Optimal Investment Strategies for DC Pension with Stochastic Salary under the Affine Interest Rate Model

Abstract

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.

1. Introduction

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. [1] and Gao [2]. In particular, we consider the following framework: (i) the optimal investment strategies are derived with CARA and CRRA utility functions; (ii) the interest rate is affine (including the CIR model and the Vasicek model); (iii) the salary follows a general stochastic process.

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., [1–5]). Some papers use the utility function that exhibits constant absolute risk aversion (CARA), that is, the exponential utility function (e.g., [6]). Some papers also adopt the CRRA and CARA utility functions simultaneously (e.g., [7, 8]). In this paper, we show the optimal investment strategies for DC pension with the CRRA and CARA utility functions.

The optimal portfolios for DC with stochastic interest rate have been widely discussed in the literatures. Some of them are by Boulier et al. [3], Battocchio and Menoncin [6], and Cairns et al. [1], where the interest rate is assumed to be of the Vasicek model. However, in the works of Deelstra et al. [4] and Gao [2], the interest rate has an affine structure, which includes the Cox-Ingersoll-Ross (CIR) model and the Vasicek model. In the Vasicek model, the volatility of interest rate is only a constant. It can generate a negative interest rate, which is not in accord with the facts. But in the CIR model, the volatility of interest rate is modified by the square of interest rate, which more tallies with practice. Obviously, the affine interest rate model does not only contain the Cox-Ingersoll-Ross (CIR) model and the Vasicek model, but also more accords with practice.

Meanwhile, Deelstra et al. [4] assumed that the stochastic interest rates followed the affine dynamics, described the contribution flow by a nonnegative, progressive measurable and square-integrable process, and then studied optimal investment strategies for different examples of guarantees and contributions. Battocchio and Menoncin [6] took into account two background risks (the salary risk and the inflation) in the Vasicek framework and analyzed in detail the behavior of the optimal portfolio with respect to salary and inflation. Cairns et al. [1] incorporated asset, salary (labor-income), and interest-rate risk (the Vasicek model), used the member’s final salary as a numeraire, and then discussed various properties and characteristics of the optimal asset-allocation strategy both with and without the presence of nonhedgeable salary risk. However, except for them, the studies related with DC generally suppose that the salary is a constant, but the assumption is difficult to be accepted for the pension investment. In fact, the optimal investment for a pension fund involves quite a long period, generally from 20 to 40 years. The pension investment is considered to be a long-term investment problem. During the period, the salary switches violently; so it becomes crucial to take into account the salary risk. As a result, we consider the salary risk and use the member’s final salary as a numeraire based on the work of Cairns et al. [1].

In addition, under the logarithmic utility function, Gao [2] just studied the portfolio problem of DC with the affine interest rate but did not consider the stochastic salary. The contribution of this paper: (i) extends the research of Gao [2] to the case of the power (CRRA) and exponential (CARA) utility functions under the stochastic salary; (ii) extends the research of Cairns et al. [1] to the case of the plan member with the CRRA and CARA utility functions under the affine interest rate model (including the CIR model and the Vasicek model). We consider that the financial market consists of three assets: a risk-less asset (i.e., cash), a zero-coupon bond, and a single risky asset (i.e., stock). Applying the maximum principle, we derive a nonlinear second-order partial differential equation (PDE) for the value function of the optimization problem. However, it is difficult to characterize the solution structure, especially under the framework of stochastic interest rates and stochastic salary. But the primary problem can be changed into a dual one by applying a Legendre transform. The transform methods can be found from the works of Xiao et al. [5] and Gao [2, 8].

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 2, we introduce the mathematical model including the financial market, the stochastic salary, and the wealth process. In Section 3, we propose the optimization problems. In Section 4, we transform the nonlinear second partial differential equation into a linear partial differential equation by the Legendre transform and dual theory. In Section 5, we obtain the explicit solutions for the CRRA and CARA utility functions, respectively. In Section 6, we draw the conclusions.

2. Mathematical Model

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 [0,𝑇], where 𝑇>0 denotes the retirement time of a representative shareholder.

2.1. The Financial Market

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 (Ω,𝐹,𝑃), where Ω is the real space, and 𝑃 is the probability measure. {𝑊_𝑟(𝑡),𝑊_𝑠(𝑡)∶𝑡≥0} is a standard, two-dimensional Brownian motion defined on a complete probability space (Ω,𝐹,𝑃). The filtration 𝐹={𝐹_𝑡}_{𝑡∈[0,𝑇]} is a right continuous filtration of sigma-algebras on this space and denotes the information structure generated by the Brownian motions.

We denote the price of the risk-free asset (i.e., cash) at time 𝑡 by 𝑆₀(𝑡), which evolves according to the following equation: 𝑑𝑆₀(𝑡)=𝑟(𝑡)𝑆₀(𝑡)𝑑𝑡,𝑆₀(0)=1, (1) where the dynamics of the short interest rate process 𝑟(𝑡) are described by the following stochastic differential equation: 𝑑𝑟(𝑡)=(𝑎−𝑏𝑟(𝑡))𝑑𝑡−𝜎_𝑟𝑑𝑊_𝑟(𝑡),𝜎_𝑟=𝑘₁𝑟(𝑡)+𝑘₂,𝑡≥0, (2) with the coefficients 𝑎,𝑏,𝑟(0),𝑘₁, and 𝑘₂ being positive real constants.

Notes that the dynamics recover, as a special case, the Vasicek [9] (resp., Cox et al. [10]) dynamics, when 𝑘₁ (resp., 𝑘₂) is equal to zero. So under these dynamics, the term structure of the interest rates is affine, which has been studied by Duffie and Kan [11], Deelstra et al. [4], and Gao [2].

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 𝑡 by 𝑆(𝑡),𝑡≥0. The dynamics of 𝑆(𝑡) are given by 𝑑𝑆(𝑡)𝑆(𝑡)=𝑟(𝑡)𝑑𝑡+𝜎_𝑠𝑑𝑊_𝑠(𝑡)+𝜆₁𝑑𝑡+𝜂₁𝜎_𝑟𝑑𝑊_𝑟(𝑡)+𝜆₂𝜎_𝑟𝑑𝑡,𝑆(0)=𝑆₀, (3) with 𝜆₁,𝜆₂ (resp., 𝜎_𝑠,𝜂₁) being constants (resp., positive constants) (see Deelstra et al. [4] and Gao [2]). Here, the two Brownian motions, 𝑊_𝑟(𝑡) and 𝑊_𝑠(𝑡), are supposed to be orthogonal.

The last asset is a zero-coupon bond with maturity 𝑇, whose price at time 𝑡 is denoted by 𝐵(𝑡,𝑇),𝑡≥0, which is described by the following stochastic differential equation (c.f. [2, 4]): 𝑑𝐵(𝑡,𝑇)𝐵()(𝑡)𝑡,𝑇=𝑟𝑑𝑡+𝜎_𝐵((𝑡×𝑇−𝑡,𝑟))𝑑𝑊_𝑟(𝑡)+𝜆₂𝜎_𝑟𝑑𝑡,𝐵(𝑇,𝑇)=1, (4) where 𝜎_𝐵(𝑇−𝑡,𝑟(𝑡))=𝑓(𝑇−𝑡)𝜎_𝑟 with 𝑓(𝑡)=2𝑒^𝑚𝑡−1𝑚−𝑏−𝑘₁𝜆₂+𝑒^𝑚𝑡𝑚+𝑏−𝑘₁𝜆₂,𝑚=𝑏−𝑘₁𝜆₂²+2𝑘₁. (5)

2.2. The Stochastic Salary

Based on the works of Deelstra et al. [4], Battocchio and Menoncin [6], and Cairns et al. [1], we denote the salary at time 𝑡 by 𝐿(𝑡) which is described by 𝑑𝐿(𝑡)𝐿(𝑡)=𝜇_𝐿(𝑡,𝑟(𝑡))𝑑𝑡+𝜂₂𝜎_𝑟𝑑𝑊_𝑟(𝑡)+𝜂₃𝜎_𝑠𝑑𝑊_𝑠(𝑡),𝐿(0)=𝐿₀, (6) where 𝜂₂,𝜂₃ are real constants, which are two volatility scale factors measuring how the risk sources of interest rate and stock affect the salary. That is to say, the salary volatility is supposed to a hedgeable volatility whose risk source belongs to the set of the financial market risk sources. This assumption is in accordance with that of Deelstra et al. [4], but is different from those of Battocchio and Menoncin [6] and Cairns et al. [1] who also assumed that the salary was affected by nonhedgeable risk source (i.e., non-financial market). Moreover, we assume that the instantaneous mean of the salary is such that 𝜇_𝐿(𝑡,𝑟(𝑡))=𝑟(𝑡)+𝑚_𝐿, where 𝑚_𝐿 is a real constant.

2.3. Pension Wealth Process

According to the viewpoint of Cairns et al. [1], we consider that the contributions are continuously into the pension fund at the rate of 𝑘𝐿(𝑡). Let 𝑉_𝑡 denote the wealth of pension fund at time 𝑡∈[0,𝑇]. 𝜋_𝐵(𝑡) and 𝜋_𝑆(𝑡) are denoted, respectively, by the proportion of the pension fund invested in the bond and the stock; so 𝜋₀(𝑡)=1−𝜋_𝐵(𝑡)−𝜋_𝑆(𝑡) is the proportion of the pension fund invested in the risk-free asset. The dynamics of the pension wealth are given by 𝑑𝑉(𝑡)=1−𝜋_𝐵−𝜋_𝑆𝑉(𝑡)𝑑𝑆₀(𝑡)𝑆₀(𝑡)+𝜋_𝐵𝑉(𝑡)𝑑𝐵(𝑡,𝑇)𝐵(𝑡,𝑇)+𝜋_𝑠𝑉(𝑡)𝑑𝑆(𝑡)𝑆(𝑡)+𝑘𝐿(𝑡)𝑑𝑡, (7) where 𝑉(0)=𝑉₀ stands for an initial wealth.

Taking into (1), (3), and (4), the evolution of pension wealth can be rewritten as 𝑑𝑉(𝑡)=𝑉(𝑡)𝑟(𝑡)+𝜋_𝐵𝜆₂𝜎_𝑟𝜎_𝐵+𝜋_𝑆𝜆₁𝜎_𝑆+𝜆₂𝜂₁𝜎²_𝑟(𝑡)(𝑡)𝜋𝑑𝑡+𝑘𝐿𝑑𝑡+𝑉_𝐵𝜎_𝐵+𝜋_𝑆𝜂₁𝜎_𝑟𝑑𝑊_𝑟(𝑡)+𝑉(𝑡)𝜋_𝑆𝜎_𝑆𝑑𝑊_𝑠(𝑡). (8)

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 [1]. Considering the plan member’s salary as a numeraire, we define a new state variable 𝑋(𝑡)=𝑉(𝑡)/𝐿(𝑡) (i.e., the relative wealth).

Taking into (6) and (8), by applying product law and Ito’s formula, the stochastic differential equation for 𝑋(𝑡) is 𝑑𝑋(𝑡)=𝑋(𝑡)𝑟(𝑡)−𝜇_𝐿+𝜂²₂𝜎²_𝑟+𝜂²₃𝜎²_𝑆+𝜋_𝐵𝜎_𝑟𝜎_𝐵𝜆₂−𝜂₂+𝜋_𝑆𝜆₁𝜎_𝑆+𝜂₃𝜎²_𝑆+𝜆₂𝜂₁𝜎²_𝑟−𝜂₁𝜂₂𝜎²_𝑟(𝑡)𝜋𝑑𝑡+𝑘𝑑𝑡+𝑋_𝐵𝜎_𝐵+𝜋_𝑆𝜂₁𝜎_𝑟−𝜂₂𝜎_𝑟𝑑𝑊_𝑟(𝑡)+𝑋(𝑡)𝜋_𝑆−𝜂₃𝜎_𝑆𝑑𝑊_𝑠(𝑡),𝑋(0)=𝑉(0)𝐿(0)=𝑉₀𝐿₀. (9)

In the remainder, therefore, we will focus on 𝑋(𝑡) alone.

3. The Optimization Program

The plan member will retire at time 𝑇 and is risk averse; so the utility function 𝑈(𝑥) is typically increasing and concave (𝑈^′′(𝑥)<0). In this section, we are interested in maximizing the utility of the plan member’s terminal relative wealth.

Let us denote a strategy 𝜋_𝑡 which is described by a dynamic process (𝜋_𝐵(𝑡),𝜋_𝑆(𝑡)). For a strategy 𝜋_𝑡, we define the utility attained by the plan member from state 𝑥 at time 𝑡 as 𝐻_𝜋𝑡(𝑡,𝑟,𝑥)=𝐸_𝜋𝑡[𝑈(𝑋(𝑇))∣𝑟(𝑡)=𝑟,𝑋(𝑡)=𝑥]. (10)

Our objective is to find the optimal value function: 𝐻(𝑡,𝑟,𝑥)=sup_{𝜋𝑡∈𝜋}𝐻_𝜋𝑡(𝑡,𝑟,𝑥), (11) and the optimal strategy is 𝜋^∗_𝑡=(𝜋^∗_𝐵(𝑡),𝜋^∗_𝑆(𝑡)) such that 𝐻_𝜋^∗_𝑡(𝑡,𝑟,𝑥)=𝐻(𝑡,𝑟,𝑥).

The Hamilton-Jacobi-Bellman (HJB) equation associated with the optimization problem is 𝐻_𝑡+(𝑎−𝑏𝑟)𝐻_𝑟+12𝜎²_𝑟𝐻_𝑟𝑟+max_{𝜋𝑡∈𝜋}𝑥𝛼₁+𝜋_𝐵𝛼₂+𝜋_𝑆𝛼₃𝐻_𝑥+𝑘𝐻_𝑥+12𝑥²𝜋_𝐵𝜎_𝐵+𝜋_𝑆𝜂₁𝜎_𝑟−𝜂₂𝜎_𝑟²𝐻_𝑥𝑥+12𝑥²𝜋_𝑆−𝜂₃²𝜎²_𝑆𝐻_𝑥𝑥−𝑥𝜎_𝑟𝜋_𝐵𝜎_𝐵+𝜋_𝑆𝜂₁𝜎_𝑟−𝜂₂𝜎_𝑟𝐻_𝑟𝑥=0, (12) with 𝛼₁=𝑟−𝜇_𝐿+𝜂²₂𝜎²_𝑟+𝜂²₃𝜎²_𝑆,𝛼₂=𝜎_𝑟𝜎_𝐵𝜆₂−𝜂₂,𝛼₃=𝜆₁𝜎_𝑆+𝜂₃𝜎²_𝑆+𝜆₂𝜂₁𝜎²_𝑟−𝜂₁𝜂₂𝜎²_𝑟,𝐻(𝑇,𝑟,𝑥)=𝑈(𝑥), (13) where 𝐻_𝑡,𝐻_𝑟,𝐻_𝑥,𝐻_𝑟𝑥,𝐻_𝑟𝑟, and 𝐻_𝑥𝑥 denote partial derivatives of first and second orders with respect to time, short interest rate, and relative wealth.

The first-order maximizing conditions for the optimal strategies 𝜋^∗_𝐵 and 𝜋^∗_𝑆 are 𝛼₂𝐻_𝑥+𝑥𝜎_𝐵𝜋^∗_𝐵𝜎_𝐵+𝜋^∗_𝑆𝜂₁𝜎_𝑟−𝜂₂𝜎_𝑟𝐻_𝑥𝑥−𝜎_𝑟𝜎_𝐵𝐻_𝑟𝑥𝛼=0,₃𝐻_𝑥+𝑥𝜂₁𝜎_𝑟𝜋^∗_𝐵𝜎_𝐵+𝜋^∗_𝑆𝜂₁𝜎_𝑟−𝜂₂𝜎_𝑟𝐻_𝑥𝑥+𝑥𝜎_𝑆𝜋^∗_𝑆−𝜂₃𝐻_𝑥𝑥−𝜂₁𝜎²_𝑟𝐻_𝑟𝑥=0. (14)

We have 𝜋^∗_𝑆=𝜂₃−𝜆₁+𝜂₃𝜎²_𝑆𝑥𝜎_𝑆𝐻_𝑥𝐻_𝑥𝑥,𝜋^∗_𝐵=𝜎_𝑟𝜂₂−𝜂₁𝜂₃𝜎_𝐵+𝛼₄𝜎_𝑟𝑥𝜎_𝐵𝐻_𝑥𝐻_𝑥𝑥+𝜎_𝑟𝑥𝜎_𝐵𝐻_𝑟𝑥𝐻_𝑥𝑥,𝛼₄=𝜂₂𝜎_𝑆+𝜆₁𝜂₁+𝜂₁𝜂₃𝜎²_𝑆−𝜆₂𝜎_𝑆𝜎_𝑆, (15)

Putting this in (12), we obtain a partial differential equation (PDE) for the value function 𝐻: 𝐻_𝑡+(𝑎−𝑏𝑟)𝐻_𝑟+12𝜎²_𝑟𝐻_𝑟𝑟+𝑘+𝑥𝛽₀𝐻_𝑥+𝛽₁−12𝜆₂−𝜂₂²𝜎²_𝑟𝐻²_𝑥𝐻_𝑥𝑥+𝜆₂−𝜂₂𝜎²_𝑟𝐻_𝑥𝐻_𝑟𝑥𝐻_𝑥𝑥−12𝜎²_𝑟𝐻²_𝑟𝑥𝐻_𝑥𝑥𝛽=0,₀=𝜆₁𝜂₃𝜎_𝑆+𝜆₂𝜂₂𝜎²_𝑟+2𝜂²₃𝜎²_𝑆−𝑚_𝐿,𝛽₁=𝜂₃𝜎_𝑆12𝜂₃𝜎³_𝑆−𝜂₃𝜎²_S−𝜆₁−12𝜆²₁. (16) with 𝐻(𝑇,𝑟,𝑥)=𝑈(𝑥).

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 (16) for the value function 𝐻 and replace it in (15) in order to obtain the optimal investment strategies.

4. The Legendre Transform

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.

Theorem 1. Let 𝑓∶𝑅^𝑛→𝑅 be a convex function, for 𝑧>0, define the Legendre transform: 𝐿(𝑧)=max_𝑥𝑓(𝑥)−𝑧𝑥. (17)

The function 𝐿(𝑧) is called the Legendre dual of the function 𝑓(𝑥) (c.f. [12]).

If 𝑓(𝑥) is strictly convex, the maximum in the above equation will be attained at just one point, which we denote by 𝑥₀. It is attained at the unique solution to the first-order condition, namely, 𝑑𝑓(𝑥)/𝑑𝑥−𝑧=0.

So, we may rewrite 𝐿(𝑧)=𝑓(𝑥₀)−𝑧𝑥₀.

According to Theorem 1, we can take advantage of the assumed convexity of the value function 𝐻(𝑡,𝑟,𝑥) to define the Legendre transform: 𝐻(𝑡,𝑟,𝑧)=sup_𝑥>0{𝐻(𝑡,𝑟,𝑥)−𝑧𝑥∣0<𝑥<∞},0<𝑡<𝑇, (18) where 𝑧>0 denotes the dual variable to 𝑥, which is the same as those of Xiao et al. [5] and Gao [2, 8].

The value of 𝑥 where this optimum is attained is denoted by 𝑔(𝑡,𝑟,𝑧), so that 𝑔(𝑡,𝑟,𝑧)=inf_𝑥>0𝑥∣𝐻(𝑡,𝑟,𝑥)≥𝑧𝑥+𝐻(𝑡,𝑟,𝑧),0<𝑡<𝑇. (19)

The two functions 𝑔(𝑡,𝑟,𝑧) and 𝐻(𝑡,𝑟,𝑧) are closely related, and we will refer to either one of them as the dual of 𝐻. In this paper, we will work mainly with the function 𝑔, as it is easier to compute numerically and suffices for the purpose of computing optimal investment strategies.

This leads to 𝐻()𝑔()𝑡,𝑟,𝑧=𝐻𝑡,𝑟,𝑔−𝑧𝑔,𝑡,𝑟,𝑧=𝑥,𝐻_𝑥=𝑧. (20)

So the function 𝐻 is related to 𝑔 by 𝑔=−𝐻_𝑧.

At the terminal time, we denote 𝑈(𝑧)=sup_𝑣>0{𝑈(𝑣)},𝐺(𝑧)−𝑧𝑣∣0<𝑣<∞=sup_𝑣>0𝑣∣𝑈(𝑣)≥𝑧𝑣+𝑈(𝑧). (21)

As a result, 𝐺(𝑧)=(𝑈^′)⁻¹(𝑧).

Generally speaking, 𝐺 is referred to as the inverse of marginal utility. Note that 𝐻(𝑇,𝑟,𝑥)=𝑈(𝑥), and then at the terminal time 𝑇, we can define 𝑔(𝑇,𝑟,𝑧)=inf_𝑥>0(𝑥)𝐻(,𝐻()𝑥∣𝑈≥𝑧𝑥+𝑇,𝑟,𝑧)𝑇,𝑟,𝑧=sup_𝑥>0{𝑈(𝑥)−𝑧𝑥}. (22)

So 𝑔(𝑇,𝑟,𝑧)=(𝑈^′)⁻¹(𝑧).

By differentiating (20) with respect to 𝑡, 𝑟, and 𝑧, the transformation rules for the derivatives of the value function 𝐻 and the dual function 𝐻 can be given by (e.g., [2, 5, 8, 12]): 𝐻_𝑥=𝑧,𝐻_𝑡=𝐻_𝑡,𝐻_𝑟=𝐻_𝑟,𝐻_𝑟𝑟=𝐻_𝑟𝑟−𝐻²_𝑟𝑧𝐻_𝑧𝑧,𝐻_𝑟𝑥𝐻=−^𝑟𝑧𝐻_𝑧𝑧,𝐻_𝑥𝑥1=−𝐻_𝑧𝑧. (23)

Substituting the expression (23), we rewrite (16) and obtain the following partial differential equation: 𝐻_𝑡+(𝑎−𝑏𝑟)𝐻_𝑟+12𝜎²_𝑟𝐻_𝑟𝑟+𝑘+𝑥𝛽₀𝑧−𝛽₁−12𝜆₂−𝜂₂²𝜎²_𝑟𝑧²𝐻_𝑧𝑧+𝜆₂−𝜂₂𝜎²_𝑟𝑧𝐻_𝑟𝑧𝛽=0,₀=𝜆₁𝜂₃𝜎_𝑆+𝜆₂𝜂₂𝜎²_𝑟+2𝜂²₃𝜎²_𝑆−𝑚_𝐿,𝛽₁=𝜂₃𝜎_𝑆12𝜂₃𝜎³_𝑆−𝜂₃𝜎²_𝑆−𝜆₁−12𝜆²₁. (24)

Combining with 𝑥=𝑔=−𝐻_𝑧 and differentiating the above equation for 𝐻 with respect to 𝑧, we derive 𝑔_𝑡+(𝑎−𝑏𝑟)𝑔_𝑟+12𝜎²_𝑟𝑔_𝑟𝑟−𝑘−𝛽₀𝑔−𝛽₀𝑧𝑔_𝑧+𝜆₂−𝜂₂𝜎²_𝑟𝑔_𝑟+𝜆₂−𝜂₂𝜎²_𝑟𝑧𝑔_𝑟𝑧−2𝛽₁−12𝜆₂−𝜂₂²𝜎²_𝑟𝑧𝑔_𝑧−𝛽₁−12𝜆₂−𝜂₂²𝜎²_𝑟𝑧²𝑔_𝑧𝑧𝛽=0,₀=𝜆₁𝜂₃𝜎_𝑆+𝜆₂𝜂₂𝜎²_𝑟+2𝜂²₃𝜎²_𝑆−𝑚_𝐿,𝛽₁=𝜂₃𝜎_𝑆12𝜂₃𝜎³_𝑆−𝜂₃𝜎²_𝑆−𝜆₁−12𝜆²₁. (25)

Here, we notice that the non-linear second-order partial differential equation (16) has been transformed into a linear partial differential equation (25) by using the Legendre transform and dual theory. Under the given utility function, it is easy to find the solution of (25) by the classical variable decomposition approach.

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 𝑔, they are given by 𝜋₀(𝑡)=1−𝜋_𝐵(𝑡)−𝜋_𝑆(𝑡),𝜋^∗_𝑆=𝜂₃+𝜆₁+𝜂₃𝜎²_𝑆𝑥𝜎_𝑆𝑧𝐻_𝑧𝑧=𝜂₃−𝜆₁+𝜂₃𝜎²_𝑆𝑥𝜎_𝑆𝑧𝑔_𝑧,𝜋^∗_𝐵=𝜂₂−𝜂₁𝜂₃𝑓()−𝛼𝑇−𝑡⁴𝑧𝐻_𝑧𝑧()+𝐻𝑥𝑓𝑇−𝑡(^𝑟𝑧)=𝜂𝑥𝑓𝑇−𝑡₂−𝜂₁𝜂₃𝑓()+𝛼𝑇−𝑡𝑥⁴𝑧𝑔_𝑧−𝑔_𝑟(),𝛼𝑥𝑓𝑇−𝑡₄=𝜂₂𝜎_𝑆+𝜆₁𝜂₁+𝜂₁𝜂₃𝜎²_𝑆−𝜆₂𝜎_𝑆𝜎_𝑆,𝑓(𝑡)=2𝑒^𝑚𝑡−1𝑚−𝑏−𝑘₁𝜆₂+𝑒^𝑚𝑡𝑚+𝑏−𝑘₁𝜆₂𝑚=𝑏−𝑘₁𝜆₂²+2𝑘₁. (26)

The problem is now to solve the linear partial differential equation (25) for 𝑔 and to replace these solutions in (26) in order to obtain the optimal strategies.

5. Optimal Investment Strategies with Some Specific Utilities

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

5.1. The Explicit Solution for The CRRA Utility Function

Assume that the plan member takes a power utility function 𝑈(𝑥)=𝑥^𝑝𝑝,with𝑝<1,𝑝≠0. (27)

The relative risk aversion of a decision maker with the utility described in (27) is constant, and (27) is a CRRA utility.

According to 𝑔(𝑇,𝑟,𝑧)=(𝑈^′)⁻¹(𝑧) and the CRRA utility function, we obtain 𝑔(𝑇,𝑟,𝑧)=𝑧^1/(𝑝−1). (28)

Therefore, we conjecture a solution to (25) with the following form: 𝑔(𝑡,𝑟,𝑧)=𝑧^1/(𝑝−1)ℎ(𝑡,𝑟)+𝑎(𝑡), (29) with the boundary conditions given by 𝑎(𝑇)=0,ℎ(𝑇,𝑟)=1.

Then, 𝑔_𝑡=ℎ_𝑡𝑧^1/(𝑝−1)+𝑎^′(𝑡),𝑔_𝑟=ℎ_𝑟𝑧^1/(𝑝−1),𝑔_𝑧ℎ=−𝑧1−𝑝^{(1/(𝑝−1))−1},𝑔_𝑟𝑟=ℎ_𝑟𝑟𝑧^1/(𝑝−1),𝑔_𝑟𝑧ℎ=−^𝑟𝑧1−𝑝^{(1/(𝑝−1))−1},𝑔_𝑧𝑧=2−𝑝ℎ1−𝑝²𝑧^{(1/(𝑝−1))−2}. (30)

Introducing these derivatives in (25), we derive ℎ_𝑡+(𝑎−𝑏𝑟)ℎ_𝑟−𝜆₂−𝜂₂𝑝𝜎²_𝑟ℎ1−𝑝^𝑟+12𝜎²_𝑟ℎ_𝑟𝑟+𝛽₀𝑝1−𝑝ℎ−𝑝ℎ1−𝑝²𝛽₁−12𝜆₂−𝜂₂²𝜎²_𝑟𝑧^1/(𝑝−1)+𝑎^′(𝑡)−𝛽₀𝑎(𝑡)𝛽−𝑘=0,₀=𝜆₁𝜂₃𝜎_𝑆+𝜆₂𝜂₂𝜎²_𝑟+2𝜂²₃𝜎²_𝑆−𝑚_𝐿,𝛽₁=𝜂₃𝜎_𝑆12𝜂₃𝜎³_𝑆−𝜂₃𝜎²_𝑆−𝜆₁−12𝜆²₁. (31)

We can split (31) into two equations in order to eliminate the dependence on 𝑧^1/(𝑝−1): 𝑎^′(𝑡)−𝛽₀𝑎(𝑡)ℎ−𝑘=0,_𝑡+(𝑎−𝑏𝑟)ℎ_𝑟−𝜆₂−𝜂₂𝑝𝜎²_𝑟ℎ1−𝑝^𝑟+12𝜎²_𝑟ℎ_𝑟𝑟+𝛽₀𝑝1−𝑝ℎ−𝑝ℎ1−𝑝²𝛽₁−12𝜆₂−𝜂₂²𝜎²_𝑟=0. (32)(33)

Taking into account the boundary condition 𝑎(𝑇)=0, the solution to (32) is 𝑎(𝑡)=−𝑘1−𝑒^{−𝛽0(𝑇−𝑡)}𝛽₀,𝛽₀=𝜆₁𝜂₃𝜎_𝑆+𝜆₂𝜂₂𝜎²_𝑟+2𝜂²₃𝜎²_𝑆−𝑚_𝐿, (34) where 𝑎^__𝑇−𝑡|=(1−𝑒^{−𝛽0(𝑇−𝑡)})/𝛽₀ is a continuous annuity of duration 𝑇−𝑡, and 𝛽₀ is the continuous technical rate.

Noting that (33) is a linear second-order PDE, we find the solution by the classical variable decomposition approach.

Let ℎ(𝑡,𝑟)=𝐴(𝑡)𝑒^{𝐵(𝑡)𝑟} (35) with the boundary conditions: 𝐴(𝑇)=1,𝐵(𝑇)=0. Introducing this in (33), we obtain 𝐴_𝑡𝐴+𝑎−𝜆₂−𝜂₂𝑘₁+𝑎𝑝11−𝑝𝐵+2𝑘₂𝐵²+𝑝𝛽₀−𝛽₁−𝑝𝛽₀1−𝑝²+𝜆₂−𝜂₂²𝑝𝑘₂21−𝑝²+𝑟𝐵_𝑡−𝑏+𝜆₂−𝜂₂𝑘₁−𝑏𝑝𝐵+11−𝑝2𝑘₁𝐵²+𝜆₂−𝜂₂²𝑝𝑘₁21−𝑝²𝛽=0,₀=𝜆₁𝜂₃𝜎_𝑆+𝜆₂𝜂₂𝜎²_𝑟+2𝜂²₃𝜎²_𝑆−𝑚_𝐿,𝛽₁=𝜂₃𝜎_𝑆12𝜂₃𝜎³_𝑆−𝜂₃𝜎²_𝑆−𝜆₁−12𝜆²₁. (36)

We can decompose (36) into two conditions in order to eliminate the dependence on 𝑟 and 𝑡: 𝐴_𝑡𝐴+𝑎−𝜆₂−𝜂₂𝑘₁+𝑎𝑝11−𝑝𝐵+2𝑘₂𝐵²+𝑝𝛽₀−𝛽₁−𝑝𝛽₀1−𝑝²+𝜆₂−𝜂₂²𝑝𝑘₂21−𝑝²𝐵=0,_𝑡−𝑏+𝜆₂−𝜂₂𝑘₁−𝑏𝑝𝐵+11−𝑝2𝑘₁𝐵²+𝜆₂−𝜂₂²𝑝𝑘₁21−𝑝²=0. (37)

Taking into account the boundary conditions, the solutions to (37) are 𝐵(𝑡)=𝑚₁−𝑚₁𝑒^{(1/2)𝑘1(𝑚1−𝑚2)(𝑇−𝑡)}1−𝑚₁/𝑚₂𝑒^{(1/2)𝑘1(𝑚1−𝑚2)(𝑇−𝑡)},𝐴(𝑡)𝜆=exp²−𝜂₂𝑘₁+𝑎𝑝−𝑎𝐵(𝑡)−11−𝑝𝑑𝑡2𝑘₂𝐵²(𝑡)−𝑝𝛽𝑑𝑡₀−𝛽₁−𝑝𝛽₀1−𝑝²𝑡+𝐶,𝐴(𝑇)=1, (38) where 𝑚_1,2=𝑏+𝜆₂−𝜂₂𝑘₁𝑝±−𝑏𝑏+𝜆₂−𝜂₂𝑘₁−𝑏𝑝²−𝜆₂−𝜂₂²𝑘²₁𝑝×1−𝑝𝑘₁⁻¹. (39)

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

Proposition 2. The optimal investment strategies are given by 𝜋₀(𝑡)=1−𝜋_𝐵(𝑡)−𝜋_𝑆(𝑡),𝜋^∗_𝑆=𝜂₃+𝜆₁+𝜂₃𝜎²_𝑆1−𝑝𝜎_𝑆𝐼(),𝜋𝑡,𝑟_∗𝐵=1𝑓()𝜂𝑇−𝑡²−𝜂₁𝜂₃−𝛼₄𝐼()𝐽(𝑡,1−𝑝𝑡,𝑟) (40) where 𝐼(𝑡,𝑟)=1+𝑘𝑙𝑣^_𝑎_𝑇−𝑡|,_{_}𝑎_𝑇−𝑡|=1−𝑒^{−𝛽0(𝑇−𝑡)}𝛽₀,𝐽(𝑡)𝐵(𝑡)=1+1−𝑝𝛼₄,𝐵(𝑡)=𝑚₁−𝑚₁𝑒^{(1/2)𝑘1(𝑚1−𝑚2)(𝑇−𝑡)}1−𝑚₁/𝑚₂𝑒^{(1/2)𝑘1(𝑚1−𝑚2)(𝑇−𝑡)},𝑓(𝑡)=2𝑒^𝑚𝑡−1𝑚−𝑏−𝑘₁𝜆₂+𝑒^𝑚𝑡𝑚+𝑏−𝑘₁𝜆₂,𝑚=𝑏−𝑘₁𝜆₂²+2𝑘₁,𝛽₀=𝜆₁𝜂₃𝜎_𝑆+𝜆₂𝜂₂𝜎²_𝑟+2𝜂²₃𝜎²_𝑆−𝑚_𝐿,𝛼₄=𝜂₂𝜎_𝑆+𝜆₁𝜂₁+𝜂₁𝜂₃𝜎²_𝑆−𝜆₂𝜎_𝑆𝜎_𝑆,𝑚_1,2=𝑏+𝜆₂−𝜂₂𝑘₁𝑝±−𝑏𝑏+𝜆₂−𝜂₂𝑘₁−𝑏𝑝²−𝜆₂−𝜂₂²𝑘²₁𝑝×1−𝑝𝑘₁⁻¹. (41)(42)(43)

Remark 3. Note that the power utility function (27) will degenerate into a logarithmic utility function 𝑈(𝑥)=ln𝑥 as the limit 𝑝→0 (e.g., [7, 13, 14]). Meanwhile, in (6), if 𝜂₂=0,𝜂₃=0, the salary is not stochastic; so the contributions are not stochastic. If we further assume that 𝑙=1, the model is the same as the model of Gao [2]. From Proposition 2, we find that as the limit 𝑝→0, the coefficients 𝑚_1,2 will reduce to 2𝑏/𝑘₁ and zero, respectively. In this case, the coefficients 𝐵(𝑡) and 𝐽(𝑡) will, respectively, reduce to zero and one. As a result, the optimal investment strategies for a logarithmic utility function are 𝜋^∗_𝑆=𝜆₁𝜎_𝑆𝑘1+𝑣^_𝑎_{𝑇−𝑡|𝑟},𝜋^∗_𝐵=𝜎_𝑟𝜆₂𝜎_𝑆−𝜆₁𝜂₁𝜎_𝐵𝜎_𝑆𝑘1+𝑣^_𝑎_{𝑇−𝑡|𝑟}, (44) where 𝜋^∗_𝑆 is the same as the result of Gao [2], but 𝜋^∗_𝐵 is different from that result because Gao [2] made mistakes in calculation.

In this section, to make it easier for us to discuss the parameters’ effect on the optimal investment strategies, we suppose that 𝛽₀>0,𝜆₁>0, and 𝜆₂>0, where the assumption is generally in line with reality.

Lemma 4. Consider𝐼()()𝑡,𝑟>0,𝑑𝐼𝑡,𝑟𝑑𝑡<0,𝑑𝜋^∗_𝑆𝑑𝑡<0. (45)

Proof. Since 𝑝<1,𝑘>0,𝛽₀>0,𝜆₁>0,𝜂₃>0, and 𝜎_𝑆>0, by differentiating 𝐼(𝑡,𝑟) with the respect to 𝑡, we have 𝑎^__𝑇−𝑡|=1−𝑒^{−𝛽0(𝑇−𝑡)}𝛽₀()()>0,𝐼𝑡,𝑟>0,𝑑𝐼𝑡,𝑟=𝑑𝑡𝑘𝑙𝑣𝑑𝑎^__𝑇−𝑡|𝑑𝑡=−𝑘𝑙𝑣𝑒^{−𝛽0(𝑇−𝑡)}<0,𝑑𝜋^∗_𝑆=𝜆𝑑𝑡¹+𝜂₃𝜎²_𝑆1−𝑝𝜎_𝑆𝑑𝐼(𝑡,𝑟)𝑑𝑡<0. (46)

Lemma 5. Consider𝑑𝐽(𝑡)=,,𝐽(𝑡)=,.𝑑𝑡>0,𝑝<0<0,0<𝑝<1≤1,𝑝<0≥1,0<𝑝<1 (47)

Proof. Since 𝑝<1, we have 𝑚₁×𝑚₂=𝜆₂−𝜂₂²𝑝1−𝑝²=,.<0,𝑝<0>0,0<𝑝<1 (48) Here, we just consider the condition of 𝛼₄>0. Differentiating 𝐵(𝑡) with the respect to 𝑡, we have 𝑑𝐵(𝑡)=−𝑚𝑑𝑡¹−𝑚₂²𝑘₁𝑚₁/2𝑚₂𝑒^{(1/2)𝑘1(𝑚1−𝑚2)(𝑇−𝑡)}1−𝑚₁/𝑚₂𝑒^{(1/2)𝑘1(𝑚1−𝑚2)(𝑇−𝑡)2}=,(𝑡)>0,𝑝<0<0,0<𝑝<1𝑑𝐽=𝑑𝑡1−𝑝𝛼₄𝑑𝐵(𝑡)=,𝑑𝑡.>0,𝑝<0<0,0<𝑝<1 (49)
In addition, noting that 𝐵(𝑇)=0 and 𝐽(𝑇)=1, we get 𝐽(𝑡)=,.≤1,𝑝<0≥1,0<𝑝<1 (50)

Lemma 6. Consider𝑓()()𝑇−𝑡>0,𝑑𝑓𝑇−𝑡𝑑𝑡<0. (51)

Proof. Since 𝑇−𝑡>0, and 𝑘₁>0, we have, 𝑚=𝑏−𝑘₁𝜆₂²+2𝑘₁>𝑏−𝑘₁𝜆₂𝑒>0,_{𝑚(𝑇−𝑡)}𝑓()=2𝑒>1,𝑇−𝑡_{𝑚(𝑇−𝑡)}−1𝑚−𝑏−𝑘₁𝜆₂+𝑒^{𝑚(𝑇−𝑡)}𝑚+𝑏−𝑘₁𝜆₂()>0,𝑑𝑓𝑇−𝑡𝑑𝑡=−4𝑚²𝑒^{𝑚(𝑇−𝑡)}𝑚−𝑏−𝑘₁𝜆₂+𝑒^{𝑚(𝑇−𝑡)}𝑚+𝑏−𝑘₁𝜆₂²<0. (52)

Lemma 7. Whether 𝑑𝜋^∗_𝐵/𝑑𝑡 is positive or negative or neither is not established, and it is affected by the coefficient of relative risk aversion 𝑝 and the other parameters.

Proof. By differentiating 𝜋^∗_𝐵 with the respect to 𝑡, we have𝑑𝜋^∗_𝐵=𝑑𝑡−1𝑓²()()𝑇−𝑡𝑑𝑓𝑇−𝑡×𝜂𝑑𝑡₂−𝜂₁𝜂₃−𝛼₄𝐼()𝐽(𝑡−𝛼1−𝑝𝑡,𝑟)₄𝑓()𝐼()(𝑡)1−𝑝𝑇−𝑡𝑡,𝑟𝑑𝐽+()𝑑𝑡𝑑𝐼𝑡,𝑟𝐽(𝑡.d𝑡) (53)
On the bases of Lemmas 4 and 6, we get 𝐼()()𝑡,𝑟>0,𝑑𝐼𝑡,𝑟𝑓()()𝑑𝑡<0,𝑇−𝑡>0,𝑑𝑓𝑇−𝑡𝑑𝑡<0. (54)
Meanwhile, based on Lemma 5, we get 𝑑𝐽(𝑡)=,,𝐽(𝑡)=,.𝑑𝑡>0,𝑝<0<0,0<𝑝<1≤1,𝑝<0≥1,0<𝑝<1 (55)
Therefore, whether 𝑑𝜋^∗_𝐵/𝑑𝑡 is positive or negative or neither is very complicated.

Lemma 8. Consider𝑑𝜋^∗_𝑆𝑑𝑙>0,𝑑𝜋^∗_𝐵=,𝑑𝑙.−,𝑝<0<0,0<𝑝<1 (56)

Proof. Since 𝑝<1,𝑘>0,𝛽₀>0,𝜆₁>0,𝜂₃>0, and 𝜎_𝑆>0, therefore 𝑎^__𝑇−𝑡|()>0,𝑑𝐼𝑡,𝑟=𝑘𝑑𝑙𝑣^_𝑎_𝑇−𝑡|>0𝑑𝜋^∗_𝑆=𝜆𝑑𝑙¹+𝜂₃𝜎²_𝑆1−𝑝𝜎_𝑆𝑑𝐼(𝑡,𝑟)𝑑𝑙>0. (57)
According to Lemmas 5 and 6, we get 𝐽(𝑡)=,,𝑓()≤1,𝑝<0≥1,0<𝑝<1𝑇−𝑡>0. (58) Similarly, we just consider the condition of 𝛼₄>0. So, 𝑑𝜋^∗_𝐵𝛼𝑑𝑙=−⁴𝐽(𝑡)𝑓()()1−𝑝𝑇−𝑡𝑑𝐼𝑡,𝑟=,𝑑𝑙.−,𝑝<0<0,0<𝑝<1 (59)

Remark 9. The parameter 𝑝 is the coefficient of the relative risk aversion. Hence, the plan member would like to avoid risk strongly if they get high 𝑝.
Lemma 4 shows that the optimal proportion invested in stock 𝜋^∗_𝑆 depends on the time 𝑡 and is a monotone decreasing function with respect to time 𝑡, but the trend is not affected by 𝑝. The stock is regarded as high risk, whose purpose is to satisfy the risk appetite of the plan member and hedge the risk. So as the retirement date approaches, the risk appetite begins to decrease so that the optimal proportion invested in stock is monotonically decreasing. It is concluded that as the retirement date approaches, there is a gradual switch from high-risk investment (i.e., stock) into low-risk investment (i.e., cash and bonds).
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 7 indicates that the effect of the time 𝑡 on 𝜋^∗_𝐵 depends on the risk aversion coefficient 𝑝 and the other parameters under the power utility. Consequently, as the retirement date approaches, how to invest between cash and bonds mainly depends on the risk aversion coefficient 𝑝 and the other parameters.
In agreement with Cairns et al. [1], instead of switching from high-risk assets into low-risk assets, in the stochastic interest rate framework, the optimal investment strategies involve a switch between different types of low-risk assets (i.e., cash and bonds).
Lemma 8 reveals that the optimal proportion invested in stock 𝜋^∗_𝑆 is a monotone increasing function with respect to the salary numeraire 𝑙, which means that the plan member will be more reluctant to invest in stock when the salary numeraire 𝑙 becomes larger, but the trend is not affected by 𝑝. However, the effect of 𝑙 on the optimal proportion invested in bonds 𝜋^∗_𝐵 depends on the risk aversion coefficient 𝑝 under the power utility. When 0<𝑝<1, 𝜋^∗_𝐵 is a monotone decreasing function with respect to 𝑙. Because the plan members would like to avoid risk strongly if they get high 𝑝, they invest in cash more as 𝑙 increases. But when the risk aversion coefficient 𝑝<0, 𝜋^∗_𝐵 depends on the risk aversion coefficient 𝑝 and the other parameters.

5.2. The Explicit Solution for The CARA Utility Function

Assume that the plan member takes an exponential utility function: 𝑈(𝑥)1=−𝑞𝑒^−𝑞𝑥,with𝑞>0. (60)

The absolute risk aversion of a decision maker with the utility described in (60) is constant, and (60) is a CARA utility.

According to 𝑔(𝑇,𝑟,𝑧)=(𝑈^′)⁻¹(𝑧) and the CARA utility function, we obtain 𝑔()1𝑇,𝑟,𝑧=−𝑞ln𝑧. (61)

So, we conjecture a solution to (25) with the following form: 𝑔()1𝑡,𝑟,𝑧=−𝑞[𝑏(𝑡)(ln𝑧+𝑚(𝑡,𝑟))]+𝑎(𝑡), (62) with the boundary conditions given by 𝑏(𝑇)=1,𝑎(𝑇)=0,𝑚(𝑇,𝑠)=0.

Therefore, 𝑔_𝑡1=−𝑞𝑏^′(𝑡)(ln𝑧+𝑚(𝑡,𝑟))+𝑏(𝑡)𝑚_𝑡+𝑎^′(𝑡),𝑔_𝑟1=−𝑞𝑏(𝑡)𝑚_𝑟,𝑔_𝑧𝑏(𝑡)=−,𝑔𝑞𝑧_𝑧𝑧=𝑏(𝑡)𝑞𝑧²,𝑔_𝑟𝑟1=−𝑞𝑏(𝑡)𝑚_𝑟𝑟,𝑔_𝑟𝑧=0. (63)

Putting these derivatives into (25), we derive 𝛽₀𝑏(𝑡)−𝑏^′(𝑡)ln𝑧+𝑎^′(𝑡)−𝛽₀𝑎(𝑡)𝑞−𝑚−𝑘_𝑡+12𝜎²_𝑟𝑚_𝑟𝑟−𝛽₀𝑚+(𝑎−𝑏𝑟)𝑚_𝑟+𝜆₂−𝜂₂𝜎²_𝑟𝑚_𝑟−𝛽₀+𝛽₁+12𝜆₂−𝜂₂²𝜎²_𝑟+𝑏^′(𝑡)𝑏(𝑡)𝑚𝑏(𝑡)𝛽=0,₀=𝜆₁𝜂₃𝜎_𝑆+𝜆₂𝜂₂𝜎²_𝑟+2𝜂²₃𝜎²_𝑆−𝑚_𝐿,𝛽₁=𝜂₃𝜎_𝑆12𝜂₃𝜎³_𝑆−𝜂₃𝜎²_𝑆−𝜆₁−12𝜆²₁. (64)

Again we can split this equation into three equations: 𝛽₀𝑏(𝑡)−𝑏^′(𝑡)𝑎=0,_′(𝑡)−𝛽₀𝑎(𝑡)𝑚−𝑘=0,_𝑡+12𝜎²_𝑟𝑚_𝑟𝑟−𝛽₀𝑚+(𝑎−𝑏𝑟)𝑚_𝑟+𝜆₂−𝜂₂𝜎²_𝑟𝑚_𝑟−𝛽₀+𝛽₁+12𝜆₂−𝜂₂²𝜎²_𝑟+𝑏^′(𝑡)𝑏(𝑡)𝑚=0. (65)(66)(67)

Combining with the account boundary conditions: 𝑏(𝑇)=1 and 𝑎(𝑇)=0, the solutions to (65) and (66) are 𝑏(𝑡)=𝑒^{𝛽0(𝑡−𝑇)},𝑎(𝑡)=−𝑘1−𝑒^{−𝛽0(𝑇−𝑡)}𝛽₀. (68)

We conjecture a solution of (67) with the following structure: 𝑚(𝑡,𝑟)=𝐴(𝑡)+𝐵(𝑡)𝑟, (69) with the boundary conditions: 𝐴(𝑇)=0 and 𝐵(𝑇)=0.

Putting this into (67), we obtain 𝐴_𝑡+𝑎𝐵+𝜆₂−𝜂₂𝑘₂1𝐵+2𝜆₂−𝜂₂²𝑘₂−𝛽₀+𝛽₁+𝑟𝐵_𝑡−𝑏𝐵+𝜆₂−𝜂₂𝑘₁1𝐵+2𝜆₂−𝜂₂²𝑘₁=0. (70)

By matching coefficients, we can decompose (70) into two conditions: 𝐵_𝑡−𝑏𝐵+𝜆₂−𝜂₂𝑘₁1𝐵+2𝜆₂−𝜂₂²𝑘₁𝐴=0,_𝑡+𝑎𝐵+𝜆₂−𝜂₂𝑘₂1𝐵+2𝜆₂−𝜂₂²𝑘₂−𝛽₀+𝛽₁=0. (71)

Taking into account the boundary conditions, the solutions to (71) are 𝐵(𝑡)=𝜃₃𝜃₁1−𝑒^{𝜃1(𝑡−𝑇)},𝐴(𝑡)=𝜃₂𝜃₃𝜃₁+𝜃₄−𝛽₀−𝛽₁()+𝜃𝑇−𝑡²𝜃₃𝜃²₁𝑒^{𝜃1(𝑡−𝑇)}−1, (72) where 𝜃₁=𝑏−𝜆₂−𝜂₂𝑘₁,𝜃₂=𝑎+𝜆₂−𝜂₂𝑘₂𝜃₃=12𝜆₂−𝜂₂²𝑘₁,𝜃₄=12𝜆₂−𝜂₂²𝑘₄,𝛽₀=𝜆₁𝜂₃𝜎_𝑆+𝜆₂𝜂₂𝜎²_𝑟+2𝜂²₃𝜎²_𝑆−𝑚_𝐿,𝛽₁=𝜂₃𝜎_𝑆12𝜂₃𝜎³_𝑆−𝜂₃𝜎²_𝑆−𝜆₁−12𝜆²₁. (73)

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

Proposition 10. The optimal investment strategies are given by 𝜋₀(𝑡)=1−𝜋_𝐵(𝑡)−𝜋_𝑆(𝑡),𝜋^∗_𝑆=𝜂₃+𝜆₁+𝜂₃𝜎²_𝑆𝑙𝑞𝜎_𝑆𝑣𝑏(𝑡),𝜋^∗_𝐵=1𝑓()𝜂𝑇−𝑡²−𝜂₁𝜂₃+𝑙𝛼𝑞𝑣⁴+𝐵(𝑡)𝑏(𝑡), (74) where 𝑏(𝑡)=𝑒^{𝛽0(𝑡−𝑇)}(𝑡)=𝜃,𝐵³𝜃₁1−𝑒^{𝜃1(𝑡−𝑇)},𝛽₀=𝜆₁𝜂₃𝜎_𝑆+𝜆₂𝜂₂𝜎²_𝑟+2𝜂²₃𝜎²_𝑆−𝑚_𝐿,𝜃₁=𝑏−𝜆₂−𝜂₂𝑘₁,𝜃₃=12𝜆₂−𝜂₂²𝑘₁𝛼₄=𝜂₂𝜎_𝑆+𝜆₁𝜂₁+𝜂₁𝜂₃𝜎²_𝑆−𝜆₂𝜎_𝑆𝜎_𝑆,𝑓(𝑡)=2𝑒^𝑚𝑡−1𝑚−𝑏−𝑘₁𝜆₂+𝑒^𝑚𝑡𝑚+𝑏−𝑘₁𝜆₂,𝑚=𝑏−𝑘₁𝜆₂²+2𝑘₁. (75)

Lemma 11. Consider𝑑𝑏(𝑡)𝑑𝑡>0,𝑑𝜋^∗_𝑆𝑑𝑡>0. (76)

Proof. Since 𝛽₀>0,𝑞>0,𝜆₁>0,𝜂₃>0,and  𝜎_𝑆>0, by differentiating 𝑏(𝑡) with the respect to 𝑡, we have 𝑑𝑏(𝑡)𝑑𝑡=𝛽₀𝑒^{𝛽0(𝑡−𝑇)}>0,𝑑𝜋^∗_𝑆=𝜆𝑑𝑡¹+𝜂₃𝜎²_𝑆𝑙𝑞𝜎_𝑆𝑣𝑑𝑏(𝑡)𝑑𝑡>0. (77)

Lemma 12. Consider𝑑𝜋^∗_𝑆𝑑𝑙>0. (78)

Proof. Since 𝑟>0,𝛽₀>0,𝑞>0,𝜆₁>0,𝜂₃>0, and 𝜎_𝑆>0, we obtain 𝑏(𝑡)=𝑒^{𝛽0(𝑡−𝑇)}>0,𝑑𝜋^∗_𝑆=𝜆𝑑𝑙¹+𝜂₃𝜎²_𝑆𝑏(𝑡)𝑞𝜎_𝑆𝑣>0. (79)

Remark 13. Lemma 11 shows that the effect of 𝑡 on 𝜋^∗_𝑆 is then different from the situation of the power utility. The optimal proportion invested in stock 𝜋^∗_𝑆 depends on the time 𝑡 and is a monotone increasing function with respect to time 𝑡. Meanwhile, we cannot find the monotone increasing or decreasing effect of 𝑡 on 𝜋^∗_𝐵. So under the exponential utility, as the retirement date approaches, the plan member will distribute more assets to invest in stock or less asset to invest in low-risk assets (i.e., cash and bonds).
This can be explained by the risky tolerance, namely, −𝑈^′(𝑥)/𝑈^′′(𝑥)=𝑞⁻¹, which is only a constant. This indicates that for an exponential utility, due to the independence of a risk tolerance coefficient on wealth, the optimal proportion invested in stock 𝜋^∗_𝑆 is independent of the profitability of risky assets and the wealth. As the wealth gives an insight into the accumulated profit gained from risky assets, the plan member will buy more risky assets as the wealth increases.
Lemma 12 reveals that the optimal proportion invested in stock 𝜋^∗_𝑆 is a monotone increasing function with respect to the salary numeraire 𝑙, which is the same as the situation of the power utility. However, the regular change in the effect of 𝑙 on 𝜋^∗_𝐵 is not found.
Nevertheless, the change trend of 𝑡 or 𝑙 on 𝜋^∗_𝑆 is not affected by the absolute risk aversion coefficient 𝑝, which is the same as the power utility.

6. Conclusions

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 [2] and Cairns et al. [1].

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.

Acknowledgments

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).