Optimal consumption–investment strategy under the Vasicek model: HARA utility and Legendre transform

Accepted Manuscript Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform Hao Chang, Kai Chang PII: DOI: Reference:

S0167-6687(15)30239-0 http://dx.doi.org/10.1016/j.insmatheco.2016.10.014 INSUMA 2295

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Insurance: Mathematics and Economics

Received date: October 2015 Revised date: October 2016 Accepted date: 26 October 2016 Please cite this article as: Chang, H., Chang, K., Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform. Insurance: Mathematics and Economics (2016), http://dx.doi.org/10.1016/j.insmatheco.2016.10.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform∗ Hao Chang1,2 ,

Kai Chang3

1. School of Science, Tianjin Polytechnic University, Tianjin 300387, China. 2. College of Management and Economics, Tianjin University, Tianjin 300072, China. 3. School of Finance, Zhejiang University of Finance & Economics, Hangzhou 310018, China. Abstract: This paper studies the optimal consumption-investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the individuals is to seek an optimal consumption-investment strategy to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. In the utility theory, Hyperbolic Absolute Risk Aversion (HARA) utility consists of CRRA utility, CARA utility and Logarithmic utility as special cases. In addition, HARA utility is seldom studied in continuous-time portfolio selection theory due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the individuals. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solution to the optimal consumption-investment strategy in a complete market. Moreover, some special cases are also discussed in detail. Finally, a numerical example is given to illustrate our results. Key–Words: consumption-investment problem; the Vasicek model; HARA utility; dynamic programming principle; Legendre transform; closed-form solution;

1

Introduction

The consumption-investment problems are the classical portfolio selection problems with consumption behavior in the mathematical finance theory, which were first studied by Merton (1969, 1971). In addition, Merton first used stochastic optimal control theory to study the explicit solution to the optimal portfolio-consumption strategy and laid a solid foundation for the extensive application of stochastic optimal control theory. Since Merton, the consumption-investment problems have been paid great attention to by many scholars such that many research results have been achieved. To sum up, these results consist of the following five aspects. (i) The consumptioninvestment problems with borrowing constraints. One can refer to the works of Fleming and Zariphopoulou (1991) and Vila and Zariphopoulou (1997). Fleming and Zariphopoulou (1991) studied a consumption-investment decision problem for a single agent and used the principle of stochastic dynamic programming to obtain the explicit solution for power utility. In addition, this paper analyzed the optimal portfolio and consumption behavior under the assumption of borrowing rate exceeding the return rate of the risk-free asset. Vila and Zariphopoulou (1997) also used stochastic dynamic programming to study the intertemporal consumption and portfolio choice with borrowing constraint and displayed that how borrowing constraint affected the consumption and portfolio choice. Moreover, Vila and Zariphopoulou (1997) presented a methodology on viscosity solution of the HJB equation, which could be used to analyze a very wide range of consumption/investment problems. (ii) The consumptioninvestment problems with transaction costs. The interested readers can see the papers of Dumas and Luciano (1991), Shreve and Soner (1994), Dai et al. (2009) and so on. Dumas and Luciano (1991) derived the exact solution to the optimal portfolios with transaction cost. Shreve and Soner (1994) used the method of viscosity ∗

This research is supported by National Natural Science Foundation of China (No. 71671122, 71673236), China Postdoctoral Science Foundation Funded Project (No. 2014M560185, 2016T90203), Humanities and Social Science Research Planning Foundation of Ministry of Education of China (No. 11YJC790006, 16YJA790004) and Tianjin Natural Science Foundation of China (No. 15JCQNJC04000).

1

solution to obtain the optimal consumption-investment policy with transaction cost and provided a verification theorem. Dai et al. (2009) used singular stochastic control theory to deal with the consumption-investment problem with proportional transaction cost and finite time horizon and analyzed the dynamic behavior of the optimal buying and selling strategies. (iii) The consumption-investment problems with stochastic interest rate. Representative works include Fleming and Pang (2004), Munk and Sørensen (2004), etc. Fleming and Pang (2004) focused on the consumption-investment policy with stochastic interest rate on infinite time horizon, and used the sub-supersolution method to prove the existence of the solution of the HJB equation. Munk and Sørensen (2004) characterized the solution to the consumption-investment problem with stochastic interest rates and numerically illustrated an important conclusion that the hedge portfolio is more sensitive to the form of term structure of interest rates than the dynamics of interest rates. (iv) The consumption-investment problems with stochastic volatility. The interested readers can refer to the works of Fleming and Hernandez-hernandez (2003), Chacko and Viceira (2005), Chang et al. (2013), etc. Fleming and Hernandez-hernandez (2003) investigated the optimal consumption problem with stochastic volatility on the infinite horizon and its volatility is an observable economic factor. Chacko and Viceira (2005) supposed the instantaneous volatility of stock price to follow a mean-reverting square-root process and systematically discussed the optimal consumption and portfolio choice of long horizon investors with volatility risks. Chang et al. (2013) assumed the instantaneous volatility of stock price to be related with stock price itself and study the optimal consumption-investment strategy with constant elasticity of variance (CEV) model. (v) The consumption-investment problems with stochastic interest rate and stochastic volatility. Main research results of this aspect include the works of Liu (2007), Noh and Kim (2011), Chang and Rong (2013). The main distinction of these three papers lies in the different framework of problem formulation with stochastic interest rate and stochastic volatility. Liu (2007) assumed that interest rate, the return rate and volatility of stock price were expressed as a function of the stochastic factor, which followed a Markovian diffusion process. Noh and Kim (2011) supposed that the dynamics of interest rate and volatility is generally linearly correlated with the dynamics of stock price. Chang and Rong (2013) assumed that there is no correlationship between the dynamics of stock price and interest rate. These models studied many consumption-investment problems with different market assumptions and different investment environments and greatly extended the works of Merton. But these results were almost studied under the assumption of Constant Relative Risk Aversion (CRRA) utility (i.e. power utility) or logarithmic utility. It is all well-known that CRRA utility, Constant Absolute Risk Aversion (CARA) utility (i.e. exponential utility) and logarithmic utility are all special cases of HARA utility. As a matter of fact, the investors should choose different utility function according to the different degree of risk preference. Therefore, it is very necessary to investigate the optimal consumption-investment strategy under HARA utility. However, due to the complicated structure of HARA utility, there was little works on the portfolio selection problems with HARA utility in the decade years. Fortunately, we can find that some important results have been achieved in the existing literatures. There are two representative papers in this aspect. One is the work of Grasselli (2003), who presented martingale method to deal with an investment problem with stochastic interest rate and obtained the explicit solutions for HARA utility. Meantime, he also proved the fact that the optimal investment policy under HARA utility converged almost surely to the one under exponential utility and logarithm utility. The other is the work of Jung and Kim (2012), who applied Legendre transform technique to tackle an investment problem under the constant elasticity of variance (CEV) model and achieved the closed-form solution for HARA utility. Apart from these works, Chang and Rong (2014) studied the optimal consumption and investment policy with HARA preference in the constant interest rate environment and obtained the explicit expression by using Legendre transform technique. Chang et al. (2014) used dynamic programming principle along with Legendre transform to investigate the optimal investment problem with random liability and affine interest rate in the HARA framework, and obtained the closed-form solution to the optimal investment strategy. Inspired by these works, we found out that it was feasible to use Legendre transform technique to deal with the consumption-investment problem with stochastic interest rate under HARA utility, which will be discussed in detail in this paper. In the recent years, Legendre transform technique has been widely used to deal with some complicated portfolio optimization problems, for example, Jonsson and Sircar (2002), Xiao et al. (2007), Gao (2009, 2010) and so on. Precisely speaking, Jonsson and Sircar (2002) presented Legendre transform-dual theory in solving continuoustime portfolio selection problems. Xiao et al. (2007) successfully used Legendre transform-dual theory to obtain the optimal policy for logarithmic utility under a CEV model. Gao (2009) further solved the optimal policy for power utility and exponential utility under the framework of Xiao et al. (2007). Gao (2010) provided Legendre transform-dual rules with four state variables and laid solid foundation for solving more complicated portfolio selection problems. Considering the fact that interest rate is uncertain in the real-world environments, and more and more scholars 2

began to concern the portfolio selection problems with stochastic interest rates and make the optimal investment strategy obtained more practical. Representative woks in this aspect included Korn and Kraft (2001, 2004), Deelstra et al. (2003), Josa-Fombellida and Rinc´on-Zapatero (2010), Guan and Liang (2014) and so on. Korn and Kraft (2001, 2004) studied the portfolio optimization problems with stochastic interest rate and presented a verification theorem. Moreover, they also discussed some sufficient conditions ensuring the proper application on the principle of stochastic dynamic programming. Deelstra et al. (2003) assumed short rate to be driven by affine interest rate and presented a special stock price model incorporating the effect of interest rate. In addition, they provided the martingale method to study the pension management problem with a minimum guarantee. Josa-Fombellida and Rinc´on-Zapatero (2010) explored the optimal strategy for pension fund with stochastic interest rate and actuarial liability. Guan and Liang (2014) introduced inflation risk into the pension management problem with stochastic interest rate and obtained the explicit solution to the optimal policy. Nevertheless, these models only studied the optimal investment strategies for CRRA utility or logarithmic utility. Meantime, our analysis suggested that there be many difficulties if we considered consumption behavior in the above literatures and only used stochastic dynamic programming or the martingale method to solve them in the HARA framework. As far as methodology, most literatures are the univariate optimization problems(for example Grasselli (2003), Jung and Kim (2012)) while the consumption-investment problems are bivariate optimization ones. So our objective function is more complicated, moreover, if we only used the martingale method used by Grasselli (2003), one of the greatest difficulties in dealing with our model lies in the construction of the suitable exponential martingale. If only stochastic dynamic programming was used, we need to directly conjecture the form of the solution to the HJB equation. Due to the complicated nonlinear structure of HARA utility, it is difficult to construct the candidate solution. Compared with the martingale method used by Grasselli (2003), Legendre transform-dual theory is easier to understand and is more convenient to use. By the analysis on the advantages of Legendre transform, we unexpectedly find that Legendre transform can convert the original nonlinear HJB equation into a linear dual equation, more importantly, its boundary condition under HARA utlity is of the linear structure. Those advantages make the solving process of our model much easier. Therefore, this paper will use stochastic dynamic programming along with Legendre transform to deal with our model. In this paper, we assume that interest rate is driven by the Vasicek model (1977) and the financial market is composed of one risk-free asset and multiple risky assets. The objective of the investor is to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. Assume that risk preference of the investor is described by HARA utility. Legendre transform method is used to change the original nonlinear HJB equation into its dual one, whose solution is more easy to conjecture in the HARA utility framework. By applying variable change technique, we obtain the explicit forms of the optimal consumptioninvestment strategy in a complete market. Some important special cases are also discussed in detail, specially, the optimal consumption-investment strategy under CARA utility is also achieved, which hasn’t been studied in the existing literatures. Finally, in order to illustrate our results and analyze economic implications of parameters, we give a numerical example. To sum up, this paper consists of the following four innovations: (i) we studied the consumption-investment problem with the Vasicek model in the HARA utility framework; (ii) we provided Legendre transform method to deal with our problem; (iii) we obtained the explicit expression of the optimal consumption-investment strategy with HARA preference; (iv) some special cases are systematically discussed, especially, the optimal consumption-investment strategy for CARA utility is also derived explicitly. The rest of this paper is organized as follows. Section 2 formulates the consumption-investment problem with the Vasicek model in the HARA utility framework. Section 3 uses Legendre transform to change the original HJB equation into its dual one. The explicit expressions of the optimal consumption-investment strategy with HARA utility are derived in Section 4. Section 5 systematically discusses some special cases and Section 6 provides a numerical example. Finally, Section 7 concludes the paper.

2

The model

This section presents the consumption-investment problem under the Vasicek model. Throughout the paper, we denote of a matrix or a sector by (·)′ , denote the norm of a sector √ √ the transpose ′ x = (x1 , x2 , · · · , xn ) by ∥x∥ = x′ x = x21 + x22 + · · · + x2n , denote the expectation by E(·), denote the finite investment horizon by [0, T ]. Assume that (W0 (t), W1 (t), · · · , Wn (t))′ is a n + 1-dimensional independent standard Brownian motion defined on complete probability space (Ω, Ft , P, {Ft }06t6T ), where {Ft }06t6T is the information filtration available generated by (W0 (t), W1 (t), · · · , Wn (t))′ . 3

Assume that the financial market consists of one risk-free asset and n risky assets, which can be traded continuously. The price process S0 (t) of the risk-free asset (e.g. a bond) evolves according to the ordinary differential equation (ODE): dS0 (t) = r(t)S0 (t)dt, S0 (0) = 1, (1) where r(t) is the short rate. In this paper, suppose that r(t) can be described by the Vasicek model (1977): ¯ (t), dr(t) = k1 (k2 − r(t))dt + bdW

r(0) = r0 > 0,

(2)

¯ (t) is a one-dimensional standard Brownian motion defined on where k1 , k2 and b are positive constants, and W (Ω, Ft , P, {Ft }06t6T ). The price process of the ith risky asset (e.g. stock) is denoted by Si (t), i = 1, 2, · · · , n, then Si (t) can be supposed to follow geometric Brownian motion:   n ∑ σij (t)dWj (t) , Si (0) = si > 0, (3) dSi (t) = Si (t) (µi (t) + r(t))dt + j=1

where b(t) = (b1 (t), b2 (t), · · · , bn (t))′ , µ(t) = (µ1 (t), µ2 (t), · · · , µn (t))′ and I = (1, 1, · · · , 1)′ , then µ(t) + r(t)I and σ(t) = (σij (t))n×n represent the appreciation sector and volatility matrix of the stocks respectively. Moreover, σ(t) = (σij (t))n×n satisfies the non-degenerated condition: σ(t)σ ′ (t) > 0, ∀t ∈ [0, T ]. Considering the effect of interest rate on the prices of the stocks, we assume that there is the linear correlation ¯ (t) and Wj (t), whose correlation coefficient is denoted by ρj (t), ρj (t) ∈ [−1, 1], j = 1, 2, · · · , n. Let between W ¯ (t) can be linearly expressed ρ(t) = (ρ1 (t), ρ2 (t), · · · , ρn (t))′ and W (t) = (W1 (t), W2 (t), · · · , Wn (t))′ , then W as: √ ¯ (t) = ρ′ (t)W (t) + 1 − ∥ρ(t)∥2 W0 (t). W Further, (2) can be rewritten as

√ dr(t) = k1 (k2 − r(t))dt + bρ (t)dW (t) + b 1 − ∥ρ(t)∥2 dW0 (t), ′

r(0) = r0 > 0.

(4)

Remark 1. In the equation (4), there are three special cases as follows. (i) If ρj (t) = 0, j = 1, 2, · · · , n, i.e. ¯ (t) is equal to W0 (t). (ii) If ∥ρ(t)∥2 < 1, it means that the financial market is incomplete. From the ρ(t) = 0, W views of real-world investment, it implies that the risks resulted from the randomization of interest rate can’t be completely hedged by the risky assets. (iii) If ∥ρ(t)∥2 = 1, it displays that the stock price dynamics and interest rate dynamics are driven by the same source of randomness. It indicates that interest rate risk can be completely hedged by the risky assets. Assume that the amount invested in the ith stock is denoted by πi (t), i = 1, 2, · · · , n, then the amount n ∑ invested in the bond is π0 (t) = X(t) − πi (t), where X(t) represents the wealth of the individuals at time t. Let i=1

π(t) = (π1 (t), π2 (t), · · · , πn (t))′ and C(t) be the consumption amount, then the wealth process X(t) satisfies the following stochastic differential equation (SDE): ( ) dX(t) = r(t)X(t) + π ′ (t)µ(t) − C(t) dt + π ′ (t)σ(t)dW (t), X(0) = x0 > 0. (5) The market price of the risk is defined as

θ(t) = σ −1 (t)µ(t). Definition 1(Admissible strategy). A consumption-investment strategy (π(t), C(t)) is said to be admissible if the following conditions are satisfied: 4

∫T ∫T 2 (i) (π(t), C(t)) is F −measurable, and satisfies ∥π(t)∥ dt < ∞ and t 0 0 C(t)dt < ∞; (∫ ) T 2 (ii) E 0 ∥π ′ (t)σ(t)∥ dt < ∞; (iii) the SDE (5) has a unique strong solution corresponding to any (π(t), C(t)). Assume that the set of all the admissible strategies is denoted by Γ = {(π(t), C(t)) : 0 6 t 6 T }, and the aim of the individuals is maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. Mathematically, the objective function can be expressed as ( ∫ T ) −βt −βT M aximize E α e U1 (C(t))dt + (1 − α)e U2 (X(T )) , (6) (π(t),C(t))∈Γ

0

where α and β are constants, α (0 < α < 1) represents weight coefficient between intermediate consumption and terminal wealth, β > 0 represents the discount rate, U1 (·)and U2 (·) represent utility functions. In this paper, we will choose Hyperbolic Absolute Risk Aversion (HARA) utility function for our analysis. In the utility theory, HARA utility function with parameters η, p and q can be written as ( )p 1−p q U1 (x) = U2 (x) = U (η, p, q, x) = x + η , q > 0, p < 1, p ̸= 0. (7) qp 1−p In reality, HARA utility function recovers CRRA utility, CARA utility and logarithmic utility as special cases. (i) Let η = 0 and q = 1 − p, then we have U (0, p, 1 − p, x) =

xp = UCRRA (x). p

(ii) Assume that η = 1 and p → −∞, then we arrive at U (1, p, q, x) = −

e−qx = UCARA (x). q

(iii) Suppose that η = 0, p → 0 and q → 1, then we get U (0, p, q, x) = ln x = Ulog (x). Remark 2. From section 2, we can see the following some main differences between our model and existing literatures. (i) Compared with the model of Grasselli et al. (2003), risky preference and objective function of our model are different, i.e. Grasselli et al. (2003) studied the optimal investment problem for pension fund with minimum guarantee under power utility, while our model studied not only the optimal investment problem but also the optimal consumption one under HARA utility. (ii) Compared with the model of Jung and Kim (2012), our model studied the consumption-investment problem with stochastic interest rate, which is a bivariate optimization problem. But Jung and Kim (2012) focused on the optimal investment problem under the CEV model, which is a univariate optimization problem. (iii) The methodology is different between our paper and Grasselli (2003). To be precise, Grasselli (2003) used the martingale method, while our paper used dynamic programming principle along with Legendre transform-dual theory.

3

HJB equation and Legendre transform Taking the problem (6) as a stochastic optimal control problem, we may define the value function as ( ∫ T ) −βt −βT V (t, r, x) = sup E α e U1 (C(t))dt + (1 − α)e U2 (X(T )) |X(t) = x, r(t) = r , (π(t),C(t))∈Γ

0

with the boundary condition given by V (T, r, x) = (1 − α)e−βT U2 (x).

5

Assume that V (t, r, x) ∈ C([0, T ] × R+ × R+ ), then we can define the following variational operator on any function V (t, r, x): Aπ,C V (t, r, x) =Vt + (rx + π ′ (t)µ(t) − C(t))Vx +

1

π ′ (t)σ(t) 2 Vxx + (k1 k2 − k1 r)Vr 2

1 + b2 Vrr + bπ ′ (t)σ(t)ρ(t)Vxr + αe−βt U1 (C(t)), 2

where Vt , Vx , Vxx , Vr , Vrr and Vrx represent first-order and second-order partial derivatives with respect to the variables t, x, r respectively. According to the principle of stochastic dynamic programming, we can easily derive the following HJB equation: { π,C } sup A V (t, r, x) = 0, V (T, r, x) = (1 − α)e−βT U2 (x). (8) (π(t),C(t))∈Γ

Assume that J(t, r, x) ∈ C([0, T ] × R+ × R+ ) is a candidate solution for the problem (8), then using the first-order condition for the maximizing problem yields π ∗ (t) = −(σ(t)σ ′ (t))−1 µ(t)

Jx Jxr − b(σ ′ (t))−1 ρ(t) , Jxx Jxx

U˙ 1 (C ∗ (t)) =

Jx . αe−βt

(9)

Substituting (9) back into (8), we get J2 Jx Jxr 1 J2 1 ∥θ(t)∥2 x − bθ′ (t)ρ(t) − b2 ∥ρ(t)∥2 xr 2 Jxx Jxx 2 Jxx 1 2 + (k1 k2 − k1 r)Jr + b Jrr − C ∗ (t)Jx + αe−βt U1 (C ∗ (t)) = 0. 2

Jt + rxJx −

(10)

with the terminal condition J(T, r, x) = (1 − α)e−βT U2 (x).

For HARA utility function, it is very difficult to directly conjecture the structure of the solution to the equation (10) because of its complicated structure. In order to obtain the closed-form solution to the optimal consumptioninvestment strategy, we introduce the following Legendre transform to convert (10) into its dual one. Definition 2. Let f : Rn → R be a convex function. Legendre transform can be defined as follows: L(z) = max{f (x) − zx}, x

(11)

then the function L(z) is called Legendre dual function of f (x) (cf. Jonsson and Sircar (2002), Xiao et al. (2007), Gao (2009, 2010) and Chang et al. (2014)). If f (x) is strictly convex, the maximum for the equation (11) will be attained at just one point, which we denote by x0 . We can attain at the unique solution by the first-order condition: df (x) − z = 0. dx So we have

L(z) = f (x0 ) − zx0 .

Following Jonsson and Sircar (2002), Xiao et al. (2007), Gao (2009, 2010) and Chang et al. (2014), we define Legendre transform as follows ˆ r, z) = sup{J(t, r, x) − zx}, J(t, (12) x>0

where z > 0 denotes the dual variable to x. The value of x which this optimum is attained at is denoted by g(t, r, z), so we have ˆ r, z)}. g(t, r, z) = inf { x| J(t, r, x) > zx + J(t, (13) x>0

6

ˆ r, z) and g(t, r, z) can be determined by The relationship between J(t, g(t, r, z) = −Jˆz (t, r, z).

(14)

ˆ r, z) as the dual function of J(t, r, x). Hence, we can choose either one of the two functions g(t, r, z) and J(t, Here, we choose g(t, r, z). Moreover, we have Jx = z,

ˆ r, z) = J(t, r, g) − zg, J(t,

g(t, r, z) = x.

(15)

Differentiating (15) with respect to t, r, z, we get Jt = Jˆt ,

Jx = z,

Jxx = −

1 , Jˆzz

Jr = Jˆr ,

Jˆ2 Jrr = Jˆrr − rz , Jˆzz

Jxr = −

Jˆrz . Jˆzz

(16)

Notice that J(T, r, x) = (1−α)e−βT U2 (x), then at the terminal time T , we can define the following Legendre transform: ˆ r, z) = sup{J(T, r, x) − zx}, J(T, x>0

In addition, we get g(T, r, z) = (U˙ 2 )−1 Plugging (16) into (10), we get

(

ˆ r, z)}. g(T, r, z) = inf { x| J(T, r, x) > zx + J(T,

z (1−α)e−βT

x>0

) , where (U˙ 2 )−1 (·) is taken as the inverse of marginal utility.

1 1 Jˆt + rzg + ∥θ(t)∥2 z 2 Jˆzz − bθ′ (t)ρ(t)z Jˆrz + (k1 k2 − k1 r)Jˆr + b2 Jˆrr 2 2 ) Jˆ2 1 2( + b ∥ρ(t)∥2 − 1 rz − C ∗ (t)z + αe−βt U1 (C ∗ (t)) = 0. 2 Jˆzz

(17)

Differentiating both sides of (17) with respect to z and using (14), we obtain 1 ∥θ(t)∥2 z 2 gzz + (k1 k2 − k1 r − bθ′ (t)ρ(t))gr − bθ′ (t)ρ(t)zgrz 2 1 1 g 2 gzz − 2gr grz gz ∂(C ∗ (t)z) ∂(αe−βt U1 (C ∗ (t))) + b2 grr + b2 (1 − ∥ρ(t)∥2 ) r − = 0, + 2 2 gz2 ∂z ∂z ( ) z where the boundary condition is given by g(T, r, z) = (U˙ 2 )−1 (1−α)e . −βT gt − rg + (∥θ(t)∥2 − r)zgz +

4

(18)

The optimal consumption-investment strategy Under HARA utility, we get g(T, r, z) =

1 1 1−p 1−p − 1 (1 − α) p−1 (eβT ) p−1 z p−1 − η. q q

So we can conjecture a solution of (18) with the following structure g(t, r, z) =

1 1 1−p 1−p − 1 (1 − α) p−1 (eβt ) p−1 z p−1 f (t, r) − ηh(t, r), q q

with boundary conditions given by f (T, r) = 1 and h(T, r) = 1. Moreover, the partial derivatives of g(t, r, z) are derived as follows. gt =

1 1 1−p 1−p 1 − 1 (1 − α) p−1 (eβt ) p−1 z p−1 (ft + βf ) − ηht , q p−1 q

7

(19)

1 1 1−p 1 − 1 −1 (1 − α) p−1 (eβt ) p−1 z p−1 f , q p−1 1 1 1−p 2−p − 1 −2 gzz = (1 − α) p−1 (eβt ) p−1 z p−1 f , q (p − 1)2 1 1 1−p 1−p − 1 gr = (1 − α) p−1 (eβt ) p−1 z p−1 fr − ηhr , q q 1 1 1−p 1 − 1 −1 grz = (1 − α) p−1 (eβt ) p−1 z p−1 fr , q p−1 1 1 1−p 1−p − 1 grr = (1 − α) p−1 (eβt ) p−1 z p−1 frr − ηhrr . q q

gz =

According to U˙ 1 (C ∗ (t)) =

Jx αe−βt

=

z , αe−βt

C ∗ (t) =

we obtain

1 1 1 1 − p − p−1 1−p α (eβt ) p−1 z p−1 − η. q q

(20)

Substituting all the partial derivatives and (20) back into (18) and separating the variables, we can derive the following equation under the condition of ∥ρ(t)∥2 = 1. ( ( ) 1 1 1 1−p 1 p p − p−1 2 βt p−1 p−1 (1 − α) (e ) z ft + β− r+ ∥θ(t)∥ f q p−1 p−1 2(p − 1)2 ) ( )− 1 ) ( p−1 1 α p bθ′ (t)ρ(t) fr + b2 frr + + k1 k2 − k1 r − p−1 2 1−α ( ) 1−p 1 2 ′ − η ht − rh + (k1 k2 − k1 r − bθ (t)ρ(t))hr + b hrr + 1 = 0. q 2 Eliminating the dependence on z and η, we get the following two equations. (

( )− 1 ) p−1 1 p p α 1 2 2 ft + β− r+ ∥θ(t)∥ f + b frr + 2 p−1 p−1 2(p − 1) 2 1−α ) ( p bθ′ (t)ρ(t) fr = 0, f (T, r) = 1; + k1 k2 − k1 r − p−1

(21)

1 (22) ht − rh + (k1 k2 − k1 r − bθ′ (t)ρ(t))hr + b2 hrr + 1 = 0, h(T, r) = 1. 2 The equations (21) and (22) are a family of special equations. Inspired by the work of Liu (2007) and Chang et al. (2014), we summarize the solving process of (21) and (22) in the following Lemma 1-4. ( )− 1 ∫ p−1 T ˆ α ˆ ˆ Lemma 1. Assume that f (t, r) = 1−α t f (s, r)ds + f (t, r) is a solution of (21), then f (t, r) satisfies the following equation: ( ) 1 p p 1 2 ˆ ft + β− r+ ∥θ(t)∥ fˆ + b2 fˆrr 2 p−1 p−1 2(p − 1) 2 ( ) (23) p + k1 k2 − k1 r − bθ′ (t)ρ(t) fˆr = 0, fˆ(T, r) = 1. p−1 Proof. Introducing the following differential operator ∇ on any function f (t, r): ( ) 1 p p 2 ∇f (t, r) = β− r+ ∥θ(t)∥ f p−1 p−1 2(p − 1)2 ) ( 1 p ′ bθ (t)ρ(t) fr + b2 frr , + k1 k2 − k1 r − p−1 2 8

we can rewrite (21) as

( )− 1 p−1 ∂f (t, r) α + ∇f (t, r) + = 0. ∂t 1−α )− 1 ∫ ( p−1 T ˆ α ˆ Further, according to f (t, r) = 1−α t f (s, r)ds + f (t, r), we get

(24)

( )− 1 p−1 ∂f (t, r) α ∂ fˆ(t, r) =− fˆ(t, r) + ∂t 1−α ∂t ) ( )− 1 (∫ T ˆ p−1 ∂ f (s, r) α ∂ fˆ(t, r) = ds − fˆ(T, r) + , 1−α ∂s ∂t t ∇f (t, r) =

(

α 1−α

)−

1 p−1

∫

T

t

∇fˆ(s, r)ds + ∇fˆ(t, r).

So (24) can be rewritten as ) ) ( ) ( )− 1 (∫ T ( ˆ ˆ(t, r) p−1 ∂ f (s, r) α ∂ f + ∇fˆ(s, r) ds − fˆ(T, r) + 1 + + ∇fˆ(t, r) = 0. 1−α ∂s ∂t t Finally, we obtain

∂ fˆ(t, r) + ∇fˆ(t, r) = 0, ∂t Therefore, we complete the proof of Lemma 1. 

fˆ(T, r) = 1.

Lemma 2. Suppose that the solution of (23) is of the structure fˆ(t, r) = eD1 (t)+D2 (t)r , with terminal conditions given by D1 (T ) = 0 and D2 (T ) = 0, then D1 (t) and D2 (t) are given by ( ) p D2 (t) = e−k1 (T −t) − 1 , (25) k1 (p − 1) ) p 1 2 D1 (t) = β+ ∥θ(s)∥ ds p−1 2(p − 1)2 t ) ∫ T( ∫ T p 1 bθ′ (s)ρ(s) D2 (s)ds + b2 D22 (s)ds. + k1 k2 − p − 1 2 t t ∫

T

(

(26)

Proof. Inserting fˆ(t, r) = eD1 (t)+D2 (t)r into (23), we get { ( ) 1 p p 2 D1 (t)+D2 (t)r ′ e D˙ 1 (t) + β+ ∥θ(t)∥ + k1 k2 − bθ (t)ρ(t) D2 (t) p−1 2(p − 1)2 p−1 ( )} 1 2 2 p ˙ + b D2 (t) + r D2 (t) − k1 D2 (t) − = 0. 2 p−1

Matching the coefficient on the both sides, we arrive at

D˙ 2 (t) − k1 D2 (t) − D˙ 1 (t) +

p = 0, p−1

( ) p p 1 1 2 ′ β+ ∥θ(t)∥ + k1 k2 − bθ (t)ρ(t) D2 (t) + b2 D22 (t) = 0. 2 p−1 2(p − 1) p−1 2

Considering terminal conditions D1 (T ) = 0 and D2 (T ) = 0 and solving the above two equations, we can obtain (25) and (26).  9

∫T

Lemma 3. Provided that h(t, r) =

t

ˆ r)ds + h(t, ˆ r) is a solution of (22), then h(t, ˆ r) satisfies h(s,

ˆ t − rh ˆ + (k1 k2 − k1 r − bθ′ (t)ρ(t))h ˆ r + 1 b2 h ˆ rr = 0, h 2

ˆ h(T, r) = 1.

(27)

Proof. The proof is similar to the process of Lemma 1.  ˆ r) = eD3 (t)+D4 (t)r is a solution of (27), with terminal conditions D3 (T ) = 0 and Lemma 4. Suppose that h(t, D4 (T ) = 0, then D3 (t) and D4 (t) are given by D4 (t) = D3 (t) =

∫

T

t

(

) 1 ( −k1 (T −t) e −1 , k1

(28)

) 1 k1 k2 − bθ′ (s)ρ(s) D4 (s)ds + b2 2

Proof. The solving process is similar to Lemma 2. 

∫

T

t

D42 (s)ds.

(29)

According to (16) and (19), we have Jx = −z Jˆzz = zgz Jxx 1 1 1−p 1 − 1 = (1 − α) p−1 (eβt ) p−1 z p−1 f q p−1 ( ) 1−p 1 g+ ηh(t, r) = p−1 q ( ) 1 1−p =− x+ ηh(t, r) , 1−p q Jxr ˆ rz = −gr =H Jxx 1 1 1−p 1−p − 1 =− (1 − α) p−1 (eβt ) p−1 z p−1 fr + ηhr q q ( ) 1−p fr 1−p =− x+ ηh + ηhr . q f q

Meantime, (20) can be rewritten as ∗

C (t) =

(

α 1−α

)−

1 p−1

( ) 1−p 1−p x+ ηh f −1 − η. q q

In addition, by using g(t, r, z) = x and (19), we obtain z = (1 − α)e

−βt

(

q x + ηh(t, r) 1−p

)p−1

f 1−p (t, r),

Considering Jx = z, by integration, we get ∗ JHARA (t, r, x)

= (1 − α)e

−βt 1

−p qp

(

q x + ηh(t, r) 1−p

)p

f 1−p (t, r).

(30)

Summarizing what is mentioned above, we can obtain the optimal consumption-investment strategy of the problem (6) under HARA utility.

10

Theorem 1. For HARA utility (7), under the condition of ∥ρ(t)∥2 = 1, the optimal consumption-investment strategy of the problem (6) is given by ( ) 1 1−p ∗ ′ −1 πHARA (t) = (σ(t)σ (t)) µ(t) X(t) + ηh(t, r) 1−p q (( ) ) (31) 1−p fr 1−p ′ −1 + b(σ (t)) ρ(t) X(t) + ηh(t, r) − ηhr , q f (t, r) q ∗ CHARA (t)

=

(

α 1−α

)−

1 p−1

( ) 1−p 1−p X(t) + ηh(t, r) f −1 (t, r) − η, q q

(32)

and the corresponding optimal value function is given by (30). Here (

)− 1 ∫ T p−1 α f (t, r) = eD1 (s)+D2 (s)r ds + eD1 (t)+D2 (t)r , 1−α t ( )− 1 ∫ T p−1 ∂f (t, r) α fr = = D2 (s)eD1 (s)+D2 (s)r ds + D2 (t)eD1 (t)+D2 (t)r , ∂r 1−α t ∫ T h(t, r) = eD3 (s)+D4 (s)r ds + eD3 (t)+D4 (t)r ,

(33) (34) (35)

t

∂h(t, r) hr = = ∂r

∫

T

D4 (s)eD3 (s)+D4 (s)r ds + D4 (t)eD3 (t)+D4 (t)r .

(36)

t

Remark 3. In the Theorem 1, we need to notice that it is very necessary to make the condition ∥ρ(t)∥2 = 1 hold. Theoretically, the explicit solutions (31) and (32) to the problem (6) can be only obtained under the condition of ∥ρ(t)∥2 = 1. Practically, ∥ρ(t)∥2 = 1 indicates that the risks caused by interest rate can be completely hedged under the optimal portfolios (31) and (32). Theorem 2. Assume that J(t, r, x) ∈ C 1,2,2 ([0, T ]×R+ ×R+ ) is a solution of the HJB equation (8), i.e. J(t, r, x) satisfies the following HJB equation: { π,C } sup A J(t, r, x) = 0, J(T, r, x) = (1 − α)e−βT U2 (x), (π(t),C(t))∈Γ

then we have V (t, r, x) 6 J(t, r, x) for an arbitrary admissible policy (π(t), C(t)) ∈ Γ. Moreover, suppose that ∗ ∗ there exists an optimal strategy (πHARA (t), CHARA (t)) ∈ Γ such that { } ∗ ∗ (πHARA (t), CHARA (t)) ∈ arg sup Aπ,C J(s, r(s), X(s)) ,

∗ ∗ ∗ ∗ then when (π(t), C(t)) = (πHARA (t), CHARA (t)), we get J(t, r, x) = V (t, r, x). It displays that (πHARA (t), CHARA (t)) given by Theorem 1 is indeed the optimal consumption and portfolio decisions for the problem (6). Proof. See Appendix.

5

Some special cases

If η = 0 and q = 1 − p, HARA utility is reduced to CRRA utility. Obviously, it is very easy for us to obtain the following optimal consumption-investment strategy under CRRA utility. Corollary 1. Under the condition of ∥ρ(t)∥2 = 1, the optimal investment-consumption strategy of the problem (6) for CRRA utility is given by ∗ πCRRA (t) =

1 fr (σ(t)σ ′ (t))−1 µ(t)X(t) + b(σ ′ (t))−1 ρ(t)X(t) , 1−p f (t, r) 11

(37)

∗ CCRRA (t)

=

and the optimal value function is expressed as

(

α 1−α

)−

1 p−1

X(t)f −1 (t, r),

(38)

xp 1−p f (t, r). p

(39)

∗ (t, r, x) = (1 − α)e−βt JCRRA

Here f (t, r) and fr are given by (33) and (34) respectively. More importantly, we can also derive the optimal investment-consumption strategy under CARA utility. Corollary 2. Under the condition of ∥ρ(t)∥2 = 1, the optimal investment strategy for the problem (6) with CARA utility is given by 1 Jxr ∗ πCARA (t) = (σ(t)σ ′ (t))−1 µ(t)h(t, r) − b(σ ′ (t))−1 ρ(t) , q Jxx and the optimal consumption strategy is determined by ( ) 1 1−α ∗ CCARA (t) = − ln + a(t) − D4 (t)r + h−1 (t, r)X(t) q α ( ) ∫ T 1 −1 α − h (t, r) h(s, r) ln + a(s) − D4 (s)r ds. q 1−α t

(40)

(41)

Further, the optimal value function is given by

{ 1 ∗ JCARA (t, r, x) = −(1 − α)e−βt h(t, r) · exp −qxh−1 (t, r) q ( ) ∫ T α −1 h(s, r) ln + h (t, r) + a(s) − D4 (s)r ds + a(t) − D4 (t)r} 1−α t

(42)

xr Here h(t, r), hr and JJxx are given by (35), (36) and (46) respectively. Proof. If η = 1 and p → −∞, we find that HARA utility is reduced to CARA utility. Further, we derive that

D2 (t) → D4 (t),

(30) can be rewritten as

∗ JHARA (t, r, x)

D1 (t) → D3 (t),

f (t, r) → h(t, r),

f r → hr .

( )p −p q = (1 − α)e x + ηh(t, r) f 1−p (t, r) qp 1−p ( )p ( )p q −1 −βt 1 − p 1+ xh (t, r) f (t, r) f −1 (t, r)h(t, r) . = (1 − α)e qp 1−p −βt 1

According to Theorem 1, taking the limitation p → −∞, we have

∗ ∗ JCARA (t, r, x) = lim JHARA (t, r, x) p→−∞

( )p ( )p 1−p q −1 = (1 − α)e · lim · lim 1 + xh (t, r) · lim f (t, r) f −1 (t, r)h(t, r) p→−∞ qp p→−∞ p→−∞ 1−p p −1 1 −1 = −(1 − α)e−βt · e−qxh (t,r) · lim f (t, r) · lim eln(f (t,r)h(t,r)) . p→−∞ p→−∞ q −βt

On the other hand, we find

( ) )p ln f −1 (t, r)h(t, r) lim ln f (t, r)h(t, r) = lim p→−∞ p→−∞ 1/p ) ( ∫ T α −1 + a(s) − D4 (s)r ds + a(t) − D4 (t)r. = h (t, r) h(s, r) ln 1−α t (

−1

12

(43)

where a(t) =

∫

T t

( ) ∫ T ∫ T ( ) 1 2 ′ 2 −k1 k2 + 2bθ (z)ρ(z) D4 (z)dz − b D42 (z)dz. −β − ∥θ(z)∥ dz + 2 t t

(44)

As a matter of fact, the second limitation in the equation (43) is a 00 . By using the L’Hˆopital’s rule, we can obtain (43). Therefore, the optimal value function (42) for CARA utility is obtained. By applying (42), we derive Jx 1 = − h(t, r), (45) Jxx q ( ) ∫ T hr hr α Jxr =− x+ h(s, r) ln + a(s) − D4 (s)r ds Jxx h(t, r) qh(t, r) t 1−α ( ) ∫ α 1 T (46) hr ln + a(s) − D4 (s)r ds − q t 1−α ∫ 1 T 1 + h(s, r)D4 (s)ds + D4 (t)h(t, r). q t q Putting (45) and (46) into (9), we get (40) and (41). 

If η = 0, p → 0 and q → 1, HARA utility is degenerated to logarithmic utility. Meantime, we have ( ) α 1 f (t, r) = (1 − e−β(T −t) ) + e−β(T −t) , f (t), 1−α β

(47)

∂f (t, r) = 0. (48) ∂r Obviously, we can directly obtain the following optimal strategy under logarithmic utility. Corollary 3. Under the condition of ∥ρ(t)∥2 = 1, the optimal investment-consumption strategy of the problem (6) under logarithmic utility is given by D1 (t) = −β(T − t),

D2 (t) = 0,

fr =

∗ πlog (t) = (σ(t)σ ′ (t))−1 µ(t)X(t), ∗ Clog (t) =

α β (1

and the optimal value function is written as

α X(t). − e−β(T −t) ) + (1 − α)e−β(T −t)

∗ Jlog (t, r, x) = (1 − α)e−βt f (t) ln x.

(49) (50)

(51)

Here f (t) is given by (43). Remark 4. We can come to the following conclusions from Corollaries 2-3: (i) Corollary 2 tells us that the optimal consumption-investment strategy under CARA utility can be obtained explicitly, to our knowledge, which wasn’t seldom investigated in the existing literatures; (ii) it can be seen from Corollary 3 that the consumption-investment strategy under logarithm utility doesn’t depend on the parameters of interest rate, moreover, which is same as that under constant interest rate. If α → 0 and β → 0, the problem (6) is reduced to the following dynamic asset allocation problem: M aximize E (U2 (X(T ))) . π(t)∈Γ

(52)

In order to compare our results with those of Grasselli (2003), we introduce the following zero-coupon bond in the financial market of this paper.

13

Proposition 1. The price of the zero-coupon bond can be defined as B(t, T ) = em(t,T )−n(t,T )r , with the conditions m(T, T ) = 0 and n(T, T ) = 0, then m(t.T ) and n(t, T ) respectively satisfy the following differential system: n(t, ˙ T ) = k1 n(t, T ) − 1,

n(T, T ) = 0;

1 m(t, ˙ T ) = (k1 k2 − bθ′ (t)ρ(t))n(t, T ) − b2 n2 (t, T ), 2

m(T, T ) = 0;

and their solutions are given by ) 1 ( −k1 (T −t) n(t, T ) = − e − 1 = −D4 (t), k1 ∫ T ∫ T 1 m(t, T ) = − (k1 k2 − bθ′ (s)ρ(s))n(s, T )ds + b2 n2 (s, T )ds = D3 (t). 2 t t

In addition, the volatility of the bond is given by σB (t, T ) = −bn(t, T ) = bD4 (t). Meantime, we have ˆ r). B(t, T ) = h(t, The solving process of the optimal portfolios for the problem (52) is analogous to those of the problem (6). We directly give some solutions of important equations. Because the consumption behavior isn’t considered, the original HJB equation (10) can be changed into 1 J2 Jx Jxr 1 J2 ∥θ(t)∥2 x − bθ′ (t)ρ(t) − b2 ∥ρ(t)∥2 xr 2 Jxx Jxx 2 Jxx 1 2 + (k1 k2 − k1 r)Jr + b Jrr = 0, 2

Jt + rxJx −

(53)

with the terminal condition J(T, r, x) = U2 (x). Its dual equation (18) is reduced to 1 ∥θ(t)∥2 z 2 gzz + (k1 k2 − k1 r − bθ′ (t)ρ(t))gr − bθ′ (t)ρ(t)zgrz 2 2 1 2 1 2 2 gr gzz − 2gr grz gz + b grr + b (1 − ∥ρ(t)∥ ) = 0, 2 2 gz2

gt − rg + (∥θ(t)∥2 − r)zgz +

(54)

where the boundary condition is given by g(T, r, z) =

1 1 − p p−1 1−p z η. − q q

Using the similar solving process as the equation (18), we get the solution to the equation (54) as follows: g(t, r, z) =

1 1 − p p−1 1−p ˆ z fˆ(t, r) − η h(t, r), q q

Further, according to (30), we can get the solution to the equation (53) given by: ( )p 1−p q ˆ r) fˆ1−p (t, r). J ∗ (t, r, x) = x + η h(t, qp 1−p

(55)

(56)

ˆ r) in (55) and (56) are given by Lemma 2 and Lemma 4 respectively. where fˆ(t, r) and h(t, As a result, we can directly get the optimal strategy with HARA utility as follows. Corollary 4. For HARA utility, under the condition of ∥ρ(t)∥2 = 1, the optimal investment strategy of the problem (52) is ( ) 1 1−p ˆ ∗ ′ −1 π (t) = (σ(t)σ (t)) µ(t) X(t) + η h(t, r) 1−p q ) ) (( (57) 1−p ˆ 1−p ˆ ′ −1 η h(t, r) D2 (t) − η hr , + b(σ (t)) ρ(t) X(t) + q q 14

with the optimal value function given by (56). Remark 5. According to Proposition 1, (57) can be rewritten as ˆ r) π ∗ (t) = C1V (t)X(t) + C2V (t)h(t, = C1V (t)X(t) + C2V (t)B(t, T ), with 1 (σ ′ (t))−1 (σ −1 (t)µ(t) − pρ(t)bD4 (t)), 1−p η C2V (t) = (σ ′ (t))−1 (σ −1 (t)µ(t) − ρ(t)bD4 (t)). q

C1V (t) =

This is consistent with the Proposition A.3 of Grasselli (2003). Apart from Corollary 4, we can also directly obtain the optimal strategies with CRRA utility and CARA utility. Corollary 5. For CRRA utility, under the condition of ∥ρ(t)∥2 = 1, the optimal investment strategy of the problem (52) is given by ( ) 1 p π ∗ (t) = (σ(t)σ ′ (t))−1 µ(t)X(t) + e−k1 (T −t) − 1 b(σ ′ (t))−1 ρ(t)X(t), (58) 1−p k1 (p − 1) with the optimal value function given by

1 p ˆ1−p x f (t, r). p

J ∗ (t, r, x) =

(59)

Corollary 6. For CARA utility, under the condition of ∥ρ(t)∥2 = 1, the optimal strategy of the problem (52) is expressed as ( ) 1 1ˆ ∗ ′ −1 ′ −1 ˆ π (t) = (σ(t)σ (t)) µ(t)h(t, r) + b(σ (t)) ρ(t) X(t) − h(t, r) D4 (t). (60) q q with the optimal value function given by 1 ˆ −1 (t, r) + ϕ1 (t)}. J ∗ (t, r, x) = − exp{−qxh q where 1 ϕ1 (t) = − 2

∫

t

T

1 ∥θ(z)∥ dz − b2 2 2

∫

T

t

D42 (z)dz

+

∫

T

bθ′ (z)ρ(z)D4 (z)dz.

t

Proof. Taking the limitation p → −∞ in the equation (57), we have

1 1 ˆ r) (σ(t)σ ′ (t))−1 µ(t)X(t) + (σ(t)σ ′ (t))−1 µ(t)h(t, 1−p q ( )) 1 − p (ˆ ′ −1 ˆ + b(σ (t)) ρ(t) lim X(t)D2 (t) + lim h(t, r)D2 (t) − hr . p→−∞ p→−∞ q

π ∗ (t) = lim

p→−∞

Easy calculations yield lim

p→−∞

1 (σ(t)σ ′ (t))−1 µ(t)X(t) = 0, 1−p

lim X(t)D2 (t) = X(t)D4 (t),

p→−∞

) ˆ ˆ 1 − p (ˆ ˆ r = 1 lim h(t, r)D2 (t) − hr = − 1 D4 (t)h(t, ˆ r). h(t, r)D2 (t) − h p→−∞ q q p→−∞ 1/(1 − p) q lim

15

(61)

For the optimal value function under CARA utility, the solving steps is similar to (42). When η = 1 and ˆ r). We derive the optimal value function as follows. p → −∞, we have fˆ(t, r) → h(t, )p ( 1−p q ˆ r) fˆ1−p (t, r) J ∗ (t, r, x) = lim x + η h(t, p→−∞ qp 1−p (62) p 1 −qxhˆ −1 (t,r) ˆ ˆ ln(fˆ−1 (t,r)h(t,r) ) . =− e h(t, r) · lim e p→−∞ q On the other hand, we find ( ) ˆ r) ( )p ln fˆ−1 (t, r)h(t, ˆ r) = lim lim ln fˆ−1 (t, r)h(t, p→−∞ p→−∞ 1/p = a(t) − D4 (t)r.

(63)

Substituting (63) back into (62), after some combinations, we can get (61).  Remark 6. We can rewrite (60) as the following form: ˆ r) π ∗ (t) = QV1 (t)X(t) + QV2 (t)h(t, = QV1 (t)X(t) + QV2 (t)B(t, T ), with QV1 (t) = (σ ′ (t))−1 ρ(t)bD4 (t), 1 QV2 (t) = (σ ′ (t))−1 (σ −1 (t)µ(t) − ρ(t)bD4 (t)). q This conclusion also conforms to the Proposition A.4 of Grasselli (2003). If k1 = k2 = b = 0, interest rate is degenerated into a constant. And it leads to ) ∫ T( 1 p p 2 D1 (t) = β+ ∥θ(s)∥ ds, D2 (t) = − (T − t), 2 p−1 2(p − 1) p−1 t D3 (t) = 0,

D4 (t) = −(T − t),

(64) (65)

Therefore, we can conclude the optimal consumption-investment strategy of the problem (6) with constant interest rate in the following Corollary 7. Corollary 7. For HARA utility, the optimal strategy of the problem (6) with constant interest rate is given by ( ) 1 1−p ∗ ′ −1 π (t) = (σ(t)σ (t)) µ(t) X(t) + ηh(t, r) , (66) 1−p q ∗

C (t) = where

(

α 1−α

a(t) =

)−

1 p−1

( ) 1−p 1−p X(t) + ηh(t, r) f −1 (t, r) − η, q q

) p p 1 2 β+ ∥θ(s)∥ − r(s) ds, p−1 2(p − 1)2 p−1 t ( )− 1 ∫ T p−1 α f (t, r) = ea(u) du + ea(t) , 1−α t ∫ T ∫T ∫T h(t, r) = e− s r(u)du ds + e− t r(s)ds .

∫

T

(

t

16

(67)

Remark 7. We find that (66) and (67) is consistent with the optimal consumption-investment strategies obtained by Chang and Rong (2014). To sum up, we find out that many important results in the extant literatures have been special cases of our model. Therefore, our model greatly extended and enriched the theory and methodology on the consumptioninvestment model. More importantly, we successfully derived the optimal consumption-investment strategy with exponential preference and obtained the explicit expression of the optimal value function. Moreover, if the consumption behavior isn’t considered, the optimal portfolios with the Vasicek model under HARA utility and CARA utility is consistent with those of Grasselli (2003), who find the optimal portfolios under HARA utility converging almost surely to those under CARA utility. This provided theoretical basis for our conclusions.

6

Numerical example

In this section, we provide a numerical example to illustrate the optimal consumption-investment strategy under HARA utility. Throughout this section, unless otherwise stated, the basic model parameters are given by k1 = 0.02, k2 = 0.05, b = 0.02, r(0) = 0.05, ρ(t) = (0.6, 0.8)′ , η = 0.6, p = −0.5, q = 3, α = 0.7, β = 0.05, t = 0, T = 1, X(0) = 100. Meantime, we assume that there are two stocks in the financial market and their appreciate rate and volatility matrix are given by ( ) 0.35 0.25 µ(t) = (0.12, 0.18)′ , σ(t) = . 0.25 0.45 In the following two subsections, we analyze the effect of model parameters on the optimal consumption and investment strategy respectively, especially, the parameters of interest rate and HARA utility. Notice that the 2 ∑ optimal investment strategy in Fig.1-Fig.4 represents the total amount πi (t) invested in the two stocks. i=1

6.1

Sensitivity analysis on the optimal investment strategy We can draw the following conclusions from Fig.1-Fig.4. 2 ∑ (a1) πi (t) decreases with respect to the parameter b. From the equation (2), we can see that the parameter i=1

b represents the volatility of short rate. It means that the bigger the value of b is, the bigger the volatility resulted from interest rate is. Due to the positive correlation between interest rate dynamics and stock price dynamics, the volatility of stock price will become larger. It implies that the risk resulted from the volatility of stock price will become larger when the risk of interest rate becomes larger. Therefore, in order to avoid risks the investor will decrease the amount invested the stocks. This conclusion conforms to economic implication of b. 2 ∑ (a2) πi (t) increases with respect to the parameter k1 . In the Vasicek model, the expectation of short rate i=1

r(t) decreases with respect to k1 , which displays that the expected short rate will descend as the value of k1 ascends. It would lead to that the risk from interest rate will decrease. Because of the positive coefficient between interest rate dynamics and stock price dynamics, the risk from stock volatility would also decrease. Under this situation investors are very willing to raise the amount invested in the stocks in order to get more revenue. This conclusion is also consistent with economic implication of k1 . 2 ∑ (a3) πi (t) will increase as p becomes larger. In the utility theory, risk aversion degree of investors is i=1

q determined by their absolute risk aversion coefficient (ARAC). For HARA utility, their ARAC is given by 1−p x + η. According to this ARAC, we find that the ARAC decreases as the value of p increases. It means that risk aversion degree of investors will decrease. Therefore, under this environment the individuals are more willing to invest more money in the stocks. This conclusion is consistent with our intuition. 2 ∑ πi (t) decreases with the value of q. For HARA utility, when the value of q becomes larger, we find (a4) i=1

that its ARAC will become bigger. It means that the individuals aren’t willing to invest more amount in the stocks. 17

the effect of b on the optimal investment strategy 40 35

optimal investment strategy

30 25 20 15 10 5 0 −5 −10

0

0.2

0.4

0.6

0.8

1

b

Figure 1: The effect of b on

2 ∑

πi (t).

i=1

the effect of k1 on the optimal investment strategy 37.16 37.14

optimal investment strategy

37.12 37.1 37.08 37.06 37.04 37.02 37

0

0.1

0.2

0.3

0.4

0.5

k1

Figure 2: The effect of k1 on

2 ∑

πi (t).

i=1

the effect of p on the optimal investment strategy 110 100

optimal investment strategy

90 80 70 60 50 40 30 20 10 −3

−2.5

−2

−1.5

−1

−0.5

0

p

Figure 3: The effect of p on

2 ∑

i=1

18

πi (t).

0.5

the effect of q on the optimal investment strategy 50

optimal investment strategy

48

46

44

42

40

38

36

0

0.5

1

1.5 q

2

Figure 4: The effect of q on

2.5

2 ∑

3

πi (t).

i=1

the effect of b on the optimal consumption strategy 65.2 65.1

optimal consumption strategy

65 64.9 64.8 64.7 64.6 64.5 64.4 64.3 64.2

0

0.2

0.4

0.6

0.8

1

b

∗ Figure 5: The effect of b on CHARA (t).

the effect of k on the optimal consumption strategy 1

65.121

optimal consumption strategy

65.1208 65.1206 65.1204 65.1202 65.12 65.1198 65.1196 65.1194

0

0.1

0.2

0.3

0.4

k1

∗ Figure 6: The effect of k1 on CHARA (t).

19

0.5

the effect of p on the optimal consumption strategy 85

optimal consumption strategy

80

75

70

65

60

55 −3

−2.5

−2

−1.5

−1

−0.5

0

0.5

p

∗ Figure 7: The effect of p on CHARA (t).

the effect of q on the optimal consumption strategy 66

optimal consumption strategy

65

64

63

62

61

60

0

0.5

1

1.5 q

2

2.5

∗ Figure 8: The effect of q on CHARA (t).

20

3

Further, in order to avoid the risk from the stocks, they will reduce the amount in the stocks. This conclusion conforms to economic implication of q as well.

6.2

Sensitivity analysis on the optimal consumption strategy

∗ Fig.5-Fig.8 show how the optimal consumption strategy CHARA (t) changes with the parameters of interest rate and utility function. We can conclude the results as follows. 2 ∑ ∗ ∗ (b1) CHARA (t) decreases as the value of b increases. This displays that CHARA (t) and πi (t) have the i=1

same trend. When the value of b becomes larger, we get that the optimal amount in the stocks will become smaller. So it will lead to that the wealth of individuals will also become smaller, and it implies that the optimal amount to be consumed will decreases. ∗ (t) increases in k1 . From (a2) in the previous subsection, we arrive at the conclusion that the (b2) CHARA bigger the value of k1 is, the more the optimal amount in the stocks is. Meantime, the wealth of investors will rise accordingly and this will cause that the amount to be consumed will rise. 2 ∑ ∗ ∗ (b3) CHARA (t) increases with respect to p. We can see that CHARA (t) has the same trend as πi (t). It i=1

implies that the wealth will also increase as the value of p becomes larger. Therefore, the optimal amount which can be consumed will ascend. 2 ∑ ∗ (b4) CHARA (t) increases in q. This trend is opposite to that of πi (t). It tells us that the optimal amount i=1

that the individuals can consume will rise as the value of q becomes larger while the optimal amount invested in stocks will decrease, which will lead to the decreasing wealth. It means that the trend of consumption and wealth is opposite. This conclusion is surprising. We need to keep this conclusion in mind in the real-world environments.

7

Conclusions

This paper investigates the consumption-investment problem with the Vasicek model under HARA utility. Considering the difficulties in conjecturing the form of the value function under HARA utility, we present Legendre transform method to obtain the explicit solution of the optimal consumption and investment strategy. Finally, we discuss some special cases in detail and give a numerical example to illustrate our results. More importantly, we illustrate that how the parameters of interest rate and risk aversion affect the optimal consumption-investment policies. Due to our work, we find out some special advantages on Legendre transform: (i) the solution of the dual equation under HARA utility is much easier to conjecture than that of the original HJB equation; (ii) by using Legendre transform, we can obtain the optimal consumption and portfolio decision under HARA utility; (iii) we can obtain the explicit expression for the optimal consumption-investment strategy with CARA utility, which is seldom studied in the consumption-investment problems; (iv) the same methodology as this paper can be also used to deal with the consumption-investment problems with other more complicated stochastic environments under 2 ∑ πi (t) and HARA utility. Meantime, we draw some important results by the numerical example as well: (i) i=1

∗ CHARA (t) have the same trend in b, k1 and p; (ii)

q.

2 ∑

i=1

∗ πi (t) and CHARA (t) have the opposite trend with respect to

In future research, we can introduce stochastic income of the individuals into the consumption-investment problems and investigate the optimal portfolios with other more complicated stochastic environments, such as those with interest rate risk and inflation risk, or with affine interest rate and Markov-regime switching model, or with affine interest rate and stochastic volatility. We leave these problems to future research. Appendix The proof of Theorem 2. Let Q = [0, ∞) × [0, ∞), we choose a sequence of bounded open sets Qi satisfying ∞ ∪ Qi ⊂ Qi+1 ⊂ Q, i = 1, 2, · · · , and Q = Qi . For (r, x) ∈ Qi , assume that the exit time of (r(t), X(t)) from i=1

Qi is denoted by τi . when i → ∞, we get τi ∧ T → T .

21

(i) Consider an arbitrary admissible strategy (π(t), C(t)). By applying Itoˆ′ s formula for J(t, r, x) on [t, T ], we obtain ∫

T

J(T, r(T ), X(T )) =J(t, r, x) + Aπ,C J(s, r(s), X(s))ds t ∫ T π ′ (s)σ(s)Jx (s, r(s), X(s))dW (s) + t ∫ T + bρ′ (s)Jr (s, r(s), X(s))dW (s) t ∫ T √ + b 1 − ∥ρ(s)∥2 Jr (s, r(s), X(s))dW0 (s). t

Taking

sup

{

} Aπ,C J(t, r, x) = 0 into consideration, which displays that the variational inequality

(π(t),C(t))∈Γ π,C A J(t, r, x) 6 0, we have

∫

T

J(T, r(T ), X(T )) 6J(t, r, x) + π ′ (s)σ(s)Jx (s, r(s), X(s))dW (s) t ∫ T + bρ′ (s)Jr (s, r(s), X(s))dW (s) t ∫ T √ b 1 − ∥ρ(s)∥2 Jr (s, r(s), X(s))dW0 (s). + t

The last three terms on the right hand of the above inequality are square-integrable martingales and their expectations are equal to zero. Hence, we get E (J(T, r(T ), X(T )) |X(t) = x, r(t) = r ) 6 J(t, r, x). Further, taking the supremum, we obtain sup (π(t),C(t))∈Γ

and it implies that

E (J(T, r(T ), X(T )) |X(t) = x, r(t) = r ) 6 J(t, r, x). V (t, r, x) 6 J(t, r, x).

∗ ∗ (ii) E (J(τi ∧ T, r(τi ∧ T ), X(τi ∧ T ))) < ∞ for a specific policy (πHARA (t), CHARA (t)). ′ ˆ Applying Ito s formula to J(t, r, x) on [0, τi ∧ T ] once again, we have

∫

τi ∧T

∗

∗

J(τi ∧ T, r(τi ∧ T ), X(τi ∧ T )) =J(0, r0 , x0 ) + Aπ ,C J(s, r(s), X(s))ds 0 ∫ τi ∧T + π ′ (s)σ(s)Jx (s, r(s), X(s))dW (s) 0 ∫ τi ∧T + bρ′ (s)Jr (s, r(s), X(s))dW (s) 0 ∫ τi ∧T √ + b 1 − ∥ρ(s)∥2 Jr (s, r(s), X(s))dW0 (s). 0

∗

∗

∗ ∗ For a specific strategy (πHARA (t), CHARA (t)) satisfies (8), i.e. Aπ ,C J(s, r(s), X(s)) = 0, and the last three terms are also square-integrable martingales. Hence, taking the expectation on both sides on the above equation, we obtain E (J(τi ∧ T, r(τi ∧ T ), X(τi ∧ T ))) = J(0, r0 , x0 ) < ∞.

22

∗ ∗ (t), CHARA (t)). (iii) V (t, r, x) = J(t, r, x) for a specific strategy (πHARA ′ ˆ Using Ito s formula for J(t, r, x) on [t, τi ∧ T ] once more, similarly, we derive

∫

τi ∧T

∗

∗

J(τi ∧ T, r(τi ∧ T ), X(τi ∧ T )) =J(t, r, x) + Aπ ,C J(s, r(s), X(s))ds t ∫ τi ∧T + π ′ (s)σ(s)Jx (s, r(s), X(s))dW (s) t ∫ τi ∧T + bρ′ (s)Jr (s, r(s), X(s))dW (s) t ∫ τi ∧T √ + b 1 − ∥ρ(s)∥2 Jr (s, r(s), X(s))dW0 (s). t

Taking the conditional expectation, we yield J(t, r, x) = E (J(τi ∧ T, r(τi ∧ T ), X(τi ∧ T )) |r(t) = t, X(t) = x ) . Taking the limitation once more, we get J(t, r, x) = lim E (J(τi ∧ T, r(τi ∧ T ), X(τi ∧ T )) |X(t) = x, r(t) = r ) . i→∞

In addition, we derive V (t, r, x) =

sup (π(t),C(t))∈Γ

( ∫ E α

0

T

e

−βt

U1 (C(t))dt + (1 − α)e

−βT

U2 (X(T )) |X(t) = x, r(t) = r

= lim E (J(τi ∧ T, r(τi ∧ T ), X(τi ∧ T )) |X(t) = x, r(t) = r )

)

i→∞

= J(t, r, x). ∗ ∗ Therefore, it implies that (πHARA (t), CHARA (t)) is indeed the optimal investment-consumption policy for the problem (6). 

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