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

Authors: Ling Zhang; Zhongfei Li
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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. Section2 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 periodt+1 (the time interval from time t to time t+1) are denoted by rt0 and rt, respectively. It is assumed that rt0 is a constant and rt is a (t+1)-measurable random variable. The risky asset will not degenerate into the riskless asset at any period, and its return rates {rt,t=0,1,…} are correlated, that is, the value of rt is dependent on the values of rs, 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 Et, 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 wealthx0, 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 p̃t=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)pt:=Pr{T∧τ=t}={p̃t,t=1,…,T-1,1-∑t=1T-1p̃t,t=T.Letut be the amount invested in the risky asset at the beginning of period t+1. The investment series over T periods, u∶={u0,u1,…,uT-1}, is called an investment strategy. Define the excess return of risky asset at period t+1(t=0,1,…,T-1) as Rt=rt-rt0, 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 xt 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(2.2)xt+1=rt0xt+Rtut,t=0,1,…,T-1.The multi-period mean-variance portfolio selection problem with uncertain exit time and serially correlate returns now can be formulated as(2.3)P(ω){max⁡uE0(xT∧τ)-ωVar0(xT∧τ)s.t.xt+1=rt0xt+Rtut,t=0,1,…,T-1,
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 modelP(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:(2.4)A(λ,ω){max⁡uE0(λxT∧τ-ωxT∧τ2)s.t.xt+1=rt0xt+Rtut,t=0,1,…,T-1,
for a given constant λ>0.LetΨA(λ,ω) and ΨP(ω) be the optimal solution sets of problem A(λ,ω) and P(ω), respectively, namely,(2.5)ΨA(λ,ω)={u∣uisanoptimalsolutionofA(λ,ω)},ΨP(ω)={u∣uisanoptimalsolutionofP(ω)}.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 solutionu* of ΨP(ω), u* is the optimal solution of ΨA(λ*,ω) with λ*=1+2ωE0(xT∧τ)|u*.Theorem 2.2.
Ifu*∈ΨA(λ*,ω), a necessary condition for u*∈ΨP(ω) is λ*=1+2ωE0(xT∧τ)|u*.


## 3. Analytical Solution of Auxiliary ProblemA(λ,ω)

In this section, we translate the auxiliary problemA(λ,ω) into a portfolio selection problem with certain exit time and then solve it by using the dynamic programming approach.Since(3.1)E0[λxT∧τ-ωxT∧τ2]=∑t=1TE0[λxT∧τ-ωxT∧τ2∣T∧τ=t]P(T∧τ=t)=E0[∑t=1T(λxt-ωxt2)pt],
problem A(λ,ω) can be written equivalently as(3.2)A(λ,ω){max⁡uE0[∑t=1T(λxt-ωxt2)pt]s.t.xt+1=rt0xt+Rtut,t=0,1,…,T-1.
Define the value function(3.3)ft*(xt)=max⁡ut⁡ft(xt)=max⁡ut⁡Et[∑s=tT(λxs-ωxs2)ps]
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(3.4)fT*(xT)=(λxT-ωxT2)pT.According to the dynamic programming principle, we have the Bellman equation(3.5)ft*(xt)=max⁡utft(xt)=max⁡utEt[(λxt-ωxt2)pt+ft+1*(xt+1)],
for t=0,1,…,T-1.First, we give the following notations:(3.6)θt=Et2(λt+1Rt)Et(ωt+1Rt2),t=0,1,…,T-1,(3.7)Ξt=Et[λ24ω∑s=tT-1θs],t=0,1,…,T-1,(3.8)ωt=pt+(rt0)2[Et(ωt+1)-Et2(ωt+1Rt)Et(ωt+1Rt2)],t=0,1,…,T-1,ωT=pT,(3.9)λt=pt+rt0[Et(λt+1)-Et(ωt+1Rt)Et(λt+1Rt)Et(ωt+1Rt2)],t=0,1,…,T-1,λT=pT.
For notational simplicity, we define ∑j=st(·)j=0 and ∏j=st(·)j=1 if s>t.Note thatRt and Rt+1 are not independent of each other for t=0,1,…,T-1, so both ωt+1 and λt+1 are dependent on the risky asset return at period t+1, Rt. Then, for t=0,1,…,T-1,(3.10)Et(ωt+1Rt)≠Et(ωt+1)Et(Rt),Et(λt+1Rt)≠Et(λt+1)Et(Rt).The following lemma comes from Xu and Li [11]. For the completeness, we provide its proof here.Lemma 3.1.
Letx be a nondegenerated random variable, and let ξ be a positive random variable under the information at time t, then Et(x2ξ)Et(ξ)>(Et(xξ))2.Proof.
Sinceξ is a positive random variable, we can define a new probability measure Q as
(3.11)dQ≜ξEt(ξ)dP,
where P is the original measure. Since x is a nondegenerated random variable, we have, under measure Q,
(3.12)VartQ(x)=EtQ(x2)-(EtQ(x))2>0.
Transforming the above inequality to under measure P, we obtain
(3.13)Et(x2ξEt(ξ))-(Et(xξEt(ξ)))2>0.
Multiplying both sides by (Et(ξ))2 in the above inequality produces
(3.14)Et(x2ξ)Et(ξ)>(Et(xξ))2.
This completes the proof.Theorem 3.2.
Fort=0,1,…,T-1,ωt>0,θt≥0, and Ξt≥0.Proof.
We use induction. Fort=T-1, since the return of the risky asset at period T, RT-1, is a nondegenerated random variable, then
(3.15)VarT-1(RT-1)=ET-1(RT-12)-(ET-1(RT-1))2>0,ET-1(RT-12)>0. So
(3.16)0≤θT-1=ET-12(pTRT-1)ET-1(pTRT-12)=pTET-12(RT-1)ET-1(RT-12)<pT,ΞT-1=ET-1(λ24ωθT-1)≥0.
Therefore,
(3.17)ωT-1=pT-1+(rT-10)2(pT-ET-12(pTRT-1)ET-1(pTRT-12))>0.
Suppose thatωs>0,θs≥0,and Ξs≥0 hold true for s=t+1,…,T-2,T-1, then for period t,
(3.18)θt=Et2(λt+1Rt)Et(ωt+1Rt2)≥0.
By Lemma 3.1, we can easily see that
(3.19)Et(ωt+1)>Et2(ωt+1Rt)Et(ωt+1Rt2).
Hence, we obtain
(3.20)Ξt=Et(∑s=tT-1λ24ωθs)=λ24ωθt+Et(Ξt+1)≥0,ωt=pt+(rt0)2[Et(ωt+1)-Et2(ωt+1Rt)Et(ωt+1Rt2)]>0.
By induction, it shows that for t=0,1,…,T-1,ωt>0,θt≥0,and Ξt≥0.The analytical optimal strategy and the value function of problemA(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 problemA(λ,ω) are, respectively, given by
(3.21)ut*=λ2ωEt(λt+1Rt)Et(ωt+1Rt2)-Et(ωt+1Rt)Et(ωt+1Rt2)rt0xt,t=0,1,…,T-1,(3.22)ft*(xt)=-ωωtxt2+λλtxt+Ξt,t=0,1,…,T-1,
where Ξt,ωt, and λt 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 conditionfT(xT)=(λxT-ωxT2)pT. Note that for t=T-1,
(3.23)fT-1*(xT-1)=max⁡uT-1⁡fT-1(xT-1)=max⁡uT-1⁡ET-1[(λxT-1-ωxT-12)pT-1+fT*(xT)]=max⁡uT-1⁡ET-1[(λxT-1-ωxT-12)pT-1+(λxT-ωxT2)pT]=max⁡uT-1⁡(λxT-1-ωxT-12)pT-1+λ[rT-10pTxT-1+ET-1(pTRT-1)uT-1]-ω[(rT-10)2pTxT-12+2rT-10ET-1(pTRT-1)uT-1xT-1+ET-1(pTRT-12)uT-12].
Since ET-1(RT-12)>0 by assumption, the function fT-1(xT-1) is a concave function of uT-1. The first-order condition gives
(3.24)λET-1(pTRT-1)-2ω[rT-10ET-1(pTRT-1)xT-1+ET-1(pTRT-12)uT-1]=0,
which yields the optimal solution uT-1* as
(3.25)uT-1*=λ2ωET-1(pTRT-1)ET-1(pTRT-12)-ET-1(pTRT-1)ET-1(pTRT-12)rT-10xT-1.
Substituting uT-1* back into fT-1(xT-1), it follows that
(3.26)fT-1*(xT-1)=(λxT-1-ωxT-12)pT-1+λ[rT-10pTxT-1+ET-1(pTRT-1)(λ2ωET-1(pTRT-1)ET-1(pTRT-12)-ET-1(pTRT-1)ET-1(pTRT-12)rT-10xT-1)]-ωλ2ωET-1(pTRT-1)ET-1(pTRT-12)[(λ2ωET-1(pTRT-1)ET-1(pTRT-12)-ET-1(pTRT-1)ET-1(pTRT-12)rT-10xT-1)2(rT-10)2pTxT-12+2rT-10ET-1(pTRT-1)×(λ2ωET-1(pTRT-1)ET-1(pTRT-12)-ET-1(pTRT-1)ET-1(pTRT-12)rT-10xT-1)xT-1+ET-1(pTRT-12)(λ2ωET-1(pTRT-1)ET-1(pTRT-12)-ET-1(pTRT-1)ET-1(pTRT-12)rT-10xT-1)2]=-ω[pT-1+(rT-10)2(pT-ET-12(pTRT-1)ET-1(pTRT-12))]xT-12+λ[pT-1+rT-10(pT-ET-12(pTRT-1)ET-1(pTRT-12))]xT-1+λ24ωET-12(pTRT-1)ET-1(pTRT-12)=-ωωT-1xT-12+λλT-1xT-1+λ24ωθT-1=-ωωT-1xT-12+λλT-1xT-1+ΞT-1.
Hence, the conclusion holds true for t=T-1.
Now we assume that the conclusion holds true for timet+1, in other words,
(3.27)ft+1*(xt+1)=-ωωt+1xt+12+λλt+1xt+1+Ξt+1,ut+1*=λ2ωEt+1(λt+2Rt+1)Et+1(ωt+2Rt+12)-Et+1(ωt+2Rt+1)Et+1(ωt+2Rt+12)rt+10xt+1,
then the optimization problem at time t for given state xt is
(3.28)ft*(xt)=max⁡ut⁡ft(xt)=max⁡ut⁡Et[(λxt-ωxt2)pt+ft+1*(xt+1)]=max⁡ut⁡Et[(λxt-ωxt2)pt-ωωt+1xt+12+λλt+1xt+1+Ξt+1]=max⁡ut⁡{(λxt-ωxt2)pt-ω[(rt0)2xt2Et(ωt+1)+2rt0Et(ωt+1Rt)utxt+Et(ωt+1Rt2)ut2]+λ(rt0xtEt(λt+1)+Et(λt+1Rt)ut)+Et(Ξt+1)[(rt0)2xt2Et(ωt+1)+2rt0Et(ωt+1Rt)utxt+Et(ωt+1Rt2)ut2]}
Noting that Et(ωt+1Rt2)>0 by Theorem 3.2, the function ft(xt) is also a concave function of ut. The first-order condition yields
(3.29)ut*=λ2ωEt(λt+1Rt)Et(ωt+1Rt2)-Et(ωt+1Rt)Et(ωt+1Rt2)rt0xt.
Therefore, for the above given ut*,
(3.30)ft*(xt)=(λxt-ωxt2)pt+(λ2ωEt(λt+1Rt)Et(ωt+1Rt2)-Et(ωt+1Rt)Et(ωt+1Rt2)rt0xt)λ[rt0xtEt(λt+1)+Et(λt+1Rt)(λ2ωEt(λt+1Rt)Et(ωt+1Rt2)-Et(ωt+1Rt)Et(ωt+1Rt2)rt0xt)]-ω[(λ2ωEt(λt+1Rt)Et(ωt+1Rt2)-Et(ωt+1Rt)Et(ωt+1Rt2)rt0xt)2(rt0)2xt2Et(ωt+1)+2rt0Et(ωt+1Rt)(λ2ωEt(λt+1Rt)Et(ωt+1Rt2)-Et(ωt+1Rt)Et(ωt+1Rt2)rt0xt)xt+Et(ωt+1Rt2)(λ2ωEt(λt+1Rt)Et(ωt+1Rt2)-Et(ωt+1Rt)Et(ωt+1Rt2)rt0xt)2]+Et(Ξt+1)=-ω[pt+(rt0)2(Et(ωt+1)-Et2(ωt+1Rt)Et(ωt+1Rt2))]xt2+λ[pt+rt0(Et(λt+1)-Et(ωt+1Rt)Et(λt+1Rt)Et(ωt+1Rt2))]xt+λ24ωθt+Et(Ξt+1)=-ωωtxt2+λλtxt+Ξt.
Hence, the conclusion is true for t. By induction, the theorem is true.


## 4. Optimal Strategy and the Efficient Frontier of the Original ProblemP(ω)

If we insert the optimal strategy given in Theorem3.3 into the dynamic process of wealth, xT and xT2 can be expressed as(4.1)xT=rT-10xT-1+RT-1uT-1*=[1-RT-1ET-1(ωTRT-1)ET-1(ωTRT-12)]rT-10xT-1+λ2ωRT-1ET-1(λTRT-1)ET-1(ωTRT-12),xT2=[1-2RT-1ET-1(ωTRT-1)ET-1(ωTRT-12)+RT-12ET-12(ωTRT-1)ET-12(ωTRT-12)](rT-10)2xT-12+λω[RT-1ET-1(λTRT-1)ET-1(ωTRT-12)-RT-12ET-1(ωTRT-1)ET-1(λTRT-1)ET-12(ωTRT-12)]rT-10xT-1+λ24ω2RT-12ET-12(λTRT-1)ET-12(ωTRT-12),(4.2)λTxT=[λT-λTRT-1ET-1(ωTRT-1)ET-1(ωTRT-12)]rT-10xT-1+λ2ωλTRT-1ET-1(λTRT-1)ET-1(ωTRT-12),ωTxT2=[ωT-2ωTRT-1ET-1(ωTRT-1)ET-1(ωTRT-12)+ωTRT-12ET-12(ωTRT-1)ET-12(ωTRT-12)](rT-10)2xT-12+λω[ωTRT-1ET-1(λTRT-1)ET-1(ωTRT-12)-ωTRT-12ET-1(ωTRT-1)ET-1(λTRT-1)ET-12(ωTRT-12)]rT-10xT-1+λ24ω2ωTRT-12ET-12(λTRT-1)ET-12(ωTRT-12).
Taking expectations on both sides of (4.2) based on the information available at time T-1, we conclude that(4.3)ET-1(λTxT)=λT-1xT-1-pT-1xT-1+λ2ωθT-1,(4.4)ET-1(ωTxT2)=ωT-1xT-12-pT-1xT-12+λ24ω2θT-1,
where(4.5)θT-1=ET-12(λTRT-1)ET-1(ωTRT-12).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(4.6)E0(xT∧τ)=∑t=1TptE0(xt)=λ0x0+λ2ω∑t=1TE0(θt-1)=λ0x0+λ2ωΘ,(4.7)E0(xT∧τ2)=∑t=1TptE0(xt2)=ω0x02+λ24ω2∑t=1TE0(θt-1)=ω0x02+λ24ω2Θ,
where(4.8)θt=Et2(λt+1Rt)Et(ωt+1Rt2),Θ=∑t=1TE0(θt-1).With the results (4.6) and (4.7), the variance of the terminal wealth xT∧τ under the optimal strategy (3.21) can be written as(4.9)Var0(xT∧τ)=E0(xT∧τ2)-[E0(xT∧τ)]2=E0(∑t=1Tptxt2)-[E0(∑t=1Tptxt)]2=ω0x02+λ24ω2Θ-[λ0x0+λ2ωΘ]2=λ24ω2(Θ-Θ2)-λωλ0Θx0+(ω0-λ02)x02.Lemma 4.1.
0<Θ<1,ω0-λ02/(1-Θ)>0.Proof.
First of all, we claim thatVar0(xT∧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(xT∧τ) can be reduced to
(4.10)Var0(xT∧τ)=λ24ω2(Θ-Θ2)>0,
and it is easy to show that 0<Θ<1.
The expression ofVar0(xT∧τ) can be further converted into
(4.11)Var0(xT∧τ)=(Θ-Θ2)[λ2ω-λ01-Θx0]2+(ω0-λ021-Θ)x02>0.
Since we know that 0<Θ<1, the above inequality implies ω0-λ02/(1-Θ)>0, and we finish the proof of Lemma 4.1.According to Theorem2.2, 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(4.12)λ*=1+2ωE0(xT∧τ)∣u*=1+2ω(λ0x0+λ*2ωΘ).
We can easily obtain(4.13)λ*=1+2ωλ0x01-Θ.
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 problemP(ω), the optimal strategy is given by
(4.14)ut*=1+2ωλ0x02ω(1-Θ)Et(λt+1Rt)Et(ωt+1Rt2)-Et(ωt+1Rt)Et(ωt+1Rt2)rt0xt,t=0,1,…,T-1,
where Θ,λ0,ωt+1,and λt+1 are given as defined.Referring to (4.6),(4.15)λ*2ω=E0(xT∧τ)-λ0x0Θ.
Substituting (4.15) back into (4.11), the relationship between Var0(xT∧τ) and E0(xT∧τ) can be shown as follows:(4.16)Var0(xT∧τ)=(1-Θ)Θ[E0(xT∧τ)-λ0x01-Θ]2+(ω0-λ021-Θ)x02.
Therefore, the efficient frontier of the original problem P(ω) is given by (4.16) for(4.17)E0(xT∧τ)∈[λ0x01-Θ,+∞).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 pt and Rt 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=⋯=pT-1=0, pT=1, and rt0=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 aren risky assets and one riskless asset with period-t+1 returns rti(i=1,2,…,n) and rt0, respectively. Define eti=rti-rt0, et=(et1,et2,…,etn)′ and Ut=(ut1,ut2,…,utn)′ for i=1,2,…,n and t=0,1,…,T-1, where uti is the amount invested in the ith risky asset at time t. In this case, the wealth dynamics is described by(5.1)xt+1=rt0xt+et′Ut,t=0,1,…,T-1.Accordingly, the multi-period mean-variance portfolio selection problem with an uncertain exit time and serial correlations can be formulated as(5.2)P̂(ω){max⁡UE0(xT∧τ)-ωVar0(xT∧τ)s.t.xt+1=rt0xt+et′Ut,t=0,1,…,T-1,
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 problemP̂(ω), the optimal investment strategy is given by
(5.3)Ut*=1+2ωλ̃0x02ω(1-Θ̃)Et-1(ω̃t+1etet′)Et(λ̃t+1et)-Et-1(ω̃t+1etet′)Et(ω̃t+1et)rt0xt,
for t=0,1,…,T-1, and the efficient frontier is given by
(5.4)Var0(xT∧τ)=(1-Θ̃)Θ̃[E0(xT∧τ)-λ̃0x01-Θ̃]2+(ω̃0-λ̃021-Θ̃)x02,
where
(5.5)θ̃t=Et(λ̃t+1et′)Et-1(ω̃t+1etet′)Et(λ̃t+1et),Θ̃=∑t=1TE0(θ̃t-1),ω̃t=pt+(rt0)2[Et(ω̃t+1)-Et(ω̃t+1et′)Et-1(ω̃t+1etet′)Et(ω̃t+1et)],ω̃T=pT,λ̃t=pt+rt0[Et(λ̃t+1)-Et(λ̃t+1et′)Et-1(ω̃t+1etet′)Et(ω̃t+1et)],λ̃T=pT,
for t=0,1,…,T-1.Remark 5.2.
When the returns rates of then 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) progress(6.1)rt=μ+ρ(rt-1-μ)+1-ρ2σyt,
where μ is the unconditional expectation of rt, ρ∈(-1,1) is the first-order autocorrelation coefficient, σ is the unconditional standard deviation of rt, yt is a random variable with standard normal distribution, and yt is independent of ys(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 timet=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 coefficientρ: 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.



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*Source: 216891-2012-05-07.xml*