), the optimal amount of life insurance premium relies on the amount of the wage earner’s human capital. Wage earners with a lower level of human capital would likely pay less insurance premium. Hence, we use the proportion of human capital being insured to measure the strength of life insurance demand. The insurance demand is determined by the wage earner’s risk aversion (γ ), the bequest motive (β), the insurance premium load (ξ ), and the relative amount of w t to ht . Another advantage of using the proportion of human capital being insured to represent the demand for life insurance is that this proportion would remain unchanged whether we express the variables in nominal terms or in real terms. Consistent with Pliska and Ye (2007), Fig. 2(a) shows that the proportion of human capital being insured is negatively related to the level of interest rates. According to Eq. (81), the proportion of human capital being insured is positively related to the value of human capital when β < 1 or ξ > 1. Since the human capital of the wage earner is equivalent to a nominal coupon bond, a higher(lower) interest rate would lead to a lower(higher) value of human capital; this implies a lower(higher) demand for life insurance. Fig. 2(b) explores the effect of inflation. In this example, we assume π = 3.57% to be the benchmark level of expected inflation. The scenarios of low and high inflation are represented

62

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

Fig. 1. This figure plots the optimal portfolio weights on the stock market fund, the nominal bond, and the inflation-indexed bond under: (a) γ = 2 and (b) γ = 5.

Fig. 2. This figure plots the effects of inflation and nominal interest rates on the demand for life insurance. We plot the life insurance demand under three different levels of nominal interest rate in (a) and the life insurance demand under three different levels of expected inflation in (b).

Fig. 3. This figure plots the effects of inflation and nominal interest rates on the demand for life insurance. In this example, the income stream is assumed to be Y (t ) = It exp (0.0043t ).

by π = 1.57% and π = 5.57%, respectively. In this example, we find that the demand for life insurance is not affected by π . This irrelevance of inflation for life insurance demand is due to the assumption of a nominal deterministic income stream, Y (t ), in the current example. In this case, the human capital is equivalent to a nominal coupon bond and its value does not affected by the level of inflation. By assuming a real deterministic income stream, i.e., an inflation-indexed stream of nominal income, the human capital is equivalent to an inflation-indexed coupon bond and its value would be affected by the level of inflation. We redo the previous

example and assume the income stream to be Y (t ) = It y(t ), where y(t ) = 40 exp(0.0043t ) denotes the real income of the wage earner. In this case, the human capital is equivalent to an inflationindexed coupon bond: Htindex =

T



s

  exp −

t

 η(u)du y(s)Gst ds.

(82)

t

The value of the human capital would decrease with Rt and increase with π . Fig. 3(a) shows that the demand for life insurance decreases with the level of interest rate, as is the case in the

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

63

Fig. 4. This figure plots the effects of (a) interest rate volatility and (b) inflation volatility on life insurance demand.

previous example. Fig. 3(b) shows that the demand for life insurance is increasing with the level of inflation under the assumption of inflation-indexed income. The next numerical example explores the effects of inflation volatility and interest rate volatility on life insurance demand. (See Fig. 4.) We use the value of σR given in Table 1 as the benchmark interest rate volatility and the benchmark inflation volatility is determined by σIR = 0.00002 and σII = 0.081. Next, we calculate the life insurance demand under a volatility which doubles (halves) the benchmark case while other parameters remain unchanged. The results show that the optimal proportion of human capital being insured decreases with the interest rate volatility and the inflation volatility. According to Eq. (80), this negative relationship could be explained by  α∗ 0 3, i.e., the risk premium earned from the risky investments. Fig. 5(a) plots the risk premium earned under the three different levels of interest rate volatility. Fig. 5(b) gives the risk premium under the three levels of inflation volatility. According to Eq. (81), a higher risk premium would lead to a higher growth of w t in the long run and the demand for life insurance would become lower due to a higher w t /ht . In the related works which ignore the risks of inflation and interest rates (see, e.g. Duarte et al., 2014; Huang and Milevsky, 2008; Pirvu and Zhang, 2012; Richard, 1975; Pliska and Ye, 2007), the nominal and the inflation-indexed bond are redundant assets and are indistinguishable from cash. In this present work, by taking the risks of inflation and interest rates into consideration, the returns on the nominal and the inflation-indexed bond would relate to the risks of interest rates and inflation. Wage earners who optimally allocate their wealth to nominal and indexed bonds could not only hedge against the risks in the investment opportunities but also earn the premium associated to the risks of interest rates and inflation. In sum, under investment opportunities with greater volatilities, wage earners benefit more from financial investments and cut their demand for life insurance.

policies under constant investment opportunities to compare the results obtained in this present research to the findings of Jensen and Steffensen (2015). Under constant investment opportunities, κ = σR = σIR = σII = σSR = σSI = 0 and the only risky asset to be traded is the stock market fund. In this case the exact solution could be obtained under the general case of ϕ ̸= 1. Proposition 5. The value function of the problem under constant investment opportunities is obtained as follows: Vt = F t

γ −1 1−ϕ

1−γ

w t , 1−γ

(83)

where Ft = δ

ϕ

T



  s    exp − l(u)du ds + exp −

t

t

5.1. Comparison to Jensen and Steffensen (2015) In a recent work, Jensen and Steffensen (2015) analyze the optimal consumption, investment, and life insurance policies under the separation between risk aversion and EIS. They separate the effects of risk aversion and EIS by forming consumption certainty equivalents and show that the solution of their model is identical to the solution under SDU in the case without mortality risk. Since a constant investment opportunity set is assumed in the work of Jensen and Steffensen (2015), we give our optimal



l(u)du ,

(84)

t

and



l(t ) ≡ ϕδ + (1 − ϕ) r +

χS2 2γ



  γ β 1/γ 1− γ1 1 + (1 − ϕ)λ(t ) ξ + ξ − . 1−γ 1−γ

(85)

The optimal policies are given by: ∗

αtS =

χ S w t + ht γ σSS wt

ct∗

w t + ht

=

(86)

δϕ

(87)

Ft

    1/γ β β 1/γ = ht + − 1 wt . η(t ) ξ ξ p∗t

Proof. See Appendix. 5. Solution under constant investment opportunities: Some discussions

T

(88)



According to the solution of Vt , the optimal consumption– wealth ratio is given by c ∗ / wt = δ ϕ /Ft . The relationship between optimal consumption–wealth ratio and the life insurance premium load is the same with that obtained in Proposition 4 since

  1 ∂ l(t ) = (1 − ϕ)λ(t ) 1 − (β/ξ ) γ ∂ξ

(89)

is of the same sign as 1 −ϕ . Next, in line with Jensen and Steffensen (2015), we analyze the optimal consumption rate with a actuarially fair insurance premium and no market risk, i.e., ξ = 1 and χS = 0. In this case, there is no investment in the stock indexed fund and therefore both the wealth w t and consumption ct∗ would be

64

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

Fig. 5. This figure plots the risk premium under different levels of (a) interest rate volatility and (b) inflation volatility.

deterministic. By direct calculation, we derive the consumption growth rate µct :

    µct dt ≡ d ln ct∗  = d ln w t − d ln Ft  ξ =1,χS =0 ξ =1,χS =0    1 − γ ϕ   1 dt . = ϕ r − δ − λ(t ) 1 − β γ ϕ(γ − 1)

(90)

It is clear that when β = 1 or γ ϕ = 1 (power utility), the consumption growth rate would be the constant r − δ over time, and the consumption curve would be upward (downward) sloping if r > δ (r < δ) and flat if r = δ . When β < 1, the consumption growth rate would depend on time whenever γ ϕ ̸= 1. A humpshaped consumption curve implies that µct is positive in the early years and negative in the late years. This would be the case only if r > δ, γ ϕ < 1, and λ(t ) increases with t. Our conclusion here is consistent with the findings in the numerical analysis of Jensen and Steffensen (2015). In this present work, we give the necessary conditions for the consumption to be hump-shaped and confirm that the framework of Jensen and Steffensen (2015) is qualitatively equivalent to the SDU framework even under mortality risk.

time t ∈ [0, T ], the optimal demand for life insurance increases with risk aversion γ when ht and wt are fixed as long as ξ > 1 or β < 1. Furthermore, since the optimal investment depends on the extent of risk aversion, the measure of relative risk aversion would affect the realizations of the state variable wt and therefore p∗t . This implies that the measure of relative risk aversion would also have an intertemporal effect on the optimal life insurance demand. In the following numerical examples, we investigate the effects of EIS and relative risk aversion on optimal life insurance demand. Table 2 lists the benchmark parameters used in the following numerical examples. In the first place, we set χS = 0 to be consistent with Jensen and Steffensen (2015). In this case,

w t = w 0 exp  ht =

T

e−



t 0

δϕ r− − l(s)

s

t r +η(u)du

    β 1/γ − 1 η(s)ds , ξ

y(s)ds

(91)

(92)

t

and the optimal life insurance could be obtained as

5.2. The effects of EIS and risk aversion on life insurance demand

  1  β γ = −1 w t + ht . η(t ) ξ

Jensen and Steffensen (2015) show that the separation between risk aversion and EIS under mortality risk could generate a humpshaped consumption, which cannot be obtained by the standard SDU without mortality risk or the power utility under uncertain lifetime. In this section, we extend their work to analyze the respective effects of risk aversion and EIS on the lifetime life insurance demand. In previous sections, we find that the optimal life insurance demand at any time t ∈ [0, τ ∧ T ), relies on γ , β, ξ , and the ratio of financial-to-human capitals. EIS has no explicit effect on optimal life insurance demand when ht and wt are fixed. In this case, we say that EIS has no cross-sectional effect on optimal life insurance. However, it is inappropriate to conclude that EIS would not affect the demand for life insurance over the planning horizon. We note that the optimal life insurance p∗t relies on the state variable wt . Since the realizations of wt over the wage earner’s lifetime would depend on the optimal consumption choice, EIS would affect the lifetime demand for life insurance implicitly through ct∗ . We refer to this effect as the intertemporal effect on optimal life insurance. Since the intertemporal effect of EIS on life insurance demand cannot be directly determined by the results given in the previous propositions, some numerical examples are presented to show how EIS affects the optimal demand for life insurance. In contrast with EIS, the measure of relative aversion has a cross-sectional effect on optimal life insurance demand. At any

Fig. 6 depicts the effect of EIS on optimal insurance coverage = 2 for t ∈ [10, 30]. The optimal life insurance demand decreases with EIS. As noted earlier, EIS could affect optimal demand for life insurance indirectly through the realizations of the state variable wt . This is because as EIS increases, wage earners become increasingly more willing to substitute consumptions across time. Wage earners with higher EIS would consume a smaller proportion out of their current wealth and, as a result, save more for future consumptions. Since the savings account balance acts as a substitute for life insurance in the utility of bequest, we conclude that a higher EIS would lead to a weaker demand for life insurance over the planning horizon. Fig. 7 shows the effect of risk aversion on optimal life insurance coverage during t = 10 to 30 under ϕ = 0.5. We find that the optimal life insurance increases with the coefficient of relative risk aversion γ . In fact, the measure of relative risk aversion has a twofold effect on optimal life insurance. First, when wt and ht are given, the wage earner who is more risk averse would purchase more life insurance to hedge against the possible losses under mortality risk; this is the so-called cross-sectional effect. Next, the wage earner who buys more life insurance would save less for the future consumptions. This implies that the savings of the wage earner would grow at a lower rate. Because the savings account balance and the life insurance purchased are perfect substitutes in the utility of bequest, a smaller savings account balance would

p∗t

(p∗t /η(t )) under γ

(93)

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

65

Table 2 Benchmark parameters used in the examples under constant investment opportunities. Planning horizon Subjective time preference Volatility of risky asset returns Real labor income Force of mortality

T = 40 δ = 0.03 σS = 0.167 y(t ) = 40e0.03t λ(t ) = 101.5 e(t −63.18)/10.5

W0 = 1 r = 0.03 ξ = e0 . 1 β=1

Initial wealth Real rate Loading factor Bequest motive

Table 3 Optimal life insurance coverage p∗t /η(t ) at t = 20.

χS = 0

χS = .36

Risk Aversion

EIS

0.8 0.4 0.2 0.1 0.01

Risk Aversion

1.25

2.5

5

10

100

1.25

2.5

5

10

100

1304.2 1322.0 1325.5 1326.5 1327.2

1354.3 1363.4 1365.2 1365.7 1366.1

1380.1 1384.7 1385.6 1385.9 1386.1

1392.2 1395.5 1396.0 1369.1 1396.2

1405.1 1405.3 1405.4 1405.4 1405.4

700.1 980.3 1058.8 1090.6 1116.2

1269.2 1308.7 1319.4 1323.7 1327.2

1363.7 1373.6 1376.0 1377.0 1377.7

1389.6 1393.0 1393.8 1394.0 1394.2

1405.1 1405.3 1405.4 1405.4 1405.4

Fig. 6. The effect of EIS on optimal insurance coverage during t = 10 to 30. Relative risk aversion is set to be γ = 2.

Fig. 7. The effect of relative risk aversion on optimal insurance coverage during t = 10 to 30. EIS is set to be ϕ = 0.5.

result in a greater demand for life insurance to maintain the level of the utility of bequest. This is the intertemporal effect. Previous studies based on power utility have consistently concluded that the optimal life insurance demand would increase with the measure of risk aversion (see, among others Pliska and Ye, 2007; Zhu, 2007). However, under the assumption of power utility, the individual’s EIS is represented by the reciprocal of the measure of risk aversion, and an increase in the measure of risk aversion can be interpreted as a decrease in EIS as well. As a result, the

effect of risk aversion on the life insurance demand is not separated from the effect of EIS. Applying SDU to represent the individual’s preferences, we successfully separate the effects of risk aversion and EIS. We show that the optimal life insurance demand increases with the measure of risk aversion when EIS remains constant but decreases with EIS when risk aversion remains unchanged. To show whether the measure of relative risk aversion or the elasticity of intertemporal substitution is a more important determinant of life insurance demand, we provide the optimal life insurance coverage at t = 20 in Table 3. The values of EIS and risk aversion that we use in this example vary across a range spanning 0.01 to 0.8 for EIS and 1.25 to 100 for relative risk aversion. From the left panel of Table 3, we find that the effect of risk aversion is obviously stronger than the EIS effect. For example, starting from a benchmark case of γ = 1.25 and ϕ = 0.8, the power utility function implies that doubling risk aversion to γ = 2.5 and simultaneously halving EIS to ϕ = 0.4 would increase the optimal life insurance coverage (p∗t /η(t )) by an amount of $59.2 (4.53% in terms of percentage changes). Now, doubling γ from 1.25 to 2.5 while keeping ϕ = 0.8 unchanged, the life insurance demand increases by $50.1 (3.84%), an increment that is comparable to that of the benchmark power utility case. On the other hand, if EIS drops from ϕ = 0.8 to ϕ = 0.4 while γ is fixed at γ = 1.25, the optimal life insurance increases by $17.8 (1.36%), which is less than one-third the increment of the benchmark case. Moreover, we find that the effect of EIS on the optimal demand for life insurance becomes less significant as the measure of relative risk aversion increases. For a highly risk-averse wage earner, say γ = 100, the effect of EIS is almost negligible. To explain why the effect of EIS diminishes as the individual becomes more risk averse, we reference Eq. (88). In this equation, we see that the sensitivity of p∗t with respect to wt is decided by

 β  γ1  ∂ p∗t = − 1 η(t ). ∂wt ξ

(94)

The absolute value of ∂ p∗t /∂wt decreases with γ as long as β < 1 or ξ > 1. Since EIS affects the optimal insurance demand indirectly via the realizations of wt , the optimal life insurance demand becomes less sensitive to EIS as γ grows. In an extreme case of γ → ∞, the wage earner would buy full insurance against the possible loss of human capital, that is, p∗t /η(t ) → ht as γ → ∞. In this case, the only determinant of life insurance demand is the wage earner’s human capital, which is irrelevant to the wage earner’s attitude toward intertemporal substitution. We redo the exercise when χS = 0.36, implying a 6% excess return on the stock market fund, and report the results on the

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right panel of Table 3. From this example, we see that with the investment in the risky asset, the life insurance demand becomes smaller in most cases. Investment in the risky asset provides the wage earner with the risk premium and increases the growth of financial savings in the long run. Since the amount of financial savings substitutes for the life insurance, the life insurance demand would decrease with the investment in the risky asset with positive risk premium. It is also noted that the life insurance demand becomes more sensitive to both the parameters ϕ and especially γ , than in the case of χS = 0. When doubling γ from 1.25 to 2.5 while keeping ϕ = 0.8, the life insurance demand increases by an amount of $569.1; an increment of 81.3%. This is because when the wage earner is more risk averse, he or she would invest a smaller proportion of his or her savings in the risky asset and, as a result, would earn less risk premium in the long run. This would reduce the growth of financial savings and increase the demand for life insurance. Besides, in these two examples, we find that optimal life insurance is less sensitive to the measure of relative risk aversion when the wage earner is less willing to substitute consumptions across time. As noted previously, the measure of risk aversion has an intertemporal effect on optimal life insurance demand, i.e., the extent of risk aversion could affect the wage earner’s portfolio decision and thus affect the life insurance demand indirectly through the realizations of wt . If the amount of savings is small, the portfolio decision would not significantly affect the optimal demand for life insurance since the demand for life insurance is now dominated by the human capital ht and, as a result, the optimal life insurance demand would be less sensitive to the measure of risk aversion. Because a low EIS leads to a low saving rate and thus reduces the savings account balance, the optimal life insurance demand would be less sensitive to the extent of risk aversion when EIS is lower. Finally, we note that unlike the effect of EIS, which is almost negligible for an extremely risk-averse wage earner, the life insurance demand still increases significantly by 5.89% (25.90%) in the case of χS = 0 (χS = 0.36) as γ varies from 1.25 to 100 for individuals with the least EIS (ϕ = 0.01) in our numerical exercises.

Acknowledgments We thank two anonymous reviewers for their constructive comments and suggestions. Appendix Proof of Proposition 2. When γ ϕ = 1, the solution of Ft satisfies the following partial differential equation:

    ∂F 1 1 ⊤ + κ(¯r − r ) − 1 − σ r 3 Fr + σ ⊤ r σ r Frr ∂t γ 2     1 1 − Φ (t ) + 1 − r F + δ γ = 0. γ

(A.1)

Applying the Feynman Kac theorem, the solution of Ft could be expressed as

   s   1 Φ (u) + 1 − exp − Xu du ds E δ γ t t    T    1  Φ (u) + 1 − + exp − Xu du Xt = rt , γ t 

1

γ



T

(A.2)

such that





dXt = κ(¯r − Xt ) −

1−

1

γ



  σ⊤ 3 dt + σR dZt , r

(A.3)

where Zt represents a standard Brownian motion. Next, we define a new variable Yt ≡ (1 − 1/γ )Xt . The dynamics of Yt could be expressed as dYt = κ(Y¯ − Yt )dt + σY dZt ,

(A.4)

where Y¯ ≡

 1−



1

 r¯ −

γ

σY ≡ 1 −

1

γ



 γ − 1 ⊤ σr 3 κγ

σR .

(A.5) (A.6)

s

6. Conclusions A wage earner with a bequest motive would purchase life insurance to protect against the loss of human capital when a premature death occurs before retirement. In an intertemporal framework, the life insurance demand over the planning horizon would be affected by the wage earner’s consumption–saving and portfolio decisions since the financial savings and the life insurance act as substitutes for each other. In this study, we solve the optimal policies for life insurance, consumption, and portfolio selection under interest rate and inflation risks. Stochastic differential utility is employed to disentangle the effects of EIS and relative risk aversion. We show that the demand for life insurance decreases with the level of nominal interest rates. If the wage earner’s income is indexed to inflation, the demand for life insurance would increase with inflation. We also find that the volatilities of interest rates and inflation would decrease the demand for life insurance. An analysis of EIS and risk aversion examines the respective effects of EIS and risk aversion on life insurance demand. With low risk aversion and high EIS, the wage earner’s lifetime savings would increase and, as a result, the demand for life insurance over the planning horizon would decrease. The effect of risk aversion on life insurance demand is more significant than the effect of EIS. Finally, the optimal consumption–wealth ratio would be affected by the life insurance premium load and the direction is dependent with the size of EIS relative to unity.

Denoting Uts ≡ E [e− t Yu du |Yu = y], then Uts would be the solution of the following partial differential equation with boundary condition Uss = 1:

∂U 1 + κ(Y¯ − y)Uy + σY2 Uyy = yU . ∂t 2

(A.7)

It is straightforward to verify that Uts given in Eq. (62) solves Eq. (A.6). Proof of Proposition 4. According to Eqs. (70), (72) and (58), the function ν1 is independent of the parameter ξ and this implies that

  1 ∂ Φ (s) ∂φ(s) = = (1 − ϕ)λ(s) 1 − (β/ξ ) γ ∂ξ ∂ξ

(A.8)

is of the same sign as (1 − ϕ) since ξ ≥ 1 and 0 ≤ β ≤ 1. Because the optimal consumption–wealth ratio is reversely related to Ft and, as a result, positively related to φ , we conclude that the consumption–wealth ratio increases (decreases) with ξ when ϕ < 1(> 1). Proof of Proposition 5. Under constant investment opportunities, the nominal interest rate is a constant R, the inflation rate is π , and the real rate is R − π . The only source of financial risk is the stock market risk ZtS with market price of risk χS . Since σr = 0, the partial differential equation in Eq. (58) reduces to an ordinary differential equation: dF dt

= l(t )F − δ ϕ ,

(A.9)

N.-W. Han, M.-W. Hung / Insurance: Mathematics and Economics 73 (2017) 54–67

with l(t ) defined in Eq. (85). Direct calculation yields the solution of F : Ft = δ

ϕ

T

 t

T

     s l(u)du ds + exp − exp − t



l(u)du . (A.10) t

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