# Time Consistent Strategies for Mean-Variance Asset-Liability Management Problems

Authors: Hui-qiang Ma; Meng Wu; Nan-jing Huang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper studies the optimal time consistent investment strategies in multiperiod asset-liability management problems under mean-variance criterion. By applying time consistent model of Chen et al. (2013) and employing dynamic programming technique, we derive two-time consistent policies for asset-liability management problems in a market with and without a riskless asset, respectively. We show that the presence of liability does affect the optimal strategy. More specifically, liability leads a parallel shift of optimal time-consistent investment policy. Moreover, for an arbitrarily risk averse investor (under the variance criterion) with liability, the time-diversification effects could be ignored in a market with a riskless asset; however, it should be considered in a market without any riskless asset.


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## Body

## 1. Introduction

By using variance as a risk measure, Markowitz [1] proposed the classic mean-variance portfolio selection model, which has become the theoretical foundation of modern finance theory and has been extended in several directions. One of the main extensions for portfolio selection is to study the optimal policy in a multiperiod setting. For example, Li and Ng [2] and Zhou and Li [3] employed an embedding technique to derive analytical solutions to multiperiod and continuous-time mean-variance models, respectively. In most of the studies in the multiperiod environment, there has been a common assumption that an investor has no long-term liability. However, in reality, many investment institutions (e.g., pension funds, insurance company, and banks) have paid great attention to their portfolios while taking into account their liabilities. Further, it has been shown by Kell and Muller [4] and Sharpe and Tint [5] that liability does affect the optimal policy. More specifically, in a single-period setting, liability leads a parallel shift of mean-variance optimal investment policy and affects the mean-variance efficient frontier. Due to both theoretical interest and practical importance of asset-liability management, the research on mean-variance asset-liability management has attracted recent attentions. For example, among others, Leippold et al. [6] considered the multiperiod asset-liability management problem where the liability is exogenous and fixed and derived an analytical optimal policy and an efficient frontier. Further, they extended their research to the case where the liability is endogenous and controllable in [7]. Chiu and Li [8] and Xie et al. [9] studied continuous-time mean-variance asset-liability management problems, respectively. Furthermore, Xie [10] studied mean-variance model with stochastic liability in a Markovian regime switching financial market, and Zeng and Li [11] investigated asset-liability management problem in a jump diffusion market.In most of the literature, the popular approaches of dealing with dynamic mean-variance asset-liability management problem are embedding technique which was developed by Li and Ng [2] and dual method. However, since the iterated-expectation property does not hold for the variance operator, the optimal asset-liability management policy (called precommitment strategy) derived by both the approaches does not satisfy Bellman's optimality principle and is time inconsistent. The main reason is that, the precommitment strategy, for time interval [t,T], computed at time t will not necessarily coincide with the strategy which computed at time t+Δt. As a result, at time t+Δt, the strategy computed at time t will not be implemented by the investor, unless there exists some commitment mechanism. Strotz [12] first formalized time inconsistency and pointed out that the conflict could be solved by a time-consistent strategy. Very recently, much more scholars have paid their attentions on constructing a time-consistent mean-variance portfolio choice. Among others, Basak and Chabakauri [13] provided a fully analytical characterization of the optimal time-consistent mean-variance portfolio within a general incomplete market economy. Wang and Forsyth [14] developed a numerical scheme for determining the optimal asset allocation strategy for time-consistent, continuous time, mean-variance optimization. By allowing the trade-off between the mean and the variance of the terminal wealth varying over time, Cui et al. [15] proposed a weak time consistency to compare with Bellman's optimality principle and derived an optimal mean-variance portfolio strategy. In all the literature mentioned above, the studies mainly referred to Bellman's optimality principle. However, the requirement (for short, REQ) that local optimum is also globally optimum is not necessarily satisfied, which is an essential requirement in solving the relevant optimal portfolio problem by the dynamic programming technique. In order to make up this shortfall, Chen et al. [16], by using a time consistent dynamic risk measure, proposed a separable dynamic mean-variance model and showed that the relevant optimal investment policy satisfies not only the Bellman's optimality principle but also the REQ.Although asset-liability management is an important issue in modern finance theory, the time-consistent asset-liability management problem has not attracted enough attention. Recently, Li et al. [17] reported the time-consistent asset-liability management problem in the continuous-time setting. They employed Basak and Chabakauri's [13] model to study the continuous-time asset-liability management problem. They derived the time-consistent optimal strategy and showed that the time-consistent efficient frontier with liability is below that without liability. However, the derived time-consistent policy does not satisfy the REQ. As the aforementioned importance of the REQ, in this paper, we employ the model of Chen et al. [16] to analysis the multiperiod asset-liability management problem. We derive time-consistent optimal investment policies in a market with and without a riskless asset, respectively. After comparing the optimal time-consistent policies with myopic strategies, we show that, for an arbitrarily risk averse investor, if there is a riskless asset in the market, the time-diversification effects arising from multiperiod optimization can be ignored, otherwise, the effects should be considered.This paper proceeds as follows. In the next section, we formulate a time-consistent asset-liability management model. In Section3, we derive the time-consistent optimal policy for a market without riskless asset. Section 4 derives the time-consistent optimal policy for a market with both riskless and risky assets. Section 5 performs numerical examples to illustrate our results. The paper is concluded in Section 6.


## 2. Model Formulation

Throughout this paper, we assume thatT∈ℕ is a fixed and finite time horizon and trading only takes place at time k=0,1,…,T. Let (Ω,ℱ,p) be a probability space, and let a σ-field ℱt be the available information at time t.Consider a security market consisting of one riskless asset andn risky assets. The return of the riskless asset at the tth investment period is assumed to be st. The return of the ith risky asset at the tth investment period is denoted by Rti, and the relevant random return vector is denoted by Rt=(Rt1,…,Rtn)′ which is ℱt+1-measurable.Consider an investor with an initial endowmentx0 and a liability l0. We assume that the liability cannot be controlled and denote by lt the accumulative liability at time t. Let Qt be the return of the liability at the tth investment period which is ℱt+1-measurable. It is clear that lt+1=Qtlt. Assume that (Rt′,Qt)′, t=0,…,T-1, are statistically independent. Denote the expected return vector by μt=E(Rt) and the variance-covariance matrix by Σt=(Cov(Rti,Rtj))n×n which is assumed to be positive definite throughout this paper. It is clear that E(RtRt′)=Σt+μtμt′ is also positive definite. Denote by Σt0=(Cov(Qt,Rt1),…,Cov(Qt,Rtn))′ the covariance vector. The investor begins his/her investment at time 0 and invests the cash amount uti in the ith risky asset at the beginning of the tth investment period, where t=0,…,T-1. ut=(ut1,…,utn)′ is called an investment decision at the tth investment period and u=(u0,…,uT-1) is an investment policy during the entire investment horizon. We denote the wealth and the surplus of the investor at time t by xt and St=xt-lt, respectively.A single-period conditional risk mapping is defined as(1)ρt(St+1∣ℱt)=Vart(St+1)-λtEt(St+1),
where Et(St+1)=E(St+1∣ℱt) and Vart(St+1)=Var(St+1∣ℱt) are the single-period conditional expectation of St+1 and its conditional variance, respectively. Tradeoff parameter λt as a constant, which is defined on [0,∞], is a weight which presents the relative importance of expected profit compared to the risk. Note that, if λt=0, then the investor is arbitrary risk averse who only focus on the risk; if λt=∞, then the investor is risk neutral who only concerns maximizing their expected profit; if λt∈(0,∞), the investor is risk averse who considers both expected profit and the risk in his decision.In multiperiod portfolio selection, two main optimal investment policies are the myopic and time-consistent strategies.Myopic Strategy is a strategy whereby at each time t the investor determines their optimal investment decision assuming the instantaneous moments of assets returns will remain fixed at their current values for the remainder of the investment horizon. More specifically, for any t∈{0,…,T-1}, the myopic strategy utmy is a solution to the following problem:
(2)minut∈ΠtVart(St+1)-λtEt(St+1),
where Πt is a set of all permit policies at time t. Intuitively, a myopic investor only cares about the mean and variance of the surplus at the current period.To measure the total risk of an investor among multiperiods (after timet), we employ a separable expected conditional mapping which is defined as (see Chen et al. [16])
(3)∑j=t+1T‍Et(ρj-1(Sj∣ℱj-1)),
which reflects all the risk in the future. Following this assumption, a separable dynamic mean-variance problem is defined as
(4)minu∈Π∑j=1T‍E(ρj-1(Sj∣ℱj-1)),
where Π is a set of all permit policies. The optimal policy of problem (4), which satisfies both Bellman's optimality principle and requirement REQ, is called Time-Consistent Strategy. Note that both Bellman's optimality principle and requirement REQ could be proved by following the methodology of Chen et al. [16]. Thus, problem (4) can be recursively solved by the dynamic programming technique. Applying the iterated-expectation property of the expected operator, that is, E(E(·∣ℱj)∣ℱk)=E(·∣ℱk) for j>k, we have
(5)∑j=1T‍E(ρj-1(Sj∣ℱj-1))=ρ0(S1)+E(+ET-3(ρT-2(ST-1)+ET-2(ρT-1(ST)))…)+ET-2(ρT-1(ST)))…)ρ1(S2)+E1((ρT-2(ST-1)+ET-2(ρT-1(ST)))…ρ2(S3)+⋯+ET-3(+ET-2(ρT-1(ST))ρT-2(ST-1)+ET-2(ρT-1(ST)))…))
(for more details see Chen et al. [16]). Then, problem (4) is equivalent to
(6)minu∈Πρ0(S1)+E(+ET-2(ρT-1(ST)))…)ρ1(S2)+E1(+ET-2(ρT-1(ST)))ρ2(S3)vvvvvvvvvvvvvvv+⋯+ET-3((ρT-1(ST))ρT-2(ST-1)vvvvvvvvvvvvvvvvvvvvvvvv+ET-2(ρT-1(ST)))…)).
It follows from Bellman's optimality principle and (6) that (4) is equivalent to find an optimal strategy to satisfy the following problem:
(7)minu0(+ET-2minuT-1(ρT-1(ST))…))ρ0(S1)+Eminu1(+ET-2minuT-1(ρT-1(ST))…))ρ1(S2)+E1minu2(+ET-2minuT-1(ρT-1(ST))…))ρ2(S3)+⋯+ET-3minuT-2(+ET-2minuT-1(ρT-1(ST))…))ρT-2(ST-1)+ET-2minuT-1(ρT-1(ST))+ET-2minuT-1(ρT-1(ST))…)))…+ET-2minuT-1(ρT-1(ST))…))))).We solve this problem in the following sections. In order to discuss the impact of riskless asset, the market is considered in two cases: with and without riskless asset. We demonstrate these results in Sections3 and 4, respectively.


## 3. Time Consistent Optimal Strategy without Riskless Asset

Consider a market consisting of onlyn risky assets and assume that the wealth process xt is in a self-financing fashion. We list the notations of this section in Table 1. The wealth process xt could be described as follows:
(8)xt+1=Rt′ut,I′ut=xt,t=0,1,…,T-1,
where I=(1,…,1)′∈ℝn. In this setting, problem (6) can be written as follows:
(9)minρ0(S1)+E0(+ET-3(ρT-2(ST-1)+ET-2(ρT-1(ST)))…)ρ1(S2)+E1(+ET-3(ρT-2(ST-1)+ET-2(ρT-1(ST)))…ρ2(S3)+⋯v+ET-3(+ET-2(ρT-1(ST))ρT-2(ST-1)vvvvvvvvvvvvvvvvvvvvvvv+ET-2(ρT-1(ST)))…))s.t.Rt′ut=xt+1,I′ut=xt,lt+1=Qtlt,St+1=xt+1-lt+1,t=0,1,…,T-1.Table 1
Model notations of Section3.
Fort=T-1:
Fort=T-2,…,0:
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]SinceΣT-11 is positive definite, we have aT-1>0. Further, it follows from the nonnegative definiteness of E(RT-1RT-1′) that ΣT-21 is also positive definite and aT-2>0. By using mathematical induction, we conclude that, for any t=0,…,T-1, Σt1 is also positive definite and at>0.By applying Bellman's optimality principle, the time-consistent optimal investment policy of problem (9) is given in the following theorem.Theorem 1.
The time-consistent optimal investment policy of problem (9) is given by
(10)ut*=(Σt1)-1Iatxt+λt12(Σt1)-1(μt-qtI)+(Σt1)-1(Σt0-γt~I+γ~t+1E(QtRt))lt,vvvvvvvvvvvvvvvvvvfort=0,…,T-1.Proof.
Whent=T-1, for given wealth xT-1 and liability lT-1 at the beginning of the (T-1)th  period, problem (9) can be expressed as follows:
(11)minVarT-1(ST)-λT-1ET-1(ST)s.t.vRT-1′uT-1=xT,vvvvvvI′uT-1=xT-1,vvvvvvlT=QT-1lT-1,vvvvvvST=xT-lT.
Substituting the binding constraints into the objective function, we have
(12)minVT-1(xT-1,lT-1)≔uT-1′ΣT-11uT-1-2[(ΣT-10)′lT-1+λT-12uT-1′]uT-1+Var(QT-1)lT-12+λT-1E(QT-1)lT-1s.t.I′uT-1=xT-1,
which is a linear-quadratic program. By using the Lagrange multiplier technique and letting ωT be the Lagrange multiplier, the Lagrange function is defined as
(13)L(uT-1)≔uT-1′ΣT-11uT-1-2[(ΣT-10)′lT-1+λT-12μT-1′]uT-1+Var(QT-1)lT-12+λT-1E(QT-1)lT-1-ωT(xT-1-I′uT-1).
By using the first-order necessary optimality condition, we have
(14)2ΣT-11uT-1-2(ΣT-10lT-1+λT-12μT-1)+ωTI=0,(15)xT-1-I′uT-1=0.
From (14), we can easily have
(16)uT-1*=(ΣT-11)-1(ΣT-10lT-1+λT-12μT-1)-12ωT(ΣT-11)-1I,(17)I′uT-1*=I′(ΣT-11)-1(ΣT-10lT-1+λT-12μT-1)-12ωTI′(ΣT-11)-1I=dT-1lT-1+λT-12bT-1-12aT-1ωT=xT-1,
which implies that the Lagrange multiplier ωT is
(18)ωT=2dT-1aT-1lT-1+λT-1bT-1aT-1-21aT-1xT-1=2γT-1lT-1+λT-1qT-1-21aT-1xT-1.
Substituting ωT into (16), we have
(19)uT-1*=(ΣT-11)-1IaT-1xT-1+(ΣT-11)-1[ΣT-10-γT-1I]lT-1+λT-12(ΣT-11)-1[μT-1-qT-1I].
Further, by substituting uT-1* into the objective function of problem (12) (see Appendix A for more details), we have
(20)VT-1*(xT-1,lT-1)=1aT-1xT-12-λT-1qT-1xT-1-2γT-1xT-1lT-1+β1,T-1lT-1+β2,T-1lT-12-(λT-1)24αT-1.
Whent=T-2, for given wealth xT-2 and liability lT-2 at the beginning of the (T-2)th period, the corresponding optimal investment problem is given as follows:
(21)minVarT-2(ST-1)-λT-2ET-2(ST-1)+ET-2(VT-1*(xT-1,lT-1))s.t.RT-2′uT-2=xT-1,I′uT-2=xT-2,lT-1=QT-2lT-2,ST-1=xT-1-lT-1.
It follows from (20) that
(22)ET-2(VT-1*(xT-1,lT-1))=1aT-1ET-2(xT-12)-λT-1qT-1ET-2(xT-1)-2γT-1ET-2(xT-1lT-1)+β1,T-1ET-2(lT-1)+β2,T-1ET-2(lT-12)-(λT-1)24αT-1.
Substituting ET-2(VT-1*(xT-1,lT-1)) and the binding constraints into the objective function of problem (21), we have
(23)minVT-2(xT-2,lT-2)s.t.I′uT-2=xT-2,
where
(24)VT-2(xT-2,lT-2)≔uT-2′ΣT-21uT-2-2{+λT-212uT-2′[(ΣT-20)′+γT-1E(QT-2RT-2′)]lT-2V+λT-212μT-2′}uT-2+[Var(QT-2)+β2,T-1E(QT-22)]lT-22+[λT-2E(QT-2)+β1,T-1E(QT-2)]lT-1-(λT-1)24αT-1.
By setting ωT-1 be the Lagrange multiplier, the Lagrange function for problem (23) is
(25)L(uT-2)≔uT-2′ΣT-21uT-2-2ΦT-2′uT-2+ΨT-2-ωT-1(xT-2-I′uT-2),
where
(26)ΦT-2≔[ΣT-20+γT-1E(QT-2RT-2)]lT-2+λT-212μT-2,ΨT-2≔[Var(QT-2)+β2,T-1E(QT-22)]lT-22+[λT-2E(QT-2)+β1,T-1E(QT-2)]lT-1-(λT-1)24αT-1.
From the first-order necessary optimality condition, we have
(27)2ΣT-21uT-2-2ΦT-2+ωT-1I=0,(28)xT-2-I′uT-2=0.
From (27), we have
(29)uT-2*=(ΣT-21)-1ΦT-2-12ωT-1(ΣT-21)-1I,(30)I′uT-2*=I′(ΣT-21)-1ΦT-2-12ωT-1I′(ΣT-21)-1I=I′(ΣT-21)-1ΦT-2-12ωT-1aT-2=xT-2,
which implies
(31)ωT-1=2I′(ΣT-21)-1ΦT-2aT-2-21aT-2xT-2.
Substituting ωT-1 into (29), we get
(32)uT-2*=(ΣT-21)-1IaT-2xT-2+(ΣT-21)-1ΦT-2-I′(ΣT-21)-1ΦT-2aT-2(ΣT-21)-1I.
Taking ΦT-2 into account, we have
(33)uT-2*=(ΣT-21)-1IaT-2xT-2+λT-212(ΣT-21)-1(μT-2-qT-2I)+(ΣT-21)-1[ΣT-20+γT-1E(QT-2RT-2)-γ~T-2I]lT-2.
Substituting uT-2* into (23) gives (see Appendix B for more details)
(34)VT-2*(xT-2,lT-2)=1aT-2xT-22-λT-21qT-2xT-2-2γ~T-2xT-2lT-2+β1,T-2lT-2+β2,T-2lT-22-(λT-1)24αT-1-(λT-21)24αT-2.
Next, by using mathematical induction, we show that both (10) and
(35)Vt*(xt,lt)=1atxt2-λt1qtxt-2γ~txtltvvvvvvvvv+β1,tlt+β2,tlt2-∑i=tT-1‍(λi1)24αi
hold. Suppose that (10) and (35) hold for time t,t+1,…,T-1. At the beginning of (t-1)th period, for given wealth xt-1 and liability lt-1, the corresponding optimal investment problem is
(36)minVart-1(St)-λt-1Et-1(St)+Et-1(Vt*(xt,lt))s.t.Rt-1′ut-1=xt,I′ut-1=xt-1,lt=Qt-1lt-1,St=xt-lt.
It follows from (35) that
(37)Et-1(Vt*(xt,lt))=1atEt-1(xt2)-λt1qtEt-1(xt)-2γ~tEt-1(xtlt)+β1,tEt-1(lt)+β2,tEt-1(lt2)-∑i=tT-1‍(λi1)24αi.
Substituting Et-1(Vt*(xt,lt)) and the binding constraints into the objective function of problem (36), we have
(38)minVt-1(xt-1,lt-1)≔uT-1′Σt-11ut-1-2Φt-1′ut-1+Ψt-1s.t.I′ut-1=xt-1,
where
(39)Φt-1≔[Σt-10+γ~tE(Qt-1Rt-1)]lt-1+λt-112μt-1,Ψt-1≔[Var(Qt-1)+β2,tE(Qt-12)]lt-12+[λt-1+β1,t]E(Qt-1)lt-1-∑i=tT-1‍(λi1)24αi.
Letting ωt be the Lagrange multiplier, the Lagrange function for problem (38) is given by
(40)L(ut-1)≔uT-1′Σt-11ut-1-2Φt-1′ut-1+Ψt-1-ωt(xt-1-I′ut-1).
It follows from the first-order necessary optimality condition that
(41)2Σt-11ut-1-2Φt-1+ωtI=0,xt-1-I′ut-1=0.
Thus, we have
(42)ut-1*=(Σt-11)-1Φt-1-12ωt(Σt-11)-1I,(43)I′ut-1*=I′(Σt-11)-1Φt-1-12ωtI′(Σt-11)-1I=I′(Σt-11)-1Φt-1-12ωtat-1=xt-1,
which implies
(44)ωt=2I′(Σt-11)-1Φt-1at-1-21at-1xt-1.
It follows from (42) that
(45)ut-1*=(Σt-11)-1Iat-1xt-1-I′(Σt-11)-1Φt-1at-1(Σt-11)-1I+(Σt-11)-1Φt-1=(Σt-11)-1Iat-1xt-1+λt-112(Σt-11)-1(μt-1-qt-1I)+(Σt-11)-1[Σt-10-γ~t-1I+γt~E(Qt-1Rt-1)]lt-1.
Substituting ut-1* into the objective function of problem (38) (see Appendix C for more details), we have
(46)Vt-1*(xt-1,lt-1)=1at-1xt-12-λt-11qt-1xt-1-2γ~t-1xt-1lt-1+β1,t-1lt-1+β2,t-1lt-12-∑i=t-1T-1‍(λi1)24αi.
This completes the proof.Remark 2.
If an investor does not have any liability, that is,lt≡0 for any t∈{0,1,…,T-1}, then the optimal time-consistent investment strategy can be simplified as follows:
(47)ut*=(Σt1)-1Iatxt+λt12(Σt1)-1(μt-qtI),t=0,…,T-1,
which is exactly the same as that in [16]. This implies that the result of Chen et al. [16] is a special case of Theorem 1. Therefore, Theorem 1 generalizes their result.Corollary 3.
If the returns of liability and risky assets are uncorrelated, that is,Σt0=0 for any t∈{0,1,…,T-1}, then the optimal investment policy for problem (9) is
(48)ut*=(Σt1)-1Iatxt+λt12(Σt1)-1(μt-qtI),t=0,…,T-1.Proof.
SinceΣt0=0, it is easy to verify that dt=I′(Σt1)-1Σt0=0, γt=dt/at=0 and γ~t=0. Substituting them into (10) gives
(49)ut*=(Σt1)-1Iatxt+λt12(Σt1)-1(μt-qtI),t=0,…,T-1.
This completes the proof.Remark 4.
After comparing Corollary3 and Remark 2, it is quite clear that, if the return of liability is uncorrelated with that of risky asset, then the liability does not affect the time-consistent optimal policy in a market without riskless asset.Remark 5.
If the return of liability is correlated to those of risky assets, then the occurrence of liability leads to a parallel shift of the optimal investment policy and the shift is(50)(Σt1)-1(Σt0-γt~I+γ~t+1E(QtRt))lt,
which depends on the current value of liability, lt, and the covariance between the returns of liability and risky assets, Σt0.Next, we compare the time-consistent strategy with the myopic strategy in a market without riskless asset. In such a market, problem (2) can be expressed as follows:
(51)minutVart(St+1)-λtEt(St+1)s.t.Rt′ut=xt+1,I′ut=xt,lt+1=Qtlt,St+1=xt+1-lt+1.
By using the same method in the proof of Theorem 1 for time T-1, the myopic strategy is given by
(52)utmy≔(Σt)-1Ia¯txt+λt2(Σt)-1(μt-q¯tI)+(Σt)-1(Σt0-γ¯tI)lt,t=0,…,T-1,
where a¯t=I′(Σt)-1I, q¯t=I′(Σt)-1μt/a¯t, γ¯t=I′(Σt)-1Σt0/a¯t. It is clear that the difference between two strategies enters into all of the three parts. More specifically, the following feature holds; if the investor is arbitrarily risk averse, that is, λt→0, then both the time consistent optimal strategy and myopic strategy reduce to
(53)ut*=(Σt1)-1Iatxt+(Σt1)-1(Σt0-γt~I+γ~t+1E(QtRt))lt,vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvt=0,…,T-1,utmy=(Σt)-1Ia¯txt+(Σt)-1(Σt0-γ¯tI)lt,t=0,…,T-1,
respectively. After comparing these two strategies, we find that, if an investor is arbitrarily risk averse, then he/she should concern about the time-diversification effects arising from multiperiod optimization.


## 4. Time Consistent Optimal Strategy with Riskless Asset

In this section, we consider a market which is consisting of one riskless asset andn risky assets and assume that the wealth process xt is also in a self-financing fashion. We list the notations of this section in Table 2. The wealth process xt can be described as follows:
(54)xt=Pt-1′ut-1+st-1xt-1,
where Pt=Rt-stI. In this setting, problem (6) can be written as follows:
(55)minρ0(S1)+E0(v+ET-3(ρT-2(ST-1)+ET-2(ρT-1(ST)))…)ρ1(S2)+E1((ρT-2(ST-1)+ET-2(ρT-1(ST)))…ρ2(S3)+⋯+ET-3(ρT-2(ST-1)+ET-2(ρT-1(ST)))…))s.t.xt+1=Pt′ut+stxt,lt+1=Qtlt,St+1=xt+1-lt+1,t=0,1,…,T-1.Table 2
Model notations of Section4.
Fort=T-1:
Fort=T-2,…,0:
λ
T
-
1
1
=
λ
T
-
1
λ
t
1
=
λ
t
+
λ
t
+
1
1
s
t
+
1
β
¯
1
,
T
-
1
=
λ
T
-
1
[
E
(
Q
T
-
1
)
-
ϵ
T
-
1
′
(
Σ
T
-
1
)
-
1
φ
T
-
1
]
β
¯
1
,
t
=
(
λ
t
+
β
1
,
t
+
1
)
E
(
Q
t
)
-
λ
t
1
ϵ
t
′
(
Σ
t
)
-
1
φ
t
β
¯
2
,
T
-
1
=
Var
(
Q
T
-
1
)
-
φ
T
-
1
′
(
Σ
T
-
1
)
-
1
φ
T
-
1
β
¯
2
,
t
=
Var
(
Q
t
)
+
β
2
,
t
+
1
E
(
Q
t
2
)
-
φ
t
′
(
Σ
t
)
-
1
φ
tBy applying Bellman's optimality principle, the time-consistent optimal investment policy of problem (55) is given in the following theorem.Theorem 6.
The optimal investment strategy of problem (55) is given by
(56)ut*=Σt-1φtlt+λt12Σt-1ϵt,t=0,…,T-1,
where φt=E(PtQt)-E(Pt)E(Qt) and ϵt=E(Pt).Proof.
Whent=T-1, for given wealth xT-1 and liability lT-1 at the beginning of the (T-1)th period, problem (55) reduces to
(57)minuT-1VarT-1(ST)-λT-1ET-1(ST)s.t.vxT=sT-1xT-1+PT-1′uT-1,vvvvvvlT=QT-1lT-1,vvvvvvST=xT-lT.
Substituting the binding constraints into the objective function, we have
(58)minuT-1VT-1(xT-1,lT-1),
where
(59)VT-1(xT-1,lT-1)≔uT-1′ΣT-1uT-1-2(φT-1′lT-1+λT-12ϵT-1′)uT-1+Var(QT-1)lT-12-λT-1sT-1xT-1+λT-1E(QT-1)lT-1.
It is clear that problem (58) is an unconstrained convex program problem. By using the first-order necessary optimality condition, we have
(60)d(VT-1(xT-1,lT-1))d(uT-1)=2ΣT-1uT-1-2(φT-1lT-1+λT-12ϵT-1)=0,
which implies
(61)uT-1*=(ΣT-1)-1[φT-1lT-1+λT-12ϵT-1].
Substituting uT-1* into the objective function of problem (58) gives (see Appendix D for more details)
(62)VT-1*(xT-1,lT-1)=β¯2,T-1lT-12+β¯1,T-1lT-1-λT-1sT-1xT-1-(λT-1)24MT-1,
where MT-1=ϵT-1′ΣT-1-1ϵT-1.
Whent=T-2, for given wealth xT-2 and liability lT-2 at the beginning of the (T-2)th period, the corresponding optimal investment problem can be expressed as follows:
(63)minVarT-2(ST-1)-λT-2ET-2(ST-1)+ET-2(VT-1*(xT-1,lT-1))s.t.xT-1=sT-2xT-2+PT-2′uT-2,vlT-1=QT-2lT-2,vST-1=xT-1-lT-1.
From (62), we can easily have
(64)ET-2(VT-1*(xT-1,lT-1))=β¯2,T-1E(QT-22)lT-22+β¯1,T-1E(QT-2)lT-2-λT-1sT-1sT-2xT-2-λT-1sT-1ϵT-2′uT-2-(λT-1)24MT-1.
Substituting ET-2(VT-1*(xT-1,lT-1)) and the binding constraints into the objective function of problem (63), we have
(65)minuT-2VT-2(xT-2,lT-2),
where
(66)VT-2(xT-2,lT-2)=uT-2′ΣT-2uT-2-2(φT-2′lT-2+λT-212ϵT-2′)uT-2+[Var(QT-2)+β¯2,T-1E(QT-22)]lT-22+[λT-2E(QT-2)+β¯1,T-1E(QT-2)]lT-1-λT-21sT-2xT-2-(λT-1)24MT-1.
The first-order necessary optimality condition implies
(67)d(VT-2(xT-2,lT-2))d(uT-2)=2ΣT-2uT-2-2(φT-2lT-2+λT-212ϵT-2)=0.
Thus,
(68)uT-2*=(ΣT-2)-1(φT-2lT-2+λT-212ϵT-2)=(ΣT-2)-1φT-2lT-2+λT-212(ΣT-2)-1ϵT-2.
Substituting uT-2* into the objective function of problem (65) (see Appendix E for more details), we have
(69)VT-2*(xT-2,lT-2)=β¯2,T-2lT-22+β¯1,T-2lT-2-λT-21sT-2xT-2-(λT-1)24MT-1-(λT-21)24MT-2,
where MT-2=ϵT-2′ΣT-2-1ϵT-2.
Next, by using mathematical induction, we show that both (56) and
(70)Vt*(xt,lt)=β¯2,tlt2+β¯1,tlt-λt1stxt-∑i=tT-1‍(λi1)24Mi
hold, where Mi=ϵi′Σi-1ϵi with i=0,1,…,T-1. Suppose that (56) and (70) are true for time t,t+1,…,T-1. At the beginning of the (t-1)th period, for given wealth xt-1 and liability lt-1, the corresponding optimal investment problem is
(71)minVart-1(St)-λt-1Et-1(St)+Et-1(Vt*(xt,lt))s.t.xt=st-1xt-1+Pt-1′ut-1,lt=Qt-1lt-1,St=xt-lt.
It follows from (70) that
(72)Et-1(Vt*(xt,lt))=β¯2,tE(Qt-12)lt-12+β¯1,tE(Qt-1)lt-λt1stst-1xt-1-λt1stϵt-1′ut-1-∑i=tT-1‍(λi1)24Mi.
Substituting Et-1(Vt*(xt,lt)) and the binding constraints into the objective function of problem (71), we have
(73)minut-1Vt-1(xt-1,lt-1),
where
(74)Vt-1(xt-1,lt-1)=ut-1′Σt-1ut-1-2(φt-1′lt-1+λt-112ϵt-1′)ut-1+[Var(Qt-1)+β¯2,tE(Qt-12)]lt-12+[λt-1E(Qt-1)+β¯1,tE(Qt-1)]lt-1-λt-11st-1xt-1-∑i=tT-1‍(λi1)24Mi.
The first-order necessary optimality condition gives
(75)d(Vt-1(xt-1,lt-1))d(ut-1)=2Σt-1ut-1-2(φt-1lt-1+λt-112ϵt-1)=0,
which implies
(76)ut-1*=Σt-1-1φt-1lt-1+λt-112Σt-1-1ϵt-1.
Substituting ut-1* into the objective function of problem (73) (see Appendix F for more details), we have
(77)Vt-1*(xt-1,lt-1)=β¯2,t-1lt-12+β¯1,t-1lt-1-λt-11st-1xt-1-∑i=t-1T-1‍(λi1)24Mi.
This completes the proof.Remark 7.
From Theorem6, it is clear that, if there is a riskless asset in the market, then the time-consistent optimal investment policy is wholly independent of the current wealth xt. However, Theorem 1 gives an opposite conclusion. This implies that the riskless asset does affect the optimal strategy. Therefore, an investor should carefully select the market they invested.Remark 8.
If there is no liability, that is,lt≡0 for any t∈{0,1,…,T-1}, then the time-consistent optimal investment policy reduces to
(78)ut*=λt12Σt-1ϵt,t=0,…,T-1,
which is the same as that in [16]. This implies that the result of Chen et al. [16] is a special case of Theorem 6. Therefore, Theorem 6 generalizes their result.Corollary 9.
If the return of liability,Qt, is uncorrelated with those of risky assets, Rt, that is, Σt0=0 for any t∈{0,1,…,T-1}, then the optimal policy for problem (55) is
(79)ut*=λt12Σt-1ϵt,t=0,…,T-1.Proof.
SinceΣt0=0, it is easy to have
(80)φt=E(PtQt)-E(Pt)E(Qt)=E((Rt-stI)Qt)-E(Rt-stI)E(Qt)=E(RtQt)-E(Rt)E(Qt)=Σt0=0.
Substituting φt=0 into (56) gives
(81)ut*=λt12Σt-1ϵt,t=0,…,T-1.
This completes the proof.Remark 10.
After comparing Corollary9 and Remark 8, it is quite clear that, if the return of liability is uncorrelated with those of risky assets, then the occurrence of liability does not affect the time-consistent optimal investment policy in a market with riskless asset.Remark 11.
If the return of liability is correlated with those of risky assets, then the occurrence of liability leads to a parallel shift of the optimal investment policy and the shift is(82)Σt-1φtlt,
which depends on the current value of the liability.Now, we compare the time-consistent policy with the myopic strategy in a market with a riskless asset. In such a market, problem (2) can be further expressed as follows:
(83)minutVart(St+1)-λtEt(St+1)s.t.xt+1=stxt+Pt′ut,lt+1=Qtlt,St+1=xt+1-lt+1.
By using the same method in the proof of Theorem 6 for time T-1, the myopic strategy is given by
(84)utmy≔Σt-1φtlt+λt2Σt-1ϵt,t=0,…,T-1.
Different with the market without riskless asset, the difference between the time-consistent optimal strategy and the myopic strategy only enters in the part depending on the risk aversion λt. Further, the following feature holds; if an investor is arbitrarily risk averse, that is, λt→0, then both the time-consistent optimal investment policy and the myopic strategy reduce to
(85)ut*=Σt-1φtlt,t=0,…,T-1.
This implies that, if an investor is arbitrarily risk averse, then he/she could ignore the time-diversification effects arising from multiperiod optimization. Further, if the investor does not have any liability, then both two strategies suggest that he/she should leave the market.Remark 12.
After comparing the results of these two different markets, we find that, for an arbitrarily risk averse investor, if there is a riskless asset in the market, the time-diversification effects could be ignored, otherwise, the effects should be considered.


## 5. Numerical Illustration

In this section, we present numerical examples to gain insights regarding the impact of time diversification and of liability on the optimal time-consistent strategies. To make it easy to analysis, we assumeT=2 and all parameters at different periods are the same. Considering a market with three risky assets whose corresponding expected return vector and the variance-covariance matrices are given as μt=(1.162,1.246,1.228) and
(86)Σt=(0.01460.01870.01450.01870.08540.01040.01450.01040.0289),
respectively. The expected return of the liability E(Qt) is 1.136, the corresponding variance Var(Qt) is 0.01, and the covariance vector Σt0 is (0.0006,0.0149,0.0050)′.Table3 illustrates how the time-consistent strategy depends on the liability. From Table 3, if an investor has a liability, then he/she could adjust their investment strategy which results in a parallel shift of the optimal time-consistent strategy. Thus, the investor should take account for the impact of liability.Table 3
Time-consistent strategies with and without liability forλ0=λ1=0.5.
t
Time-consistent  strategy with  liabilityut*
Time-consistent  strategy without  liabilityu~t*
0
(
1.194
l
0
+
0.903
x
0
+
3.736
0.107
l
0
-
0.066
x
0
+
0.203
-
0.577
l
0
+
0.163
x
0
-
3.939
)
(
0.903
x
0
+
3.736
-
0.066
x
0
+
0.203
0.163
x
0
-
3.939
)
1
(
-
0.372
l
1
+
1.018
x
1
+
1.743
0.198
l
1
-
0.060
x
1
+
0.095
-
0.633
l
1
+
0.042
x
1
-
1.838
)
(
1.018
x
1
+
1.743
-
0.060
x
1
+
0.095
0.042
x
1
-
1.838
)Tables4 and 5 show the time-consistent strategy and the myopic strategy in a market without riskless asset for λt=0.5 and λt=0, respectively. In Table 4, we find that the two strategies are different and the difference between them enters into all of the three parts. Table 5 figures out that the two strategies are still very different even if the investor is arbitrarily risk averse. Further, Tables 4 and 5 imply that the investor can not ignore the time diversification effects in a market without riskless asset.Table 4
Investment strategies in a market without riskless asset forλ0=λ1=0.5.
t
Time-consistent strategyut*
Myopic strategyutmy
0
(
0.903
x
0
+
1.194
l
0
+
3.736
-
0.066
x
0
+
0.107
l
0
+
0.203
0.163
x
0
-
0.577
l
0
-
3.939
)
(
1.018
x
0
+
0.449
l
0
+
1.743
-
0.060
x
0
+
0.150
l
0
+
0.095
0.042
x
0
-
0.599
l
0
-
1.838
)
1
(
1.018
x
1
-
0.372
l
1
+
1.743
-
0.060
x
1
+
0.198
l
1
+
0.095
0.042
x
1
-
0.633
l
1
-
1.838
)
(
1.018
x
1
+
0.449
l
1
+
1.743
-
0.060
x
1
+
0.150
l
1
+
0.095
0.042
x
1
-
0.599
l
1
-
1.838
)Table 5
Investment strategies in a market without riskless asset forλ0=λ1=0.
t
Time-consistent strategyut*
Myopic strategyutmy
0
(
0.903
x
0
+
1.194
l
0
-
0.066
x
0
+
0.107
l
0
0.163
x
0
-
0.577
l
0
)
(
1.018
x
0
+
0.449
l
0
-
0.060
x
0
+
0.150
l
0
0.042
x
0
-
0.599
l
0
)
1
(
1.018
x
1
-
0.372
l
1
-
0.060
x
1
+
0.198
l
1
0.042
x
1
-
0.633
l
1
)
(
1.018
x
1
+
0.449
l
1
-
0.060
x
1
+
0.150
l
1
0.042
x
1
-
0.599
l
1
)Next, we consider a market consisting of both risky assets and a riskless asset. Suppose that the return of the riskless assetst is 1.04. Then we have ϵt=μt-stI=(0.122,0.206,0.188)′ and φt=Σt0=(0.0006,0.0149,0.0050)′.Tables6 and 7 show the time-consistent strategy and the myopic strategy in a market with a riskless asset for λt=0.5 and λt=0, respectively. From Table 6, it is clear that the two strategies are different which is consistent with the results in a market without riskless asset. However, they may be exactly the same if the investor is arbitrarily risk averse. This implies that the investor who is arbitrarily risk averse can ignore the time diversification effects.Table 6
Investment strategies in a market with a riskless asset forλ0=λ1=0.5.
t
Time-consistent strategyut*
Myopic strategyutmy
0
(
0.444
l
0
+
7.889
0.150
l
0
-
0.062
-
0.599
l
0
-
3.572
)
(
0.444
l
0
+
3.867
0.150
l
0
-
0.031
-
0.599
l
0
-
1.751
)
1
(
0.444
l
1
+
3.867
0.150
l
1
-
0.031
-
0.599
l
1
-
1.751
)
(
0.444
l
1
+
3.867
0.150
l
1
-
0.031
-
0.599
l
1
-
1.751
)Table 7
Investment strategies in a market with a riskless asset forλ0=λ1=0.
t
Time-consistent strategyut*
Myopic strategyutmy
0
(
0.444
l
0
0.150
l
0
-
0.599
l
0
)
(
0.444
l
0
0.150
l
0
-
0.599
l
0
)
1
(
0.444
l
1
0.150
l
1
-
0.599
l
1
)
(
0.444
l
1
0.150
l
1
-
0.599
l
1
)


## 6. Conclusion

In this paper, we consider the time-consistency of the optimal asset-liability management policies in a market with and without a riskless asset, respectively. By employing the dynamic programming technique, we give the optimal time-consistent investment policies. After comparing the optimal time-consistent policies with myopic strategies, we show that, a risk averse investor should concern the time diversification effects. Further, an arbitrarily risk averse investor could ignore these effects in a market with a riskless asset.



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*Source: 709129-2013-10-09.xml*