Precommitted Investment Strategy versus Time-Consistent Investment Strategy for a Dual Risk Model

Abstract

We are concerned with optimal investment strategy for a dual risk model. We assume that the company can invest into a risk-free asset and a risky asset. Short-selling and borrowing money are allowed. Due to lack of iterated-expectation property, the Bellman Optimization Principle does not hold. Thus we investigate the precommitted strategy and time-consistent strategy, respectively. We take three steps to derive the precommitted investment strategy. Furthermore, the time-consistent investment strategy is also obtained by solving the extended Hamilton-Jacobi-Bellman equations. We compare the precommitted strategy with time-consistent strategy and find that these different strategies have different advantages: the former can make value function maximized at the original time 𝑡=0 and the latter strategy is time-consistent for the whole time horizon. Finally, numerical analysis is presented for our results.

1. Introduction

We start from a dual risk model; namely, the surplus process of a company is given by 𝑅(𝑡)=𝑥+^𝑁(𝑡)_𝑗=1𝑍_𝑗−𝑐𝑡, (1) where 𝑥 is the initial capital and 𝑐 is the rate of expenses; the positive incomes or profits 𝑍_𝑗(𝑗=1,2,3,…) arrive as a Poisson process {𝑁(𝑡)}_𝑡≥0 with intensity 𝜆; 𝑍_𝑗(𝑗=1,2,3,…) are independent and identically distributed (i.i.d) nonnegative random variables with the first and second moments 𝜇_𝑧 and 𝜎²_𝑧. The expected increase of the surplus per unit time satisfies the positive loading condition: 𝑝=𝜆𝜇_𝑧−𝑐>0. The model is called dual as opposed to the Cramer-Lundberg risk model with applications to insurance.

The dual risk model can be described as the amount of capital of a company engaging in research and development. The company pays expenses and gets occasional revenues. Revenues are interpreted as the values of future gains from an invention or discovery, while the decrease of surplus can represent costs of production, payments to employees, maintenance of equipment, and so forth. This is also the case for a portfolio of life annuities, where the risk consists of survival and the event death leads to gains. Furthermore, many scholars investigated the dual risk model under different criterions. Stochastic control theory and the related methodologies are the main tools for finding the related minimum probability of ruin or the maximum value for the expected utility of terminal wealth or the maximum value for the expected present value of the dividends minus capital injections. For related works see, for example, Avanzi et al. [1, 2], Zhu and Yang [3], Dai et al. [4], and Yao et al. [5] and references therein.

In this paper, we are concerned with optimal investment strategy for the dual risk model under mean-variance criterion. Our objective is to find the optimal investment strategy such that the expected terminal wealth is maximized and the variance of the terminal wealth is minimized. According to the biobjective optimization theory, the alternative objective is to find a strategy which maximizes the expected terminal wealth minus the variance of the terminal wealth; namely, (𝐸MV)^0,𝑥(𝑋(𝑇−𝛾))2Var_0,𝑥(𝑋(𝑇))⟶max𝑙∈𝑈, (2) where 𝛾 is a prespecified risk aversion coefficient, 𝐸_0,𝑥[⋅]=𝐸[⋅∣𝑋^𝑙(0)=𝑥], and Var_0,𝑥[⋅]=Var[⋅∣𝑋^𝑙(0)=𝑥].

It is well known that this criterion lacks the iterated-expectation property, so problem (MV) is time-inconsistent which means that Bellman Optimality Principle does not hold. The lack of time consistency leads to conceptual as well as computational problems. Recently there are two approaches to the conceptual problem.(i)Fix one initial point (0,𝑥₀) and then try to find a strategy 𝑙^∗ which maximizes 𝐸_0,𝑥0(𝑋(𝑇))−𝛾Var_0,𝑥0(𝑋(𝑇)). We then simply disregard the fact that at later points in time the strategy will not be optimal. Such a strategy is known as a precommitted strategy. Zhou and Li [6] and Li and Ng [7] made big breakthrough works and proposed an embedding method to derive the precommitted strategy for problem (MV).(ii)We take the time-inconsistency seriously and formulate the problem within a game theoretic framework. Problem (MV) is viewed as a game, where the players are the future incarnations of our own preferences and Nash equilibrium points can be derived in order to address the general time-inconsistency. See more references in Björk and Murgoci [8], Ekeland and Lazrak [9], and Kryger and Steffensen [10].

Thus, problem (MV) can be reduced to a resolvable problem by virtue of some techniques including the embedding technique [6, 7] and the game theoretical technique [8]. Our contributions are as follows. Firstly, we take three steps to deal with the precommitted investment problem. This method is different from the embedding technique in Zhou and Li [6] and Li and Ng [7]. Secondly, the time-consistent strategy from the game theoretical perspective is also derived and we also present economics implications and sensitivity analysis for our results.

This paper is organized as follows. Section 2 formulates problem (MV) and gives the main results. Section 3 provides numerical analysis for our results.

2. Model and Main Results

This section starts with a filtered complete probability space (Ω,ℱ,{ℱ_𝑡}_{0≤𝑡≤𝑇},𝒫), where 𝑇 represents the fixed time horizon and ℱ_𝑡 stands for the information available at time 𝑡. We recall that the dual risk model and the surplus process of the company are subject to the following stochastic process: 𝑅(𝑡)=𝑥+^𝑁(𝑡)_𝑗=1𝑍_𝑗−𝑐𝑡. (3)

The company invests its wealth into a Black-Scholes market consisting of a risk-free asset and a risky asset. Denote the amount of money invested in the risky asset at time 𝑡 by 𝑙(𝑡), and the allowance of short-selling and borrowing money implies 𝑙(𝑡)∈ℝ. The price of the risk-free asset 𝑆₀(𝑡) is modeled by 𝑑𝑆₀(𝑡)=𝑟₀(𝑡)𝑆₀(𝑡)𝑑𝑡,𝑆₀(0)=𝑠₀ (4) and the price of the risky asset 𝑆₁(𝑡) evolves according to the dynamics 𝑑𝑆₁(𝑡)=𝑆₁(𝑡)𝑟₁(𝑡)𝑑𝑡+𝜎(𝑡)𝑑𝑊(𝑡),𝑆₁(0)=𝑠₁, (5) where 𝑊(𝑡) is a standard Brownian motion which is independent of ∑^𝑁(𝑡)_𝑗=1𝑍_𝑗, the appreciation rate 𝑟₁(𝑡) and risk-free rate 𝑟₀(𝑡) satisfy 𝑟₁(𝑡)>𝑟₀(𝑡), 𝜎(𝑡) is the volatility coefficient, and they are all bounded continuous functions. Denote the resulting surplus process after incorporating strategy 𝑙 into (3) by 𝑋^𝑙(𝑡), and 𝑋^𝑙(𝑡) is subject to the following stochastic equation: 𝑑𝑋^𝑙(𝑡)=𝑟₀(𝑡)𝑋^𝑙(𝑡)(𝑡)𝑙(𝑡)(𝑡)𝑙(𝑡)(𝑡)+𝑟−𝑐𝑑𝑡+𝜎𝑑𝑊+𝑑^𝑁(𝑡)_𝑗=1𝑍_𝑗, (6) where 𝑟(𝑡)=𝑟₁(𝑡)−𝑟₀(𝑡). Denote that 𝐶^1,2(𝑄)={𝜙(𝑡,𝑥)∣𝜙(𝑡,⋅) is once continuously differentiable on [0,𝑇] and 𝜙(⋅,𝑥) is twice continuously differentiable on ℝ}.

According to the theory of stochastic calculus, the infinitesimal operator of the surplus process 𝑋^𝑙(𝑡) is 𝒜𝜙(𝑡,𝑥)=𝜙_𝑡(𝑡,𝑥)+𝜙_𝑥(𝑡,𝑥)𝑟₀(𝑡)(𝑡)𝑙(𝑡)+1𝑥−𝑐+𝑟2𝜙_𝑥𝑥(𝑡,𝑥)𝑙²(𝑡)𝜎²(𝑡)()(,()+𝜆𝐸𝜙𝑡,𝑥+𝑍−𝜙𝑡,𝑥)for𝜙𝑡,𝑥∈𝐶^1,2(𝑄). (7)

In order to solve problem (MV), we firstly give the definition of admissible strategy for the dual risk process 𝑋^𝑙(𝑡).

Definition 1. A strategy 𝑙={𝑙(𝑡),𝑡≥0} is said to be admissible if(1)𝑙∶[0,∞]×Ω→ℝ is an ℱ-adapted process;(2)𝑙(𝑡) satisfies the integrability condition: 𝐸∫^𝑡₀𝑙²(𝑡)𝑑𝑠<∞ almost surely, for all 𝑡≥0;(3)SDE (6) has a unique solution corresponding to 𝑙.

Denote the set of all admissible strategies over the time horizon [0,𝑇] by 𝑈. Our aim is to find a strategy which maximizes the expected terminal wealth minus the variance of the terminal wealth; in other words, we aim to solve problem (MV). In the next subsection we will give the optimal precommitted investment strategy and the time-consistent strategy.

2.1. The Optimal Precommitted Investment Strategy for Problem (MV)

This subsection puts forward an idea to solve problem (MV).

Firstly, problem (MV) can be transformed into a constrained problem. For 𝑑∈ℝ, define the following problem with a constrained expectation of terminal wealth: ()𝐸CMV^0,𝑥𝑋^𝑙(𝑇)−𝛾2Var_0,𝑥𝑋^𝑙(𝑇)𝐸⟶max_0,𝑥𝑋^𝑙(𝑇)=𝑑,𝑙∈𝑈. (8)

Considering the terminal condition 𝐸_0,𝑥(𝑋^𝑙(𝑇))=𝑑 sufficiently, problem (CMV) equals the following problem: −𝛾CMV2𝐸_0,𝑥𝑋^𝑙(𝑇)²𝛾+𝑑+2𝑑²𝐸⟶max_0,𝑥𝑋^𝑙(𝑇)=𝑑,𝑙∈𝑈. (9)𝑉(𝑥,𝑑) and 𝑉(𝑥) represent the value functions for problem (CMV) and problem (MV), respectively, and they have the following relationship 𝑉(𝑥)=sup_𝑑∈𝑅𝑉(𝑥,𝑑).

Secondly, problem (CMV) is solved by Lagrange technique. By introducing a Lagrange multiplier 𝑦∶=𝑦(𝑥,𝑑)∈ℝ, define a quadratic problem without the constrained expectation of the terminal wealth: ()𝐸BM^0,𝑥𝑦𝑋^𝑙(𝑇)−𝛾2𝑋^𝑙(𝑇)²+𝛾+𝑑2𝑑²−𝑑𝑦⟶max𝑙∈𝑈; (10)𝑉(𝑥,𝑑,𝑦) represents the value function for problem (BM). The duality theory implies that the value function for problem (CMV) satisfies that 𝑉(𝑥,𝑑)=inf_𝑦∈ℝ𝑉(𝑥,𝑑,𝑦); then we have 𝑉(𝑥)sup_𝑑∈𝑅inf_𝑦∈ℝ𝑉𝑥,𝑑,𝑦. (11)

We will take three steps to derive the precommitted investment strategy for problem (MV). Based on the discussion in Appendix A, we can derive the precommitted investment strategy and the value function.

Theorem 2. For problem (𝑀𝑉), the optimal precommitted investment strategy is given by 𝑙^∗(𝑡)=𝑟(𝑡)𝜎²(𝑡)𝑒^{−∫𝑇𝑡𝑟0(𝑠)𝑑𝑠}𝑝^𝑡₀𝑒^{∫𝑇𝑠𝑟0(𝑣)𝑑𝑣}𝑑𝑠+𝑥𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}+𝑒^{∫𝑇0𝜉(𝑠)𝑑𝑠}𝛾−𝑥(𝑡), (12) and the value function is given by 𝑉(𝑥)=𝑥𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}+𝑝^𝑇₀𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}−𝑑𝑠𝛾𝜆𝜎²_𝑧2^𝑇₀𝑒^{∫𝑇𝑠(2𝑟0(𝑢)−𝜉(𝑢))𝑑𝑢}𝑒𝑑𝑠+^{∫𝑇0𝜉(𝑠)𝑑𝑠}−1.2𝛾 (13)

Furthermore, the efficient frontier of problem (MV) at initial state (0,𝑥) is given by𝐸_0,𝑥𝑋^𝑙∗(𝑇)=𝑝^𝑇₀𝑒^{∫𝑇𝑠𝑟0(𝑣)𝑑𝑣}𝑑𝑠+𝑥𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}+Var_0,𝑥𝑋^𝑙∗(𝑇)−𝜆𝜎²_𝑧^𝑇₀𝑒^{∫𝑇𝑠(2𝑟0(𝑢)−𝜉(𝑢))𝑑𝑢}𝑑𝑠𝑒^{∫𝑇0𝜉(𝑠)𝑑𝑠}−1. (14)

Remark 3. Our method is different from the embedding technique proposed by Zhou and Li [6]. Problem (MV) can be embedded into a class of auxiliary stochastic linear-quadratic (LQ) problems, and the precommitted strategy for problem (MV) was derived by solving the LQ problems. Correspondingly, we take three steps to derive the precommitted strategy. Compared with the embedding method, our method is simple but easy to implement for solving problem (MV).

2.2. The Time-Consistent Investment Strategy for Problem (MV)

This subsection provides the time-consistent investment strategy for problem (MV). Firstly, define a dynamic problem 𝐽()MV𝑡,𝑥,𝑙=𝐸_𝑡,𝑥𝑋^𝑙(𝑇−𝛾)2Var_𝑡,𝑥𝑋^𝑙(𝑇)⟶max𝑙∈𝑈. (15)

According to the theory stated in Björk and Murgoci [8], we can convert this time-inconsistent problem into a time-consistent problem. The equilibrium strategy is defined by the same way in Björk and Murgoci [8] or Zeng et al. [11].

Definition 4 (equilibrium strategy). For any fixed chosen initial state (𝑡,𝑥)∈𝑄∶=[0,𝑇]×𝑅, consider an admissible strategy 𝑙_∗(𝑡,𝑥). Choose two fixed real numbers 𝑙̃>0 and 𝜀>0 and define the following strategy: 𝑙^𝜀()=̃()∈[]𝑙𝑠,𝑥𝑙,for𝑠,𝑥𝑡,𝑡+𝜀×ℝ_∗(𝑠,𝑥),for(𝑠,𝑥)∈[𝑡+𝜀,𝑇]×ℝ. (16)
If liminf_𝜖→0𝐽𝑡,𝑥,𝑙_∗−𝐽𝑡,𝑥,𝑙^𝜀𝜀≥0,∀𝑙̃∈ℝ⁺,(𝑡,𝑥)∈𝑄, (17) then 𝑙_∗(𝑡,𝑥) is called an equilibrium strategy and the corresponding equilibrium value function 𝜓(𝑡,𝑥) for problem (MV) is given by 𝜓(𝑡,𝑥)=𝐽𝑡,𝑥,𝑙_∗=𝐸_𝑡,𝑥𝑋^𝑙∗(𝑇−𝛾)2Var_𝑡,𝑥𝑋^𝑙∗(𝑇). (18)

By Definition 4, we know that the equilibrium strategy is time-consistent. So the equilibrium strategy 𝑙_∗(𝑡) and the equilibrium value function 𝜓(𝑡,𝑥) are called optimal time-consistent strategy and optimal equilibrium value function for problem (MV), respectively. It is easy to see that the equilibrium value function 𝑉^∗(𝑥) for problem (MV) satisfies 𝑉^∗(𝑥)=𝜓(0,𝑥). Based on the discussion in Appendix B, the time-consistent investment strategy and the equilibrium value function for problem (MV) are given by the following theorem.

Theorem 5. For the dual model, the optimal time-consistent strategy 𝑙_∗(𝑡) is given by 𝑙_∗(𝑡)=𝑟(𝑡)𝛾𝜎²(𝑡)𝑒^{−∫𝑇𝑡𝑟0(𝑠)𝑑𝑠} (19) and the optimal equilibrium value function is given by 𝑉^∗(𝑥)=𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}𝑥+𝑝^𝑇₀𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}−𝑑𝑠𝛾𝜆𝜎²_𝑧2^𝑇₀𝑒^{2∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}1𝑑𝑠+2𝛾^𝑇₀𝜉(𝑠)𝑑𝑠. (20)

Furthermore, the efficient frontier under time-consistent strategy is given by the following equation:𝐸_0,𝑥𝑋^𝑙∗(𝑇)=𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}𝑥+𝑝^𝑇₀𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}+𝑑𝑠Var_0,𝑥𝑋^𝑙∗(𝑇)−𝜆𝜎²_𝑧^𝑇₀𝑒^{2∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}𝑑𝑠^𝑇₀𝜉(𝑠)𝑑𝑠. (21)

Remark 6. The time-consistent strategy is not affected by the initial information and the current wealth which is different from the precommitted strategy. While the time-consistent strategy is time deterministic, the precommitted strategy is stochastically dependent on the current wealth. We exploit Monte Carlo methods to simulate the precommitted investment strategy. We compare the average of 1000 tracks of the precommmitted investment strategy with the time-consistent strategy. Figure 1(a) shows that the time-consistent investment strategy is smaller than the average of the precommitted investment strategy; that is to say, the company should invest more money into the risky asset under the precommitted investment strategy.

(a) The comparison of optimal strategies

(b) The comparison of optimal value functions

(c) The comparison of different efficient frontiers

Figure 1 

The comparisons between precommitted strategy and time-consistent strategy with parameters 𝑟0=0.06, 𝑟1=0.15, 𝑐=0.6, 𝑥=10, 𝛾=0.6, 𝜆=1, 𝜇𝑧=1, 𝜎𝑧=1.1, 𝜎=0.3, and 𝑇=10.

Remark 7. The optimal value function 𝑉(𝑥) and the equilibrium value function 𝑉^∗(𝑥) are not the same for problem (MV). From (13) and (20), it is not hard to calculate that 𝑉(𝑥)>𝑉^∗(𝑥) which is illustrated by Figure 1(b). It follows from (14) and (21) that the efficient frontiers for problem (MV) are not straight lines but hyperbolas in mean-standard variance plane and that the expectation of terminal wealth under the time-consistent strategy is never bigger than the one under the precommitted strategy for the same variance of terminal wealth which is illustrated by Figure 1(c).

Remark 8. From Remark 7, we can see that the value function 𝑉(𝑥) is larger than the equilibrium value function 𝑉^∗(𝑥) and that the efficient frontier under the time-consistent equilibrium strategy is never above the efficient frontier under the precommitted strategy. But the conclusion that the precommitted strategy is prior to the time-consistent equilibrium strategy is not right, for the precommitted strategy is a global optimal strategy only at 𝑡=0 and the time-consistent strategy is a suboptimal strategy for all 𝑡≥0.

3. Numerical Analysis

This section provides some numerical examples to illustrate the effect of parameters on the optimal strategies and the corresponding value functions. For convenience but without loss of generality, all the parameters involved are constants. For the following numerical illustrations, unless otherwise stated, the basic parameters are given by 𝑟₀=0.06, 𝑟₁=0.15, 𝑐=0.6, 𝑥=10, 𝛾=0.6, 𝜆=1, 𝜇_𝑧=1, 𝜎_𝑧=1.1, 𝜎=0.3, and 𝑇=10.

3.1. Analysis of the Precommitted Strategy and the Corresponding Value Function

This subsection will provide numerical examples to show how the parameters effect the precommitted strategy and the value function.

Because the precommitted investment strategy depends on the current wealth, we will investigate the effect of all the parameters for the precommitted strategy in the same sample trajectory by stochastic simulations. In order to model the trajectory, we assume that {𝑁(𝑡)}_𝑡≥0 is a Poisson process with intensity 𝜆=1 and the profit or the income 𝑍_𝑗(𝑗=1,2,3,…) is exponentially distributed with mean 1. From (12), we can see that the precommitted investment strategy increases when the current wealth decreases; namely, if the current wealth is smaller, the company should invest more money in the risky asset. Figure 1 shows how the coefficients involved impact on the optimal precommitted investment strategy. From Figure 2, we can conclude the following: (1) the precommitted investment strategy is decreasing with respect to 𝛾 and 𝜎 which shows that the more the company dislikes risk or the larger the market’s risk is, the less amount the company invests in the risky asset; (2) the precommitted investment strategy is also decreasing with respect to 𝑐 which shows that the more the company’s expenditure is, the less amount the company invests in the risky asset; (3) the precommitted investment strategy has a more complex relation with 𝑟₁ and 𝑟₀ because the increase of 𝑟₁ or 𝑟₀ can increase the deterministic part and the stochastic part (current wealth) of the precommited strategy which results in the uncertainty of their difference.

(a) The effect of 𝛾 on the optimal precommitted strategy

(b) The effect of 𝑟0 on the optimal precommitted strategy

(c) The effect of 𝑟1 on the optimal precommitted strategy

(d) The effect of 𝑐 on the optimal precommitted strategy

(e) The effect of 𝜎 on the optimal precommitted strategy

Figure 2 

The effect of parameters on the optimal precommitted strategy.

Secondly, Figure 3 shows how the coefficients involved impact on the value function. Figure 3(a) illustrates that the value function is decreasing with respect to 𝛾, namely, the larger risk aversion the company has, the smaller the value function is; Figure 3(b) reveals when the risk-free rate is small enough, the value function decreases and when the risk-free rate is close to the the appreciation rate, the value function increases; Figure 3(c) illustrates that the value function is increasing with respect to 𝑟₁, namely, the bigger the appreciation rate is, the bigger the value function is; Figure 3(d) reveals that the value function is decreasing with respect to 𝜎, namely, the bigger the volatility of the market’s risky asset is, the smaller the value function is. Recalling 𝑝=𝜆𝜇−𝑐, Figure 3(e) illustrates that the value function is increasing with respect to 𝑝, namely, the larger the expectation of the positive income is or the smaller the expense rate of the company is or the bigger the intensity of the jumps of the profit is, the bigger the value function becomes; Figure 3(f) shows the value function is increasing with respect to 𝜎_𝑧, namely, the smaller the second moment of the positive income is, the bigger the value function becomes.

(a) The effect of 𝛾 on the optimal value function

(b) The effect of 𝑟0 on the optimal value function

(c) The effect of 𝑟1 on the optimal value function

(d) The effect of 𝜎 on the optimal value function

(e) The effect of 𝑝 on the optimal value function

(f) The effect of 𝜎𝑧 on the optimal value function

Figure 3 

The effect of parameters on the optimal value function.

3.2. Analysis of the Time-Consistent Strategy and the Equilibrium Value Function

This subsection will work on numerical analysis of the time-consistent strategy and the equilibrium value function.

Firstly, we will show how the coefficients involved impact on the time-consistent investment strategy. From (19) we can see that the time-consistent investment strategy is independent of the current wealth. Figure 4(a) illustrates that the time-consistent investment strategy is decreasing with respect to 𝛾, namely, the more the company dislikes risk, the less amount the company invests in the risky asset; Figure 4(b) reveals that the time-consistent investment strategy is decreasing with respect to 𝑟₀, namely, the smaller the risk-free rate is, the more amount the company invests in the risky asset; Figure 4(c) reveals that the time-consistent investment strategy is increasing with respect to 𝑟₁, namely, when the appreciation rate 𝑟₁ increases, the company should invest more money in the risky asset; Figure 4(d) tells that the time-consistent investment strategy is decreasing with respect to 𝜎, namely, when volatility of the risky asset increases, the company should invest more money in the risk-free asset.

(a) The effect of 𝛾 on the optimal time-consistent strategy

(b) The effect of 𝑟0 on the optimal time-consistent strategy

(c) The effect of 𝑟1 on the optimal time-consistent strategy

(d) The effect of 𝜎 on the optimal time-consistent strategy

Figure 4 

The effect of parameters on the optimal time-consistent strategy.

Secondly, Figure 5 shows how the coefficients involved impact on the equilibrium value function. The parameters 𝛾, 𝑟₁, 𝜎, 𝑝, and 𝜎_𝑧 have the similar effect on the equilibrium value function as their effect on the value function with precommitment discussed in Section 3.1. Furthermore, the equilibrium value function is increasing with respect to the risk-free rate.

(a) The effect of 𝛾 on the optimal equilibrium value function

(b) The effect of 𝑟0 on the optimal equilibrium value function

(c) The effect of 𝑟1 on the optimal equilibrium value function

(d) The effect of 𝜎 on the optimal equilibrium value function

(e) The effect of 𝑝 on the optimal equilibrium value function

(f) The effect of 𝜎𝑧 on the optimal equilibrium value function

Figure 5 

The effect of parameters on the optimal equilibrium value function.

4. Conclusion

In this paper, optimal investment strategies for a dual risk model are explored under mean-variance criterion. We assume that a company can invest into a finance market which consists of a risk-free asset and a risky asset. We have derived the optimal precommitted strategy and the time-consistent equilibrium strategy for problem (MV). In the end, numerical analysis is given for optimal investment strategies and the value functions. From the comparisons on the value functions and efficient frontiers, it seems to be that the precommitted strategy is better than the time-consistent strategy. Unfortunately, the precommitted strategy is not time-consistent. Thus these investment strategies have different advantages. These strategies are all important for the company.

Appendices

A.

In this appendix, we will take three steps to derive the precommitted investment strategy for problem (MV).

Step 1. The closed expressions for the optimal investment strategy and value function are derived by solving the related Hamilton-Jacobi-Bellman (HJB) equation for problem (BM). Based on similar arguments in Fleming and Soner [12], we obtain the HJB equation which the value function 𝑉(𝑥,𝑑,𝑦) satisfies.

Lemma A.1 (verification theorem). If a function 𝑊(𝑡,𝑥)∈𝐶^1,2(𝑄) and a function 𝑙^∗(𝑡)∈𝑈 satisfy the following HJB equation: sup_𝑙∈𝑈𝒜^𝑙𝑊(𝑡,𝑥)=0,𝑊()𝛾𝑇,𝑥=−2𝑥²𝛾+𝑦𝑥+2𝑑²+𝑑−𝑦𝑑,𝑙^∗(𝑡)=argsup𝒜^𝑙𝑊(𝑡,𝑥), (A.1)(A.2)(A.3) then 𝑉(𝑥,𝑑,𝑦)=𝑊(0,𝑥) and 𝑙^∗(𝑡) is the optimal investment strategy.

Next, we will construct the solution to problem (BM). Assume that there exists a real function 𝑊(𝑡,𝑥) which satisfies the condition in Lemma A.1. Inserting (7) into (A.1), we have sup_𝑙∈𝑈𝑊_𝑡(𝑡,𝑥)+𝑊_𝑥(𝑡,𝑥)𝑟₀(𝑡)(𝑡)𝑙(𝑡+1𝑥−𝑐+𝑟)2𝑊_𝑥𝑥(𝑡,𝑥)𝑙²(𝑡)𝜎²(𝑡)+𝜆𝐸[𝑊(𝑡,𝑥+𝑍)−𝑊(𝑡,𝑥)]=0. (A.4)

Because the structure of (A.4) and the boundary condition of 𝑊(𝑡,𝑥) given by (A.2) are quadratic in 𝑥, it is natural to conjecture that 𝑊(𝑡,𝑥)=𝐴(𝑡)𝑥²+𝐵(𝑡)𝑥+𝐶(𝑡),𝐴(𝑇)𝛾=−2(𝑇)(𝑇)=𝛾,𝐵=𝑦,𝐶2𝑑²+𝑑−𝑦𝑑. (A.5)(A.6)

The corresponding partial derivatives are given by the following equations: 𝑊_𝑡(𝑡,𝑥)=𝐴̇(𝑡)𝑥²+𝐵̇(𝑡)𝐶̇(𝑡),𝑊𝑥+_𝑥()(𝑡)(𝑡),𝑊𝑡,𝑥=2𝐴𝑥+𝐵_𝑥𝑥(𝑡,𝑥)=2𝐴(𝑡). (A.7)

Inserting (A.5)–(A.7) into (A.4) yields sup_𝑙∈𝑈𝐴̇(𝑡)𝑥²+𝐵̇(𝑡)𝐶̇(𝑡)+((𝑡)(𝑡×𝑟𝑥+2𝐴𝑥+𝐵))₀(𝑡)𝑥−𝑐+𝑟(𝑡)𝑙(𝑡)+𝜆𝜇_𝑧+𝐴(𝑡)𝑙²(𝑡)𝜎²(𝑡)+𝜆𝜎²_𝑧=0. (A.8)

Differentiating the function in the left bracket of (A.8) with respect to 𝑙 and setting the derivative to zero, we get 𝑙(𝑡)((𝑡)(𝑡𝑟(𝑡)=−2𝐴𝑥+𝐵))2𝐴(𝑡)𝜎²(𝑡). (A.9)

Inserting (A.9) into (A.8), we have the following equation: 𝐴̇(𝑡)+𝐴(𝑡)2𝑟₀(𝑡)−𝜉(𝑡)𝑥²+𝐵̇(𝑡)+𝐵(𝑡)𝑟₀(𝑡)(𝑡)𝑥+𝐶̇(𝑡)−𝜉+2𝐴𝑝+𝜆𝜎²_𝑧𝐴(𝑡)(𝑡)−𝐵+𝑝𝐵²(𝑡)(𝑡)𝜉(𝑡)4𝐴=0, (A.10) where 𝜉(𝑡)=𝑟²(𝑡)𝜎²(𝑡),𝑝=𝜆𝜇_𝑧−𝑐>0. (A.11)

To ensure the above equation holds, we require that 𝐴̇(𝑡)+𝐴(𝑡)2𝑟₀(𝑡)−𝜉(𝑡)=0,𝐴(𝑇)=−𝛾2,𝐵̇(𝑡)+𝐵(𝑡)𝑟₀(𝑡)(𝑡)(𝑇)𝐶̇(𝑡)−𝜉+2𝐴𝑝=0,𝐵=𝑦,+𝜆𝜎²_𝑧𝐴(𝑡)(𝑡)−𝐵+𝑝𝐵²(𝑡)(𝑡)𝜉(𝑡)𝐶(𝑇)=𝛾4𝐴=0,2𝑑²+𝑑−𝑦𝑑. (A.12)

By solving the system of equations, we have 𝐴(𝑡)𝛾=−2𝑒^{∫𝑇𝑡(2𝑟0(𝑠)−𝜉(𝑠))𝑑𝑠},𝐵(𝑡)=𝑒^{∫𝑇𝑡(𝑟0(𝑠)−𝜉(𝑠))𝑑𝑠}−𝑝𝛾^𝑇_𝑡𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢},𝐶(𝑡)𝑑𝑠+𝑦=−𝛾𝜆𝜎²_𝑧2^𝑇_𝑡𝑒^{∫𝑇𝑠(2𝑟0(𝑢)−𝜉(𝑢))𝑑𝑢}−1𝑑𝑠𝑒2𝛾^{−∫𝑇𝑡𝜉(𝑢)𝑑𝑢}−𝑝𝛾^𝑇_𝑡𝑒^{∫𝑇𝑠𝑟0(𝑣)𝑑𝑣}𝑑𝑠+𝑦²+𝛾2𝑑²𝑦+𝑑−𝑑𝑦+².2𝛾 (A.13)

Substituting (A.13) into (A.9), we have the precommitted investment strategy for problem (MV): 𝑙(𝑡)=𝑟(𝑡)𝜎²(𝑡)𝑒^{−∫𝑇𝑡𝑟0(𝑠)𝑑𝑠}−𝑝^𝑇_𝑡𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}𝑦𝑑𝑠+𝛾−𝑥(𝑡). (A.14)

Correspondingly, the value function for problem (BM) is given by 𝑉𝑥,𝑑,𝑦=−𝛾𝜆𝜎²_𝑧2^𝑇₀𝑒^{∫𝑇𝑠(2𝑟0(𝑢)−𝜉(𝑢))𝑑𝑢}−𝛾𝑑𝑠2𝑒^{−∫𝑇0𝜉(𝑠)𝑑𝑠}𝑥𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}+𝑝^𝑇₀𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}𝑦𝑑𝑠−𝛾²+𝛾2𝑑²𝑦+𝑑−𝑑𝑦+².2𝛾 (A.15)

Step 2. Problem (CMV) is solved by virtue of the relationship of the value functions for problem (BM) and problem (CMV). Differentiating 𝑉(𝑥,𝑑,𝑦) with respect to 𝑦 yields 𝜕𝑉𝜕𝑦=𝑒^{−∫𝑇0𝜉(𝑠)𝑑𝑠}𝑥𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}+𝑝^𝑇₀𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}𝑦𝑑𝑠−𝛾𝑦−𝑑+𝛾,𝜕²𝑉𝜕𝑦²=1−𝑒^{−∫𝑇0𝜉(𝑠)𝑑𝑠}𝛾>0. (A.16)(A.17)
Setting the derivative of 𝑉(𝑥,𝑑,𝑦) to zero yields 𝑦^∗=𝑝∫^𝑇₀𝑒^{∫𝑇𝑠𝑟0(𝑣)𝑑𝑣}𝑑𝑠+𝑥𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}−𝑑𝑒^{∫𝑇0𝜉(𝑠)𝑑𝑠}𝛾1−𝑒^{∫𝑇0𝜉(𝑠)𝑑𝑠}. (A.18)
By virtue of the condition of a stationary point becoming an extreme point, we conclude that 𝑦^∗ is the point which minimizes 𝑉(𝑥,𝑑,𝑦). The optimal investment strategy 𝑙^∗(𝑡) and the value function 𝑉(𝑥,𝑑) for problem (CMV) can be derived by inserting (A.17) into (A.14)-(A.15) as follows: 𝑙^∗(𝑡)=𝑟(𝑡)𝜎²(𝑡)𝑒^{−∫𝑇𝑡𝑟0(𝑠)𝑑𝑠}×−𝑝^𝑇_𝑡𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}+𝑝𝑑𝑠_𝑇0𝑒^{∫𝑇𝑠𝑟0(𝑣)𝑑𝑣}𝑑𝑠+𝑥𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}−𝑑𝑒^{∫𝑇0𝜉(𝑠)𝑑𝑠}×1−𝑒^{∫𝑇0𝜉(𝑠)𝑑𝑠−1}(𝑡,𝑉()−𝑥)𝑥,𝑑=−𝛾𝜆𝜎²_𝑧2^𝑇₀𝑒^{∫𝑇𝑠(2𝑟0(𝑢)−𝜉(𝑢))𝑑𝑢}+𝛾𝑑𝑠21−𝑒^{∫𝑇0𝜉(𝑠)𝑑𝑠}×𝑥𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}+𝑝^𝑇₀𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}𝑑𝑠−𝑑²+𝑑. (A.19)

Step 3. Finally problem (MV) is solved by virtue of 𝑉(𝑥)=sup_𝑑∈𝑅𝑉(𝑥,𝑑). Differentiating 𝑉(𝑥,𝑑) at 𝑑, we can conclude that 𝜕𝑉(𝑥,𝑑)=𝛾𝜕𝑑1−𝑒^{∫𝑇0𝜉(𝑠)𝑑𝑠}×𝑑−𝑥𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}−𝑝^𝑇₀𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}𝜕𝑑𝑠+1,₂𝑉(𝑥,𝑑)𝜕𝑑²=𝛾1−𝑒^{∫𝑇0𝜉(𝑠)𝑑𝑠}<0. (A.20)
From the extreme value theory, the optimal expected terminal wealth 𝑑^∗ does exist. 𝑑^∗ can be derived by 𝜕𝑉(𝑥,𝑑)/𝜕𝑑=0 and is given by the following equation 𝑑^∗=𝑝^𝑇₀𝑒^{∫𝑇𝑠𝑟0(𝑣)𝑑𝑣}𝑑𝑠+𝑥𝑒^{∫𝑇0𝑟0(𝑠)𝑑𝑠}+𝑒^{∫𝑇0𝜉(𝑠)𝑑𝑠}−1𝛾. (A.21)
By inserting (A.21) into (A.19), Theorem 2 is proved.

B.

This section will prove Theorem 5. Due to the relationship of the equilibrium value function for problem (MV) and problem (MV), we firstly derive the time-consistent investment strategy for problem (MV). From standard arguments as in Björk and Murgoci [8], we obtain the extended HJB equations for problem (MV).

Lemma B.1 (verification theorem). If the two functions 𝑊(𝑡,𝑥) and ℎ(𝑡,𝑥)∈𝐶^1,2(𝑄) satisfy the following extended HJB equations: sup_𝑙∈𝑈𝒜^𝑙𝑊(𝑡,𝑥)−𝒜^𝑙𝛾2ℎ²(𝑡,𝑥)+𝛾ℎ(𝑡,𝑥)𝒜^𝑙ℎ(𝑡,𝑥)=0,𝑊(𝑇,𝑥)=𝑥,𝒜^𝑙∗ℎ(𝑡,𝑥)=0,ℎ(𝑇,𝑥)=𝑥, (B.1)(B.2)(B.3)(B.4) where 𝑙_∗(𝑡)()𝛾=argsup𝒜𝑊𝑡,𝑥−𝒜2ℎ²()()(,𝑡,𝑥+𝛾ℎ𝑡,𝑥𝒜ℎ𝑡,𝑥) (B.5) then 𝜓(𝑡,𝑥)=𝑊(𝑡,𝑥), 𝐸_𝑡,𝑥(𝑋^𝑙∗(𝑡))=ℎ(𝑡,𝑥), and 𝑙_∗(𝑡) is the optimal time-consistent strategy.

Next, we will construct the solution to problem (MV). Assume that there exist two real functions 𝑊(𝑡,𝑥) and ℎ(𝑡,𝑥) satisfying the conditions in Lemma B.1. By virtue of (7), (B.1) can be rewritten as sup_𝑙∈𝑈𝑊_𝑡(𝑡,𝑥)+𝑊_𝑥(𝑡,𝑥)𝑟₀(𝑡)(𝑡)𝑙(𝑡)+1𝑥−𝑐+𝑟2𝑊_𝑥𝑥(𝑡,𝑥)−𝛾ℎ²_𝑥(𝑡,𝑥)𝑙²(𝑡)𝜎²(𝑡)𝑊()−𝛾+𝜆𝐸𝑡,𝑥+𝑍2ℎ()×(ℎ()(𝑊()+𝛾𝑡,𝑥+𝑍𝑡,𝑥+𝑍−2ℎ𝑡,𝑥))−𝜆𝑡,𝑥2ℎ²(𝑡,𝑥)=0. (B.6)

Equation (B.3) can be also rewritten as ℎ_𝑡(𝑡,𝑥)+ℎ_𝑥(𝑡,𝑥)𝑟₀(𝑡)𝑥−𝑐+𝑟(𝑡)𝑙_∗(𝑡)+ℎ_𝑥𝑥(𝑡,𝑥)2𝑙²_∗(𝑡)𝜎²(𝑡)+𝜆𝐸[ℎ(𝑡,𝑥+𝑍)−ℎ(𝑡,𝑥)]=0, (B.7) where 𝑙_∗ is determined below.

Since the linear structure of (B.6)-(B.7) and the boundary conditions of 𝑊(𝑡,𝑥) and ℎ(𝑡,𝑥) given by (B.2) and (B.4) are linear in 𝑥, we can conjecture that 𝑊(𝑡,𝑥) and ℎ(𝑡,𝑥) have the following expressions: 𝑊(𝑡,𝑥)=𝑀(𝑡)𝑥+𝑁(𝑡),𝑀(𝑇)=1,𝑁(𝑇)=0,ℎ(𝑡,𝑥)=𝑚(𝑡)𝑥+𝑛(𝑡),𝑚(𝑇)=1,𝑛(𝑇)=0. (B.8)(B.9)

Thus, the corresponding partial derivatives are calculated as follows: 𝑊_𝑡(𝑡,𝑥)=𝑀̇(𝑡)𝑥+𝑁̇(𝑡),𝑊_𝑥()(𝑡),𝑊𝑡,𝑥=𝑀_𝑥𝑥(𝑡,𝑥)=0,ℎ_𝑡(𝑡,𝑥)=𝑚̇(𝑡)𝑥+𝑛̇(𝑡),ℎ_𝑥()(𝑡),ℎ𝑡,𝑥=𝑚_𝑥𝑥(𝑡,𝑥)=0. (B.10)(B.11)

By substituting (B.8)–(B.11) into (B.6), it yields that sup_𝑙∈𝑈𝑀̇(𝑡)𝑥+𝑁̇(𝑡)+𝑀𝑟₀(𝑡)𝑥−𝑐+𝑟(𝑡)𝑙(𝑡)+𝜆𝜇_𝑧−𝛾2𝑚²(𝑡)𝜆𝜎²_𝑧+𝑙²(𝑡)𝜎²(𝑡)=0. (B.12)

Differentiating the function in the left bracket of (B.12) with respect to 𝑙 and setting the derivative to zero, we get 𝑙_∗(𝑡)=𝑀(𝑡)𝑟(𝑡)𝛾𝑚²(𝑡)𝜎²(𝑡). (B.13)

Inserting (B.13) into (B.12) and (B.7), we have 𝑀̇(𝑡)+𝑟₀(𝑡)𝑀(𝑡)𝑥+𝑁̇(𝑡)−𝑐𝑀(𝑡)+𝜆𝜇_𝑧𝑀(𝑡)−𝛾2𝑚²(𝑡)𝜆𝜎²_𝑧+𝑀²(𝑡)𝜉(𝑡)2𝛾𝑚²(𝑡)𝑚̇(𝑡)=0,+𝑟₀(𝑡)𝑚(𝑡)𝑥+𝑛̇(𝑡)−𝑐𝑚(𝑡)+𝜆𝜇_𝑧𝑚(𝑡)+𝑀(𝑡)𝜉(𝑡)𝛾𝑚(𝑡)=0. (B.14)

In order to ensure the above equations hold, we require 𝑀̇(𝑡)+𝑟₀(𝑡)𝑀(𝑡)(𝑇)𝑁̇(𝑡)+=0,𝑀=1,𝜆𝜇_𝑧𝛾−𝑐𝑀−2𝑚²(𝑡)𝜆𝜎²_𝑧+𝑀²(𝑡)𝜉(𝑡)2𝛾𝑚²(𝑡)𝑁(𝑇)𝑚̇(𝑡)=0,=0,+𝑟₀(𝑡)𝑚(𝑡)(𝑇)𝑛̇(𝑡)+=0,𝑚=1,𝜆𝜇_𝑧𝑚(𝑡)+𝑀(𝑡)𝜉(𝑡)−𝑐(𝑡)(𝑇)𝛾𝑚=0,𝑛=0. (B.15)

By simple calculation, the solutions to (B.15) are given by the following equations: 𝑀(𝑡)=𝑒^{∫𝑇𝑡𝑟0(𝑠)𝑑𝑠},𝑁(𝑡)=𝜆𝜇_𝑧−𝑐^𝑇_𝑡𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}−𝛾𝑑𝑠2𝜆𝜎²_𝑧^𝑇_𝑡𝑒^{2∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}1𝑑𝑠+2𝛾^𝑇_𝑡𝜉(𝑠)𝑚(𝑡)𝑑𝑠,=𝑒^{∫𝑇𝑡𝑟0(𝑠)𝑑𝑠},𝑛(𝑡)=𝜆𝜇_𝑧−𝑐^𝑇_𝑡𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}1𝑑𝑠+𝛾^𝑇_𝑡𝜉(𝑠)𝑑𝑠. (B.16)

Substituting (B.16) into (B.13) and (B.8), we derive the explicit expressions for 𝑊(𝑡,𝑥) and ℎ(𝑡,𝑥) for problem (MV): 𝑙_∗(𝑡)=𝑟(𝑡)𝛾𝜎²(𝑡)𝑒^{−∫𝑇𝑡𝑟0(𝑠)𝑑𝑠},𝑊(𝑡,𝑥)=𝑒^{∫𝑇𝑡𝑟0(𝑠)𝑑𝑠}𝑥+𝑝^𝑇_𝑡𝑒^{∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}−𝑑𝑠𝛾𝜆𝜎²_𝑧2^𝑇_𝑡𝑒^{2∫𝑇𝑠𝑟0(𝑢)𝑑𝑢}1𝑑𝑠+2𝛾^𝑇_𝑡𝜉(𝑠)𝑑𝑠. (B.17)

When the original time 𝑡 is equal to 0, the time-consistent equilibrium strategy and the equilibrium value function for problem (MV) are shown in Theorem 5.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant nos. 11201335, 11301376, and 71071111) and the Research Project of the Social Science and Humanity on Young Fund of the Ministry of Education (Grant no. 11YJC910007).