Time-Consistent Strategies for a Multiperiod Mean-Variance Portfolio Selection Problem

Abstract

It remained prevalent in the past years to obtain the precommitment strategies for Markowitz's mean-variance portfolio optimization problems, but not much is known about their time-consistent strategies. This paper takes a step to investigate the time-consistent Nash equilibrium strategies for a multiperiod mean-variance portfolio selection problem. Under the assumption that the risk aversion is, respectively, a constant and a function of current wealth level, we obtain the explicit expressions for the time-consistent Nash equilibrium strategy and the equilibrium value function. Many interesting properties of the time-consistent results are identified through numerical sensitivity analysis and by comparing them with the classical pre-commitment solutions.

1. Introduction

Since the pioneering work of [1] in a single period, mean-variance formulation has been one of focused topics of portfolio selection optimization and has stimulated hundreds of extensions and applications. The interested readers can refer to [2, 3] for detailed information. The objective of quite a number of existing mean-variance portfolio selection models is seeking an optimal strategy 𝜋(⋅) which maximizes the mean-variance utility 𝐸[𝑋^𝜋_𝑇]−𝜔Var[𝑋^𝜋_𝑇], where 𝑋^𝜋_𝑇 is the terminal wealth. But it is well known that this mean-variance criterion lacks of iterated-expectation property, which gives rise to time-inconsistent investment strategy in the sense that Bellman optimality principle is not available any more. The so-called time-inconsistent strategy means that optimal strategy obtained at time 𝑛 does not agree with optimal strategy derived at time 𝑚 where 𝑚>𝑛. Therefore, the optimal strategy in the classical time-inconsistent models is just optimal from the viewpoint of the initial time, and decision makers at any time 𝑘 after the initial time must commit themselves to the initial optimal strategy even if it is not optimal at time 𝑘. So, the time-inconsistent optimal strategy in the classical mean-variance model is called the precommitment strategy. But this precommitment has been criticized for lacking rationality. For one simple example, investment psychology and tastes will often change over time, and the decision maker at later time may not commit themselves to following a strategy which is not optimal at their current time. The work in [4] analyzed the incentives which induce the investor to revise her optimal strategy at subsequent dates under mean-variance criterion.

For this reason, we want to find an optimal strategy with time consistency which is necessary for a rational individual. The analysis of inconsistency can be traced back to [5] which pointed out that “optimal plan of the present moment is generally one which will not be obeyed" and the time-inconsistent problem can be solved by precommitment strategy or alternatively time-consistent strategy. The authors of [5, 6] devoted themselves to identifying an intertemporal consumption programme that would be “the best plan that an agent would actually follow." The work in [7] questioned the generality of the existence of Strotz-Pollak equilibrium and gave an alternative criterion of Nash equilibrium. More recently, it is of interest to study time-inconsistent problems. The work of [8, 9] investigated a time-consistent strategy for a consumption and investment problem with nonexponential discounting. The work in [10] gave general approaches to handle time-inconsistent problems by viewing them as a game theoretic framework and looking for Nash subgame perfect equilibrium points. They formally defined the continuous time equilibrium concept and derived the extension of HJB equation and its verification theorem for a very general objective functions. The work in [11] studied a continuous-time mean-variance portfolio optimization model on the assumption that the risk aversion factor depended dynamically on the current wealth. In view of the extension of HJB equation developed in [10], they obtained the time-consistent equilibrium control and equilibrium value function. The work in [12] provided explicit solutions to a series of cases, including mean-standard deviation in continuous-time setting. The work in [13] investigated optimal mean-variance time-consistent investment and reinsurance policies for an insurer under continuous-time setting. The work in [14] developed a fully numerical scheme to determine time-consistent mean-variance strategy based on piecewise constant policy technique. As for the discrete-time mean-variance models, the work in [15] gave a complicated backwards recursive relationship about time-consistent investment strategy but had not found analytical expression for the strategy.

To the best of our knowledge, no existing literature has given time-consistent equilibrium strategy and equilibrium value function in closed form for discrete-time mean-variance asset allocation. Our research will fill the gap. We view this decision-making process as a noncooperative game and suppose that there is one decision maker, referred to as “decision maker 𝑛", for each point of time 𝑛. This assumption is reasonable. On one hand, in the real world, there are often quite different persons who will join in the decision-making process especially when the investment horizon is long; on the other hand, we can image decision-maker 𝑛 as the future incarnation of themselves at time 𝑛 considering that the tastes of the decision maker will change over time. So our work in this paper is listed as follows: (a) derive the analytical expressions for the time-consistent equilibrium strategy and equilibrium value function when the risk aversion is assumed to be a constant and a function of current wealth, respectively; (b) when risk aversion factor is a constant, compare our time-consistent results with the precommitment ones in [2] and present the particular properties of the time-consistent results; (c) study the problem in discrete-time setting with nonconstant risk aversion which is a function of current wealth and identify the properties of the investment proportion by numerical analysis.

The rest of the paper is organized as follows. In Section 2, the problem formulation is presented, and the recursive formula of the equilibrium value function is derived. In Section 3, equilibrium strategy and equilibrium value function for mean-variance model with constant risk aversion are obtained. Comparison of our time-consistent results with the precommitment ones in [2] is also given in this section. In Section 4, the equilibrium results are investigated on the assumption that the risk aversion depends dynamically on the current wealth and numerical analysis is given to demonstrate the properties of investment proportion. Section 5 presents our conclusions.

2. Problem Formulation

In this paper, we assume that investors join the market at time 0 with an initial wealth 𝑥₀ and plan to process the investment in 𝑇 consecutive time periods. There are one risk-free asset and 𝑚₁ risky assets in the market. The 𝑚₁ risky assets have random returns 𝑅_𝑛=(𝑅_𝑛,1,𝑅_𝑛,2,…,𝑅_𝑛,𝑚1)^′ at period 𝑛 (time interval [𝑛,𝑛+1)) where 𝑅_𝑛,𝑘 denotes the random return of the 𝑘th asset at period 𝑛 and superscript “′” stands for the transpose of a matrix or vector. Denote by 𝑟^𝑓_𝑛 the return of the risk-free asset at period 𝑛, and denote by 𝑋_𝑛 and 𝜋_𝑛=(𝜋_𝑛,1,…,𝜋_𝑛,𝑚1)^′ the wealth available for investment and the amounts invested in 𝑚₁ risky asset at time 𝑛, respectively. Then the wealth dynamics is 𝑋_𝑛+1=𝑋_𝑛−^𝑚1_𝑘=1𝜋_𝑛,𝑘𝑟^𝑓_𝑛+^𝑚1_𝑘=1𝜋_𝑛,𝑘𝑅_𝑛,𝑘=𝑋_𝑛𝑟^𝑓_𝑛+𝜋^′_𝑛𝑅^𝑒_𝑛, (1) where 𝑅^𝑒_𝑛,𝑘=𝑅_𝑛,𝑘−𝑟^𝑓_𝑛,𝑅^𝑒_𝑛=(𝑅^𝑒_𝑛,1,𝑅^𝑒_𝑛,2,…,𝑅^𝑒_𝑛,𝑚1)^′ for 𝑛=0,1,…,𝑇−1.

As we know, the classic mean-variance optimization problem is as follows: max_{𝜋0,𝜋1,…,𝜋𝑇−1}𝐸𝑋^𝜋_𝑇−𝜔Var𝑋^𝜋_𝑇, (2)

which results in a time-inconsistent strategy, that is, precommitment strategy. Therefore, as mentioned in Section 1, this paper aims to solve this problem from another perspective and to look for the time-consistent Nash equilibrium investment strategy. To this end, we first give the definition of Nash equilibrium strategy according to [10] and the references therein.

Let 𝜋(𝑛)={𝜋_𝑛,𝜋_𝑛+1,…,𝜋_𝑇−1} be the policy made at time 𝑛 and 𝐽_𝑛𝔽_𝑛,𝜋(𝑛)=𝐸𝑋_𝑇^𝜋(𝑛)∣𝔽_𝑛−𝜔𝔽_𝑛Var𝑋_𝑇^𝜋(𝑛)∣𝔽_𝑛, (3) where 𝑋_𝑇^𝜋(𝑛) is the terminal wealth corresponding to the investment strategy 𝜋, and 𝔽_𝑛 is the information at time 𝑛, such as wealth level. A natural assumption is that risk aversion 𝜔 is a function of 𝔽_𝑛.

Definition 1. Let 𝜋 be a fixed control law. For an arbitrary point 𝑛(𝑛=0,1,…,𝑇−1), one selects an arbitrary control value 𝜋_𝑛 and define the strategy 𝜋(𝑛)=(𝜋_𝑛,𝜋_𝑛+1,…,𝜋_𝑇−1).
Then 𝜋 is said to be a subgame perfect Nash equilibrium strategy (or simply equilibrium strategy) if for all 𝑛<𝑇, it satisfies max_𝜋𝑛𝐽_𝑛𝔽_𝑛;𝜋(𝑛)=𝐽_𝑛𝔽_𝑛;𝜋(𝑛). (4)
In addition, if equilibrium strategy 𝜋 exists, the equilibrium value function is defined as 𝑉_𝑛𝔽_𝑛=𝐽_𝑛𝔽_𝑛;𝜋(𝑛). (5)

Let 𝜋(𝑛) be the Nash equilibrium strategy at time 𝑛, then Definition 1 makes it possible to solve the problem by the following procedure:(a)𝜋(𝑇−1)=𝜋_𝑇−1=argmax_𝜋𝑇−1[𝐸(𝑋^𝜋𝑇−1_𝑇∣𝔽_𝑇−1)−𝜔(𝔽_𝑇−1)Var(𝑋^𝜋𝑇−1_𝑇∣𝔽_𝑇−1)];(b)given that the decision maker 𝑇−1 will use 𝜋_𝑇−1, 𝜋_𝑇−2 is the optimal control that optimizes objective function 𝐽_𝑇−2(𝔽_𝑇−2;(𝜋_𝑇−2,𝜋_𝑇−1));(c)generally, 𝜋_𝑛 is obtained by letting decision maker 𝑛 choose 𝜋_𝑛 to maximize 𝐽_𝑛 given that the forthcoming decision makers 𝑛+1,…,𝑇−1 will choose the strategy 𝜋(𝑛+1)=(𝜋_𝑛+1,…,𝜋_𝑇−1); that is, 𝜋_𝑛=argmax_𝜋𝑛𝐽_𝑛𝔽_𝑛;𝜋_𝑛,𝜋_𝑛+1,…,𝜋_𝑇−1. (6)

Now we try to derive the time-consistent Nash equilibrium strategy and value function, but first we need to give the following notations and assumptions throughout this paper:(N1)𝑟^𝑒_𝑛=𝐸(𝑅^𝑒_𝑛),𝑛=0,1,…,𝑇−1;(N2)𝜎_𝑛=𝐸(𝑅^𝑒_𝑛𝑅^𝑒_𝑛^′)−𝑟^𝑒_𝑛𝑟^𝑒_𝑛^′,𝑛=0,1,…,𝑇−1;(N3)𝜅_𝑛=𝑟^𝑒_𝑛^′(𝜎_𝑛)⁻¹𝑟^𝑒_𝑛,𝑛=0,1,…,𝑇−1;(N4)𝑔_𝑛^𝜋(𝑛)(𝔽_𝑛)=𝐸[𝑋_𝑇^𝜋(𝑛)∣𝔽_𝑛],𝑔_𝑛(𝔽_𝑛)=𝐸[𝑋𝜋_𝑇(𝑛)∣𝔽_𝑛],𝑛=0,1,…,𝑇−1;(N5)ℎ_𝑛^𝜋(𝑛)(𝔽_𝑛)=𝐸[(𝑋_𝑇^𝜋(𝑛))²∣𝔽_𝑛],ℎ_𝑛(𝔽_𝑛)=𝐸[(𝑋𝜋_𝑇(𝑛))²∣𝔽_𝑛],𝑛=0,1,…,𝑇−1.(A1) The distribution function of the random returns 𝑅_𝑛 is 𝐹_𝑛, and {𝑅_𝑛,𝑛=0,1,…,𝑇−1} is assumed to be statistically independent.(A2)𝜎_𝑛 is assumed to be positive definite.(A3) Short selling is allowed for all risky assets in all periods. Unlimited borrowing and lending are permitted. Transaction costs are not taken into account.(A4) Capital additions or withdrawals are forbidden for all assets in all periods.

With the notations above, we can obtain the recursions of 𝐽_𝑛 and 𝑉_𝑛. For the sake of convenience, we define 𝐸_{𝑛,𝔽𝑛}(⋅)=𝐸[⋅∣𝔽_𝑛] and Var_{𝑛,𝔽𝑛}(⋅)=Var(⋅∣𝔽_𝑛), then𝐽_𝑛𝔽_𝑛;𝜋(𝑛)=𝐸_{𝑛,𝔽𝑛}𝐽_𝑛+1𝔽_𝑛+1()−𝐸;𝜋𝑛+1_{𝑛,𝔽𝑛}𝐸_{𝑛+1,𝔽𝑛+1}𝑋_𝑇^𝜋(𝑛+1)−𝜔𝔽_𝑛+1Var_{𝑛+1,𝔽𝑛+1}𝑋_𝑇^𝜋(𝑛+1)−𝐸_{𝑛,𝔽𝑛}𝑋_𝑇^𝜋(𝑛)+𝜔𝔽_𝑛Var_{𝑛,𝔽𝑛}𝑋_𝑇^𝜋(𝑛)=𝐸_{𝑛,𝔽𝑛}𝐽_𝑛+1𝔽_𝑛+1(−𝐸;𝜋𝑛+1)_{𝑛,𝔽𝑛}𝐸_{𝑛+1,𝔽𝑛+1}𝑋_𝑇^𝜋(𝑛+1)−𝐸_{𝑛,𝔽𝑛}𝑋_𝑇^𝜋(𝑛)+𝐸_{𝑛,𝔽𝑛}𝜔𝔽_𝑛+1Var_{𝑛+1,𝔽𝑛+1}𝑋_𝑇^𝜋(𝑛+1)−𝜔𝔽_𝑛Var_{𝑛,𝔽𝑛}𝑋_𝑇^𝜋(𝑛)=𝐸_{𝑛,𝔽𝑛}𝐽_𝑛+1𝔽_𝑛+1;𝜋(𝑛+1)+𝐸_{𝑛,𝔽𝑛}𝜔𝔽_𝑛+1Var_{𝑛+1,𝔽𝑛+1}𝑋_𝑇^𝜋(𝑛+1)−𝜔𝔽_𝑛Var_{𝑛,𝔽𝑛}𝑋_𝑇^𝜋(𝑛)=𝐸_{𝑛,𝔽𝑛}𝐽_𝑛+1𝔽_𝑛+1;𝜋(𝑛+1)+𝐸_{𝑛,𝔽𝑛}𝜔𝔽_𝑛+1𝐸_{𝑛+1,𝔽𝑛+1}𝑋_𝑇^{𝜋(𝑛+1)2}−𝜔𝔽_𝑛+1𝐸_{𝑛+1,𝔽𝑛+1}𝑋_𝑇^{𝜋(𝑛+1)2}−𝜔𝔽_𝑛𝐸_{𝑛,𝔽𝑛}𝑋_𝑇^𝜋(𝑛)2−𝐸_{𝑛,𝔽𝑛}𝑋_𝑇^𝜋(𝑛)2=𝐸_{𝑛,𝔽𝑛}𝐽_𝑛+1𝔽_𝑛+1;𝜋(𝑛+1)+𝐸_{𝑛,𝔽𝑛}𝜔𝔽_𝑛+1𝐸_{𝑛+1,𝔽𝑛+1}𝑋_𝑇^{𝜋(𝑛+1)2}−𝜔𝔽_𝑛𝐸_{𝑛,𝔽𝑛}𝑋_𝑇^𝜋(𝑛)2−𝐸_{𝑛,𝔽𝑛}𝜔𝔽_𝑛+1𝐸_{𝑛+1,𝔽𝑛+1}𝑋_𝑇^{𝜋(𝑛+1)2}+𝜔𝔽_𝑛𝐸_{𝑛,𝔽𝑛}𝐸_{𝑛+1,𝔽𝑛+1}𝑋_𝑇^{𝜋(𝑛+1)2}=𝐸_{𝑛,𝔽𝑛}𝐽_𝑛+1𝔽_𝑛+1;𝜋(𝑛+1)+𝐸_{𝑛,𝔽𝑛}𝜔𝔽_𝑛+1ℎ^𝜋(𝑛+1)_𝑛+1𝔽_𝑛+1−𝜔𝔽_𝑛𝐸_{𝑛,𝔽𝑛}ℎ^𝜋(𝑛+1)_𝑛+1𝔽_𝑛+1−𝐸_{𝑛,𝔽𝑛}𝜔𝔽_𝑛+1𝑔^𝜋(𝑛+1)_𝑛+1𝔽_𝑛+1²+𝜔𝔽_𝑛𝐸_{𝑛,𝔽𝑛}𝑔^𝜋(𝑛+1)_𝑛+1𝔽_𝑛+1². (7)

By Definition 1, we have 𝑉_𝑛+1(𝔽_𝑛+1)=𝐽_𝑛+1(𝔽_𝑛+1;𝜋_𝑛+1), and hence (7) implies the following recursion for equilibrium value function 𝑉_𝑛(𝔽_𝑛): 𝑉_𝑛𝔽_𝑛=max_𝜋𝑛𝐸_{𝑛,𝔽𝑛}𝑉_𝑛+1𝔽_𝑛+1+𝐸_{𝑛,𝔽𝑛}𝜔𝔽_𝑛+1ℎ_𝑛+1𝔽_𝑛+1−𝜔𝔽_𝑛𝐸_{𝑛,𝔽𝑛}ℎ_𝑛+1𝔽_𝑛+1−𝐸_{𝑛,𝔽𝑛}𝜔𝔽_𝑛+1𝑔_𝑛+1𝔽_𝑛+1²+𝜔𝔽_𝑛𝐸_{𝑛,𝔽𝑛}𝑔_𝑛+1𝔽_𝑛+1²,𝑉𝑛=0,1,…,𝑇−1,_𝑇𝔽_𝑛=𝑥, (8)(9) where the recursions of 𝑔_𝑛(𝔽_𝑛) and ℎ_𝑛(𝔽_𝑛) are as follows: ℎ_𝑛𝔽_𝑛=𝐸_{𝑛,𝔽𝑛}𝑋𝜋_𝑇(𝑛)²=𝐸_{𝑛,𝔽𝑛}𝐸_{𝑛+1,𝔽𝑛+1}𝑋𝜋_𝑇^(𝑛)2=𝐸_{𝑛,𝔽𝑛}ℎ_𝑛+1𝔽_𝑛+1ℎ,𝑛=0,1,…,𝑇−1,_𝑇𝔽_𝑇=𝑥²,𝑔_𝑛𝔽_𝑛=𝐸_{𝑛,𝔽𝑛}𝑋𝜋_𝑇(𝑛)=𝐸_{𝑛,𝔽𝑛}𝐸_{𝑛+1,𝔽𝑛+1}𝑋𝜋_𝑇(𝑛)=𝐸_{𝑛,𝔽𝑛}𝑔_𝑛+1𝔽_𝑛+1,𝑔𝑛=0,1,…,𝑇−1,_𝑇𝔽_𝑇=𝑥. (10)

3. Nash Equilibrium Strategy with Constant Risk Aversion

3.1. Equilibrium Strategy and Equilibrium Value Function

When the risk aversion is a constant, 𝐽_𝑛 is of the form 𝐽_𝑛(𝑥;𝜋(𝑛))=𝐸𝑋_𝑇^𝜋(𝑛)∣𝑋_𝑛=𝑥−𝜔Var𝑋_𝑇^𝜋(𝑛)∣𝑋_𝑛𝑉=𝑥_𝑛𝔽_𝑛=𝑉_𝑛𝑋_𝑛=𝑥=𝐽_𝑛(𝑥;𝜋(𝑛)). (11)

According to (8), the recursion of equilibrium value function 𝑉_𝑛(𝑥) is simplified as 𝑉_𝑛(𝑥)=max_𝜋𝑛𝐸_𝑛,𝑥𝑉_𝑛+1𝑋^𝜋𝑛_𝑛+1−𝜔Var_𝑛,𝑥𝑔_𝑛+1𝑋^𝜋𝑛_𝑛+1,𝑉𝑛=0,1,…,𝑇−1,_𝑇(𝑥)=𝑥. (12)(13)

Recursion (12) indicates that 𝑉_𝑛(𝑥) does not depend on ℎ_𝑛, and then we only need to find the explicit expression of 𝑔_𝑛 by the following recursion: 𝑔_𝑛(𝑥)=𝐸_𝑛,𝑥𝑋𝜋_𝑇(𝑛)=𝐸_𝑛,𝑥𝑔_𝑛+1𝑋𝜋_𝑛𝑛+1𝑔,𝑛=0,1,…,𝑇−1,_𝑇(𝑥)=𝑥. (14)(15)

The following theorem gives the explicit expressions of 𝜋 and 𝑉_𝑛(𝑥)

Theorem 2. When the risk aversion is a constant, the Nash equilibrium strategy is given by 𝜋_𝑛=112𝜔∏^𝑇−1_𝑘=𝑛+1𝑟^𝑓_𝑘𝜎_𝑛⁻¹𝑟^𝑒_𝑛,𝑛=0,1,…,𝑇−1. (16)
The corresponding equilibrium value function is 𝑉_𝑛(𝑥)=𝑥^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘+14𝜔^𝑇−1_𝑘=𝑛𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘𝑔,𝑛=0,1,…,𝑇,_𝑛(𝑥)=𝑥^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘+12𝜔^𝑇−1_𝑘=𝑛𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘,𝑛=0,1,…,𝑇. (17)(18)

Proof. Obviously (17) and (18) hold true for 𝑛=𝑇. Then for 𝑛=𝑇−1, 𝑉_𝑇−1(𝑥)=max_𝜋𝑇−1𝐸_{𝑇−1,𝑥}𝑉_𝑇𝑋^𝜋𝑇−1_𝑇−𝜔Var_{𝑇−1,𝑥}𝑔_𝑇𝑋^𝜋𝑇−1_𝑇=max_𝜋𝑇−1𝐸_{𝑇−1,𝑥}𝑋^𝜋𝑇−1_𝑇−𝜔Var_{𝑇−1,𝑥}𝑋^𝜋𝑇−1_𝑇=𝑥𝑟^𝑓_𝑇−1+sup_𝜋𝑇−1𝜋^′_𝑇−1𝑟^𝑒_𝑇−1−𝜔𝜋^′_𝑇−1𝜎_𝑇−1𝜋_𝑇−1. (19)
Since 𝜎_𝑇−1 is positive definite, the optimal solution exists and is given by 𝜋_𝑇−1=1𝜎2𝜔^𝑇−1−1𝑟^𝑒_𝑇−1. (20)
Substituting (20) into (19) yields 𝑉_𝑇−1(𝑥)=𝑥𝑟^𝑓_𝑇−1+1𝑟4𝜔^{𝑒′𝑇−1}𝜎⁻¹_𝑇−1𝑟^𝑒_𝑇−1,𝑔_𝑇−1(𝑥)=𝐸_{𝑇−1,𝑥}𝑋𝜋_𝑇−1𝑇=𝑥𝑟^𝑓_𝑇−1+1𝑟2𝜔^{𝑒′𝑇−1}𝜎⁻¹_𝑇−1𝑟^𝑒_𝑇−1. (21)
Hence (16), (17), and (18) hold true for 𝑛=𝑇−1. Now we assume that (17) and (18) are true for 𝑛+1, then for 𝑛, 𝑉_𝑛(𝑥)=max_𝜋𝑛𝐸_𝑛,𝑥𝑉_𝑛+1𝑋^𝜋𝑛_𝑛+1−𝜔Var_𝑛,𝑥𝑔_𝑛+1𝑋^𝜋𝑛_𝑛+1=max_𝜋𝑛𝐸_𝑛,𝑥𝑥𝑟^𝑓_𝑛+𝑅^𝑒_𝑛^′𝜋_𝑛^𝑇−1_𝑘=𝑛+1𝑟^𝑓_𝑘+14𝜔^𝑇−1_𝑘=𝑛+1𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘−𝜔Var_𝑛,𝑥𝑥𝑟^𝑓_𝑛+𝑅^𝑒_𝑛^′𝜋_𝑛^𝑇−1_𝑘=𝑛+1𝑟^𝑓_𝑘+12𝜔^𝑇−1_𝑘=𝑛+1𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘=max_𝜋𝑛𝑥^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘+𝑟^𝑒_𝑛^′𝜋_𝑛^𝑇−1_𝑘=𝑛+1𝑟^𝑓_𝑘+14𝜔^𝑇−1_𝑘=𝑛+1𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘−𝜔^𝑇−1_𝑘=𝑛+1𝑟^𝑓_𝑘²𝜋^′_𝑛𝜎_𝑛𝜋_𝑛. (22)
It is obvious that the optimal solution of (22) exits and is given by 𝜋_𝑛=112𝜔∏^𝑇−1_𝑘=𝑛+1𝑟^𝑓_𝑘𝜎_𝑛⁻¹𝑟^𝑒_𝑛. (23)
Substituting (23) into (22), we obtain 𝑉_𝑛(𝑥)=𝑥^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘+14𝜔^𝑇−1_𝑘=𝑛𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘, (24)
and according to (14), 𝑔_𝑛(𝑥)=𝐸_𝑛,𝑥𝑔_𝑛+1𝑋𝜋_𝑛𝑛+1=𝐸_𝑛,𝑥𝑋𝜋_𝑛𝑛+1^𝑇−1_𝑘=𝑛+1𝑟^𝑓_𝑘+12𝜔^𝑇−1_𝑘=𝑛+1𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘=𝑥𝑟^𝑓_𝑛+𝜋_𝑛𝑟^𝑒_𝑛^𝑇−1_𝑘=𝑛+1𝑟^𝑓_𝑘+12𝜔^𝑇−1_𝑘=𝑛+1𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘=𝑥^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘+1𝑟2𝜔^𝑒𝑛′𝜎_𝑛⁻¹𝑟^𝑒_𝑛+12𝜔^𝑇−1_𝑘=𝑛+1𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘=𝑥^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘+12𝜔^𝑇−1_𝑘=𝑛𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘. (25)
Equations (23), (24), and (25) mean that (16), (17), and (18) hold true for 𝑛. By induction, the proof of Theorem 2 is complete.

3.2. Comparison to the Precommitment Results

3.2.1. About the Value Function

In view of (17), we know that the equilibrium value function at initial time 0 is 𝑉₀𝑥₀=𝑥₀^𝑇−1_𝑘=0𝑟^𝑓_𝑘+14𝜔^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘, (26)

and the precommitment value function of [2] is 𝐸_0,𝑥0𝑋_𝑇−𝜔Var_0,𝑥0𝑋_𝑇=𝑥₀^𝑇−1_𝑘=0𝑟^𝑓_𝑘+14𝜔1−∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘. (27)

The relationship between (26) and (27) is summarized in the following lemma.

Lemma 3. Consider the following:𝐸_0,𝑥0𝑋_𝑇−𝜔Var_0,𝑥0𝑋_𝑇≥𝑉₀𝑥₀. (28)

Proof. First of all, we have 𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′=𝜎_𝑘⁻¹−𝜎_𝑘⁻¹𝑟^𝑒_𝑘𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹1+𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘, (29)
then, 𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘=𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹−𝜎_𝑘⁻¹𝑟^𝑒_𝑘𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹1+𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘𝑟^𝑒_𝑘=𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘𝑟1−^𝑒𝑘′𝜎_𝑘⁻¹𝑟^𝑒_𝑘1+𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘=𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘1+𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘. (30)
Now, 𝐸_0,𝑥0𝑋_𝑇−𝜔Var_0,𝑥0𝑋_𝑇−𝑉₀𝑥₀=14𝜔1−∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘−^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘=1∏4𝜔1−^𝑇−1_𝑘=01/1+𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘∏^𝑇−1_𝑘=01/1+𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘−14𝜔^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘=14𝜔^𝑇−1_𝑘=01+𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘−14𝜔^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘−1=14𝜔4𝜔1+𝑟^𝑒₀^′𝜎₀⁻¹𝑟^𝑒₀⋯1+𝑟^𝑒′_𝑇−1𝜎_𝑇−1⁻¹𝑟^𝑒_𝑇−1−^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘−1≥0. (31)

This proof gives an important inequality needed in the later analysis as 1−∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘−^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘=1+𝑟^𝑒₀^′𝜎₀⁻¹𝑟^𝑒₀⋯1+𝑟^𝑒′_𝑇−1𝜎_𝑇−1⁻¹𝑟^𝑒_𝑇−1−^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘−1≥0. (32)

Lemma 3 shows that the precommitment value function is greater than the equilibrium value function. This is a fair game of God. The Nash equilibrium strategy gains the time consistency but at the same time destroys the welfare of the whole decision procedure because of the inability to precommit. Referring to the proof of Lemma 3, we also realize that the distance between these two value functions is amplified at longer time horizon. Specially, when 𝑇=1, these two value functions coincide with each other. When 𝑇=1, the time-consistent results should be and are actually the same as the precommitment ones.

3.2.2. About the Investment Strategy

The time-consistent strategy at each period is 𝜋_𝑛=112𝜔∏^𝑇−1_𝑘=𝑛+1𝑟^𝑓_𝑘𝜎_𝑛⁻¹𝑟^𝑒_𝑛,𝑛=0,1,…,𝑇−1. (33)

Referring to [2], the precommitment strategy at each period is 𝜋^∗_𝑛=−𝑟^𝑓_𝑛𝐸⁻¹𝑅^𝑒_𝑛𝑅^𝑒_𝑛^′𝑟^𝑒_𝑛𝑥_𝑛+^𝑇−1_𝑘=0𝑟^𝑓_𝑘𝑥₀+12𝜔∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘×^𝑇−1_𝑘=𝑛+11𝑟^𝑓_𝑘𝐸⁻¹𝑅^𝑒_𝑛𝑅^𝑒_𝑛^′𝑟^𝑒_𝑛,𝑛=0,1,…,𝑇−1. (34)

The significant differences between 𝜋 and 𝜋^∗ are as follows.(a)Since the time-consistent strategy at time 𝑛 will not be affected by the initial information, then it has nothing to do with the initial wealth 𝑥₀ in contrast with precommitment strategy.(b)the time consistent is time deterministic but the precommitment one is stochastically dependent on the current wealth.

3.2.3. About the Efficient Frontier

In this Section, we want to compare our efficient frontier with the one in [2]. But first of all, we need the results in Lemma 4.

Lemma 4. Under the time-consistent strategy (16), 𝐸𝑋𝜋_𝑛=𝑥₀^𝑛−1_𝑘=0𝑟^𝑓_𝑘+112𝜔∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘^𝑛−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘,𝐸𝑋𝜋_𝑛²=𝑥₀^𝑛−1_𝑘=0𝑟^𝑓_𝑘²+1𝜔∑^𝑛−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘²𝑥₀^𝑇−1_𝑘=0𝑟^𝑓_𝑘+14𝜔²1∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘^{2𝑛−1𝑘=0}𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘×1+^𝑘−1_𝑚=0𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚+^𝑘_𝑚=0𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚. (35)(36)

Proof. Substituting (16) into (1) yields 𝑋𝜋_𝑛=𝑋𝜋_𝑛−1𝑟^𝑓_𝑛−1+112𝜔∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘×𝑅^𝑒′_𝑛−1𝜎_𝑛−1⁻¹𝑟^𝑒_𝑛−1,𝑛=1,2,…,𝑇, (37)
and for 𝑛=1,2,…,𝑇, 𝑋𝜋_𝑛²=𝑋𝜋_𝑛−1²𝑟^𝑓_𝑛−1²+1𝜔𝑋𝜋_𝑛−1𝑟^𝑓_𝑛−11∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘𝑅^𝑒′_𝑛−1𝜎_𝑛−1⁻¹𝑟^𝑒_𝑛−1+14𝜔²1∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘²𝑟^𝑒′_𝑛−1𝜎_𝑛−1⁻¹×𝑅^𝑒_𝑛−1𝑅^𝑒′_𝑛−1𝜎_𝑛−1⁻¹𝑟^𝑒_𝑛−1. (38)
Hence, 𝐸𝑋𝜋_𝑛=𝐸𝑋𝜋_𝑛−1𝑟^𝑓_𝑛−1+112𝜔∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘𝑟^𝑒′_𝑛−1𝜎_𝑛−1⁻¹𝑟^𝑒_𝑛−1,𝑛=1,2,…,𝑇, (39)
and for 𝑛=1,2,…,𝑇, 𝐸𝑋𝜋_𝑛²=𝐸𝑋𝜋_𝑛−1²𝑟^𝑓_𝑛−1²+1𝜔𝐸𝑋𝜋_𝑛−1𝑟^𝑓_𝑛−1×1∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘𝑟^𝑒′_𝑛−1𝜎_𝑛−1⁻¹𝑟^𝑒_𝑛−1+14𝜔²1∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘²𝑟^𝑒′_𝑛−1𝜎_𝑛−1⁻¹𝑟^𝑒_𝑛−1×1+𝑟^𝑒′_𝑛−1𝜎_𝑛−1⁻¹𝑟^𝑒_𝑛−1. (40)
By repeatedly using recursive equation (39), we can obtain (35). Substituting (35) into (40) yields 𝐸𝑋𝜋_𝑛²=𝐸𝑋𝜋_𝑛−1²𝑟^𝑓_𝑛−1²+1𝜔𝑟^𝑒′_𝑛−1𝜎_𝑛−1⁻¹𝑟^𝑒_𝑛−1∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘²𝑥₀^𝑇−1_𝑘=0𝑟^𝑓_𝑘+14𝜔²𝑟^𝑒′_𝑛−1𝜎_𝑛−1⁻¹𝑟^𝑒_𝑛−1∏^𝑇−1_𝑘=𝑛𝑟^𝑓_𝑘²×1+^𝑛−2_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘+^𝑛−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘. (41)
Repeatedly using the above recursive equation yields (36).

Then according to (35) and (36), we can obtain the efficient frontier under the time-consistent strategy.

Theorem 5. The efficient frontier under the time-consistent strategy (16) is Var𝑋𝜋_𝑇=𝐸𝑋𝜋_𝑇−𝑥₀∏^𝑇−1_𝑘=0𝑟^𝑓_𝑘²∑^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘. (42)

Proof. When 𝑛=𝑇, (35) becomes 𝐸𝑋𝜋_𝑇=𝑥₀^𝑇−1_𝑘=0𝑟^𝑓_𝑘+12𝜔^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘. (43)
Equation (43), as we expect, coincides with the expression of 𝑔₀(𝑥) in (18).
When 𝑛=𝑇, (36) becomes 𝐸𝑋𝜋_𝑇²=𝑥₀^2𝑇−1_𝑘=0𝑟^𝑓_𝑘²+1𝜔^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘𝑥₀^𝑇−1_𝑘=0𝑟^𝑓_𝑘+14𝜔^2𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘1+^𝑘−1_𝑚=0𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚+^𝑘_𝑚=0𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚=𝑥₀^𝑇−1_𝑘=0𝑟^𝑓_𝑘+12𝜔^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘²−14𝜔^2𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘²+14𝜔^2𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘1+^𝑘−1_𝑚=0𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚+^𝑘_𝑚=0𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚. (44)
Equation (44) together with (43) yields Var𝑋𝜋_𝑇=14𝜔^2𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘×1+^𝑘−1_𝑚=0𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚+^𝑘_𝑚=0𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚−^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘=14𝜔^2𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘×1+^𝑘_𝑚=0𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚−^𝑇−1_𝑚=𝑘𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚=14𝜔^2𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘×1+^𝑇−1_𝑚=𝑘𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚−^𝑇−1_𝑚=𝑘𝑟^𝑒_𝑚^′𝜎_𝑚⁻¹𝑟^𝑒_𝑚=14𝜔^2𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘. (45)
Referring to (43) and (45), we get the efficient frontier as follows: Var𝑋𝜋_𝑇=𝐸𝑋𝜋_𝑇−𝑥₀∏^𝑇−1_𝑘=0𝑟^𝑓_𝑘²∑^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘. (46)

Now we want to compare efficient frontier (42) with the one in [2].

Efficient frontier in [2] is Var𝑋^𝜋∗_𝑇=∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘1−∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘×𝐸𝑋^𝜋∗_𝑇−𝑥₀^𝑇−1_𝑘=0𝑟^𝑓_𝑘². (47)

In view of (32), we can find that ∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘1−∏^𝑇−1_𝑘=01−𝑟^𝑒_𝑘^′𝐸⁻¹𝑅^𝑒_𝑘𝑅^𝑒_𝑘^′𝑟^𝑒_𝑘≤1∑^𝑇−1_𝑘=0𝑟^𝑒_𝑘^′𝜎_𝑘⁻¹𝑟^𝑒_𝑘. (48)

Therefore, we have given a mathematical proof to the fact that the efficient frontier for the time-consistent strategy is never above the efficient frontier for the precommitment strategy. Moreover, the shorter the investment horizon 𝑇 is, the closer these two efficient frontiers are.

3.3. Numerical Analysis

In this part, we want to compare expected terminal wealth and efficient frontier under time-consistent framework with the corresponding ones, respectively, in [2]. Let initial wealth 𝑥₀=1. For convenience, we assume that there are only one risk-free asset and one risky asset in the market. Furthermore, we suppose that the risk-free return is a constant 𝑟^𝑓=1.15 during all the investment periods and risky returns {𝑅_𝑛,𝑛=0,1,…,𝑇−1} have the same distribution function with the same expectation 𝐸(𝑅_𝑛)=1.35 and variance Var(𝑅_𝑛)=0.2.

(1) Let risk aversion 𝜔=1. Here we fix other parameters and make 𝑇 increase from 1 to 8, to present the effect of investment horizon on the distance between time-consistent expected terminal wealth and precommitment one. Refering to (43) and formula (56) in [2], we obtain Figure 1, which shows that precommitment expected terminal wealth is higher than the time-consistent one and the gap between these two expectations is widened when 𝑇 increases. Actually, the gap sequences are 0,0.0506, 0.1747, 0.4051, 0.7899, 1.3985, 2.3316, and 3.7352 when 𝑇 is changed from 1 to 8. This phenomenon shows that for short time horizons, time inconsistency has a slight effect on the relative results. But for long time horizon, noncooperation of each decision maker destroys the welfare of the whole game, and the longer 𝑇 is, the larger the loss is.

Figure 1 

Effect of time horizons.

(2) Let 𝑇=8. Here we fix other parameters and make risk aversion 𝜔 increase from 1 to 8. The effect of risk aversion on the distance between time-consistent expected terminal wealth and precommitment one is showed in Figure 2, which indicates that the distance between these two kinds of expected final wealth is minor when risk aversion is big enough. This is because, by (43) and formula (56) in [2], we can find that distance between these two expectations are only affected by the risky investment income. When 𝜔 is big enough, either the time-consistent investor or the precommitment one will prefer to invest less wealth in risky asset, which leads to the conclusion in Figure 2.

Figure 2 

Effect of the risk aversion.

(3) We want to compare the precommitment efficient frontier with the time-consistent one when 𝑇=4,8 and 𝜔=2. Conclusions showed in Figure 3 coincide with our conclusions derived by mathematical analysis in Section 3.2.3 and the reason behind Figure 3 is similar to that demonstrated in Figure 1.

Figure 3 

Efficient frontier.

4. Nash Equilibrium Strategy with Wealth-Dependent Risk Aversion

In this section, we will consider an optimization problem max_𝜋(𝑛)𝐸_𝑛,𝑥𝑋_𝑇^𝜋(𝑛)−𝜔(𝑥)Var_𝑛,𝑥𝑋_𝑇^𝜋(𝑛) (49)

under the framework of Nash equilibrium, where the risk aversion 𝜔(𝑥) is a function of current wealth 𝑥. We just study a tractable situation that the risk aversion is inversely proportional to wealth according to some psychological results; that is, 𝜔(𝑥)=𝜔/𝑥 where constant 𝜔>0 is called the coefficient of risk aversion. It still remains open what the time-consistent strategy is with other forms of 𝜔(𝑥).

4.1. Nash Equilibrium Strategy and Equilibrium Value Function

Similarly, referring to (8), the recursion of 𝑉_𝑛(𝑥) is 𝑉_𝑛(𝑥)=max_𝜋𝑛𝐸_𝑛,𝑥𝑉_𝑛+1𝑋^𝜋𝑛_𝑛+1+𝐸_𝑛,𝑥𝜔𝑋^𝜋𝑛_𝑛+1ℎ_𝑛+1𝑋^𝜋𝑛_𝑛+1−𝜔(𝑥)ℎ_𝑛+1𝑋^𝜋𝑛_𝑛+1−𝐸_𝑛,𝑥𝜔𝑋^𝜋𝑛_𝑛+1𝑔_𝑛+1𝑋^𝜋𝑛_𝑛+1²+𝜔(𝑥)𝐸_𝑛,𝑥𝑔_𝑛+1𝑋^𝜋𝑛_𝑛+1²,𝑛=0,1,…,𝑇−1, (50)

with terminal condition 𝑉_𝑇(𝑥)=𝑥.

The recursions of 𝑔_𝑛(𝑥) and ℎ_𝑛(𝑥) are, respectively, 𝑔_𝑛(𝑥)=𝐸𝑋𝜋_𝑇(𝑛)∣𝑋_𝑛=𝑥=𝐸_𝑛,𝑥𝑔_𝑛+1𝑋𝜋_𝑛𝑛+1𝑔,𝑛=0,1,…,𝑇−1,_𝑇(𝑥)ℎ=𝑥,_𝑛(𝑥)=𝐸𝑋𝜋_𝑇(𝑛)²∣𝑋_𝑛=𝑥=𝐸_𝑛,𝑥ℎ_𝑛+1𝑋𝜋_𝑛𝑛+1ℎ,𝑛=0,1,…,𝑇−1,_𝑇(𝑥)=𝑥². (51)

Before giving the time-consistent results, we need to introduce the following sequences {𝑎_𝑛} and {𝑏_𝑛} and their properties: 𝑎_𝑇𝑏=1,_𝑇𝜉=1,_𝑛+1=𝑏_𝑛+1𝐸𝑅^𝑒_𝑛𝑅^𝑒_𝑛^′−𝑎_𝑛+1²𝑟^𝑒_𝑛𝑟^𝑒_𝑛^′,𝑎𝑛=𝑇−1,𝑇−2,…,0,_𝑛=𝑎_𝑛+1𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑟^𝑒_𝑛𝑏,𝑛=𝑇−1,𝑇−2,…,0,_𝑛=𝑏_𝑛+1𝐸×𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑅^𝑒_𝑛,𝑛=𝑇−1,𝑇−2,…,0. (52)

Lemma 6. For 𝑛=0,1,…,𝑇−1, 𝑏_𝑛>(𝑎_𝑛)² and 𝜉_𝑛(𝑛=𝑇,𝑇−1,…,1) is positive definite.

Proof. 𝜉_𝑇=𝐸(𝑅^𝑒_𝑇−1𝑅^𝑒′_𝑇−1)−𝑟^𝑒_𝑇−1𝑟^𝑒′_𝑇−1=𝜎_𝑇−1 is obviously positive definite. Then, 𝑏_𝑇−1=𝐸𝑟^𝑓_𝑇−1+1𝑟2𝜔^{𝑒′𝑇−1}×𝐸𝑅^𝑒_𝑇−1𝑅^𝑒′_𝑇−1−𝑟^𝑒_𝑇−1𝑟^𝑒′_𝑇−1⁻¹𝑅^𝑒_𝑇−1²,𝑎_𝑇−1=𝑟^𝑓_𝑇−1+1𝑟2𝜔^{𝑒′𝑇−1}×𝐸𝑅^𝑒_𝑇−1𝑅^𝑒′_𝑇−1−𝑟^𝑒_𝑇−1𝑟^𝑒′_𝑇−1⁻¹𝑟^𝑒_𝑇−1, (53)
and hence 𝑏_𝑇−1>(𝑎_𝑇−1)². Now for any column vector 𝑟≠𝟎, we have 𝑟^′𝜉_𝑇−1𝑟=𝑟^′𝑏_𝑇−1𝐸𝑅^𝑒_𝑇−2𝑅^𝑒′_𝑇−2−𝑎_𝑇−1²𝑟^𝑒_𝑇−2𝑟^𝑒′_𝑇−2𝑟>𝑟^′𝑎_𝑇−1²𝐸𝑅^𝑒_𝑇−2𝑅^𝑒′_𝑇−2−𝑎_𝑇−1²𝑟^𝑒_𝑇−2𝑟^𝑒′_𝑇−2𝑟=𝑎_𝑇−1²𝑟^′𝜎_𝑇−2𝑟>0, (54)
and then 𝜉_𝑇−1 is positive definite.
Now we assume that for 𝑛+1,𝑏_𝑛+1>(𝑎_𝑛+1)² and 𝜉_𝑛+1 is positive definite, and then for 𝑛, 𝑏_𝑛−𝑎_𝑛²=𝑏_𝑛+1𝐸𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑅^𝑒_𝑛²−𝑎_𝑛+1²𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑟^𝑒_𝑛²>𝑎_𝑛+1²𝐸𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑅^𝑒_𝑛²−𝑎_𝑛+1²𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑟^𝑒_𝑛²>0. (55)
Similar to the proof of 𝜉_𝑇−1 which is positive definite, we can immediately prove that 𝜉_𝑛 is positive definite given 𝑏_𝑛>(𝑎_𝑛)². By induction, we complete the proof of Lemma 6.

The recursions (50)–(51) and Lemma 6 yield the following theorem.

Theorem 7. If 𝑥>0, the time-consistent strategy is 𝜋_𝑛(𝑥)=𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛𝜉2𝜔^𝑛+1−1𝑟^𝑒_𝑛𝑥,𝑛=0,1,…,𝑇−1, (56)and the equilibrium strategy is 𝑉_𝑛(𝑥)=𝑎_𝑛𝑥+𝜔𝑎_𝑛²𝑥−𝜔𝑏_𝑛𝑔𝑥,𝑛=0,1,…,𝑇,_𝑛(𝑥)=𝑎_𝑛ℎ𝑥,𝑛=0,1,…,𝑇−1,_𝑛(𝑥)=𝑏_𝑛𝑥²,𝑛=0,1,…,𝑇−1. (57)(58)(59)
If 𝑥≤0,𝑉_𝑛(𝑥)=𝑔_𝑛(𝑥)=ℎ_𝑛(𝑥)=+∞ and the model is meaningless.

Remark 8. If the initial wealth is big enough and risky assets have steady returns, then at each period wealth level is often greater than 0 in the real-world situation. Therefore, the condition 𝑥>0 is not a severe requirement.

Proof. By (50), we obtain 𝑉_𝑇−1(𝑥)=max_𝜋𝑇−1𝐸_{𝑇−1,𝑥}𝑉_𝑇𝑋^𝜋𝑇−1_𝑇+𝐸_{𝑇−1,𝑥}𝜔𝑋^𝜋𝑇−1_𝑇ℎ_𝑇𝑋^𝜋𝑇−1_𝑇−𝜔(𝑥)ℎ_𝑇𝑋^𝜋𝑇−1_𝑇−𝐸_{𝑇−1,𝑥}𝜔𝑋^𝜋𝑇−1_𝑇𝑔_𝑇𝑋^𝜋𝑇−1_𝑇²+𝜔(𝑥)𝐸_{𝑇−1,𝑥}𝑔_𝑇𝑋^𝜋𝑇−1_𝑇²=max_𝜋𝑇−1𝐸_{𝑇−1,𝑥}𝑋^𝜋𝑇−1_𝑇−𝜔(𝑥)Var_{𝑇−1,𝑥}𝑋^𝜋𝑇−1_𝑇=max_𝜋𝑇−1𝑥𝑟^𝑓_𝑇−1+𝜋^′_𝑇−1𝑟^𝑒_𝑇−1−𝜔(𝑥)𝜋^′_𝑇−1𝜎_𝑇−1𝜋_𝑇−1. (60)
It is obvious that when 𝑥>0, 𝜋_𝑇−1(𝑥)=1(𝑥)𝜎2𝜔^𝑇−1−1𝑟^𝑒_𝑇−1=1𝜎2𝜔^𝑇−1−1𝑟^𝑒_𝑇−1𝑥. (61)
Then, 𝑔_𝑇−1(𝑥)=𝐸_{𝑇−1,𝑥}𝑋𝜋_𝑇−1𝑇=𝑥𝑟^𝑓_𝑇−1+1(𝑥)𝑟2𝜔^{𝑒′𝑇−1}𝜎_𝑇−1⁻¹𝑟^𝑒_𝑇−1=𝑟^𝑓_𝑇−1+𝜅_𝑇−12𝜔𝑥=𝑎_𝑇−1ℎ𝑥,_𝑇−1(𝑥)=𝐸_{𝑇−1,𝑥}𝑋𝜋_𝑇−1𝑇²=𝐸_{𝑇−1,𝑥}𝑥𝑟^𝑓_𝑇−1+𝜋^′_𝑇−1𝑅^𝑒_𝑇−1²=𝑥²𝑟^𝑓_𝑇−1²+𝑟^𝑓_𝑇−1𝜅_𝑇−1𝜔(𝑥)𝜅𝑥+^𝑇−11+𝜅_𝑇−14(𝜔(𝑥))²=𝑟^𝑓_𝑇−1²+𝑟^𝑓_𝑇−1𝜅_𝑇−1𝜔+𝜅_𝑇−11+𝜅_𝑇−14𝜔²𝑥²=𝑏_𝑇−1𝑥². (62)(63)
Substituting (61) into 𝑉_𝑇−1(𝑥) yields 𝑉_𝑇−1(𝑥)=𝑥𝑟^𝑓_𝑇−1+1(𝑥)𝑟4𝜔^{𝑒′𝑇−1}𝜎_𝑇−1⁻¹𝑟^𝑒_𝑇−1=𝑟^𝑓_𝑇−1+1𝜅4𝜔^𝑇−1𝑥=𝑎_𝑇−1+𝜔𝑎_𝑇−1²−𝜔𝑏_𝑇−1𝑥. (64)
Equations (61)–(64) mean that (56)–(59) are true for 𝑛=𝑇−1.
We assume the results in Theorem 7 hold true for 𝑛+1, and then according to (50) and (57), 𝑉_𝑛(𝑥)=max_𝜋𝑛𝐸_𝑛,𝑥𝑎_𝑛+1+𝜔𝑎_𝑛+1²−𝜔𝑏_𝑛+1𝑋^𝜋𝑛_𝑛+1+𝐸_𝑛,𝑥𝜔𝑏_𝑛+1𝑋^𝜋𝑛_𝑛+1−𝜔𝑥𝑏_𝑛+1𝑋^𝜋𝑛_𝑛+1²−𝐸_𝑛,𝑥𝜔𝑎_𝑛+1²𝑋^𝜋𝑛_𝑛+1+𝜔𝑥𝐸_𝑛,𝑥𝑎_𝑛+1𝑋^𝜋𝑛_𝑛+1²=max_𝜋𝑛𝐸_𝑛,𝑥𝑎_𝑛+1𝑋^𝜋𝑛_𝑛+1−𝜔𝑥𝑏_𝑛+1𝑋^𝜋𝑛_𝑛+1²+𝜔𝑥𝐸_𝑛,𝑥𝑎_𝑛+1𝑋^𝜋𝑛_𝑛+1²=max_𝜋𝑛𝐸_𝑛,𝑥𝑎_𝑛+1𝑥𝑟^𝑓_𝑛+𝜋^′_𝑛𝑅^𝑒_𝑛−𝜔𝑥𝑏_𝑛+1𝑥𝑟^𝑓_𝑛+𝜋_𝑛′𝑅^𝑒_𝑛²+𝜔𝑥𝑎_𝑛+1𝑥𝑟^𝑓_𝑛+𝜋^′_𝑛𝑟^𝑒_𝑛².=max_𝜋𝑛𝑎_𝑛+1𝑥𝑟^𝑓_𝑛+𝜋^′_𝑛𝑟^𝑒_𝑛−𝜔𝑥𝑏_𝑛+1𝑥²𝑟^𝑓_𝑛²+2𝑥𝑟^𝑓_𝑛𝜋^′_𝑛𝑟^𝑒_𝑛+𝜋^′_𝑛𝐸𝑅^𝑒_𝑛𝑅^𝑒_𝑛^′𝜋_𝑛+𝜔𝑥𝑎_𝑛+1𝑥𝑟^𝑓_𝑛+𝜋^′_𝑛𝑟^𝑒_𝑛². (65)(66)
By Lemma 6, we know that 𝜉_𝑛+1=𝑏_𝑛+1𝐸(𝑅^𝑒_𝑛𝑅^𝑒_𝑛^′)−(𝑎_𝑛+1)²𝑟^𝑒_𝑛𝑟^𝑒_𝑛^′ is positive definite. Given that 𝑥>0, we can see that the optimal solution of (66) exists and is 𝜋_𝑛(𝑥)=𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛×𝑏2𝜔_𝑛+1𝐸𝑅^𝑒_𝑛𝑅^𝑒_𝑛^′−𝑎_𝑛+1²𝑟^𝑒_𝑛𝑟^𝑒_𝑛^′−1𝑟^𝑒_𝑛𝑥. (67)
Then, 𝑔_𝑛(𝑥)=𝐸_𝑛,𝑥𝑔_𝑛+1𝑋𝜋_𝑛𝑛+1=𝐸_𝑛,𝑥𝑎_𝑛+1𝑥𝑟^𝑓_𝑛+𝜋^′_𝑛𝑅^𝑒_𝑛=𝑎_𝑛+1𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑟^𝑒_𝑛𝑥≡𝑎_𝑛ℎ𝑥,_𝑛(𝑥)=𝐸_𝑛,𝑥𝑏_𝑛+1𝑋𝜋_𝑛𝑛+1²=𝑏_𝑛+1𝐸_𝑛,𝑥𝑥𝑟^𝑓_𝑛+𝜋^′_𝑛𝑅^𝑒_𝑛²=𝑏_𝑛+1𝐸𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑅^𝑒_𝑛²𝑥²≡𝑏_𝑛𝑥². (68)
According to (65), we have 𝑉_𝑛(𝑥)=𝑎_𝑛+1𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑟^𝑒_𝑛𝑥−𝜔𝑏_𝑛+1𝐸𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑅^𝑒_𝑛²𝑥+𝜔𝑎_𝑛+1²𝑟^𝑓_𝑛+𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓_𝑛2𝜔×𝑟^𝑒_𝑛^′𝜉_𝑛+1⁻¹𝑟^𝑒_𝑛²𝑥=𝑎_𝑛𝑥+𝜔𝑎_𝑛²𝑥−𝜔𝑏_𝑛𝑥. (69)
Therefore, we complete the proof of Theorem 7.

4.2. Numerical Analysis

In this part, for convenience, we assume that there are only one risk-free asset and one risky asset in the market. Furthermore, we suppose that the risk-free return is a constant 𝑟^𝑓=1.15 during all the investment periods and risky returns {𝑅_𝑛,𝑛=0,1,…,𝑇−1} have the same distribution function with the same expectation 𝐸(𝑅_𝑛)=1.35 and variance Var(𝑅_𝑛)=0.3.

Based on the assumption above, time-consistent strategy (56) can be simplified as 𝜋_𝑛=𝑎_𝑛+1+2𝜔𝑎_𝑛+1²−𝑏_𝑛+1𝑟^𝑓𝑏_𝑛+1−𝑎_𝑛+1²−𝑏_𝑛+1(𝑟^𝑒)²/Var(𝑅^𝑒)×1𝑟2𝜔Var^𝑒(𝑅^𝑒)𝑥𝑛=0,1,…,𝑇−1, (70)

and we define 𝜌_𝑛=𝑎_𝑛+1+2𝜔𝑎_𝑛+1²𝑏_𝑛+1𝑟^𝑓𝑏_𝑛+1−𝑎_𝑛+1²−𝑏_𝑛+1(𝑟^𝑒)²/Var(𝑅^𝑒)1𝑟2𝜔Var^𝑒(𝑅^𝑒) (71)

as proportion invested in the risky asset at time 𝑛.

(1) First, we want to find the relationship between 𝜌_𝑛 and the coefficient of risk aversion 𝜔. Let 𝑇=8 and coefficient of risk aversion 𝜔=1,3,5, respectively, and then we get Figure 4. From Figure 4, we can find that the smaller the 𝜔, the larger the investment proportion. This is because coefficient 𝜔 now is an index directly proportional to risk aversion, and the higher the 𝜔 is, the more risk aversion the investor has.

Figure 4 

Effect of coefficient of risk aversion on the investment proportion.

(2) Let 𝜔 be a constant 3 and let 𝑇 take values 4, 6, 8, and 10, respectively, and then we obtain Figure 5, which indicates some interesting phenomena as follows.

Figure 5 

Investment proportions with different time horizons.

(a) In the last 4 periods, no matter whether 𝑇 is 4, 6, 8, or 10, the investment proportions at each corresponding period take the same values. For an instance, proportions in the final period are the same no matter which value 𝑇 takes. This is one important character of the time-consistent strategy since the decision at time 𝑛 under time-consistent framework just depends on the decisions of the forthcoming decision makers.

(b) In the first 4 periods, if investment time horizon 𝑇 is shorter, 𝜌_𝑛(𝑛=0,1,2,3) is higher. For an instance, when 𝑇=4, 𝜌₀ is higher than the 𝜌₀ when 𝑇=6,8, and 10. The reason is that, with shorter investment horizon, the investor will have more confidence to control the uncertainty in the future, which results in a higher investment proportion of the risky asset. In contrast, there is more uncertainty when investment time horizon 𝑇 is longer, then the investor would rather choose a relatively safer investment strategy, that is, choose a strategy with lower proportion invested in the risky asset.

5. Conclusions

It remains prevalent to obtain the pre-commitment strategy for multiperiod mean-variance portfolio selection problems, but not much is known about their time-consistent strategy. This paper aims to investigate the time-consistent Nash equilibrium strategy for a multiperiod mean-variance model. We view this decision-making process as a noncooperative game and suppose that there is one decision maker for each point of time. Two cases are considered in our paper. In the first case, we assume that the risk aversion is a constant and compare our time-consistent results with the precommitment ones. Some desired conclusions are obtained by rigorous proofs. In the second case, the risk aversion depends dynamically on the current wealth. The numerical analysis indicates that time-consistent decision at current time just depends on the decisions of the forthcoming decision makers, which is one important character of time-consistent strategy.

Acknowledgments

This research is supported by Grants of the MOE Project of the Key Research Institute of Humanities and Social Sciences at Universities of China (no. 11JJD790004), Humanity and Social Science Foundation of Ministry of Education of China (no. 12YJCZH219), and the National Natural Science Foundation of China (no. 71271223, no. 71201173).