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We study the optimal investment strategies of DC pension, with the stochastic interest rate (including the CIR model and the Vasicek model) and stochastic salary. In our model, the plan member is allowed to invest in a risk-free asset, a zero-coupon bond, and a single risky asset. By applying the Hamilton-Jacobi-Bellman equation, Legendre transform, and dual theory, we find the explicit solutions for the CRRA and CARA utility functions, respectively.

There are two radically different methods to design a pension fund: defined-benefit plan (hereinafter DB) and defined-contribution plan (hereinafter DC). In DB, the benefits are fixed in advance by the sponsor and the contributions are adjusted in order to maintain the fund in balance, where the associated financial risks are assumed by the sponsor agent; in DC, the contributions are fixed and the benefits depend on the returns on the assets of the fund, where the associated financial risks are borne by the beneficiary. Historically, DB is the more popular. However, in recent years, owing to the demographic evolution and the development of the equity markets, DC plays a crucial role in the social pension systems.

Our main objective in this paper is to find the optimal investment strategies for DC, which is a common model in the employment system. The paper extends the previous works of Cairns et al. [

Because the member of DC has some freedom in choosing the investment allocation of her pension fund in the accumulation phase, she has to solve an optimal investment strategies’ problem. Traditionally, the usual method to deal with it has been the maximization of expected utility of final wealth. Consistently with the economics and financial literature, the most widely used utility function exhibits constant relative risk aversion (CRRA), that is, the power or logarithmic utility function (e.g., [

The optimal portfolios for DC with stochastic interest rate have been widely discussed in the literatures. Some of them are by Boulier et al. [

Meanwhile, Deelstra et al. [

In addition, under the logarithmic utility function, Gao [

The most novel feature of our research is the application of affine interest rate model and stochastic salary under the CRRA and CARA utility functions, which has not been reported in the existing literature. We assume that the term structure of the interest rates is affine, not a constant and the salary volatility is a hedgeable volatility whose risk source belongs to the set of the financial market risk sources. Consequently, a complicated nonlinear second-order partial differential equation is derived by using the methods of stochastic optimal control. However, we find that it is difficult to determine an explicit solution, and then we transform the primary problem into the dual one by applying a Legendre transform and derive a linear partial differential equation. Furthermore, we obtain the explicit solutions for the optimal strategies under the CRRA or CARA utility functions.

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

In this section, we introduce the market structure and define the stochastic dynamics of the asset values and the salary.

We consider a complete and frictionless financial market which is continuously open over the fixed time interval

We suppose that the market is composed of three kinds of financial assets: a risk-free asset, a zero-coupon bond, and a single risky asset, and the investor can buy or sell continuously without incurring any restriction as short sales constraint or any trading cost. For the sake of simplicity, we will only consider a risky asset which can indeed represent the index of the stock market.

Let us begin with a complete probability space

We denote the price of the risk-free asset (i.e., cash) at time

Notes that the dynamics recover, as a special case, the Vasicek [

We assume that the price of the risky asset is a continuous time stochastic process. We denote the price of the risky asset (i.e., stock) at time

The last asset is a zero-coupon bond with maturity

Based on the works of Deelstra et al. [

According to the viewpoint of Cairns et al. [

Taking into (

At the time of retirement, the plan member will be concerned about the preservation of his standard of living so he will be interested in his retirement income relative to his preretirement salary [

Taking into (

In the remainder, therefore, we will focus on

The plan member will retire at time

Let us denote a strategy

Our objective is to find the optimal value function:

The Hamilton-Jacobi-Bellman (HJB) equation associated with the optimization problem is

The first-order maximizing conditions for the optimal strategies

We have

Putting this in (

Here, we notice that the stochastic control problem described in the previous section has been transformed into a PDE. The problem is now to solve (

In this section, we transform the non-linear second partial differential equation into a linear partial differential equation via the Legendre transform and dual theory.

Let

The function

If

So, we may rewrite

According to Theorem

The value of

The two functions

This leads to

So the function

At the terminal time, we denote

As a result,

Generally speaking,

So

By differentiating (

Substituting the expression (

Combining with

Here, we notice that the non-linear second-order partial differential equation (

Similarly, we can compute the optimal investment strategies as the feedback formulas in terms of derivatives of the value function. In terms of the dual function

The problem is now to solve the linear partial differential equation (

This section provides the explicit solutions for the CRRA and CARA utility functions.

Assume that the plan member takes a power utility function

The relative risk aversion of a decision maker with the utility described in (

According to

Therefore, we conjecture a solution to (

Then,

Introducing these derivatives in (

We can split (

Taking into account the boundary condition

Noting that (

Let

We can decompose (

Taking into account the boundary conditions, the solutions to (

From the above calculation, we finally obtain the optimal investment strategies under the CRRA utility.

The optimal investment strategies are given by

Note that the power utility function (

In this section, to make it easier for us to discuss the parameters’ effect on the optimal investment strategies, we suppose that

Consider

Since

Consider

Since

In addition, noting that

Consider

Since

Whether

By differentiating

On the bases of Lemmas

Meanwhile, based on Lemma

Therefore, whether

Consider

Since

According to Lemmas

The parameter

Lemma

Thus it can be seen that, as the retirement date approaches, the plan member will think more about how to invest between cash and bonds. However, Lemma

In agreement with Cairns et al. [

Lemma

Assume that the plan member takes an exponential utility function:

The absolute risk aversion of a decision maker with the utility described in (

According to

So, we conjecture a solution to (

Therefore,

Putting these derivatives into (

Again we can split this equation into three equations:

Combining with the account boundary conditions:

We conjecture a solution of (

Putting this into (

By matching coefficients, we can decompose (

Taking into account the boundary conditions, the solutions to (

From the above calculation, we finally obtain the optimal investment strategies under the CARA utility.

The optimal investment strategies are given by

Consider

Since

Consider

Since

Lemma

This can be explained by the risky tolerance, namely,

Lemma

Nevertheless, the change trend of

We have analyzed an investment problem for a defined contribution pension plan with stochastic salary under the affine interest rate model. In view of the related literatures, we have adopted the CRRA and CARA utility functions. And then, the problem of the maximization of the terminal relative wealth’s utility has been solved analytically by the Legendre transform and dual theory. As above mentioned, we have analyzed the effect of different parameters on the optimal investment strategies under the CRRA and CARA utility functions, respectively, and compared their differences. So, this paper extends the research of Gao [

The further research on the stochastic optimal control of DC mainly spread our work under the more generalized situation: (i) assuming the salary to be affected by non-hedgeable risk source under the research framework; (ii) assuming the risky asset to follow a constant elasticity of variance (CEV) model, and so forth. It is noteworthy that the optimal solution with the extended framework is very difficult. Nevertheless, the above methodology cannot be applied to the extended framework, which will result in a more sophisticated nonlinear partial differential equation and cannot tackle it at present.

The authors are grateful to an anonymous referee for careful reading of the paper and helpful comments and suggestions. X. Rong was supported by the Natural Science Foundation of Tianjin under Grant no. 09JCYBJC01800. C. Zhang was supported by the Young Scholar Program of Tianjin University of Finance and Economics (TJYQ201201).