Multi-Period Mean-Variance Portfolio Selection with Uncertain Time Horizon When Returns Are Serially Correlated

Abstract

We study a multi-period mean-variance portfolio selection problem with an uncertain time horizon and serial correlations. Firstly, we embed the nonseparable multi-period optimization problem into a separable quadratic optimization problem with uncertain exit time by employing the embedding technique of Li and Ng (2000). Then we convert the later into an optimization problem with deterministic exit time. Finally, using the dynamic programming approach, we explicitly derive the optimal strategy and the efficient frontier for the dynamic mean-variance optimization problem. A numerical example with AR(1) return process is also presented, which shows that both the uncertainty of exit time and the serial correlations of returns have significant impacts on the optimal strategy and the efficient frontier.

1. Introduction

The portfolio selection problem, which is one of great importance from both theoretical and practical perspectives, aims to find the best allocation of wealth among different assets in financial market. The mean-variance analysis pioneered by Markowitz [1] is one of the most widely used frameworks dealing with portfolio selection problems. In the past few decades, the mean-variance model stimulates a great deal of extensions and applied researches under single-period setting. Due to the nonseparability of multi-period mean-variance models, only up to 2000, Li and Ng [2] develop the embedding technique and solve a multi-period mean-variance portfolio selection problem analytically. In their work, the returns of risky assets are assumed to be independent and identically distributed, and this assumption is also adopted by lots of the later literature, such as Guo and Hu [3].

A large number of empirical analyses of the assets price dynamics show that there exist salient serial correlations in the returns of financial assets, and the correlation structure is very complicated. The ARMA model is developed to study the feature of the financial assets returning with serial correlations in the field of econometrics, and it is widely used in the empirical research of financial market. Hakansson [4, 5] had already taken the impact of serial correlations into account on his portfolio selection problems and had investigated the myopic optimal portfolio strategies when there existed serial correlations of yields and not. However, due to the complexity of multi-period portfolio selection problem with serial correlations of returns, there are little relevant literature and results focused on the impact of serial correlations on the optimal portfolio selection strategy. Balvers and Mitchell [6] first derived an analytical solution for a dynamic portfolio selection problem with autocorrelation assets returns, where the utility function was a negative exponential function, and the assets returns were subject to the normal ARMA(1,1) process. Dokuchaev [7] analyzed a discrete-time portfolio selection model with serial correlations and found the correlation structure which ensured the optimal strategy being myopic for both the power and the log utility functions. Çelikyurt and Özekici [8] studied such models with the assumption that the market evolution followed a Markov chain and the states were observable, whose objective functions depended on the mean and the variance of the terminal wealth. Çanakoğlu and Özekici [9] considered the utility maximization problem with imperfect information modulated by a hidden Markov chain, and obtained the explicit characterization of the optimal strategy and the value function. Wei and Ye [10] extended the work of [8] to take the risk control over bankruptcy into consideration. Xu and Li [11] investigated a multi-period mean-variance portfolio selection problem with one risky asset whose returns were serially correlated. By using the embedding technique of Li and Ng [2] and the dynamic programming approach, they obtained the explicit optimal strategy and proposed a measure of the risky asset value. To our knowledge, up to now, quite a few papers consider serial correlations of returns under dynamic portfolio selection framework.

On the other hand, the literature mentioned above makes an important hypothesis, implicitly or explicitly, that an investor knows her/his final exit time exactly at the moment of entering the market and making investment decisions, that is, the investment horizon is deterministic, either finite or infinite. In practice, however, the investor’s exit time may be impacted by many exogenous and endogenous factors. An investor may exit from the market when she/he faces an unexpected need of huge consumption, sudden death, job loss, early retirement, investment target achieved, and so forth. Thus, it is more practical to weaken the restrictive assumption that the investment horizon is deterministic. If the exit time is uncertain, it is a random variable. As far as we know, study on the uncertain exit time can be dated back to Yarri [12], who studied an optimal consumption problem with an uncertain investment horizon. Hakansson [13] extended the work of [12] to a multi-period setting with a risky asset and an uncertain time horizon. Merton [14] addressed a dynamic optimal investment and consumption problem, and the uncertain retiring time was defined as the first jump of an independent Poisson process. Karatzas and Wang [15] considered an optimal investment problem in complete markets with the assumption that the exit time was a stopping time of the asset price filtration. Martellini and Uros̆ević [16] extended the original model of [1] to a static mean-variance model in which the exit time was dependent on asset returns. Guo and Hu [3] analyzed a multi-period mean-variance investment problem with uncertainty time of exiting. Huang et al. [17] dealt with the portfolio selection problem with uncertain time horizon by adopting the worst-case CVaR methodology. Blanchet-Scalliet et al. [18] extended the optimal investment problem of [14] to allow the investor’s time horizon to be stochastic and correlate to the returns of risky assets. Yi et al. [19] considered a multiperiod asset-liability management problem with an uncertain investment horizon under the mean-variance framework.

To the best of our knowledge, there is no work that considers the multiperiod mean-variance portfolio selection with an uncertain investment horizon and serially correlate returns at the same time. In the present paper, we try to tackle such problem. We assume that the distribution of the exit time is known, and the serial correlations of risky asset returns are settled the same as Hakansson [5] and Xu and Li [11]. We first embed our nonseparable problem, in the sense of dynamic programming, into a separable one by employing the embedding technique of Li and Ng [2]; then transform the separable problem with uncertain exit time into one with deterministic time horizon; finally solve the problem with deterministic time horizon by using the dynamic programming approach.

The rest of the paper is organized as follows. Section 2 formulates our problem and embeds it into a separable auxiliary problem. In Section 3, we solve the tractable auxiliary problem. In Section 4, we derive the optimal strategy and the efficient frontier of the original problem. Section 5 extends the results to the case of multiple risky assets. Section 6 gives a numerical simulation to show the impacts of exit time and serial correlations on the mean-variance efficient frontier. Finally, we conclude the paper in Section 7.

2. Modeling

We consider a financial market consisting of a risky asset and a riskless asset. The return rates of the riskless asset and risky asset at period 𝑡+1 (the time interval from time 𝑡 to time 𝑡+1) are denoted by 𝑟⁰_𝑡 and 𝑟_𝑡, respectively. It is assumed that 𝑟⁰_𝑡 is a constant and 𝑟_𝑡 is a (𝑡+1)-measurable random variable. The risky asset will not degenerate into the riskless asset at any period, and its return rates {𝑟_𝑡,𝑡=0,1,…} are correlated, that is, the value of 𝑟_𝑡 is dependent on the values of 𝑟_𝑠, 𝑠<𝑡, which are the realized returns of risky asset at the past periods. Thus, at time 𝑡, the expectation of a random variable, denoted by 𝐸_𝑡, is a conditional expectation based on all of the history information up to time 𝑡.

We assume that an investor, who joins the market at time 0 with the initial wealth 𝑥₀, may invest her/his wealth among the risky asset and the riskless asset within a time horizon of 𝑇 periods. At the beginning of each period 𝑡 (𝑡=1,…,𝑇), the investor may adjust the amount invested in the risky and riskless assets by transaction. However, she/he may be forced to leave the financial market at time 𝜏 before 𝑇 by some uncontrollable reasons. The uncertain exit time 𝜏 is supposed to be an exogenously random variable with the discrete probability distribution 𝑝_𝑡=Pr{𝜏=𝑡},𝑡=1,2,…. Therefore, the actual exit time of the investor is 𝑇∧𝜏∶=min{𝑇,𝜏}, and its probability distribution is𝑝_𝑡{}=𝑝∶=Pr𝑇∧𝜏=𝑡^𝑡,𝑡=1,…,𝑇−1,1−^𝑇−1_𝑡=1𝑝_𝑡,𝑡=𝑇. (2.1)

Let 𝑢_𝑡 be the amount invested in the risky asset at the beginning of period 𝑡+1. The investment series over 𝑇 periods, 𝑢∶={𝑢₀,𝑢₁,…,𝑢_𝑇−1}, is called an investment strategy. Define the excess return of risky asset at period 𝑡+1(𝑡=0,1,…,𝑇−1) as 𝑅_𝑡=𝑟_𝑡−𝑟⁰_𝑡, which is assumed to be nondegenerated before time 𝑡+1, that is, the risky asset will not degenerate into the riskless asset at period 𝑡+1. Let 𝑥_𝑡 be the wealth of the investor at time 𝑡(𝑡=0,1,…,𝑇). If the investment strategy 𝑢 is used in a self-financing way, the wealth dynamics can be described mathematically as𝑥_𝑡+1=𝑟⁰_𝑡𝑥_𝑡+𝑅_𝑡𝑢_𝑡,𝑡=0,1,…,𝑇−1. (2.2)

The multi-period mean-variance portfolio selection problem with uncertain exit time and serially correlate returns now can be formulated as𝑃(𝜔)max_𝑢𝐸₀𝑥_𝑇∧𝜏−𝜔Var₀𝑥_𝑇∧𝜏s.t.𝑥_𝑡+1=𝑟⁰_𝑡𝑥_𝑡+𝑅_𝑡𝑢_𝑡,𝑡=0,1,…,𝑇−1, (2.3) where 𝜔 is a given positive constant, representing the degree of the investor’s risk aversion, and Var₀ is the variance conditional on the information available at time 0. There are some other assumptions with respect to model 𝑃(𝜔), which are summarized as follows: (a) short selling is permitted at any periods for the risky asset; (b) transaction costs and fees are negligible; (c) the investor can borrow and lend the riskless asset at any periods without limitation.

Recall that the mean-variance model 𝑃(𝜔) is difficult to solve due to its nonseparable structure in the sense of dynamic programming, which is one of the most powerful and universal methodologies for optimization problems with separable nature. Fortunately, Li and Ng [2] propose an embedding technique, and this technique is also applicable to solve the current problem with uncertain exit time and serially correlate returns. Instead of solving problem 𝑃(𝜔) directly, we first consider the following auxiliary problem:𝐴(𝜆,𝜔)max_𝑢𝐸₀𝜆𝑥_𝑇∧𝜏−𝜔𝑥²_𝑇∧𝜏s.t.𝑥_𝑡+1=𝑟⁰_𝑡𝑥_𝑡+𝑅_𝑡𝑢_𝑡,𝑡=0,1,…,𝑇−1, (2.4) for a given constant 𝜆>0.

Let Ψ_𝐴(𝜆,𝜔) and Ψ_𝑃(𝜔) be the optimal solution sets of problem 𝐴(𝜆,𝜔) and 𝑃(𝜔), respectively, namely,Ψ_𝐴()=(,Ψ𝜆,𝜔𝑢∣𝑢isanoptimalsolutionof𝐴𝜆,𝜔)_𝑃(𝜔)=𝑢∣𝑢isanoptimalsolutionof𝑃(𝜔). (2.5)

The following two theorems can be proven by a similar method to that described in Li and Ng [2], and so their proofs are omitted.

Theorem 2.1. For any optimal solution 𝑢^∗ of Ψ_𝑃(𝜔), 𝑢^∗ is the optimal solution of Ψ_𝐴(𝜆^∗,𝜔) with 𝜆^∗=1+2𝜔𝐸₀(𝑥_𝑇∧𝜏)|_𝑢∗.

Theorem 2.2. If 𝑢^∗∈Ψ_𝐴(𝜆^∗,𝜔), a necessary condition for 𝑢^∗∈Ψ_𝑃(𝜔) is 𝜆^∗=1+2𝜔𝐸₀(𝑥_𝑇∧𝜏)|_𝑢∗.

3. Analytical Solution of Auxiliary Problem 𝐴(𝜆,𝜔)

In this section, we translate the auxiliary problem 𝐴(𝜆,𝜔) into a portfolio selection problem with certain exit time and then solve it by using the dynamic programming approach.

Since𝐸₀𝜆𝑥_𝑇∧𝜏−𝜔𝑥²_𝑇∧𝜏=^𝑇_𝑡=1𝐸₀𝜆𝑥_𝑇∧𝜏−𝜔𝑥²_𝑇∧𝜏∣𝑇∧𝜏=𝑡𝑃(𝑇∧𝜏=𝑡)=𝐸₀^𝑇_𝑡=1𝜆𝑥_𝑡−𝜔𝑥²_𝑡𝑝_𝑡, (3.1) problem 𝐴(𝜆,𝜔) can be written equivalently as𝐴(𝜆,𝜔)max_𝑢𝐸₀^𝑇_𝑡=1𝜆𝑥_𝑡−𝜔𝑥²_𝑡𝑝_𝑡s.t.𝑥_𝑡+1=𝑟⁰_𝑡𝑥_𝑡+𝑅_𝑡𝑢_𝑡,𝑡=0,1,…,𝑇−1. (3.2) Define the value function𝑓^∗_𝑡𝑥_𝑡=max_𝑢𝑡𝑓_𝑡𝑥_𝑡=max_𝑢𝑡𝐸_𝑡^𝑇_𝑠=𝑡𝜆𝑥_𝑠−𝜔𝑥²_𝑠𝑝_𝑠 (3.3) as the optimal expected utility using the optimal strategy conditional on the information available at time 𝑡(𝑡=0,1,…,𝑇−1), and the boundary condition is𝑓^∗_𝑇𝑥_𝑇=𝜆𝑥_𝑇−𝜔𝑥²_𝑇𝑝_𝑇. (3.4)

According to the dynamic programming principle, we have the Bellman equation𝑓^∗_𝑡𝑥_𝑡=max_𝑢𝑡𝑓_𝑡𝑥_𝑡=max_𝑢𝑡𝐸_𝑡𝜆𝑥_𝑡−𝜔𝑥²_𝑡𝑝_𝑡+𝑓^∗_𝑡+1𝑥_𝑡+1, (3.5) for 𝑡=0,1,…,𝑇−1.

First, we give the following notations:𝜃_𝑡=𝐸²_𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡Ξ,𝑡=0,1,…,𝑇−1,_𝑡=𝐸_𝑡𝜆²4𝜔^𝑇−1_𝑠=𝑡𝜃_𝑠𝜔,𝑡=0,1,…,𝑇−1,_𝑡=𝑝_𝑡+𝑟⁰_𝑡²𝐸_𝑡𝜔_𝑡+1−𝐸²_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡,𝑡=0,1,…,𝑇−1,𝜔_𝑇=𝑝_𝑇,𝜆_𝑡=𝑝_𝑡+𝑟⁰_𝑡𝐸_𝑡𝜆_𝑡+1−𝐸_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡,𝑡=0,1,…,𝑇−1,𝜆_𝑇=𝑝_𝑇. (3.6)(3.7)(3.8)(3.9) For notational simplicity, we define ∑^𝑡_𝑗=𝑠(⋅)_𝑗=0 and ∏^𝑡_𝑗=𝑠(⋅)_𝑗=1 if 𝑠>𝑡.

Note that 𝑅_𝑡 and 𝑅_𝑡+1 are not independent of each other for 𝑡=0,1,…,𝑇−1, so both 𝜔_𝑡+1 and 𝜆_𝑡+1 are dependent on the risky asset return at period 𝑡+1, 𝑅_𝑡. Then, for 𝑡=0,1,…,𝑇−1,𝐸_𝑡𝜔_𝑡+1𝑅_𝑡≠𝐸_𝑡𝜔_𝑡+1𝐸_𝑡𝑅_𝑡,𝐸_𝑡𝜆_𝑡+1𝑅_𝑡≠𝐸_𝑡𝜆_𝑡+1𝐸_𝑡𝑅_𝑡. (3.10)

The following lemma comes from Xu and Li [11]. For the completeness, we provide its proof here.

Lemma 3.1. Let 𝑥 be a nondegenerated random variable, and let 𝜉 be a positive random variable under the information at time 𝑡, then 𝐸_𝑡(𝑥²𝜉)𝐸_𝑡(𝜉)>(𝐸_𝑡(𝑥𝜉))².

Proof. Since 𝜉 is a positive random variable, we can define a new probability measure 𝑄 as 𝜉d𝑄≜𝐸_𝑡(𝜉)d𝑃, (3.11) where 𝑃 is the original measure. Since 𝑥 is a nondegenerated random variable, we have, under measure 𝑄, Var^𝑄_𝑡(𝑥)=𝐸^𝑄_𝑡𝑥²−𝐸^𝑄_𝑡(𝑥)²>0. (3.12) Transforming the above inequality to under measure 𝑃, we obtain 𝐸_𝑡𝑥²𝜉𝐸_𝑡(𝜉)−𝐸_𝑡𝑥𝜉𝐸_𝑡(𝜉)²>0. (3.13) Multiplying both sides by (𝐸_𝑡(𝜉))² in the above inequality produces 𝐸_𝑡𝑥²𝜉𝐸_𝑡(𝜉)>𝐸_𝑡(𝑥𝜉)². (3.14) This completes the proof.

Theorem 3.2. For 𝑡=0,1,…,𝑇−1,𝜔_𝑡>0,𝜃_𝑡≥0, and Ξ_𝑡≥0.

Proof. We use induction. For 𝑡=𝑇−1, since the return of the risky asset at period 𝑇, 𝑅_𝑇−1, is a nondegenerated random variable, then Var_𝑇−1𝑅_𝑇−1=𝐸_𝑇−1𝑅²_𝑇−1−𝐸_𝑇−1𝑅_𝑇−1²>0, (3.15)𝐸_𝑇−1(𝑅²_𝑇−1)>0. So 0≤𝜃_𝑇−1=𝐸²_𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1=𝑝_𝑇𝐸²_𝑇−1𝑅_𝑇−1𝐸_𝑇−1𝑅²_𝑇−1<𝑝_𝑇,Ξ_𝑇−1=𝐸_𝑇−1𝜆²𝜃4𝜔^𝑇−1≥0. (3.16) Therefore, 𝜔_𝑇−1=𝑝_𝑇−1+𝑟⁰_𝑇−1²𝑝_𝑇−𝐸²_𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1>0. (3.17)
Suppose that 𝜔_𝑠>0,𝜃_𝑠≥0,and Ξ_𝑠≥0 hold true for 𝑠=𝑡+1,…,𝑇−2,𝑇−1, then for period 𝑡, 𝜃_𝑡=𝐸²_𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡≥0. (3.18) By Lemma 3.1, we can easily see that 𝐸_𝑡𝜔_𝑡+1>𝐸²_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡. (3.19) Hence, we obtain Ξ_𝑡=𝐸∑_𝑡^𝑇−1_𝑠=𝑡𝜆²𝜃4𝜔^𝑠=𝜆²𝜃4𝜔^𝑡+𝐸_𝑡Ξ_𝑡+1𝜔≥0,_𝑡=𝑝_𝑡+𝑟⁰_𝑡²𝐸_𝑡𝜔_𝑡+1−𝐸²_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡>0. (3.20) By induction, it shows that for 𝑡=0,1,…,𝑇−1,𝜔_𝑡>0,𝜃_𝑡≥0,and Ξ_𝑡≥0.

The analytical optimal strategy and the value function of problem 𝐴(𝜆,𝜔) can be derived by using dynamic programming approach, which are summarized in the following theorem.

Theorem 3.3. The optimal strategy and the value functions of problem 𝐴(𝜆,𝜔) are, respectively, given by 𝑢^∗_𝑡=𝜆𝐸2𝜔^𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡−𝐸_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡𝑟⁰_𝑡𝑥_𝑡𝑓,𝑡=0,1,…,𝑇−1,_∗𝑡𝑥_𝑡=−𝜔𝜔_𝑡𝑥²_𝑡+𝜆𝜆_𝑡𝑥_𝑡+Ξ_𝑡,𝑡=0,1,…,𝑇−1, (3.21)(3.22) where Ξ_𝑡,𝜔_𝑡, and 𝜆_𝑡 are given as defined in (3.7)–(3.9).

Proof. We will show that the above recursive formulas hold true by induction starting with the boundary condition 𝑓_𝑇(𝑥_𝑇)=(𝜆𝑥_𝑇−𝜔𝑥²_𝑇)𝑝_𝑇. Note that for 𝑡=𝑇−1, 𝑓^∗_𝑇−1𝑥_𝑇−1=max_𝑢𝑇−1𝑓_𝑇−1𝑥_𝑇−1=max_𝑢𝑇−1𝐸_𝑇−1𝜆𝑥_𝑇−1−𝜔𝑥²_𝑇−1𝑝_𝑇−1+𝑓^∗_𝑇𝑥_𝑇=max_𝑢𝑇−1𝐸_𝑇−1𝜆𝑥_𝑇−1−𝜔𝑥²_𝑇−1𝑝_𝑇−1+𝜆𝑥_𝑇−𝜔𝑥²_𝑇𝑝_𝑇=max_𝑢𝑇−1𝜆𝑥_𝑇−1−𝜔𝑥²_𝑇−1𝑝_𝑇−1+𝜆𝑟⁰_𝑇−1𝑝_𝑇𝑥_𝑇−1+𝐸_𝑇−1𝑝_𝑇𝑅_𝑇−1𝑢_𝑇−1−𝜔𝑟⁰_𝑇−1²𝑝_𝑇𝑥²_𝑇−1+2𝑟⁰_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅_𝑇−1𝑢_𝑇−1𝑥_𝑇−1+𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1𝑢²_𝑇−1. (3.23) Since 𝐸_𝑇−1(𝑅²_𝑇−1)>0 by assumption, the function 𝑓_𝑇−1(𝑥_𝑇−1) is a concave function of 𝑢_𝑇−1. The first-order condition gives 𝜆𝐸_𝑇−1𝑝_𝑇𝑅_𝑇−1−2𝜔𝑟⁰_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅_𝑇−1𝑥_𝑇−1+𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1𝑢_𝑇−1=0, (3.24) which yields the optimal solution 𝑢^∗_𝑇−1 as 𝑢^∗_𝑇−1=𝜆𝐸2𝜔^𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1−𝐸_𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1𝑟⁰_𝑇−1𝑥_𝑇−1. (3.25) Substituting 𝑢^∗_𝑇−1 back into 𝑓_𝑇−1(𝑥_𝑇−1), it follows that 𝑓^∗_𝑇−1𝑥_𝑇−1=𝜆𝑥_𝑇−1−𝜔𝑥²_𝑇−1𝑝_𝑇−1+𝜆𝑟⁰_𝑇−1𝑝_𝑇𝑥_𝑇−1+𝐸_𝑇−1𝑝_𝑇𝑅_𝑇−1𝜆𝐸2𝜔^𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1−𝐸_𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1𝑟⁰_𝑇−1𝑥_𝑇−1−𝜔𝑟⁰_𝑇−1²𝑝_𝑇𝑥²_𝑇−1+2𝑟⁰_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅_𝑇−1×𝜆𝐸2𝜔^𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1−𝐸_𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1𝑟⁰_𝑇−1𝑥_𝑇−1𝑥_𝑇−1+𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1𝜆𝐸2𝜔^𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1−𝐸_𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1𝑟⁰_𝑇−1𝑥_𝑇−1²=−𝜔𝑝_𝑇−1+𝑟⁰_𝑇−1²𝑝_𝑇−𝐸²_𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1𝑥²_𝑇−1+𝜆𝑝_𝑇−1+𝑟⁰_𝑇−1𝑝_𝑇−𝐸²_𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1𝑥_𝑇−1+𝜆²𝐸4𝜔^2𝑇−1𝑝_𝑇𝑅_𝑇−1𝐸_𝑇−1𝑝_𝑇𝑅²_𝑇−1=−𝜔𝜔_𝑇−1𝑥²_𝑇−1+𝜆𝜆_𝑇−1𝑥_𝑇−1+𝜆²𝜃4𝜔^𝑇−1=−𝜔𝜔_𝑇−1𝑥²_𝑇−1+𝜆𝜆_𝑇−1𝑥_𝑇−1+Ξ_𝑇−1. (3.26) Hence, the conclusion holds true for 𝑡=𝑇−1.
Now we assume that the conclusion holds true for time 𝑡+1, in other words, 𝑓^∗_𝑡+1𝑥_𝑡+1=−𝜔𝜔_𝑡+1𝑥²_𝑡+1+𝜆𝜆_𝑡+1𝑥_𝑡+1+Ξ_𝑡+1,𝑢^∗_𝑡+1=𝜆𝐸2𝜔^𝑡+1𝜆_𝑡+2𝑅_𝑡+1𝐸_𝑡+1𝜔_𝑡+2𝑅²_𝑡+1−𝐸_𝑡+1𝜔_𝑡+2𝑅_𝑡+1𝐸_𝑡+1𝜔_𝑡+2𝑅²_𝑡+1𝑟⁰_𝑡+1𝑥_𝑡+1, (3.27) then the optimization problem at time 𝑡 for given state 𝑥_𝑡 is 𝑓^∗_𝑡𝑥_𝑡=max_𝑢𝑡𝑓_𝑡𝑥_𝑡=max_𝑢𝑡𝐸_𝑡𝜆𝑥_𝑡−𝜔𝑥²_𝑡𝑝_𝑡+𝑓^∗_𝑡+1𝑥_𝑡+1=max_𝑢𝑡𝐸_𝑡𝜆𝑥_𝑡−𝜔𝑥²_𝑡𝑝_𝑡−𝜔𝜔_𝑡+1𝑥²_𝑡+1+𝜆𝜆_𝑡+1𝑥_𝑡+1+Ξ_𝑡+1=max_𝑢𝑡𝜆𝑥_𝑡−𝜔𝑥²_𝑡𝑝_𝑡−𝜔𝑟⁰_𝑡²𝑥²_𝑡𝐸_𝑡𝜔_𝑡+1+2𝑟⁰_𝑡𝐸_𝑡𝜔_𝑡+1𝑅_𝑡𝑢_𝑡𝑥_𝑡+𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡𝑢²_𝑡+𝜆𝑟⁰_𝑡𝑥_𝑡𝐸_𝑡𝜆_𝑡+1+𝐸_𝑡𝜆_𝑡+1𝑅_𝑡𝑢_𝑡+𝐸_𝑡Ξ_𝑡+1 (3.28) Noting that 𝐸_𝑡(𝜔_𝑡+1𝑅²_𝑡)>0 by Theorem 3.2, the function 𝑓_𝑡(𝑥_𝑡) is also a concave function of 𝑢_𝑡. The first-order condition yields 𝑢^∗_𝑡=𝜆𝐸2𝜔^𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡−𝐸_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡𝑟⁰_𝑡𝑥_𝑡. (3.29) Therefore, for the above given 𝑢^∗_𝑡, 𝑓^∗_𝑡𝑥_𝑡=𝜆𝑥_𝑡−𝜔𝑥²_𝑡𝑝_𝑡+𝜆𝑟⁰_𝑡𝑥_𝑡𝐸_𝑡𝜆_𝑡+1+𝐸_𝑡𝜆_𝑡+1𝑅_𝑡𝜆𝐸2𝜔^𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡−𝐸_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡𝑟⁰_𝑡𝑥_t−𝜔𝑟⁰_𝑡²𝑥²_𝑡𝐸_𝑡𝜔_𝑡+1+2𝑟⁰_𝑡𝐸_𝑡𝜔_𝑡+1𝑅_𝑡𝜆𝐸2𝜔^𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡−𝐸_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡𝑟⁰_𝑡𝑥_𝑡𝑥_𝑡+𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡𝜆𝐸2𝜔^𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡−𝐸_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡𝑟⁰_𝑡𝑥_𝑡²+𝐸_𝑡Ξ_𝑡+1=−𝜔𝑝_𝑡+𝑟⁰_𝑡²𝐸_𝑡𝜔_𝑡+1−𝐸²_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡𝑥²_𝑡+𝜆𝑝_𝑡+𝑟⁰_𝑡𝐸_𝑡𝜆_𝑡+1−𝐸_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡𝑥_𝑡+𝜆²𝜃4𝜔^𝑡+𝐸_𝑡Ξ_𝑡+1=−𝜔𝜔_𝑡𝑥²_𝑡+𝜆𝜆_𝑡𝑥_𝑡+Ξ_𝑡. (3.30) Hence, the conclusion is true for 𝑡. By induction, the theorem is true.

4. Optimal Strategy and the Efficient Frontier of the Original Problem 𝑃(𝜔)

If we insert the optimal strategy given in Theorem 3.3 into the dynamic process of wealth, 𝑥_𝑇 and 𝑥²_𝑇 can be expressed as𝑥_𝑇=𝑟⁰_𝑇−1𝑥_𝑇−1+𝑅_𝑇−1𝑢^∗_𝑇−1=𝑅1−^𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅²_𝑇−1𝑟⁰_𝑇−1𝑥_𝑇−1+𝜆𝑅2𝜔^𝑇−1𝐸_𝑇−1𝜆_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅²_𝑇−1,𝑥²_𝑇=1−2𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅²_𝑇−1+𝑅²_𝑇−1𝐸²_𝑇−1𝜔_𝑇𝑅_𝑇−1𝐸²_𝑇−1𝜔_𝑇𝑅²_𝑇−1𝑟⁰_𝑇−1²𝑥²_𝑇−1+𝜆𝜔𝑅_𝑇−1𝐸_𝑇−1𝜆_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅²_𝑇−1−𝑅²_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜆_𝑇𝑅_𝑇−1𝐸²_𝑇−1𝜔_𝑇𝑅²_𝑇−1𝑟⁰_𝑇−1𝑥_𝑇−1+𝜆²4𝜔²𝑅²_𝑇−1𝐸²_𝑇−1𝜆_𝑇𝑅_𝑇−1𝐸²_𝑇−1𝜔_𝑇𝑅²_𝑇−1, (4.1)𝜆_𝑇𝑥_𝑇=𝜆_𝑇−𝜆_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅²_𝑇−1𝑟⁰_𝑇−1𝑥_𝑇−1+𝜆𝜆2𝜔^𝑇𝑅_𝑇−1𝐸_𝑇−1𝜆_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅²_𝑇−1,𝜔_𝑇𝑥²_𝑇=𝜔_𝑇−2𝜔_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅²_𝑇−1+𝜔_𝑇𝑅²_𝑇−1𝐸²_𝑇−1𝜔_𝑇𝑅_𝑇−1𝐸²_𝑇−1𝜔_𝑇𝑅²_𝑇−1𝑟⁰_𝑇−1²𝑥²_𝑇−1+𝜆𝜔𝜔_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜆_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅²_𝑇−1−𝜔_𝑇𝑅²_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜆_𝑇𝑅_𝑇−1𝐸²_𝑇−1𝜔_𝑇𝑅²_𝑇−1𝑟⁰_𝑇−1𝑥_𝑇−1+𝜆²4𝜔²𝜔_𝑇𝑅²_𝑇−1𝐸²_𝑇−1𝜆_𝑇𝑅_𝑇−1𝐸²_𝑇−1𝜔_𝑇𝑅²_𝑇−1. (4.2) Taking expectations on both sides of (4.2) based on the information available at time 𝑇−1, we conclude that𝐸_𝑇−1𝜆_𝑇𝑥_𝑇=𝜆_𝑇−1𝑥_𝑇−1−𝑝_𝑇−1𝑥_𝑇−1+𝜆𝜃2𝜔^𝑇−1,𝐸_𝑇−1𝜔_𝑇𝑥²_𝑇=𝜔_𝑇−1𝑥²_𝑇−1−𝑝_𝑇−1𝑥²_𝑇−1+𝜆²4𝜔²𝜃_𝑇−1, (4.3)(4.4) where𝜃_𝑇−1=𝐸²_𝑇−1𝜆_𝑇𝑅_𝑇−1𝐸_𝑇−1𝜔_𝑇𝑅²_𝑇−1. (4.5)

The above equations are recursive equations, and by taking expectations on both sides of (4.3) and (4.4) at time 𝑇−2,…,1,0 repeatedly, we obtain𝐸₀𝑥_𝑇∧𝜏=^𝑇_𝑡=1𝑝_𝑡𝐸₀𝑥_𝑡=𝜆₀𝑥₀+𝜆2𝜔^𝑇_𝑡=1𝐸₀𝜃_𝑡−1=𝜆₀𝑥₀+𝜆𝐸2𝜔Θ,₀𝑥²_𝑇∧𝜏=^𝑇_𝑡=1𝑝_𝑡𝐸₀𝑥²_𝑡=𝜔₀𝑥²₀+𝜆²4𝜔^2𝑇_𝑡=1𝐸₀𝜃_𝑡−1=𝜔₀𝑥²₀+𝜆²4𝜔²Θ, (4.6)(4.7) where𝜃_𝑡=𝐸²_𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡,Θ=^𝑇_𝑡=1𝐸₀𝜃_𝑡−1. (4.8)

With the results (4.6) and (4.7), the variance of the terminal wealth 𝑥_𝑇∧𝜏 under the optimal strategy (3.21) can be written asVar₀𝑥_𝑇∧𝜏=𝐸₀𝑥²_𝑇∧𝜏−𝐸₀𝑥_𝑇∧𝜏²=𝐸₀^𝑇_𝑡=1𝑝_𝑡𝑥²_𝑡−𝐸₀^𝑇_𝑡=1𝑝_𝑡𝑥_𝑡²=𝜔₀𝑥²₀+𝜆²4𝜔²Θ−𝜆₀𝑥₀+𝜆Θ2𝜔²=𝜆²4𝜔²Θ−Θ²−𝜆𝜔𝜆₀Θ𝑥₀+𝜔₀−𝜆²₀𝑥²₀. (4.9)

Lemma 4.1. 0<Θ<1,𝜔₀−𝜆²₀/(1−Θ)>0.

Proof. First of all, we claim that Var₀(𝑥_𝑇∧𝜏)>0, since it measures the risk of investor at the time of exiting market, and the risky asset cannot degenerate into the riskless asset. Especially, when 𝑥₀=0, Var₀(𝑥_𝑇∧𝜏) can be reduced to Var₀𝑥_𝑇∧𝜏=𝜆²4𝜔²Θ−Θ²>0, (4.10) and it is easy to show that 0<Θ<1.
The expression of Var₀(𝑥_𝑇∧𝜏) can be further converted into Var₀𝑥_𝑇∧𝜏=Θ−Θ²𝜆−𝜆2𝜔⁰𝑥1−Θ⁰²+𝜔₀−𝜆²₀𝑥1−Θ²⁰>0. (4.11) Since we know that 0<Θ<1, the above inequality implies 𝜔₀−𝜆²₀/(1−Θ)>0, and we finish the proof of Lemma 4.1.

According to Theorem 2.2, a necessary condition for the optimal solution of auxiliary problem 𝐴(𝜆^∗,𝜔) to attain the optimality of problem 𝑃(𝜔) at the same time is𝜆^∗=1+2𝜔𝐸₀𝑥_𝑇∧𝜏∣_𝑢∗=1+2𝜔𝜆₀𝑥₀+𝜆^∗Θ.2𝜔 (4.12) We can easily obtain𝜆^∗=1+2𝜔𝜆₀𝑥₀.1−Θ (4.13) Finally, substituting (4.13) back into (3.21) yields the analytically optimal strategy of the original problem 𝑃(𝜔), which is summarized in the following theorem.

Theorem 4.2. For the mean-variance problem 𝑃(𝜔), the optimal strategy is given by 𝑢^∗_𝑡=1+2𝜔𝜆₀𝑥₀()𝐸2𝜔1−Θ^𝑡𝜆_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡−𝐸_𝑡𝜔_𝑡+1𝑅_𝑡𝐸_𝑡𝜔_𝑡+1𝑅²_𝑡𝑟⁰_𝑡𝑥_𝑡,𝑡=0,1,…,𝑇−1, (4.14) where Θ,𝜆₀,𝜔_𝑡+1,and 𝜆_𝑡+1 are given as defined.

Referring to (4.6),𝜆^∗=𝐸2𝜔⁰𝑥_𝑇∧𝜏−𝜆₀𝑥₀Θ. (4.15) Substituting (4.15) back into (4.11), the relationship between Var₀(𝑥_𝑇∧𝜏) and 𝐸₀(𝑥_𝑇∧𝜏) can be shown as follows:Var₀𝑥_𝑇∧𝜏=(1−Θ)Θ𝐸₀𝑥_𝑇∧𝜏−𝜆₀𝑥₀1−Θ²+𝜔₀−𝜆²₀𝑥1−Θ²⁰. (4.16) Therefore, the efficient frontier of the original problem 𝑃(𝜔) is given by (4.16) for𝐸₀𝑥_𝑇∧𝜏∈𝜆₀𝑥₀.1−Θ,+∞ (4.17)

From the efficient frontier (4.16) of the optimal dynamic mean-variance portfolio selection problem with an uncertain exit time, when returns are serially correlated, we can obtain the trade-off between the return and the risk when investor exits from market. Since all of the parameters Θ, 𝜆₀, and 𝜔₀ are functions of 𝑝_𝑡 and 𝑅_𝑡 for 𝑡=0,1,…,𝑇−1, both the exiting time and the correlations of the risky asset returns have impacts on the optimal strategy and the efficient frontier, and this is quite different from the cases with deterministic terminal time, and the risky asset returns at different periods are independent.

Remark 4.3. In Xu and Li [11], a multi-period portfolio selection problem with serial correlation and a certain exit time is studied. If 𝑝₁=𝑝₂=⋯=𝑝_𝑇−1=0, 𝑝_𝑇=1, and 𝑟⁰_𝑡=𝑟,𝑡=0,1,…,𝑇−1 in our model, our result is exactly the same as the one of Xu and Li [11]. So we generalize the model and results of Xu and Li [11] to the case with an uncertain investment horizon.

5. Extension to the Situation with Multiple Risky Assets

The results in the previous sections can be extended to the general situation with multiple risky assets. Suppose that there are 𝑛 risky assets and one riskless asset with period-𝑡+1 returns 𝑟^𝑖_𝑡(𝑖=1,2,…,𝑛) and 𝑟⁰_𝑡, respectively. Define 𝑒^𝑖_𝑡=𝑟^𝑖_𝑡−𝑟⁰_𝑡, 𝑒_𝑡=(𝑒¹_𝑡,𝑒²_𝑡,…,𝑒^𝑛_𝑡)^′ and 𝑈_𝑡=(𝑢¹_𝑡,𝑢²_𝑡,…,𝑢^𝑛_𝑡)^′ for 𝑖=1,2,…,𝑛 and 𝑡=0,1,…,𝑇−1, where 𝑢^𝑖_𝑡 is the amount invested in the 𝑖th risky asset at time 𝑡. In this case, the wealth dynamics is described by𝑥_𝑡+1=𝑟⁰_𝑡𝑥_𝑡+𝑒^′_𝑡𝑈_𝑡,𝑡=0,1,…,𝑇−1. (5.1)

Accordingly, the multi-period mean-variance portfolio selection problem with an uncertain exit time and serial correlations can be formulated as𝑃(𝜔)max_𝑈𝐸₀𝑥_𝑇∧𝜏−𝜔Var₀𝑥_𝑇∧𝜏s.t.𝑥_𝑡+1=𝑟⁰_𝑡𝑥_𝑡+𝑒^′_𝑡𝑈_𝑡,𝑡=0,1,…,𝑇−1, (5.2) where 𝜔≥0 is a pregiven parameter, representing the degree of the investor’s risk aversion.

With the same method as in the previous section, we can show the following theorem.

Theorem 5.1. For problem 𝑃(𝜔), the optimal investment strategy is given by 𝑈^∗_𝑡=1+2𝜔𝜆₀𝑥₀Θ𝐸2𝜔1−^𝑡−1𝜔_𝑡+1𝑒_𝑡𝑒^′_𝑡𝐸_𝑡𝜆_𝑡+1𝑒_𝑡−𝐸_𝑡⁻¹𝜔_𝑡+1𝑒_𝑡𝑒^′_t𝐸_𝑡𝜔_𝑡+1𝑒_𝑡𝑟⁰_𝑡𝑥_𝑡, (5.3) for 𝑡=0,1,…,𝑇−1, and the efficient frontier is given by Var₀𝑥_𝑇∧𝜏=1−ΘΘ𝐸₀𝑥_𝑇∧𝜏−𝜆₀𝑥₀1−Θ²+𝜔₀−𝜆²₀Θ𝑥1−²⁰, (5.4) where 𝜃_𝑡=𝐸_𝑡𝜆_𝑡+1𝑒^′_𝑡𝐸_𝑡⁻¹𝜔_𝑡+1𝑒_𝑡𝑒^′_𝑡𝐸_𝑡𝜆_𝑡+1𝑒_𝑡,Θ=^𝑇_𝑡=1𝐸₀𝜃_𝑡−1,𝜔_𝑡=𝑝_𝑡+𝑟⁰_𝑡²𝐸_𝑡𝜔_𝑡+1−𝐸_𝑡𝜔_𝑡+1𝑒^′_𝑡𝐸_𝑡⁻¹𝜔_𝑡+1𝑒_𝑡𝑒^′_𝑡𝐸_𝑡𝜔_𝑡+1𝑒_𝑡,𝜔_𝑇=𝑝_𝑇,𝜆_𝑡=𝑝_𝑡+𝑟⁰_𝑡𝐸_𝑡𝜆_𝑡+1−𝐸_𝑡𝜆_𝑡+1𝑒^′_𝑡𝐸_𝑡⁻¹𝜔_𝑡+1𝑒_𝑡𝑒^′_𝑡𝐸_𝑡𝜔_𝑡+1𝑒_𝑡,𝜆_𝑇=𝑝_𝑇, (5.5) for 𝑡=0,1,…,𝑇−1.

Remark 5.2. When the returns rates of the 𝑛 risky assets are statistically independent, our results reduce to the results of Guo and Hu [3]. That is, we extend the model and results of Guo and Hu [3] to the case with serially correlate returns.

6. Numerical Example

In the previous sections, we derive the optimal strategies and the mean-variance efficient frontiers of two optimal portfolio selection problems with serial correlations and uncertain exit time. In this section, we provide a numerical example to demonstrate the impacts of the uncertainty of exit time and the serial correlations of returns on the efficient frontier. In the example, we only consider one risky asset for simplicity and assume that its return rate is subject to AR(1) progress𝑟_𝑡=𝜇+𝜌𝑟_𝑡−1−𝜇+1−𝜌²𝜎𝑦_𝑡, (6.1) where 𝜇 is the unconditional expectation of 𝑟_𝑡, 𝜌∈(−1,1) is the first-order autocorrelation coefficient, 𝜎 is the unconditional standard deviation of 𝑟_𝑡, 𝑦_𝑡 is a random variable with standard normal distribution, and 𝑦_𝑡 is independent of 𝑦_𝑠(𝑠<𝑡).

To examine the impact of the uncertainty of exit time on the efficient frontier clearly, we compare the efficient frontiers under three different probability distributions of uncertain exit time 𝑡=𝑇∧𝜏: 𝑃₁=(𝑝₀,𝑝₁,𝑝₂,𝑝₃)=(0,0.09,0.2,0.71), 𝑃₂=(𝑝₀,𝑝₁,𝑝₂,𝑝₃)=(0,0,0.5,0.5), and 𝑃₃=(𝑝₀,𝑝₁,𝑝₂,𝑝₃)=(0,0.2,0.1,0.7). The remaining parameters are set as 𝜇=0.03, 𝑏=0.5, 𝜎=0.02, 𝑥₀=1, and 𝑇=3. Figure 1 implies when the investor exits from market later, the investor enjoys more expected wealth returns at the same level of risk than the one terminates the investment earlier.

Figure 1 

Efficient frontiers with different probability distributions of exit time.

Furthermore, to test the impact of the correlation coefficient on the efficient frontier, we compare the efficient frontiers under three different settings of correlation coefficient 𝜌: 𝜌₁=0.1,𝜌₂=0.4,and𝜌₃=0.7. The other parameters are given as 𝜇=0.02, 𝜎=0.02, 𝑥₀=1, and 𝑇=3. It is obvious from Figure 2 that the investor who takes the risky asset with larger correlation coefficient will suffer less risk than the one with less correlate risky asset to achieve the same expected wealth return.

Figure 2 

Efficient frontiers with different values of correlation coefficient.

7. Conclusion

In this paper, we consider an optimal portfolio selection problem under multi-period setting and mean-variance framework for an investor, who does not know with certainty when she/he will exit the market in which the capital returns are serially correlated. The problem is much more complicated than the case with deterministic exit time and/or with serially noncorrelate assets returns. By applying the dynamic programming approach and the embedding technique of Li and Ng [2], both the optimal strategy and the efficient frontier of the problem are derived explicitly. Our results include, as special cases, the ones of Li and Ng [2], Guo and Hu [3], and Xu and Li [11]. In addition, a numerical example with AR (1) return process is presented. It shows that both the serial correlations of assets returns and the uncertainty of exit time have significant impacts on the optimal strategy and the efficient frontier.

Acknowledgments

The authors are grateful to the reviewers for their constructive comments and suggestions. This research is supported by the National Science Foundation for Distinguished Young Scholars under Grant no. 70825002, the China Postdoctoral Science Foundation Funded Project under Grant no. 2011M501351, and the Philosophy and Social Science Programming Foundation of Guangdong Province under Grant no. GD11YYJ07.