Abstract
We consider the optimal consumption and portfolio choice problem with constant absolute risk aversion (CARA) utility and a subsistence consumption constraint. A subsistence consumption constraint means there exists a positive constant minimum level for the agent’s optimal consumption. We use the dynamic programming approach to solve the optimization problem and also give the verification theorem. We illustrate the effects of the subsistence consumption constraint on the optimal consumption and portfolio choice rules by the numerical results.
1. Introduction
Following the seminal research works of Merton [1, 2], various problems of continuous-time optimal consumption and portfolio selection have been considered under various financial/economic constraints. One of the interesting research topics in a continuous-time portfolio selection problem is the optimization problem subject to a subsistence consumption constraint (or a downside consumption constraint). A subsistence constraint means that there exists a positive lower bound level for the agent’s optimal consumption rate. Thus this constraint affects the agent’s financial decision including her optimal portfolio.
Lakner and Nygren [3] have studied the portfolio optimization problem subject to a downside constraint for consumption and an insurance constraint for terminal wealth with a martingale approach. Gong and Li [4] have investigated the role of index bonds in the optimal consumption and portfolio selection problem with constant relative risk aversion (CRRA) utility and a real subsistence consumption constraint using the dynamic programming approach. Shin et al. [5] have also considered a similar problem to that of Gong and Li [4]. They have studied the portfolio selection problem with a general utility function and a downside consumption constraint using the martingale approach. Yuan and Hu [6] have investigated the optimal consumption and portfolio selection problem with a consumption habit constraint and a terminal wealth downside constraint using the martingale approach. In this paper we use the dynamic programming method based on Karatzas et al. [7] to derive the value function and the optimal policies in closed-form with a constant absolute risk aversion (CARA) utility function and a subsistence consumption constraint. Lim et al. [8] have considered a similar portfolio optimization problem combined with the voluntary retirement choice problem. Shin and Lim [9] have analyzed the effects of the subsistence consumption constraint for behavior of investment in the risky asset.
The rest of this paper proceeds as follows. Section 2 introduces the financial market. In Section 3 we consider the main optimization problem. We use the dynamic programming principle to derive the solutions in closed form with CARA utility and a subsistence consumption constraint. We also give some numerical results and the solutions derived by the martingale method. Section 4 concludes.
2. The Financial Market Setup
We assume that there are two assets in the financial market: one is a riskless asset with constant interest rate , and the other is a stock whose price process evolves according to the stochastic differential equation (SDE) where is the constant expected rate of return of the stock, is the constant volatility of the stock, and is a standard Brownian motion on a probability space endowed with the filtration which is the augmentation under of the natural filtration generated by the standard Brownian motion . We assume that so that the market price of risk is not zero: Let be an economic agent’s wealth at time , the amount of money invested in the stock at time , and the consumption rate at time . The portfolio process is adapted to and satisfies, for all , almost surely (a.s.), and the consumption rate process is a nonnegative process adapted to such that, for all , a.s., We assume that there is a subsistence consumption constraint which restricts the minimum consumption level. That is, the consumption process should satisfy where is a constant lower bound for the consumption rates. Thus the agent’s wealth process follows the SDE with an initial endowment . (We need this restriction on the initial endowment for the positive consumption rate. See Lemma 3.1 of Gong and Li [4]). A consumption-portfolio plan satisfying the above conditions is called admissible at if , for all . We let denote the class of admissible controls at .
3. The Optimization Problem
Now the agent’s optimization problem with initial wealth is to choose to maximize the following expected life-time utility: Here, is the subjective discount factor and is a constant absolute risk aversion (CARA) utility function defined by where is the agent’s coefficient of absolute risk aversion. Thus the agent’s value function is given by Bellman equation corresponding to the optimization problem for is We assume that the wealth process must satisfy a transversality condition We will find the solution , as the candidate value function, to Bellman equation (10) under the conditions that and for and for a real number . After obtaining the solution, we can check these conditions. Under these conditions, in particular, the first-order condition (FOC), with respect to , is binding if so that the maximizing in Bellman equation (10) is in this case.
Thus, from the first-order conditions (FOCs) of Bellman equation (10), we derive the candidate optimal consumption and portfolio
Remark 1. For later use, we consider two quadratic algebraic equations: with two roots and and with two roots and .
Theorem 2. Let be given by where and is determined from the algebraic equation Then it satisfies Bellman equation (10).
Proof. By using Remark 1, we can check the inequalities in (16) and (17). The inequality in (18) holds by (17). Define the function of on by
By using (16) and (17), one can check
Since the function is increasing in , it has the inverse function. Let for be the inverse function of . In particular, we have
By (19), we have
By using Remark 1, (16), (18), (22), and (23), we can show that the function defined by (15) is continuous. By using Remark 1, (20), (23), and the inverse relationship between and , we can obtain
By (15), (18), (20), (22), and (24), we get the smooth-pasting () condition
and the high-contact () condition
Thus, the function is twice continuously differentiable. Furthermore, and for and .
For , if we substitute FOCs (12) into Bellman equation (10), we obtain the changed Bellman equation
We can easily check that is the solution to (27) for .
For , we also obtain the changed Bellman equation from (10):
If we substitute (24) into Bellman equation (28), then we obtain the equation
By using (15) and (23), we can check that (29) holds.
Now we can derive the candidate optimal policies with the function in Theorem 2.
Theorem 3. The candidate optimal policies are given by such that where is determined from the optimal wealth process
Theorem 4 (verification theorem). The value function of the optimization problem (9) is equal to in Theorem 2. That is, . Consequently the candidate consumption and portfolio in Theorem 3 are the optimal policies of problem (9).
Proof. For arbitrary given consumption and portfolio plan and , we have where the inequality is obtained from Bellman equation (10), the first equality from applying Itô’s formula to , and the second equality from . Taking and using transversality condition (11), we have for arbitrary given consumption and portfolio plan ; that is, Now we consider the candidate optimal consumption and portfolio plan in Theorem 3. For , we have where the first equality is obtained from Bellman equation (10), the second from applying Itô’s formula to , and the third from . Taking and using transversality condition (11), we have Thus, from (35) and (37), we show that which is the solution to Bellman equation (10) is a real value function of the optimization problem (9).
Now we compare our solution in Theorem 3 with the Merton’s solution with CARA utility. The optimal consumption and portfolio policies without the subsistence constraint are given by respectively. Figures 1 and 2 give the numerical results for the optimal consumption and portfolio.
Remark 5. Dynamic programming principle can be also applied to the CRRA utility function with a subsistence consumption constraint following our approach. See Gong and Li [4] and Lee and Shin [10].
Remark 6. Following Shin et al. [5] we can use the martingale method to derive a similar solution with CARA utility. We will give (rough) sketch of the derivation.
From the agent’s wealth dynamics in (6), we obtain the budget constraint where the state price density is defined by A dual utility function of the CARA utility function is derived by where . The dual value function is given by where and and are two real roots of the quadratic algebraic equation The undetermined coefficients of are given by Then we use the Legendre transform inverse formula to obtain the value function as follows: where and is determined from the following algebraic equation: Actually we can show that in (45) and in (46) agree with in (15) and in (17), respectively, if we set and show that Refer to Lee and Shin [10].
Remark 7. We simplify the calculation by using the dynamic programming approach, where, instead of Legendre transformation in the martingale approach, we introduce the function in (20) whose inverse is in (23). The link between the two methods is the relation for , where the last equality is in (24).
4. Concluding Remarks
We have considered the optimal consumption and portfolio choice problem with constant absolute risk aversion (CARA) utility and a subsistence consumption constraint. The existence of a subsistence consumption constraint is realistic since one needs a minimum level of consumption to live. For example, we cannot live without necessities. We have obtained the closed form solution to optimization problem by using the dynamic programming approach. We have illustrated the effects of the subsistence consumption constraint on the optimal consumption and portfolio by the numerical results. Furthermore one can consider the optimization problem under regime switching as future research.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are indebted to two anonymous referees for their helpful comments and insightful suggestions. The research of the corresponding author (Yong Hyun Shin) was supported by Sookmyung Women’s University Research Grants 2012.