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

We study amulti-periodmean-variance portfolio selection problemwith an uncertain time horizon and serial correlations. Firstly, we embed the nonseparablemulti-period optimization problem into a separable quadratic optimization problemwith 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.


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 meanvariance portfolio selection problem analytically.In their work, the returns of risky assets are Mathematical Problems in Engineering 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 discretetime 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.C ¸elikyurt 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.C ¸anako glu 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 multiperiod 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 Urosević 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 meanvariance framework.
To the best of our knowledge, there is no work that considers the multiperiod meanvariance 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 meanvariance efficient frontier.Finally, we conclude the paper in Section 7.

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 r 0 t and r t , respectively.It is assumed that r 0 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 x 0 , 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 is

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 : {u 0 , u 1 , . . ., u T −1 }, is called an investment strategy.

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Define the excess return of risky asset at period t 1 t 0, 1, . . ., T − 1 as R t r t − r 0 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 as The multi-period mean-variance portfolio selection problem with uncertain exit time and serially correlate returns now can be formulated as where ω is a given positive constant, representing the degree of the investor's risk aversion, and Var 0 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: x t 1 r 0 t x t R t u 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,

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

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. Since

3.1
problem A λ, ω can be written equivalently as

3.2
Define the value function 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 is According to the dynamic programming principle, we have the Bellman equation First, we give the following notations: 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,

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 Proof.Since ξ is a positive random variable, we can define a new probability measure Q as dQ ξ E t ξ dP, 3.11 where P is the original measure.Since x is a nondegenerated random variable, we have, under measure Q, Transforming the above inequality to under measure P , we obtain Multiplying both sides by E t ξ 2 in the above inequality produces

3.14
This completes the proof.
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
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.

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Theorem 3.3.The optimal strategy and the value functions of problem A λ, ω are, respectively, given by 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 − ωx 2 T p T .Note that for t T − 1,

3.23
Since E T −1 R 2 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 which yields the optimal solution u * T −1 as

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, Noting that E t ω t 1 R 2 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

3.29
Therefore, for the above given u * t ,

3.30
Hence, the conclusion is true for t.By induction, the theorem is true.

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 x 2 T can be expressed as

4.2
Taking expectations on both sides of 4.2 based on the information available at time T − 1, we conclude that where 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 obtain where 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 as Proof.First of all, we claim that Var 0 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 x 0 0, Var 0 x T ∧τ can be reduced to and it is easy to show that 0 < Θ < 1.
The expression of Var 0 x T ∧τ can be further converted into Since we know that 0 < Θ < 1, the above inequality implies ω 0 − λ 2 0 / 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 is We can easily obtain 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 where Θ, λ 0 , ω t 1 , and λ t 1 are given as defined.
Referring to 4.6 , Substituting 4.15 back into 4.11 , the relationship between Var 0 x T ∧τ and E 0 x T ∧τ can be shown as follows: Therefore, the efficient frontier of the original problem P ω is given by 4.16 for 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, λ 0 , and ω 0 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.

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 Accordingly, the multi-period mean-variance portfolio selection problem with an uncertain exit time and serial correlations can be formulated as x t 1 r 0 t x t e t U 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  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.

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 progress 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: P 1 p 0 , p 1 , p 2 , p 3 0, 0.09, 0.2, 0.71 , P 2 p 0 , p 1 , p 2 , p 3 0, 0, 0.5, 0.5 , and P 3 p 0 , p 1 , p 2 , p 3 0, 0.2, 0.1, 0.7 .The remaining parameters are set as μ 0.03, b 0.5, σ 0.02, x 0 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.
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: ρ 1 0.1, ρ 2 0.4, and ρ 3 0.7.The other parameters are given as μ 0.02, σ 0.02, x 0 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.

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.

Remark 4 . 3 .
In Xu andLi 11  , a multi-period portfolio selection problem with serial correlation and a certain exit time is studied.If p 1 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.

Figure 1 :
Figure 1: Efficient frontiers with different probability distributions of exit time.

3 Figure 2 :
Figure 2: Efficient frontiers with different values of correlation coefficient.
. . ., 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 by 1 e t e t E t λ t 1 e t − E −1 t ω t 1 e t e t E t ω t 1 e t r 0 t x t , 5.3 for t 0, 1, . . ., T − 1, and the efficient frontier is given by Var