Abstract
This paper provides a Legendre transform method to deal with a class of investment and consumption problems, whose objective function 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 hyperbolic absolute risk aversion (HARA) utility function, which includes power utility, exponential utility, and logarithm utility as special cases. The optimal investment and consumption strategy for HARA utility is explicitly obtained by applying dynamic programming principle and Legendre transform technique. Some special cases are also discussed.
1. Introduction
The investment and consumption problem was originated from the seminal work of Merton [1, 2], who first used stochastic optimal control to deal with a continuous-time portfolio selection problem with consumption behavior and obtained the closed-form solution of the optimal investment and consumption strategy under power utility and logarithm utility. In the following years, many scholars began to pay attention to the investment and consumption problems and obtained many research results. One can refer to the papers of Chacko and Viceira [3], Liu [4], Noh and Kim [5], Chang and Rong [6], and Chang et al. [7]. But these results were generally achieved under power utility, which was taken as a special case of hyperbolic absolute risk aversion (HARA) utility function.
HARA utility includes power utility, exponential utility, and logarithm utility as special cases. From the expression of HARA utility, we can see that HARA utility has more complicated structure than other utility functions, which leads to that there is a little work on portfolio selection problems with HARA utility in the existing literature. As a matter of fact, Grasselli [8] investigated a portfolio selection problem with HARA utility in the stochastic interest rate environment and verified that the optimal policy corresponding to HARA utility converges to the one corresponding to exponential utility and logarithm utility. Jung and Kim [9] provided a Legendre transform method to deal with an optimal portfolio model with HARA utility.
In recent years, Legendre transform is often used to deal with the complicated portfolio selection problems, for example, the optimal investment problems under the CEV model. One can refer to the work of Jonsson and Sircar [10], Xiao et al. [11], and Gao [12, 13]. The advantage of Legendre transform is to transform the nonlinear HJB equation into a linear dual one and the structure of the solution under HARA utility is easy to conjecture. Therefore, it is very possible for us to use Legendre transform to obtain the optimal investment strategy of more complicated portfolio selection problems.
In this paper, we consider a class of investment and consumption problems, whose objective function is to maximize the expected discount utility of intermediate consumption and terminal wealth, and wish to obtain the optimal investment and consumption strategy for HARA utility. We provide a Legendre transform method to deal with this problem and obtain the explicit expression of the optimal investment and consumption strategy under HARA utility. Some special cases are also discussed. There are two main contributions in this paper. (i) We study a class of investment and consumption problems, whose objective function is more complicated than that of the optimal investment problems. In other words, we not only hope to obtain the optimal investment strategy, but also hope to obtain the optimal strategy of consumption. (ii) We obtain the explicit expression of the optimal investment and consumption strategy under HARA utility, which has rarely been studied in the optimal portfolio selection problems. It is all well known that most of portfolio selection problems are studied under power utility or exponential utility.
The structure of this paper is as follows. Section 2 formulates a class of investment and consumption problems with HARA utility, which wish to maximize the expected discount utility of intermediate consumption and terminal wealth. Section 3 uses Legendre transform to obtain the dual one of the HJB equation. Section 4 obtains the optimal investment and consumption strategy for HARA utility and Section 5 concludes the paper.
2. The Model
Throughout the paper, represents the transpose of a matrix or sector, represents the norms of a sector , represents the expectation, and represents the fixed and finite investment horizon. Assume that is a n-dimensional independent standard Brownian motion defined on complete probability space , in which represents information flow generated by .
Assume that the financial market is composed of assets, which are continuously traded on . One is the risk-free asset (i.e., a bond), whose price at time is denoted by . Then satisfies the following equation: where is the risk-free interest rate.
The other assets are risky assets (i.e., stocks), whose prices at time are denoted by , , and then satisfies the following geometric Brownian motion (GBM): where and represent the appreciation rate sector and volatility matrix of the stocks, respectively, and , . In addition, we assume that satisfies the nondegenerated condition: , for all , and , , and are Borel-measurable bounded deterministic functions on .
Assume that the amount at time invested in the th stock is denoted by , . Letting , represents the wealth at time and represents the consumption rate at time ; then, the amount invested in the bond is . Suppose that the investor cannot consider transaction cost and short-sell the stocks; then, the wealth process under trading strategy satisfies the following stochastic differential equation(SDE): where
Definition 1 (admissible strategy). An investment and consumption strategy is said to be admissible if the following conditions are satisfied:(i) and are all -measurable and satisfy and ;(ii);(iii)the SDE (3) has a pathwise unique solution according to .
Assume that the set of all admissible investment and consumption strategies is denoted by . In this paper, we assume that the investor wishes to maximize the following objective function: where and are all utility functions and is the subjective discount rate. The parameter determines the relative importance of intermediate consumption and terminal wealth. When , expected utility only depends on terminal wealth and the problem (5) is reduced to an asset allocation problem.
In this paper, we choose hyperbolic absolute risk aversion (HARA) utility function for our analysis. HARA utility function with parameters , , and is given by As a matter of fact, HARA utility function recovers power utility, exponential utility, and logarithm utility as special cases.(i)If we choose and , then we have (ii)If we choose and , then we have (iii)If we choose , , and , then we have
3. HJB Equation and Legendre Transform
We define the value function as with boundary condition given by .
Applying the principle of optimality (referring to Fleming and Soner [14]), if the value function , then satisfies the following Hamilton-Jacobi-Bellman (HJB) equation: where and , , and represent the first-order and second-order partial derivatives of with respect to the variables .
Assume that is a solution of (11); then, according to the first-order condition the optimal investment and consumption strategy is given by Letting and substituting (13) into (11), we obtain with boundary condition given by .
For HARA utility, it is very difficult to directly conjecture the form of the solution of (14). Therefore, we introduce the following Legendre transform technique.
Definition 2. Let be a convex function. Legendre transform can be defined as follows:
and then the function is called the Legendre dual function of (cf. Jonsson and Sircar [10], Xiao et al. [11], and Gao [12, 13]).
If is strictly convex, the maximum in (15) will be attained at just one point, which we denoted by . We can attain at the unique solution by the first-order condition:
So we have
Following Jonsson and Sircar [10], Xiao et al. [11], and Gao [12, 13], Legendre transform can be defined by
where denotes the dual variable to . The value of where this optimum is attained is denoted by ; so we have
The relationship between and is given by
Hence, we can choose either one of two functions and as the dual function of . In this paper we choose . Moreover, we have
Differentiating (21) with respect to and , we get Notice that ; then at the terminal time , we can define So we have , where is taken as the inverse of marginal utility.
Putting (22) into (14), we obtain Differentiating (24) with respect to and using (20), we derive with boundary condition given by .
Under HARA utility, we have
4. The Optimal Investment and Consumption Strategy
Applying (22) to (13), we get Assume that a solution of (25) with terminal condition (26) is given by Then the partial derivatives are derived as follows:
Plugging (27)–(31) into (25), we obtain Further, (32) can be decomposed into the following two equations:
Solving (33), we get where
Applying (20), (22), (28), and (30), we have Meantime, (27) can be rewritten as
On the other hand, according to and (28), we derive Considering , we obtain the following optimal value function:
Summarizing what is mentioned above and considering , we can draw the following conclusions.
Theorem 3. Under HARA utility (6), the optimal investment and consumption strategy of the problem (5) is given by where and are given by (34) and (35), respectively.
Theorem 4 (Verification theorem). Let be a solution to (11); then, for all admissible strategies , we have ; if there exists satisfies then we conclude that and given by Theorem 3 is the optimal investment and consumption strategy of the problem (5).
Proof. Considering and using formula from to for , we obtain
Considering that is a solution to (11); that is, we have . So we obtain
The last one term in (45) is a local martingale and its expected value is equal to zero. Thus, we get
Maximizing (46) for all admissible strategies , we derive
When , all inequalities become equalities; that is, , and it shows that given by Theorem 3 is the optimal investment and consumption strategy of the problem (5).
The proof is completed.
In particular, we can obtain the following three special cases.
Corollary 5. If and , HARA utility is degenerated to power utility. Therefore, the optimal investment and consumption strategy of the problem (5) under is given by where is given by (36).
Corollary 6. If and , HARA utility is reduced to exponential utility. Therefore, the optimal investment and consumption strategy of the problem (5) under is given by where is given by (35), and is given by
Proof.
The equation (41) in Theorem 3 can be rewritten as
When and , then we have
Therefore, it is obvious that (49) is proved.
Similarly, (42) can be changed into
When , we have and . Therefore, it leads to that
For the second term in (54), we get
By using L'Hopital's rule, (56) is equal to
As a result, (50) is also proved.
Corollary 7. If , , and , HARA utility is reduced to logarithm utility. It leads to . Therefore, the optimal investment and consumption strategy of the problem (5) under is given by
5. Conclusions
This paper is concerned with a class of investment and consumption problems with HARA utility. We present a Legendre transform method to deal with them and obtain the explicit expressions of the optimal investment and consumption strategy. Some special cases are also discussed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is supported by National Natural Science Foundation of China (no. 11301376), Humanities and Social Science Research Youth Foundation of Ministry of Education of China (no. 11YJC790006), and Higher School Science and Technology Development Foundation of Tianjin (no. 20100821).