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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.

Since the pioneering work of [

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 [

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

The rest of the paper is organized as follows. In Section

In this paper, we assume that investors join the market at time

As we know, the classic mean-variance optimization problem is as follows:

which results in a time-inconsistent strategy, that is, precommitment strategy. Therefore, as mentioned in Section

Let

Let

Then

In addition, if equilibrium strategy

Let

given that the decision maker

generally,

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:

The distribution function of the random returns

Short selling is allowed for all risky assets in all periods. Unlimited borrowing and lending are permitted. Transaction costs are not taken into account.

Capital additions or withdrawals are forbidden for all assets in all periods.

With the notations above, we can obtain the recursions of

By Definition

When the risk aversion is a constant,

According to (

Recursion (

The following theorem gives the explicit expressions of

When the risk aversion is a constant, the Nash equilibrium strategy is given by

The corresponding equilibrium value function is

Obviously (

Since

Substituting (

Hence (

It is obvious that the optimal solution of (

Substituting (

and according to (

Equations (

In view of (

and the precommitment value function of [

The relationship between (

Consider the following:

First of all, we have

then,

Now,

This proof gives an important inequality needed in the later analysis as

Lemma

The time-consistent strategy at each period is

Referring to [

The significant differences between

Since the time-consistent strategy at time

the time consistent is time deterministic but the precommitment one is stochastically dependent on the current wealth.

In this Section, we want to compare our efficient frontier with the one in [

Under the time-consistent strategy (

Substituting (

and for

Hence,

and for

By repeatedly using recursive equation (

Repeatedly using the above recursive equation yields (

Then according to (

The efficient frontier under the time-consistent strategy (

When

Equation (

When

Equation (

Referring to (

Now we want to compare efficient frontier (

Efficient frontier in [

In view of (

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

In this part, we want to compare expected terminal wealth and efficient frontier under time-consistent framework with the corresponding ones, respectively, in [

Effect of time horizons.

Effect of the risk aversion.

Efficient frontier.

In this section, we will consider an optimization problem

under the framework of Nash equilibrium, where the risk aversion

Similarly, referring to (

with terminal condition

The recursions of

Before giving the time-consistent results, we need to introduce the following sequences

For

and hence

and then

Now we assume that for

Similar to the proof of

The recursions (

If

If

If the initial wealth is big enough and risky assets have steady returns, then at each period wealth level is often greater than

By (

It is obvious that when

Then,

Substituting (

Equations (

We assume the results in Theorem

By Lemma

Then,

According to (

Therefore, we complete the proof of Theorem

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

Based on the assumption above, time-consistent strategy (

and we define

as proportion invested in the risky asset at time

Effect of coefficient of risk aversion on the investment proportion.

Investment proportions with different time horizons.

(a) In the last

(b) In the first

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.

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).