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

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.

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 Section 3, 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 that 𝑇∈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 ℱ_𝑡 be the available information at time 𝑡.

Consider a security market consisting of one riskless asset and 𝑛 risky assets. The return of the riskless asset at the 𝑡th investment period is assumed to be 𝑠_𝑡. The return of the 𝑖th risky asset at the 𝑡th investment period is denoted by 𝑅^𝑖_𝑡, and the relevant random return vector is denoted by 𝑅_𝑡=(𝑅¹_𝑡,…,𝑅^𝑛_𝑡)^′ which is ℱ_𝑡+1-measurable.

Consider an investor with an initial endowment 𝑥₀ and a liability 𝑙₀. We assume that the liability cannot be controlled and denote by 𝑙_𝑡 the accumulative liability at time 𝑡. Let 𝑄_𝑡 be the return of the liability at the 𝑡th investment period which is ℱ_𝑡+1-measurable. It is clear that 𝑙_𝑡+1=𝑄_𝑡𝑙_𝑡. Assume that (𝑅^′_𝑡,𝑄_𝑡)^′, 𝑡=0,…,𝑇−1, are statistically independent. Denote the expected return vector by 𝜇_𝑡=𝐸(𝑅_𝑡) and the variance-covariance matrix by Σ_𝑡=(Cov(𝑅^𝑖_𝑡,𝑅^𝑗_𝑡))_𝑛×𝑛 which is assumed to be positive definite throughout this paper. It is clear that 𝐸(𝑅_𝑡𝑅^′_𝑡)=Σ_𝑡+𝜇_𝑡𝜇^′_𝑡 is also positive definite. Denote by Σ⁰_𝑡=(Cov(𝑄_𝑡,𝑅¹_𝑡),…,Cov(𝑄_𝑡,𝑅^𝑛_𝑡))^′ the covariance vector. The investor begins his/her investment at time 0 and invests the cash amount 𝑢^𝑖_𝑡 in the 𝑖th risky asset at the beginning of the 𝑡th investment period, where 𝑡=0,…,𝑇−1. 𝑢_𝑡=(𝑢¹_𝑡,…,𝑢^𝑛_𝑡)^′ is called an investment decision at the 𝑡th investment period and 𝑢=(𝑢₀,…,𝑢_𝑇−1) is an investment policy during the entire investment horizon. We denote the wealth and the surplus of the investor at time 𝑡 by 𝑥_𝑡 and 𝑆_𝑡=𝑥_𝑡−𝑙_𝑡, respectively.

A single-period conditional risk mapping is defined as 𝜌_𝑡𝑆_𝑡+1∣ℱ_𝑡=Var_𝑡𝑆_𝑡+1−𝜆_𝑡𝐸_𝑡𝑆_𝑡+1, (1) where 𝐸_𝑡(𝑆_𝑡+1)=𝐸(𝑆_𝑡+1∣ℱ_𝑡) and Var_𝑡(𝑆_𝑡+1)=Var(𝑆_𝑡+1∣ℱ_𝑡) are the single-period conditional expectation of 𝑆_𝑡+1 and its conditional variance, respectively. Tradeoff parameter 𝜆_𝑡 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 𝜆_𝑡=0, then the investor is arbitrary risk averse who only focus on the risk; if 𝜆_𝑡=∞, then the investor is risk neutral who only concerns maximizing their expected profit; if 𝜆_𝑡∈(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 𝑢_𝑡^𝑚𝑦 is a solution to the following problem: min_{𝑢𝑡∈Π𝑡}Var_𝑡𝑆_𝑡+1−𝜆_𝑡𝐸_𝑡𝑆_𝑡+1, (2) where Π_𝑡 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 time 𝑡), we employ a separable expected conditional mapping which is defined as (see Chen et al. [16]) ^𝑇_𝑗=𝑡+1𝐸_𝑡𝜌_𝑗−1𝑆_𝑗∣ℱ_𝑗−1, (3) which reflects all the risk in the future. Following this assumption, a separable dynamic mean-variance problem is defined as min^𝑇_{𝑢∈Π𝑗=1}𝐸𝜌_𝑗−1𝑆_𝑗∣ℱ_𝑗−1, (4) 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(⋅∣ℱ_𝑗)∣ℱ_𝑘)=𝐸(⋅∣ℱ_𝑘) for 𝑗>𝑘, we have ^𝑇_𝑗=1𝐸𝜌_𝑗−1𝑆_𝑗∣ℱ_𝑗−1=𝜌₀𝑆₁+𝐸𝜌₁𝑆₂+𝐸₁𝜌₂𝑆₃+⋯+𝐸_𝑇−3𝜌_𝑇−2𝑆_𝑇−1+𝐸_𝑇−2𝜌_𝑇−1𝑆_𝑇… (5) (for more details see Chen et al. [16]). Then, problem (4) is equivalent to min_𝑢∈Π𝜌₀𝑆₁+𝐸𝜌₁𝑆₂+𝐸₁𝜌₂𝑆₃+⋯+𝐸_𝑇−3𝜌_𝑇−2𝑆_𝑇−1+𝐸_𝑇−2𝜌_𝑇−1𝑆_𝑇…. (6) It follows from Bellman's optimality principle and (6) that (4) is equivalent to find an optimal strategy to satisfy the following problem: min_𝑢0𝜌₀𝑆₁+𝐸min_𝑢1𝜌₁𝑆₂+𝐸₁min_𝑢2𝜌₂𝑆₃+⋯+𝐸_𝑇−3min_𝑢𝑇−2𝜌_𝑇−2𝑆_𝑇−1+𝐸_𝑇−2min_𝑢𝑇−1𝜌_𝑇−1𝑆_𝑇…. (7)

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 Sections 3 and 4, respectively.

3. Time Consistent Optimal Strategy without Riskless Asset

Consider a market consisting of only 𝑛 risky assets and assume that the wealth process 𝑥_𝑡 is in a self-financing fashion. We list the notations of this section in Table 1. The wealth process 𝑥_𝑡 could be described as follows: 𝑥_𝑡+1=𝑅^′_𝑡𝑢_𝑡,𝐼^′𝑢_𝑡=𝑥_𝑡,𝑡=0,1,…,𝑇−1, (8) where 𝐼=(1,…,1)^′∈ℝ^𝑛. In this setting, problem (6) can be written as follows: min𝜌₀𝑆₁+𝐸₀𝜌₁𝑆₂+𝐸₁𝜌₂𝑆₃+⋯+𝐸_𝑇−3𝜌_𝑇−2𝑆_𝑇−1+𝐸_𝑇−2𝜌_𝑇−1𝑆_𝑇…s.t.𝑅^′_𝑡𝑢_𝑡=𝑥_𝑡+1,𝐼^′𝑢_𝑡=𝑥_𝑡,𝑙_𝑡+1=𝑄_𝑡𝑙_𝑡,𝑆_𝑡+1=𝑥_𝑡+1−𝑙_𝑡+1,𝑡=0,1,…,𝑇−1. (9)

Table 1 

Model notations of Section 3.

Since Σ¹_𝑇−1 is positive definite, we have 𝑎_𝑇−1>0. Further, it follows from the nonnegative definiteness of 𝐸(𝑅_𝑇−1𝑅^′_𝑇−1) that Σ¹_𝑇−2 is also positive definite and 𝑎_𝑇−2>0. By using mathematical induction, we conclude that, for any 𝑡=0,…,𝑇−1, Σ¹_𝑡 is also positive definite and 𝑎_𝑡>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 𝑢^∗_𝑡=Σ¹_𝑡⁻¹𝐼𝑎_𝑡𝑥_𝑡+𝜆¹_𝑡2Σ¹_𝑡⁻¹𝜇_𝑡−𝑞_𝑡𝐼+Σ¹_𝑡⁻¹Σ⁰_𝑡−𝛾_𝑡𝐼+𝛾_𝑡+1𝐸𝑄_𝑡𝑅_𝑡𝑙_𝑡,𝑓𝑜𝑟𝑡=0,…,𝑇−1. (10)

Proof. When 𝑡=𝑇−1, for given wealth 𝑥_𝑇−1 and liability 𝑙_𝑇−1 at the beginning of the (𝑇−1)th period, problem (9) can be expressed as follows: minVar_𝑇−1𝑆_𝑇−𝜆_𝑇−1𝐸_𝑇−1𝑆_𝑇s.t.𝑅^′_𝑇−1𝑢_𝑇−1=𝑥_𝑇,𝐼^′𝑢_𝑇−1=𝑥_𝑇−1,𝑙_𝑇=𝑄_𝑇−1𝑙_𝑇−1,𝑆_𝑇=𝑥_𝑇−𝑙_𝑇. (11) Substituting the binding constraints into the objective function, we have min𝑉_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1∶=𝑢^′_𝑇−1Σ¹_𝑇−1𝑢_𝑇−1−2Σ⁰_𝑇−1^′𝑙_𝑇−1+𝜆_𝑇−12𝑢^′_𝑇−1𝑢_𝑇−1+Var𝑄_𝑇−1𝑙²_𝑇−1+𝜆_𝑇−1𝐸𝑄_𝑇−1𝑙_𝑇−1s.t.𝐼^′𝑢_𝑇−1=𝑥_𝑇−1, (12) which is a linear-quadratic program. By using the Lagrange multiplier technique and letting 𝜔_𝑇 be the Lagrange multiplier, the Lagrange function is defined as 𝐿𝑢_𝑇−1∶=𝑢^′_𝑇−1Σ¹_𝑇−1𝑢_𝑇−1−2Σ⁰_𝑇−1^′𝑙_𝑇−1+𝜆_𝑇−12𝜇^′_𝑇−1𝑢_𝑇−1+Var𝑄_𝑇−1𝑙²_𝑇−1+𝜆_𝑇−1𝐸𝑄_𝑇−1𝑙_𝑇−1−𝜔_𝑇𝑥_𝑇−1−𝐼^′𝑢_𝑇−1. (13) By using the first-order necessary optimality condition, we have 2Σ¹_𝑇−1𝑢_𝑇−1−2Σ⁰_𝑇−1𝑙_𝑇−1+𝜆_𝑇−12𝜇_𝑇−1+𝜔_𝑇𝑥𝐼=0,_𝑇−1−𝐼^′𝑢_𝑇−1=0. (14)(15) From (14), we can easily have 𝑢^∗_𝑇−1=Σ¹_𝑇−1⁻¹Σ⁰_𝑇−1𝑙_𝑇−1+𝜆_𝑇−12𝜇_𝑇−1−12𝜔_𝑇Σ¹_𝑇−1⁻¹𝐼𝐼,_′𝑢^∗_𝑇−1=𝐼^′Σ¹_𝑇−1⁻¹Σ⁰_𝑇−1𝑙_𝑇−1+𝜆_𝑇−12𝜇_𝑇−1−12𝜔_𝑇𝐼^′Σ¹_𝑇−1⁻¹𝐼=𝑑_𝑇−1𝑙_𝑇−1+𝜆_𝑇−12𝑏_𝑇−1−12𝑎_𝑇−1𝜔_𝑇=𝑥_𝑇−1, (16)(17) which implies that the Lagrange multiplier 𝜔_𝑇 is 𝜔_𝑇𝑑=2^𝑇−1𝑎_𝑇−1𝑙_𝑇−1+𝜆_𝑇−1𝑏_𝑇−1𝑎_𝑇−11−2𝑎_𝑇−1𝑥_𝑇−1=2𝛾_𝑇−1𝑙_𝑇−1+𝜆_𝑇−1𝑞_𝑇−11−2𝑎_𝑇−1𝑥_𝑇−1. (18) Substituting 𝜔_𝑇 into (16), we have 𝑢^∗_𝑇−1=Σ¹_𝑇−1⁻¹𝐼𝑎_𝑇−1𝑥_𝑇−1+Σ¹_𝑇−1⁻¹Σ⁰_𝑇−1−𝛾_𝑇−1𝐼𝑙_𝑇−1+𝜆_𝑇−12Σ¹_𝑇−1⁻¹𝜇_𝑇−1−𝑞_𝑇−1𝐼. (19) Further, by substituting 𝑢^∗_𝑇−1 into the objective function of problem (12) (see Appendix A for more details), we have 𝑉^∗_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1=1𝑎_𝑇−1𝑥²_𝑇−1−𝜆_𝑇−1𝑞_𝑇−1𝑥_𝑇−1−2𝛾_𝑇−1𝑥_𝑇−1𝑙_𝑇−1+𝛽_1,𝑇−1𝑙_𝑇−1+𝛽_2,𝑇−1𝑙²_𝑇−1−𝜆_𝑇−1²4𝛼_𝑇−1. (20)
When 𝑡=𝑇−2, for given wealth 𝑥_𝑇−2 and liability 𝑙_𝑇−2 at the beginning of the (𝑇−2)th period, the corresponding optimal investment problem is given as follows: minVar_𝑇−2𝑆_𝑇−1−𝜆_𝑇−2𝐸_𝑇−2𝑆_𝑇−1+𝐸_𝑇−2𝑉^∗_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1s.t.𝑅^′_𝑇−2𝑢_𝑇−2=𝑥_𝑇−1,𝐼^′𝑢_𝑇−2=𝑥_𝑇−2,𝑙_𝑇−1=𝑄_𝑇−2𝑙_𝑇−2,𝑆_𝑇−1=𝑥_𝑇−1−𝑙_𝑇−1. (21) It follows from (20) that 𝐸_𝑇−2𝑉^∗_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1=1𝑎_𝑇−1𝐸_𝑇−2𝑥²_𝑇−1−𝜆_𝑇−1𝑞_𝑇−1𝐸_𝑇−2𝑥_𝑇−1−2𝛾_𝑇−1𝐸_𝑇−2𝑥_𝑇−1𝑙_𝑇−1+𝛽_1,𝑇−1𝐸_𝑇−2𝑙_𝑇−1+𝛽_2,𝑇−1𝐸_𝑇−2𝑙²_𝑇−1−𝜆_𝑇−1²4𝛼_𝑇−1. (22) Substituting 𝐸_𝑇−2(𝑉^∗_𝑇−1(𝑥_𝑇−1,𝑙_𝑇−1)) and the binding constraints into the objective function of problem (21), we have min𝑉_𝑇−2𝑥_𝑇−2,𝑙_𝑇−2s.t.𝐼^′𝑢_𝑇−2=𝑥_𝑇−2, (23) where 𝑉_𝑇−2𝑥_𝑇−2,𝑙_𝑇−2∶=𝑢^′_𝑇−2Σ¹_𝑇−2𝑢_𝑇−2−2Σ⁰_𝑇−2^′+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅^′_𝑇−2𝑙_𝑇−2+𝜆¹_𝑇−22𝜇^′_𝑇−2𝑢_𝑇−2+Var𝑄_𝑇−2+𝛽_2,𝑇−1𝐸𝑄²_𝑇−2𝑙²_𝑇−2+𝜆_𝑇−2𝐸𝑄_𝑇−2+𝛽_1,𝑇−1𝐸𝑄_𝑇−2𝑙_𝑇−1−𝜆_𝑇−1²4𝛼_𝑇−1. (24) By setting 𝜔_𝑇−1 be the Lagrange multiplier, the Lagrange function for problem (23) is 𝐿𝑢_𝑇−2∶=𝑢^′_𝑇−2Σ¹_𝑇−2𝑢_𝑇−2−2Φ^′_𝑇−2𝑢_𝑇−2+Ψ_𝑇−2−𝜔_𝑇−1𝑥_𝑇−2−𝐼^′𝑢_𝑇−2, (25) where Φ_𝑇−2∶=Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2𝑙_𝑇−2+𝜆¹_𝑇−22𝜇_𝑇−2,Ψ_𝑇−2∶=Var𝑄_𝑇−2+𝛽_2,𝑇−1𝐸𝑄²_𝑇−2𝑙²_𝑇−2+𝜆_𝑇−2𝐸𝑄_𝑇−2+𝛽_1,𝑇−1𝐸𝑄_𝑇−2𝑙_𝑇−1−𝜆_𝑇−1²4𝛼_𝑇−1. (26) From the first-order necessary optimality condition, we have 2Σ¹_𝑇−2𝑢_𝑇−2−2Φ_𝑇−2+𝜔_𝑇−1𝑥𝐼=0,_𝑇−2−𝐼^′𝑢_𝑇−2=0. (27)(28) From (27), we have 𝑢^∗_𝑇−2=Σ¹_𝑇−2⁻¹Φ_𝑇−2−12𝜔_𝑇−1Σ¹_𝑇−2⁻¹𝐼𝐼,_′𝑢^∗_𝑇−2=𝐼^′Σ¹_𝑇−2⁻¹Φ_𝑇−2−12𝜔_𝑇−1𝐼^′Σ¹_𝑇−2⁻¹𝐼=𝐼^′Σ¹_𝑇−2⁻¹Φ_𝑇−2−12𝜔_𝑇−1𝑎_𝑇−2=𝑥_𝑇−2, (29)(30) which implies 𝜔_𝑇−1𝐼=2^′Σ¹_𝑇−2⁻¹Φ_𝑇−2𝑎_𝑇−21−2𝑎_𝑇−2𝑥_𝑇−2. (31) Substituting 𝜔_𝑇−1 into (29), we get 𝑢^∗_𝑇−2=Σ¹_𝑇−2⁻¹𝐼𝑎_𝑇−2𝑥_𝑇−2+Σ¹_𝑇−2⁻¹Φ_𝑇−2−𝐼^′Σ¹_𝑇−2⁻¹Φ_𝑇−2𝑎_𝑇−2Σ¹_𝑇−2⁻¹𝐼. (32) Taking Φ_𝑇−2 into account, we have 𝑢^∗_𝑇−2=Σ¹_𝑇−2⁻¹𝐼𝑎_𝑇−2𝑥_𝑇−2+𝜆¹_𝑇−22Σ¹_𝑇−2⁻¹𝜇_𝑇−2−𝑞_𝑇−2𝐼+Σ¹_𝑇−2⁻¹Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2−𝛾_𝑇−2𝐼𝑙_𝑇−2. (33) Substituting 𝑢^∗_𝑇−2 into (23) gives (see Appendix B for more details) 𝑉^∗_𝑇−2𝑥_𝑇−2,𝑙_𝑇−2=1𝑎_𝑇−2𝑥²_𝑇−2−𝜆¹_𝑇−2𝑞_𝑇−2𝑥_𝑇−2−2𝛾_𝑇−2𝑥_𝑇−2𝑙_𝑇−2+𝛽_1,𝑇−2𝑙_𝑇−2+𝛽_2,𝑇−2𝑙²_𝑇−2−𝜆_𝑇−1²4𝛼_𝑇−1−𝜆¹_𝑇−2²4𝛼_𝑇−2. (34)
Next, by using mathematical induction, we show that both (10) and 𝑉^∗_𝑡𝑥_𝑡,𝑙_𝑡=1𝑎_𝑡𝑥²_𝑡−𝜆¹_𝑡𝑞_𝑡𝑥_𝑡−2𝛾_𝑡𝑥_𝑡𝑙_𝑡+𝛽_1,𝑡𝑙_𝑡+𝛽_2,𝑡𝑙²_𝑡−^𝑇−1_𝑖=𝑡𝜆¹_𝑖²4𝛼_𝑖 (35) 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 𝑥_𝑡−1 and liability 𝑙_𝑡−1, the corresponding optimal investment problem is minVar_𝑡−1𝑆_𝑡−𝜆_𝑡−1𝐸_𝑡−1𝑆_𝑡+𝐸_𝑡−1𝑉^∗_𝑡𝑥_𝑡,𝑙_𝑡s.t.𝑅^′_𝑡−1𝑢_𝑡−1=𝑥_𝑡,𝐼^′𝑢_𝑡−1=𝑥_𝑡−1,𝑙_𝑡=𝑄_𝑡−1𝑙_𝑡−1,𝑆_𝑡=𝑥_𝑡−𝑙_𝑡. (36) It follows from (35) that 𝐸_𝑡−1𝑉^∗_𝑡𝑥_𝑡,𝑙_𝑡=1𝑎_𝑡𝐸_𝑡−1𝑥²_𝑡−𝜆¹_𝑡𝑞_𝑡𝐸_𝑡−1𝑥_𝑡−2𝛾_𝑡𝐸_𝑡−1𝑥_𝑡𝑙_𝑡+𝛽_1,𝑡𝐸_𝑡−1𝑙_𝑡+𝛽_2,𝑡𝐸_𝑡−1𝑙²_𝑡−^𝑇−1_𝑖=𝑡𝜆¹_𝑖²4𝛼_𝑖. (37) Substituting 𝐸_𝑡−1(𝑉^∗_𝑡(𝑥_𝑡,𝑙_𝑡)) and the binding constraints into the objective function of problem (36), we have min𝑉_𝑡−1𝑥_𝑡−1,𝑙_𝑡−1∶=𝑢^′_𝑇−1Σ¹_𝑡−1𝑢_𝑡−1−2Φ^′_𝑡−1𝑢_𝑡−1+Ψ_𝑡−1s.t.𝐼^′𝑢_𝑡−1=𝑥_𝑡−1, (38) where Φ_𝑡−1∶=Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1𝑙_𝑡−1+𝜆¹_𝑡−12𝜇_𝑡−1,Ψ_𝑡−1∶=Var𝑄_𝑡−1+𝛽_2,𝑡𝐸𝑄²_𝑡−1𝑙²_𝑡−1+𝜆_𝑡−1+𝛽_1,𝑡𝐸𝑄_𝑡−1𝑙_𝑡−1−^𝑇−1_𝑖=𝑡𝜆¹_𝑖²4𝛼_𝑖. (39) Letting 𝜔_𝑡 be the Lagrange multiplier, the Lagrange function for problem (38) is given by 𝐿𝑢_𝑡−1∶=𝑢^′_𝑇−1Σ¹_𝑡−1𝑢_𝑡−1−2Φ^′_𝑡−1𝑢_𝑡−1+Ψ_𝑡−1−𝜔_𝑡𝑥_𝑡−1−𝐼^′𝑢_𝑡−1. (40) It follows from the first-order necessary optimality condition that 2Σ¹_𝑡−1𝑢_𝑡−1−2Φ_𝑡−1+𝜔_𝑡𝑥𝐼=0,_𝑡−1−𝐼^′𝑢_𝑡−1=0. (41) Thus, we have 𝑢^∗_𝑡−1=Σ¹_𝑡−1⁻¹Φ_𝑡−1−12𝜔_𝑡Σ¹_𝑡−1⁻¹𝐼𝐼,_′𝑢^∗_𝑡−1=𝐼^′Σ¹_𝑡−1⁻¹Φ_𝑡−1−12𝜔_𝑡𝐼^′Σ¹_𝑡−1⁻¹𝐼=𝐼^′Σ¹_𝑡−1⁻¹Φ_𝑡−1−12𝜔_𝑡𝑎_𝑡−1=𝑥_𝑡−1, (42)(43) which implies 𝜔_𝑡𝐼=2^′Σ¹_𝑡−1⁻¹Φ_𝑡−1𝑎_𝑡−11−2𝑎_𝑡−1𝑥_𝑡−1. (44) It follows from (42) that 𝑢^∗_𝑡−1=Σ¹_𝑡−1⁻¹𝐼𝑎_𝑡−1𝑥_𝑡−1−𝐼^′Σ¹_𝑡−1⁻¹Φ_𝑡−1𝑎_𝑡−1Σ¹_𝑡−1⁻¹𝐼+Σ¹_𝑡−1⁻¹Φ_𝑡−1=Σ¹_𝑡−1⁻¹𝐼𝑎_𝑡−1𝑥_𝑡−1+𝜆¹_𝑡−12Σ¹_𝑡−1⁻¹𝜇_𝑡−1−𝑞_𝑡−1𝐼+Σ¹_𝑡−1⁻¹Σ⁰_𝑡−1−𝛾_𝑡−1𝐼+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1𝑙_𝑡−1. (45) Substituting 𝑢^∗_𝑡−1 into the objective function of problem (38) (see Appendix C for more details), we have 𝑉^∗_𝑡−1𝑥_𝑡−1,𝑙_𝑡−1=1𝑎_𝑡−1𝑥²_𝑡−1−𝜆¹_𝑡−1𝑞_𝑡−1𝑥_𝑡−1−2𝛾_𝑡−1𝑥_𝑡−1𝑙_𝑡−1+𝛽_1,𝑡−1𝑙_𝑡−1+𝛽_2,𝑡−1𝑙²_𝑡−1−^𝑇−1_{𝑖=𝑡−1}𝜆¹_𝑖²4𝛼_𝑖. (46) This completes the proof.

Remark 2. If an investor does not have any liability, that is, 𝑙_𝑡≡0 for any 𝑡∈{0,1,…,𝑇−1}, then the optimal time-consistent investment strategy can be simplified as follows: 𝑢^∗_𝑡=Σ¹_𝑡⁻¹𝐼𝑎_𝑡𝑥_𝑡+𝜆¹_𝑡2Σ¹_𝑡⁻¹𝜇_𝑡−𝑞_𝑡𝐼,𝑡=0,…,𝑇−1, (47) 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, Σ⁰_𝑡=0 for any 𝑡∈{0,1,…,𝑇−1}, then the optimal investment policy for problem (9) is 𝑢^∗_𝑡=Σ¹_𝑡⁻¹𝐼𝑎_𝑡𝑥_𝑡+𝜆¹_𝑡2Σ¹_𝑡⁻¹𝜇_𝑡−𝑞_𝑡𝐼,𝑡=0,…,𝑇−1. (48)

Proof. Since Σ⁰_𝑡=0, it is easy to verify that 𝑑_𝑡=𝐼^′(Σ¹_𝑡)⁻¹Σ⁰_𝑡=0, 𝛾_𝑡=𝑑_𝑡/𝑎_𝑡=0 and 𝛾_𝑡=0. Substituting them into (10) gives 𝑢^∗_𝑡=Σ¹_𝑡⁻¹𝐼𝑎_𝑡𝑥_𝑡+𝜆¹_𝑡2Σ¹_𝑡⁻¹𝜇_𝑡−𝑞_𝑡𝐼,𝑡=0,…,𝑇−1. (49) This completes the proof.

Remark 4. After comparing Corollary 3 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 Σ¹_𝑡⁻¹Σ⁰_𝑡−𝛾_𝑡𝐼+𝛾_𝑡+1𝐸𝑄_𝑡𝑅_𝑡𝑙_𝑡, (50) which depends on the current value of liability, 𝑙_𝑡, and the covariance between the returns of liability and risky assets, Σ⁰_𝑡.

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: min_𝑢𝑡Var_𝑡𝑆_𝑡+1−𝜆_𝑡𝐸_𝑡𝑆_𝑡+1s.t.𝑅^′_𝑡𝑢_𝑡=𝑥_𝑡+1,𝐼^′𝑢_𝑡=𝑥_𝑡,𝑙_𝑡+1=𝑄_𝑡𝑙_𝑡,𝑆_𝑡+1=𝑥_𝑡+1−𝑙_𝑡+1. (51) By using the same method in the proof of Theorem 1 for time 𝑇−1, the myopic strategy is given by 𝑢_𝑡^𝑚𝑦Σ∶=^𝑡−1𝐼𝑎_𝑡𝑥_𝑡+𝜆_𝑡2Σ_𝑡⁻¹𝜇_𝑡−𝑞_𝑡𝐼+Σ_𝑡⁻¹Σ⁰_𝑡−𝛾_𝑡𝐼𝑙_𝑡,𝑡=0,…,𝑇−1, (52) where 𝑎_𝑡=𝐼^′(Σ_𝑡)⁻¹𝐼, 𝑞_𝑡=𝐼^′(Σ_𝑡)⁻¹𝜇_𝑡/𝑎_𝑡, 𝛾_𝑡=𝐼^′(Σ_𝑡)⁻¹Σ⁰_𝑡/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, 𝜆_𝑡→0, then both the time consistent optimal strategy and myopic strategy reduce to 𝑢^∗_𝑡=Σ¹_𝑡⁻¹𝐼𝑎_𝑡𝑥_𝑡+Σ¹_𝑡⁻¹Σ⁰_𝑡−𝛾_𝑡𝐼+𝛾_𝑡+1𝐸𝑄_𝑡𝑅_𝑡𝑙_𝑡,𝑢𝑡=0,…,𝑇−1,_𝑡𝑚𝑦=Σ_𝑡⁻¹𝐼𝑎_𝑡𝑥_𝑡+Σ_𝑡⁻¹Σ⁰_𝑡−𝛾_𝑡𝐼𝑙_𝑡,𝑡=0,…,𝑇−1, (53) 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 and 𝑛 risky assets and assume that the wealth process 𝑥_𝑡 is also in a self-financing fashion. We list the notations of this section in Table 2. The wealth process 𝑥_𝑡 can be described as follows: 𝑥_𝑡=𝑃^′_𝑡−1𝑢_𝑡−1+𝑠_𝑡−1𝑥_𝑡−1, (54) where 𝑃_𝑡=𝑅_𝑡−𝑠_𝑡𝐼. In this setting, problem (6) can be written as follows: min𝜌₀𝑆₁+𝐸₀𝜌₁𝑆₂+𝐸₁𝜌₂𝑆₃+⋯+𝐸_𝑇−3𝜌_𝑇−2𝑆_𝑇−1+𝐸_𝑇−2𝜌_𝑇−1𝑆_𝑇…s.t.𝑥_𝑡+1=𝑃^′_𝑡𝑢_𝑡+𝑠_𝑡𝑥_𝑡,𝑙_𝑡+1=𝑄_𝑡𝑙_𝑡,𝑆_𝑡+1=𝑥_𝑡+1−𝑙_𝑡+1,𝑡=0,1,…,𝑇−1. (55)

Table 2 

Model notations of Section 4.

By 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 𝑢^∗_𝑡=Σ_𝑡⁻¹𝜑_𝑡𝑙_𝑡+𝜆¹_𝑡2Σ_𝑡⁻¹𝜖_𝑡,𝑡=0,…,𝑇−1, (56) where 𝜑_𝑡=𝐸(𝑃_𝑡𝑄_𝑡)−𝐸(𝑃_𝑡)𝐸(𝑄_𝑡) and 𝜖_𝑡=𝐸(𝑃_𝑡).

Proof. When 𝑡=𝑇−1, for given wealth 𝑥_𝑇−1 and liability 𝑙_𝑇−1 at the beginning of the (𝑇−1)th period, problem (55) reduces to min_𝑢𝑇−1Var_𝑇−1𝑆_𝑇−𝜆_𝑇−1𝐸_𝑇−1𝑆_𝑇s.t.𝑥_𝑇=𝑠_𝑇−1𝑥_𝑇−1+𝑃^′_𝑇−1𝑢_𝑇−1,𝑙_𝑇=𝑄_𝑇−1𝑙_𝑇−1,𝑆_𝑇=𝑥_𝑇−𝑙_𝑇. (57) Substituting the binding constraints into the objective function, we have min_𝑢𝑇−1𝑉_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1, (58) where 𝑉_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1∶=𝑢^′_𝑇−1Σ_𝑇−1𝑢_𝑇−1−2𝜑^′_𝑇−1𝑙_𝑇−1+𝜆_𝑇−12𝜖^′_𝑇−1𝑢_𝑇−1+Var𝑄_𝑇−1𝑙²_𝑇−1−𝜆_𝑇−1𝑠_𝑇−1𝑥_𝑇−1+𝜆_𝑇−1𝐸𝑄_𝑇−1𝑙_𝑇−1. (59) It is clear that problem (58) is an unconstrained convex program problem. By using the first-order necessary optimality condition, we have 𝑑𝑉_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1𝑑𝑢_𝑇−1=2Σ_𝑇−1𝑢_𝑇−1−2𝜑_𝑇−1𝑙_𝑇−1+𝜆_𝑇−12𝜖_𝑇−1=0, (60) which implies 𝑢^∗_𝑇−1=Σ_𝑇−1⁻¹𝜑_𝑇−1𝑙_𝑇−1+𝜆_𝑇−12𝜖_𝑇−1. (61) Substituting 𝑢^∗_𝑇−1 into the objective function of problem (58) gives (see Appendix D for more details) 𝑉^∗_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1=𝛽_2,𝑇−1𝑙²_𝑇−1+𝛽_1,𝑇−1𝑙_𝑇−1−𝜆_𝑇−1𝑠_𝑇−1𝑥_𝑇−1−𝜆_𝑇−1²4𝑀_𝑇−1, (62) where 𝑀_𝑇−1=𝜖^′_𝑇−1Σ⁻¹_𝑇−1𝜖_𝑇−1.
When 𝑡=𝑇−2, for given wealth 𝑥_𝑇−2 and liability 𝑙_𝑇−2 at the beginning of the (𝑇−2)th period, the corresponding optimal investment problem can be expressed as follows: minVar_𝑇−2𝑆_𝑇−1−𝜆_𝑇−2𝐸_𝑇−2𝑆_𝑇−1+𝐸_𝑇−2𝑉^∗_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1s.t.𝑥_𝑇−1=𝑠_𝑇−2𝑥_𝑇−2+𝑃^′_𝑇−2𝑢_𝑇−2,𝑙_𝑇−1=𝑄_𝑇−2𝑙_𝑇−2,𝑆_𝑇−1=𝑥_𝑇−1−𝑙_𝑇−1. (63) From (62), we can easily have 𝐸_𝑇−2𝑉^∗_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1=𝛽_2,𝑇−1𝐸𝑄²_𝑇−2𝑙²_𝑇−2+𝛽_1,𝑇−1𝐸𝑄_𝑇−2𝑙_𝑇−2−𝜆_𝑇−1𝑠_𝑇−1𝑠_𝑇−2𝑥_𝑇−2−𝜆_𝑇−1𝑠_𝑇−1𝜖^′_𝑇−2𝑢_𝑇−2−𝜆_𝑇−1²4𝑀_𝑇−1. (64) Substituting 𝐸_𝑇−2(𝑉^∗_𝑇−1(𝑥_𝑇−1,𝑙_𝑇−1)) and the binding constraints into the objective function of problem (63), we have min_𝑢𝑇−2𝑉_𝑇−2𝑥_𝑇−2,𝑙_𝑇−2, (65) where 𝑉_𝑇−2𝑥_𝑇−2,𝑙_𝑇−2=𝑢^′_𝑇−2Σ_𝑇−2𝑢_𝑇−2−2𝜑^′_𝑇−2𝑙_𝑇−2+𝜆¹_𝑇−22𝜖^′_𝑇−2𝑢_𝑇−2+Var𝑄_𝑇−2+𝛽_2,𝑇−1𝐸𝑄²_𝑇−2𝑙²_𝑇−2+𝜆_𝑇−2𝐸𝑄_𝑇−2+𝛽_1,𝑇−1𝐸𝑄_𝑇−2𝑙_𝑇−1−𝜆¹_𝑇−2𝑠_𝑇−2𝑥_𝑇−2−𝜆_𝑇−1²4𝑀_𝑇−1. (66) The first-order necessary optimality condition implies 𝑑𝑉_𝑇−2𝑥_𝑇−2,𝑙_𝑇−2𝑑𝑢_𝑇−2=2Σ_𝑇−2𝑢_𝑇−2−2𝜑_𝑇−2𝑙_𝑇−2+𝜆¹_𝑇−22𝜖_𝑇−2=0. (67) Thus, 𝑢^∗_𝑇−2=Σ_𝑇−2⁻¹𝜑_𝑇−2𝑙_𝑇−2+𝜆¹_𝑇−22𝜖_𝑇−2=Σ_𝑇−2⁻¹𝜑_𝑇−2𝑙_𝑇−2+𝜆¹_𝑇−22Σ_𝑇−2⁻¹𝜖_𝑇−2. (68) Substituting 𝑢^∗_𝑇−2 into the objective function of problem (65) (see Appendix E for more details), we have 𝑉^∗_𝑇−2𝑥_𝑇−2,𝑙_𝑇−2=𝛽_2,𝑇−2𝑙²_𝑇−2+𝛽_1,𝑇−2𝑙_𝑇−2−𝜆¹_𝑇−2𝑠_𝑇−2𝑥_𝑇−2−𝜆_𝑇−1²4𝑀_𝑇−1−𝜆¹_𝑇−2²4𝑀_𝑇−2, (69) where 𝑀_𝑇−2=𝜖^′_𝑇−2Σ⁻¹_𝑇−2𝜖_𝑇−2.
Next, by using mathematical induction, we show that both (56) and 𝑉^∗_𝑡𝑥_𝑡,𝑙_𝑡=𝛽_2,𝑡𝑙²_𝑡+𝛽_1,𝑡𝑙_𝑡−𝜆¹_𝑡𝑠_𝑡𝑥_𝑡−^𝑇−1_𝑖=𝑡𝜆¹_𝑖²4𝑀_𝑖 (70) hold, where 𝑀_𝑖=𝜖^′_𝑖Σ_𝑖⁻¹𝜖_𝑖 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 𝑥_𝑡−1 and liability 𝑙_𝑡−1, the corresponding optimal investment problem is minVar_𝑡−1𝑆_𝑡−𝜆_𝑡−1𝐸_𝑡−1𝑆_𝑡+𝐸_𝑡−1𝑉^∗_𝑡𝑥_𝑡,𝑙_𝑡s.t.𝑥_𝑡=𝑠_𝑡−1𝑥_𝑡−1+𝑃^′_𝑡−1𝑢_𝑡−1,𝑙_𝑡=𝑄_𝑡−1𝑙_𝑡−1,𝑆_𝑡=𝑥_𝑡−𝑙_𝑡. (71) It follows from (70) that 𝐸_𝑡−1𝑉^∗_𝑡𝑥_𝑡,𝑙_𝑡=𝛽_2,𝑡𝐸𝑄²_𝑡−1𝑙²_𝑡−1+𝛽_1,𝑡𝐸𝑄_𝑡−1𝑙_𝑡−𝜆¹_𝑡𝑠_𝑡𝑠_𝑡−1𝑥_𝑡−1−𝜆¹_𝑡𝑠_𝑡𝜖^′_𝑡−1𝑢_𝑡−1−^𝑇−1_𝑖=𝑡𝜆¹_𝑖²4𝑀_𝑖. (72) Substituting 𝐸_𝑡−1(𝑉^∗_𝑡(𝑥_𝑡,𝑙_𝑡)) and the binding constraints into the objective function of problem (71), we have min_𝑢𝑡−1𝑉_𝑡−1𝑥_𝑡−1,𝑙_𝑡−1, (73) where 𝑉_𝑡−1𝑥_𝑡−1,𝑙_𝑡−1=𝑢^′_𝑡−1Σ_𝑡−1𝑢_𝑡−1−2𝜑^′_𝑡−1𝑙_𝑡−1+𝜆¹_𝑡−12𝜖^′_𝑡−1𝑢_𝑡−1+Var𝑄_𝑡−1+𝛽_2,𝑡𝐸𝑄²_𝑡−1𝑙²_𝑡−1+𝜆_𝑡−1𝐸𝑄_𝑡−1+𝛽_1,𝑡𝐸𝑄_𝑡−1𝑙_𝑡−1−𝜆¹_𝑡−1𝑠_𝑡−1𝑥_𝑡−1−^𝑇−1_𝑖=𝑡𝜆¹_𝑖²4𝑀_𝑖. (74) The first-order necessary optimality condition gives 𝑑𝑉_𝑡−1𝑥_𝑡−1,𝑙_𝑡−1𝑑𝑢_𝑡−1=2Σ_𝑡−1𝑢_𝑡−1−2𝜑_𝑡−1𝑙_𝑡−1+𝜆¹_𝑡−12𝜖_𝑡−1=0, (75) which implies 𝑢^∗_𝑡−1=Σ⁻¹_𝑡−1𝜑_𝑡−1𝑙_𝑡−1+𝜆¹_𝑡−12Σ⁻¹_𝑡−1𝜖_𝑡−1. (76) Substituting 𝑢^∗_𝑡−1 into the objective function of problem (73) (see Appendix F for more details), we have 𝑉^∗_𝑡−1𝑥_𝑡−1,𝑙_𝑡−1=𝛽_2,𝑡−1𝑙²_𝑡−1+𝛽_1,𝑡−1𝑙_𝑡−1−𝜆¹_𝑡−1𝑠_𝑡−1𝑥_𝑡−1−^𝑇−1_{𝑖=𝑡−1}𝜆¹_𝑖²4𝑀_𝑖. (77) This completes the proof.

Remark 7. From Theorem 6, 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 𝑥_𝑡. 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, 𝑙_𝑡≡0 for any 𝑡∈{0,1,…,𝑇−1}, then the time-consistent optimal investment policy reduces to 𝑢^∗_𝑡=𝜆¹_𝑡2Σ_𝑡⁻¹𝜖_𝑡,𝑡=0,…,𝑇−1, (78) 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, 𝑄_𝑡, is uncorrelated with those of risky assets, 𝑅_𝑡, that is, Σ⁰_𝑡=0 for any 𝑡∈{0,1,…,𝑇−1}, then the optimal policy for problem (55) is 𝑢^∗_𝑡=𝜆¹_𝑡2Σ_𝑡⁻¹𝜖_𝑡,𝑡=0,…,𝑇−1. (79)

Proof. Since Σ⁰_𝑡=0, it is easy to have 𝜑_𝑡=𝐸𝑃_𝑡𝑄_𝑡−𝐸𝑃_𝑡𝐸𝑄_𝑡=𝐸𝑅_𝑡−𝑠_𝑡𝐼𝑄_𝑡−𝐸𝑅_𝑡−𝑠_𝑡𝐼𝐸𝑄_𝑡=𝐸𝑅_𝑡𝑄_𝑡−𝐸𝑅_𝑡𝐸𝑄_𝑡=Σ⁰_𝑡=0. (80) Substituting 𝜑_𝑡=0 into (56) gives 𝑢^∗_𝑡=𝜆¹_𝑡2Σ_𝑡⁻¹𝜖_𝑡,𝑡=0,…,𝑇−1. (81) This completes the proof.

Remark 10. After comparing Corollary 9 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 Σ_𝑡⁻¹𝜑_𝑡𝑙_𝑡, (82) 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: min_𝑢𝑡Var_𝑡𝑆_𝑡+1−𝜆_𝑡𝐸_𝑡𝑆_𝑡+1s.t.𝑥_𝑡+1=𝑠_𝑡𝑥_𝑡+𝑃^′_𝑡𝑢_𝑡,𝑙_𝑡+1=𝑄_𝑡𝑙_𝑡,𝑆_𝑡+1=𝑥_𝑡+1−𝑙_𝑡+1. (83) By using the same method in the proof of Theorem 6 for time 𝑇−1, the myopic strategy is given by 𝑢_𝑡^𝑚𝑦∶=Σ_𝑡⁻¹𝜑_𝑡𝑙_𝑡+𝜆_𝑡2Σ_𝑡⁻¹𝜖_𝑡,𝑡=0,…,𝑇−1. (84) 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 𝜆_𝑡. Further, the following feature holds; if an investor is arbitrarily risk averse, that is, 𝜆_𝑡→0, then both the time-consistent optimal investment policy and the myopic strategy reduce to 𝑢^∗_𝑡=Σ_𝑡⁻¹𝜑_𝑡𝑙_𝑡,𝑡=0,…,𝑇−1. (85) 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 assume 𝑇=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 𝜇_𝑡=(1.162,1.246,1.228) and Σ_𝑡=0.01460.01870.01450.01870.08540.01040.01450.01040.0289, (86) respectively. The expected return of the liability 𝐸(𝑄_𝑡) is 1.136, the corresponding variance Var(𝑄_𝑡) is 0.01, and the covariance vector Σ⁰_𝑡 is (0.0006,0.0149,0.0050)^′.

Table 3 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 for 𝜆₀=𝜆₁=0.5.

Tables 4 and 5 show the time-consistent strategy and the myopic strategy in a market without riskless asset for 𝜆_𝑡=0.5 and 𝜆_𝑡=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 for 𝜆₀=𝜆₁=0.5.

Table 5 

Investment strategies in a market without riskless asset for 𝜆₀=𝜆₁=0.

Next, we consider a market consisting of both risky assets and a riskless asset. Suppose that the return of the riskless asset 𝑠_𝑡 is 1.04. Then we have 𝜖_𝑡=𝜇_𝑡−𝑠_𝑡𝐼=(0.122,0.206,0.188)^′ and 𝜑_𝑡=Σ⁰_𝑡=(0.0006,0.0149,0.0050)^′.

Tables 6 and 7 show the time-consistent strategy and the myopic strategy in a market with a riskless asset for 𝜆_𝑡=0.5 and 𝜆_𝑡=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 for 𝜆₀=𝜆₁=0.5.

Table 7 

Investment strategies in a market with a riskless asset for 𝜆₀=𝜆₁=0.

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.

Appendices

A. Proof of (20)

Firstly, substituting 𝑢^∗_𝑇−1 into 𝑢^∗′_𝑇−1Σ¹_𝑇−1𝑢^∗_𝑇−1 gives 𝑢^∗′_𝑇−1Σ¹_𝑇−1𝑢^∗_𝑇−1=𝐼𝑎_𝑇−1𝑥_𝑇−1+Σ⁰_𝑇−1−𝛾_𝑇−1𝐼𝑙_𝑇−1+𝜆_𝑇−12𝜇_𝑇−1−𝑞_𝑇−1𝐼^′Σ¹_𝑇−1⁻¹×𝐼𝑎_𝑇−1𝑥_𝑇−1+Σ⁰_𝑇−1−𝛾_𝑇−1𝐼×𝑙_𝑇−1+𝜆_𝑇−12𝜇_𝑇−1−𝑞_𝑇−1𝐼=1𝑎_𝑇−1𝑥²_𝑇−1+Σ⁰_𝑇−1−𝛾_𝑇−1𝐼^′×Σ¹_𝑇−1⁻¹Σ⁰_𝑇−1−𝛾_𝑇−1𝐼𝑙²_𝑇−1+𝜆_𝑇−1Σ⁰_𝑇−1−𝛾_𝑇−1𝐼^′×Σ¹_𝑇−1⁻¹𝜇_𝑇−1−𝑞_𝑇−1𝐼𝑙_𝑇−1+𝜆_𝑇−1²4𝜇_𝑇−1−𝑞_𝑇−1𝐼^′×Σ¹_𝑇−1⁻¹𝜇_𝑇−1−𝑞_𝑇−1𝐼=1𝑎_𝑇−1𝑥²_𝑇−1+Σ⁰_𝑇−1^′Σ¹_𝑇−1⁻¹Σ⁰_𝑇−1−𝑎_𝑇−1𝛾²_𝑇−1𝑙²_𝑇−1+Σ⁰_𝑇−1^′Σ¹_𝑇−1⁻¹𝜇_𝑇−1−𝑏_𝑇−1𝛾_𝑇−1×𝜆_𝑇−1𝑙_𝑇−1+𝜆_𝑇−1²4𝛼_𝑇−1. (A.1)

Secondly, by substituting 𝑢^∗_𝑇−1 into 2[(Σ⁰_𝑇−1)^′𝑙_𝑇−1+(𝜆_𝑇−1/2)𝜇^′_𝑇−1]𝑢^∗_𝑇−1, we have 2Σ⁰_𝑇−1^′𝑙_𝑇−1+𝜆_𝑇−12𝜇^′_𝑇−1𝑢^∗_𝑇−1=2𝑎_𝑇−1Σ⁰_𝑇−1^′𝑙_𝑇−1+𝜆_𝑇−12𝜇^′_𝑇−1Σ¹_𝑇−1⁻¹𝐼𝑥_𝑇−1+2Σ⁰_𝑇−1^′𝑙_𝑇−1+𝜆_𝑇−12𝜇^′_𝑇−1Σ¹_𝑇−1⁻¹×Σ⁰_𝑇−1−𝛾_𝑇−1𝐼𝑙_𝑇−1+𝜆_𝑇−1Σ⁰_𝑇−1^′𝑙_𝑇−1+𝜆_𝑇−12𝜇^′_𝑇−1Σ¹_𝑇−1⁻¹×𝜇_𝑇−1−𝑞_𝑇−1𝐼=2𝛾_𝑇−1𝑙_𝑇−1𝑥_𝑇−1+𝑞_𝑇−1𝜆_𝑇−1𝑥_𝑇−1+2Σ⁰_𝑇−1^′Σ¹_𝑇−1⁻¹Σ⁰_𝑇−1−𝛾_𝑇−1𝑑_𝑇−1𝑙²_𝑇−1+2𝜆_𝑇−1Σ⁰_𝑇−1^′Σ¹_𝑇−1⁻¹𝜇_𝑇−1−𝑏_𝑇−1𝛾_𝑇−1𝑙_𝑇−1+𝜆_𝑇−1²2𝛼_𝑇−1. (A.2) Thus, 𝑉^∗_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1=1𝑎_𝑇−1𝑥²_𝑇−1−𝜆_𝑇−1𝑞_𝑇−1𝑥_𝑇−1−2𝛾_𝑇−1𝑥_𝑇−1𝑙_𝑇−1+𝜆_𝑇−1𝐸𝑄_𝑇−1−Σ⁰_𝑇−1^′Σ¹_𝑇−1⁻¹𝜇_𝑇−1+𝑏_𝑇−1𝛾_𝑇−1𝑙_𝑇−1+Var𝑄_𝑇−1−Σ⁰_𝑇−1^′Σ¹_𝑇−1⁻¹Σ⁰_𝑇−1+𝛾_𝑇−1𝑑_𝑇−1𝑙²_𝑇−1−𝜆_𝑇−1²4𝛼_𝑇−1=1𝑎_𝑇−1𝑥²_𝑇−1−𝜆_𝑇−1𝑞_𝑇−1𝑥_𝑇−1−2𝛾_𝑇−1𝑥_𝑇−1𝑙_𝑇−1+𝛽_1,𝑇−1𝑙_𝑇−1+𝛽_2,𝑇−1𝑙²_𝑇−1−𝜆_𝑇−1²4𝛼_𝑇−1. (A.3)

B. Proof of (34)

It is easy to verify that 𝐼^′𝑎_𝑇−2Σ¹_𝑇−2⁻¹Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2−𝛾_𝑇−2𝐼=𝑑_𝑇−2𝑎_𝑇−2+𝛾_𝑇−1𝐼^′Σ¹_𝑇−2⁻¹𝐸𝑄_𝑇−2𝑅_𝑇−2𝑎_𝑇−2−𝛾_𝑇−2=𝛾_𝑇−2+𝛾_𝑇−1𝐼^′Σ¹_𝑇−2⁻¹𝐸𝑄_𝑇−2𝑅_𝑇−2𝑎_𝑇−2−𝛾_𝑇−2=𝛾_𝑇−2−𝛾_𝑇−2𝐼=0,_′𝑎_𝑇−2Σ¹_𝑇−2⁻¹𝜇_𝑇−2−𝑞_𝑇−2𝐼=𝑏_𝑇−2𝑎_𝑇−2−𝑞_𝑇−2Σ=0,_0𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2^′Σ¹_𝑇−2⁻¹𝐼=𝑎_𝑇−2𝛾_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅^′_𝑇−2Σ¹_𝑇−2⁻¹𝐼𝑎_𝑇−2=𝑎_𝑇−2𝛾_𝑇−2. (B.1)

Firstly, we compute 𝑢^∗′_𝑇−2Σ¹_𝑇−2𝑢^∗_𝑇−2. It follows from the previous equalities that both coefficients of 𝑥_𝑇−2 and 𝑙_𝑇−2𝑥_𝑇−2 in 𝑢^∗′_𝑇−2Σ¹_𝑇−2𝑢^∗_𝑇−2 are 0; the coefficient of 𝑥²_𝑇−2 is 𝐼^′𝑎_𝑇−2Σ¹_𝑇−2⁻¹𝐼𝑎_𝑇−2=1𝑎_𝑇−2, (B.2) the coefficient of 𝑙_𝑇−2 is 𝜆¹_𝑇−2Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2−𝛾_𝑇−2𝐼^′×Σ¹_𝑇−2⁻¹𝜇_𝑇−2−𝑞_𝑇−2𝐼=𝜆¹_𝑇−2Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2^′×Σ¹_𝑇−2⁻¹𝜇_𝑇−2−𝑞_𝑇−2𝐼−𝜆¹_𝑇−2𝛾_𝑇−2𝐼^′Σ¹_𝑇−2⁻¹𝜇_𝑇−2−𝑞_𝑇−2𝐼=𝜆¹_𝑇−2Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2^′×Σ¹_𝑇−2⁻¹𝜇_𝑇−2−𝑞_𝑇−2𝐼, (B.3) the coefficient of 𝑙²_𝑇−2 is Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2−𝛾_𝑇−2𝐼^′⋅Σ¹_𝑇−2⁻¹⋅Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2−𝛾_𝑇−2𝐼=Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2^′×Σ¹_𝑇−2⁻¹Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2−𝑎_𝑇−2𝛾²_𝑇−2, (B.4) and the constant term is 𝜆¹_𝑇−2²4𝜇_𝑇−2−𝑞_𝑇−2𝐼^′Σ¹_𝑇−2⁻¹𝜇_𝑇−2−𝑞_𝑇−2𝐼=𝜆¹_𝑇−2²4𝛼_𝑇−2. (B.5)

Secondly, we compute Φ^′_𝑇−2𝑢^∗_𝑇−2. The coefficient of 𝑥²_𝑇−2 is 0, the coefficient of term 𝑥_𝑇−2 is 𝜆¹_𝑇−22𝜇^′_𝑇−2Σ¹_𝑇−2⁻¹𝐼𝑎_𝑇−2=𝜆¹_𝑇−2𝑞_𝑇−22, (B.6) the coefficient of 𝑙_𝑇−2𝑥_𝑇−2 is 𝛾_𝑇−2, the coefficient of 𝑙_𝑇−2 is 𝜆¹_𝑇−22Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2^′×Σ¹_𝑇−2⁻¹𝜇_𝑇−2−𝑞_𝑇−2𝐼+𝜆¹_𝑇−22𝜇^′_𝑇−2Σ¹_𝑇−2⁻¹×Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2−𝛾_𝑇−2𝐼, (B.7) the coefficient of 𝑙²_𝑇−2 is Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2^′Σ¹_𝑇−2⁻¹×Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2−𝛾_𝑇−2𝐼=Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2^′Σ¹_𝑇−2⁻¹×Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2−𝑎_𝑇−2𝛾²_𝑇−2, (B.8) and the constant term is 𝜆¹_𝑇−22²𝜇^′_𝑇−2Σ¹_𝑇−2⁻¹𝜇_𝑇−2−𝑞_𝑇−2𝐼=𝜆¹_𝑇−22²𝛼_𝑇−2. (B.9)

Then we can compute 𝑉^∗_𝑇−2(𝑥_𝑇−2,𝑙_𝑇−2) easily. The coefficient of 𝑥²_𝑇−2 is 𝐼^′𝑎_𝑇−2Σ¹_𝑇−2⁻¹𝐼𝑎_𝑇−2=1𝑎_𝑇−2, (B.10) the coefficient of 𝑥_𝑇−2 is −𝜆¹_𝑇−2𝑞_𝑇−2, the coefficient of 𝑙_𝑇−2𝑥_𝑇−2 is −2𝛾_𝑇−2, the coefficient of 𝑙²_𝑇−2 is Var𝑄_𝑇−2+𝛽_2,𝑇−1𝐸𝑄²_𝑇−2+𝑎_𝑇−2𝛾²_𝑇−2−Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2^′Σ¹_𝑇−2⁻¹×Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2=𝛽_2,𝑇−2, (B.11) the coefficient of 𝑙_𝑇−2 is −𝜆¹_𝑇−2𝜇^′_𝑇−2Σ¹_𝑇−2⁻¹×Σ⁰_𝑇−2+𝛾_𝑇−1𝐸𝑄_𝑇−2𝑅_𝑇−2−𝛾_𝑇−2𝐼+𝜆_𝑇−2𝐸𝑄_𝑇−2+𝛽_1,𝑇−1𝐸𝑄_𝑇−2=𝛽_1,𝑇−2, (B.12) and the constant term is −𝜆²_𝑇−14𝛼_𝑇−1−𝜆¹_𝑇−2²4𝛼_𝑇−2. (B.13) Thus, 𝑉^∗_𝑇−2𝑥_𝑇−2,𝑙_𝑇−2=1𝑎_𝑇−2𝑥²_𝑇−2−𝜆¹_𝑇−2𝑞_𝑇−2𝑥_𝑇−2−2𝛾_𝑇−2𝑥_𝑇−2𝑙_𝑇−2+𝛽_1,𝑇−2𝑙_𝑇−2+𝛽_2,𝑇−2𝑙²_𝑇−2−𝜆²_𝑇−14𝛼_𝑇−1−𝜆¹_𝑇−2²4𝛼_𝑇−2. (B.14)

C. Proof of (46)

It is easy to verify that 𝐼^′𝑎_𝑡−1Σ¹_𝑡−1⁻¹Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1−𝛾_𝑡−1𝐼=𝑑_𝑡−1𝑎_𝑡−1+𝛾_𝑡𝐼^′Σ¹_𝑡−1⁻¹𝐸𝑄_𝑡−1𝑅_𝑡−1𝑎_𝑡−1−𝛾_𝑡−1=𝛾_𝑡−1+𝛾_𝑡𝐼^′Σ¹_𝑡−1⁻¹𝐸𝑄_𝑡−1𝑅_𝑡−1𝑎_𝑡−1−𝛾_𝑡−1=𝛾_𝑡−1−𝛾_𝑡−1𝐼=0,_′𝑎_𝑡−1Σ¹_𝑡−1⁻¹𝜇_𝑡−1−𝑞_𝑡−1𝐼=𝑏_𝑡−1𝑎_𝑡−1−𝑞_𝑡−1Σ=0,_0𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1^′Σ¹_𝑡−1⁻¹𝐼=𝑎_𝑡−1𝛾_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅^′_𝑡−1Σ¹_𝑡−1⁻¹𝐼𝑎_𝑡−1=𝑎_𝑡−1𝛾_𝑡−1. (C.1)

Firstly, we compute 𝑢^∗′_𝑡−1Σ¹_𝑡−1𝑢^∗_𝑡−1. It follows from the previous equalities that both coefficients of 𝑥_𝑡−1 and 𝑙_𝑡−1𝑥_𝑡−1 are 0; the coefficient of 𝑥²_𝑡−1 is 𝐼^′𝑎_𝑡−1Σ¹_𝑡−1⁻¹𝐼𝑎_𝑡−1=1𝑎_𝑡−1, (C.2) the coefficient of 𝑙_𝑡−1 is 𝜆¹_𝑡−1Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1−𝛾_𝑡−1𝐼^′×Σ¹_𝑡−1⁻¹𝜇_𝑡−1−𝑞_𝑡−1𝐼=𝜆¹_𝑡−1Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1^′×Σ¹_𝑡−1⁻¹𝜇_𝑡−1−𝑞_𝑡−1𝐼−𝜆¹_𝑡−1𝛾_𝑡−1𝐼^′Σ¹_𝑡−1⁻¹𝜇_𝑡−1−𝑞_𝑡−1𝐼=𝜆¹_𝑡−1Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1^′×Σ¹_𝑡−1⁻¹𝜇_𝑡−1−𝑞_𝑡−1𝐼, (C.3) the coefficient of 𝑙²_𝑇−2 is Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1−𝛾_𝑡−1𝐼^′⋅Σ¹_𝑡−1⁻¹⋅Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1−𝛾_𝑡−1𝐼=Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1^′Σ¹_𝑡−1⁻¹×Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1−𝑎_𝑡−1𝛾²_𝑡−1, (C.4) and the constant term is 𝜆¹_𝑡−1²4𝜇_𝑡−1−𝑞_𝑡−1𝐼^′Σ¹_𝑡−1⁻¹𝜇_𝑡−1−𝑞_𝑡−1𝐼=𝜆¹_𝑡−1²4𝛼_𝑡−1. (C.5)

Secondly, we compute Φ^′_𝑡−1𝑢^∗_𝑡−1. The coefficient of 𝑥²_𝑡−1 is 0, the coefficient of 𝑥_𝑡−1 is 𝜆¹_𝑡−12𝜇^′_𝑡−1Σ¹_𝑡−1⁻¹𝐼𝑎_𝑡−1=𝜆¹_𝑡−1𝑞_𝑡−12, (C.6) the coefficient of 𝑙_𝑡−1𝑥_𝑡−1 is 𝛾_𝑡−1, the coefficient of 𝑙_𝑡−1 is 𝜆¹_𝑡−12Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1^′Σ¹_𝑡−1⁻¹×𝜇_𝑡−1−𝑞_𝑡−1𝐼+𝜆¹_𝑡−12𝜇^′_𝑡−1Σ¹_𝑡−1⁻¹×Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1−𝛾_𝑡−1𝐼, (C.7) the coefficient of 𝑙²_𝑡−1 is [Σ⁰_𝑡−1+𝛾_𝑡𝐸(𝑄_𝑡−1𝑅_𝑡−1)]^′Σ¹_𝑡−1⁻¹×Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1−𝛾_𝑡−1𝐼=[Σ⁰_𝑡−1+𝛾_𝑡𝐸(𝑄_𝑡−1𝑅_𝑡−1)]^′Σ¹_𝑡−1⁻¹×Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1−𝑎_𝑡−1𝛾²_𝑡−1, (C.8) and the constant term is 𝜆¹_𝑡−12²𝜇_𝑡−1Σ¹_𝑡−1⁻¹𝜇_𝑡−1−𝑞_𝑡−1𝐼=𝜆¹_𝑡−12²𝛼_𝑡−1. (C.9)

Then it is easy to compute 𝑉^∗_𝑡−1(𝑥_𝑡−1,𝑙_𝑡−1). The coefficient of 𝑥²_𝑡−1 is 1/𝑎_𝑡−1, the coefficient of 𝑥_𝑡−1 is −𝜆¹_𝑡−1𝑞_𝑡−1, the coefficient of 𝑙_𝑡−1𝑥_𝑡−1 is −2𝛾_𝑡−1, the coefficient of 𝑙²_𝑡−1 is Var𝑄_𝑡−1+𝛽_2,𝑡𝐸𝑄²_𝑡−1+𝑎_𝑡−1𝛾²_𝑡−1−Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1^′Σ¹_𝑡−1⁻¹×Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1=𝛽_2,𝑡−1, (C.10) the coefficient of 𝑙_𝑡−1 is −𝜆¹_𝑡−1𝜇^′_𝑡−1Σ¹_𝑡−1⁻¹Σ⁰_𝑡−1+𝛾_𝑡𝐸𝑄_𝑡−1𝑅_𝑡−1−𝛾_𝑡−1𝐼+𝜆_𝑡−1+𝛽_1,𝑡𝐸𝑄_𝑡−1=𝛽_1,𝑡−1, (C.11) and the constant term is −(1/4)∑^𝑇−1_{𝑖=𝑡−1}(𝜆¹_𝑖)²𝛼_𝑖.

Thus, 𝑉^∗_𝑡−1𝑥_𝑡−1,𝑙_𝑡−1=1𝑎_𝑡−1𝑥²_𝑡−1−𝜆¹_𝑡−1𝑞_𝑡−1𝑥_𝑡−1−2𝛾_𝑡−1𝑥_𝑡−1𝑙_𝑡−1+𝛽_1,𝑡−1𝑙_𝑡−1+𝛽_2,𝑡−1𝑙²_𝑡−1−14^𝑇−1_{𝑖=𝑡−1}𝜆¹_𝑖²𝛼_𝑖. (C.12)

D. Proof of (62)

Substituting 𝑢^∗_𝑇−1 into 𝑢^∗′_𝑇−1Σ_𝑇−1𝑢^∗_𝑇−1, we have 𝑢^∗′_𝑇−1Σ_𝑇−1𝑢^∗_𝑇−1=𝜑^′_𝑇−1𝑙_𝑇−1+𝜆_𝑇−12𝜖^′_𝑇−1×Σ_𝑇−1⁻¹𝜑_𝑇−1𝑙_𝑇−1+𝜆_𝑇−12𝜖_𝑇−1=𝜑^′_𝑇−1Σ_𝑇−1⁻¹𝜑_𝑇−1𝑙²_𝑇−1+𝜆_𝑇−1𝜖^′_𝑇−1×Σ_𝑇−1⁻¹𝜑_𝑇−1𝑙_𝑇−1+𝜆_𝑇−1²4𝑀_𝑇−1,2𝜑^′_𝑇−1𝑙_𝑇−1+𝜆_𝑇−12𝜖^′_𝑇−1𝑢^∗_𝑇−1=2𝑢^∗′_𝑇−1Σ_𝑇−1𝑢^∗_𝑇−1. (D.1) Thus, 𝑉^∗_𝑇−1𝑥_𝑇−1,𝑙_𝑇−1=−𝑢^∗′_𝑇−1Σ_𝑇−1𝑢^∗_𝑇−1+Var𝑄_𝑇−1𝑙²_𝑇−1−𝜆_𝑇−1𝑠_𝑇−1𝑥_𝑇−1+𝜆_𝑇−1𝐸𝑄_𝑇−1𝑙_𝑇−1=Var𝑄_𝑇−1−𝜑^′_𝑇−1Σ_𝑇−1⁻¹𝜑_𝑇−1𝑙²_𝑇−1+𝜆_𝑇−1𝐸𝑄_𝑇−1−𝜖^′_𝑇−1Σ_𝑇−1⁻¹𝜑_𝑇−1×𝑙_𝑇−1−𝜆_𝑇−1𝑠_𝑇−1𝑥_𝑇−1−𝜆_𝑇−1²4𝑀_𝑇−1=𝛽_2,𝑇−1𝑙²_𝑇−1+𝛽_1,𝑇−1𝑙_𝑇−1−𝜆_𝑇−1𝑠_𝑇−1𝑥_𝑇−1−𝜆_𝑇−1²4𝑀_𝑇−1. (D.2)

E. Proof of (69)

Taking 𝑢^∗_𝑇−2 into account, we have 𝑢^∗′_𝑇−2Σ_𝑇−2𝑢^∗_𝑇−2=𝜑^′_𝑇−2𝑙_𝑇−2+𝜆¹_𝑇−22𝜖^′_𝑇−2×Σ_𝑇−2⁻¹𝜑_𝑇−2𝑙_𝑇−2+𝜆¹_𝑇−22𝜖_𝑇−2=𝜑^′_𝑇−2Σ_𝑇−2⁻¹𝜑_𝑇−2𝑙²_𝑇−2+𝜆¹_𝑇−2𝜖^′_𝑇−2×Σ_𝑇−2⁻¹𝜑_𝑇−2𝑙_𝑇−2+𝜆¹_𝑇−2²4𝑀_𝑇−2,2𝜑^′_𝑇−2𝑙_𝑇−2+𝜆¹_𝑇−22𝜖^′_𝑇−2𝑢^∗_𝑇−2=2𝑢^∗′_𝑇−2Σ_𝑇−2𝑢^∗_𝑇−2. (E.1) Thus, 𝑉^∗_𝑇−2𝑥_𝑇−2,𝑙_𝑇−2=−𝑢^∗′_𝑇−2Σ_𝑇−2𝑢^∗_𝑇−2+Var𝑄_𝑇−2+𝛽_2,𝑇−1𝐸𝑄²_𝑇−2𝑙²_𝑇−2+𝜆_𝑇−2𝐸𝑄_𝑇−2+𝛽_1,𝑇−1𝐸𝑄_𝑇−2𝑙_𝑇−1−𝜆¹_𝑇−2𝑠_𝑇−2𝑥_𝑇−2−𝜆_𝑇−1²4𝑀_𝑇−1=𝛽_2,𝑇−2𝑙²_𝑇−2+𝛽_1,𝑇−2𝑙_𝑇−2−𝜆¹_𝑇−2𝑠_𝑇−2𝑥_𝑇−2−𝜆_𝑇−1²4𝑀_𝑇−1−𝜆¹_𝑇−2²4𝑀_𝑇−2, (E.2) where 𝛽_1,𝑇−2=𝜆_𝑇−2+𝛽_1,𝑇−1𝐸𝑄_𝑇−2−𝜆¹_𝑇−2𝜖^′_𝑇−2Σ_𝑇−2⁻¹𝜑_𝑇−2,𝛽_2,𝑇−2=Var𝑄_𝑇−2+𝛽_2,𝑇−1𝐸𝑄²_𝑇−2−𝜑^′_𝑇−2Σ_𝑇−2⁻¹𝜑_𝑇−2. (E.3)

F. Proof of (77)

It follows from 𝑢^∗_𝑡−1 that 𝑢^∗′_𝑡−1Σ_𝑡−1𝑢^∗_𝑡−1=𝜑^′_𝑇−1𝑙_𝑡−1+𝜆¹_𝑡−12𝜖^′_𝑇−1×Σ_𝑡−1⁻¹𝜑_𝑡−1𝑙_𝑡−1+𝜆¹_𝑡−12𝜖_𝑡−1=𝜑^′_𝑇−1Σ_𝑡−1⁻¹𝜑_𝑡−1𝑙²_𝑡−1+𝜆¹_𝑡−1𝜖^′_𝑇−1×Σ_𝑡−1⁻¹𝜑_𝑡−1𝑙_𝑡−1+𝜆¹_𝑡−1²4𝑀_𝑡−1,2𝜑^′_𝑇−1𝑙_𝑡−1+𝜆¹_𝑡−12𝜖^′_𝑇−1𝑢^∗_𝑡−1=2𝑢^∗′_𝑡−1Σ_𝑡−1𝑢^∗_𝑡−1. (F.1) Thus, 𝑉^∗_𝑡−1𝑥_𝑡−1,𝑙_𝑡−1=−𝑢^∗′_𝑡−1Σ_𝑡−1𝑢^∗_𝑡−1+Var𝑄_𝑡−1+𝛽_2,𝑡𝐸𝑄²_𝑡−1𝑙²_𝑡−1+𝜆_𝑡−1𝐸𝑄_𝑡−1+𝛽_1,𝑡𝐸𝑄_𝑡−1𝑙_𝑡−1−𝜆¹_𝑡−1𝑠_𝑡−1𝑥_𝑡−1−^𝑇−1_𝑖=𝑡𝜆¹_𝑖²4𝑀_𝑖=𝛽_2,𝑡−1𝑙²_𝑡−1+𝛽_1,𝑡−1𝑙_𝑡−1−𝜆¹_𝑡−1𝑠_𝑡−1𝑥_𝑡−1−^𝑇−1_{𝑖=𝑡−1}𝜆¹_𝑖²4𝑀_𝑖, (F.2) where 𝛽_1,𝑡−1=𝜆_𝑡−1+𝛽_1,𝑡𝐸𝑄_𝑡−1−𝜆¹_𝑡−1𝜖^′_𝑇−1Σ_𝑡−1⁻¹𝜑_𝑡−1,𝛽_2,𝑡−1=Var𝑄_𝑡−1+𝛽_2,𝑡𝐸𝑄²_𝑡−1−𝜑^′_𝑇−1Σ_𝑡−1⁻¹𝜑_𝑡−1. (F.3)

Acknowledgments

This work was supported by the National Natural Science Foundation of China (71101099, 11171237, 11101069), and by the Construction Foundation of Southwest University for Nationalities for the subject of Applied Economics (2011XWD-S0202).