Optimization Problems of Excess-of-Loss Reinsurance and Investment under the CEV Model

Abstract

We consider that the insurer purchases excess-of-loss reinsurance and invests its wealth in the constant elasticity of variance (CEV) stock market. We model risk process by Brownian motion with drift and study the optimization problem of maximizing the exponential utility of terminal wealth under the controls of excess-of-loss reinsurance and investment. Using stochastic control theory and power transformation technique, we obtain explicit expressions for the optimal polices and value function. We also show that the optimal excess-of-loss reinsurance is always better than optimal proportional reinsurance. Some numerical examples are given.

1. Introduction

Many papers deal with optimal reinsurance or optimal investment issues for diffusion approximation risk models in the past ten years. In these papers, the insurer is allowed to take reinsurance and/or invest its capital in the Black-Scholes market. Some of the problems have been dealt with through stochastic control theory and related methodologies for finding the minimum probability of ruin or the maximum expected utility of terminal wealth. Browne [1] used a Brownian motion with a drift to describe the surplus of the insurer and found the optimal investment policy to maximize the expected exponential utility of terminal wealth. Later, Schmidli [2], Taksar and Markussen [3] considered the optimal reinsurance policy which minimizes the ruin probability of the cedent.

Recently, much research on insurance optimization in the presence of both proportional reinsurance and investment has been done. Luo et al. [4] studied optimal proportional reinsurance and investment policy which minimizes the probability of ruin. Bai and Guo [5] investigated the problem of maximizing the expected exponential utility of terminal wealth with multiple risky assets and proportional reinsurance. For related works, see, for example, Promislow and Young [6], Liang and Guo [7] and references therein.

The excess-of-loss reinsurance has also attracted interest among academia and practitioners. Asmussen et al. [8] studied a dynamic choice of excess-of-loss reinsurance retention level and the dividend distribution policy which maximizes the expected present value of the dividends in a diffusion model. Zhang et al. [9] considered the problem of minimizing the probability of ruin by controlling the combinational quota-share and excess-of-loss reinsurance strategy. Meng and Zhang [10] investigated optimal risk control for the excess-of-loss reinsurance policy which minimizes the probability of ruin. Bai and Guo [11] explored optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection under the Black-Scholes market which maximizes the expected exponential utility of terminal wealth.

Although many papers are dealing with risk models with investment in the Black-Scholes market, there are analyses based on the other kinds of risk assets process in the actuarial literature. For example, Irgens and Paulsen [12] studied the optimal reinsurance and investment strategy with a jump-diffusion process risk asset market. In fact, there is strong empirical evidence that the variance (or volatility) of asset returns, particularly stock market returns, is not constant [13]. The constant elasticity of variance (CEV) process can describe stochastic volatility of the risky asset to some extent. The CEV model is expressed in terms of a stochastic diffusion process with respect to a standard Brownian motion: d𝑆(𝑡)=𝜇𝑆(𝑡)d𝑡+𝜎𝑆(𝑡)^𝛽+1d𝑊(𝑡), (1) where 𝛽≤0 is the elasticity parameter. This model is characterized by the dependence of the volatility rate, that is, 𝜎𝑆(𝑡)^𝛽+1 on the risk asset price. When the price increases, the instantaneous volatility rate decreases. This seems reasonable because the higher the stock price, the higher the equity market value, and thus the lower the proportion of liability, which results in a decrease in the risk of ruin. The volatility rate or the risk measure is thus decreased. So, the CEV model with stochastic volatility is a natural extension of the GBM (geometric Brownian motion) model. The CEV process was usually applied to calculating the theoretical price, sensitivities and implied volatility of option (see, e.g., Schroder [14]).

In this paper, we consider that the insurer purchases excess-of-loss reinsurance and invests its wealth in the CEV stock market. Although Gu et al. [15] used the CEV model to study the problems of reinsurance and investment under diffusion claim process, they get the optimal strategies for the proportional reinsurance. By the technique of stochastic control theory, we model risk process to Brownian motion with a drift to study the optimization problem of maximizing the exponential utility of terminal wealth under the controls of excess-of-loss reinsurance and investment, which is very different from those in Gu et al. The excess-of-loss reinsurance is a harder problem than proportional reinsurance from the mathematical point of view. Our contribution in this paper is to obtain explicit expressions for the optimal investment and excess-of-loss polices and value function. Moreover, we show that the excess-of-loss reinsurance dominates proportional reinsurance under our objective function. To the best of our knowledge, this is the first study to extend the research of Gu et al. to the case that the insurer purchases excess-of-loss reinsurance.

The rest of the paper is organized as follows. In Section 2, the model and assumption are given. In Section 3, we show that the optimal excess-of-loss reinsurance policy is always better than proportional reinsurance policy. The solution of the model is constructed in Section 4. In Section 5, some numerical examples are given.

2. The Model and Assumptions

2.1. Notation

Before introducing the mathematical models, some principal notations are listed. {𝑁(𝑡),𝑡≥0}: the claim arrival process; 𝑌_𝑖: the claim size; 𝑐: the premium rate of the insurer without reinsurance policy;  𝜂, 𝜃: the safety loading of the insurer and reinsurer, respectively;  𝑎(𝑡): the retention level at time 𝑡;  𝜋(𝑡): the amount invested in the risky asset at time 𝑡;  𝛼=(𝜋(𝑡),𝑎(𝑡)): the admissible policy, denoted by (𝜋,𝑎) for simplicity;  𝑐^(𝑎): the premium rate under the given reinsurance policy 𝑎;  𝑌_𝑖(𝑎): the part of the claim held by the insurer under the given reinsurance policy 𝑎; 𝜇(𝑎), 𝜎²(𝑎): the first and second moments of 𝑌_𝑖(𝑎), respectively;  {𝑆₀(𝑡),𝑡≥0}, {𝑆(𝑡),𝑡≥0}: the price process of the risk-free asset and risky asset, respectively;  {𝑋(𝑡),𝑡≥0}: the surplus process of the insurer;  𝑉(𝑡,𝑥,𝑠): the optimal value function at time 𝑡.

2.2. Problem Formulation

Let (Ω,ℱ,𝑃) be a probability space with filtration {ℱ_𝑡,𝑡≥0} containing all objects defined as follows. In the classical Cramer-Lundberg model, the reserve of an insurer at time 𝑡, denoted by 𝑃(𝑡), evolves over time as 𝑃(𝑡)=𝑥₀+𝑐𝑡−^𝑁(𝑡)_𝑖=1𝑌_𝑖, (2) where 𝑥₀ is the initial level of reserve and {𝑁(𝑡),𝑡≥0} is a Poisson process with intensity 𝜆. And 𝑌₁,𝑌₂,…, independent of {𝑁(𝑡)}, are i.i.d. random variables with common continuous distribution 𝐹 having finite first and second moments 𝜇_∞, 𝜎²_∞, respectively. The premium rate 𝑐 is assumed to be calculated via the expected principle, that is, 𝑐=1+𝜂𝜆𝜇_∞, (3) where 𝜂>0 is the relative safety loading of the insurer.

We now consider a modification of the above Cramer-Lundberg model that takes into account the presence of reinsurance. Let 𝑎 be a retention level and 𝑌_𝑖(𝑎) denote the part of the claims held by the insurer. In other words, 𝑌_𝑖−𝑌_𝑖(𝑎) is the residual part of 𝑌_𝑖 that is covered by the reinsurer. Then, for a given reinsurance policy 𝑎, the corresponding reserve process is 𝑃^(𝑎)(𝑡)=𝑥₀+𝑐^(𝑎)𝑡−^𝑁(𝑡)_𝑖=1𝑌_𝑖(𝑎), (4) where premium rate is 𝑐^(𝑎)=1+𝜂𝜆𝜇_∞−(1+𝜃)𝜆𝐸𝑌_𝑖−𝑌_𝑖(𝑎=()𝐸𝑌)1+𝜃_𝑖(𝑎)𝜆+𝜂−𝜃𝜆𝜇_∞, (5) where 𝜃 denotes the safety loading of the reinsurer, the reinsurer also used the expected value principle. In this paper, we consider noncheap reinsurance, that is, 𝜃>𝜂, which is reasonable in actuarial practice. Otherwise, the insurer could reinsure the whole claims. According to Grandell [16], 𝑃^(𝑎)(𝑡) can be approximated by the diffusion process {𝑅^(𝑎)(𝑡),𝑡≥0}: d𝑅^(𝑎)(𝑡)=𝜆𝜃𝐸𝑌_𝑖(𝑎)+𝜂−𝜃𝜇_∞+d𝑡𝜆𝐸𝑌_𝑖(𝑎)²d𝑊(𝑡), (6) where 𝑊(𝑡) is a standard Brownian motion adapted to ℱ_𝑡.

For the excess-of-loss reinsurance with retention level 𝑎 (i.e., 𝑌_𝑖(𝑎)=min(𝑌_𝑖,𝑎)=𝑌_𝑖∧𝑎), 𝜇(𝑎)=𝐸𝑌_𝑖=∧𝑎_𝑎0𝑦d𝐹𝑦+𝑎𝐹(𝑎)=^𝑎₀𝐹𝑦𝜎d𝑦,₂(𝑎)=𝐸𝑌_𝑖∧𝑎²=^𝑎₀𝑦²d𝐹𝑦+𝑎²𝐹(𝑎)=^𝑎₀2𝑦𝐹𝑦d𝑦, (7) where 𝐹(𝑦)=𝑃(𝑌_𝑖>𝑦). Without loss of generality, we assume that 𝜆=1; then the corresponding diffusion approximation claim process (6) becomes d𝑅^(𝑎)(𝑡)=𝜃𝜇(𝑎(𝑡))+𝜂−𝜃𝜇_∞d𝑡+𝜎(𝑎(𝑡))d𝑊(𝑡). (8)

We assumed that an insurer is allowed to invest its surplus in financial market consisting of a risk-free asset (bond or bank account) and a risky asset (stock or mutual fund). Specifically, the risk-free price process is given by d𝑆₀(𝑡)=𝑟𝑆₀(𝑡)d𝑡, (9) where 𝑟>0 is the risk-free interest rate.

As previously mentioned, the CEV model has advantages over the GBM model because of the stochastic volatility rate. We describe the risky asset price process by d𝑆(𝑡)=𝜇𝑆(𝑡)d𝑡+𝜎𝑆(𝑡)^𝛽+1d𝐵(𝑡), (10) where 𝜇(>𝑟) is an expected instantaneous rate of the risky asset and 𝜎𝑆^𝛽+1(𝑡) is a standard instantaneous volatility. 𝐵(𝑡) is another ℱ_𝑡-adapted standard Brownian and independent of the claim process.

Remark 1. If 𝛽<0, it can generate a distribution with heavy left tail. Empirical evidence supports the CEV model in the stock market (see, e.g., Schroder [14]). If 𝛽>0, the situation is unrealistic.

Let 𝛼={(𝜋(𝑡),𝑎(𝑡)),𝑡≥0}, denoted by (𝜋,𝑎) for simplicity, be any admissible control policy which is a two-dimensional ℱ_𝑡-adapted stochastic process, where 𝜋 represents the amount invested in the risky asset at time 𝑡, and 0≤𝑎(𝑡)≤∞ represents the excess-of-loss level at time 𝑡; the set of all admissible policies is denoted by Π.

The dynamics of resulting surplus process can be described as (𝑡)(𝑡)(𝑡)d𝑋=𝜋d𝑆𝑆(𝑡)+(𝑋(𝑡)−𝜋(𝑡))d𝑆₀(𝑡)𝑆₀(𝑡)+d𝑅^(𝑎)(𝑡)=𝑟𝑋(𝑡)+𝜇−𝑟𝜋(𝑡)+𝜃𝜇(𝑎(𝑡))+𝜂−𝜃𝜇_∞(𝑡)𝑆(𝑡)d𝑡+𝜎𝜋^𝛽d𝐵(𝑡)+𝜎(𝑎(𝑡))d𝑊(𝑡). (11)

Remark 2. In this paper, we assume that continuous trading is allowed and all assets are infinitely divisible. We allowed 𝜋(𝑡)<0 and 𝜋(𝑡)>𝑋(𝑡), that means we allowed the insurer to short sell the risky asset and borrow money from a bank for investing in the risky asset.

We are interested in maximizing the utility of the cedent’s terminal wealth, say at time 𝑇. Let 𝑢(𝑥) be the utility function with 𝑢^′>0 and 𝑢^′′<0. For 𝛼∈Π, we define the return function as 𝑉^𝛼(𝑡,𝑥,𝑠)=𝐸[𝑢(𝑋(𝑇))∣𝑋(𝑡)=𝑥,𝑆(𝑡)=𝑠]. (12) The optimal value function is defined as 𝑉(𝑡,𝑥,𝑠)=sup_𝛼∈Π𝑉^𝛼(𝑡,𝑥,𝑠). (13) Our objective is finding an optimal policy 𝛼^∗∈Π𝑉(𝑡,𝑥,𝑠)=𝑉^𝛼∗(𝑡,𝑥,𝑠). (14)

In the case of proportional reinsurance, an explicit solution to this problem was found by Gu et al. [15]. However, the excess-of-loss reinsurance is a harder problem than proportional reinsurance from the mathematical point of view: the functional relation between 𝜇(𝑎) and 𝜎(𝑎) is much more complicated even for a relatively simple distribution 𝐹 such as the exponential or uniform.

Remark 3. A variety of utility functions are studied for investment and consumption strategies by an individual; see, for example, Karatzas [17] and references therein. We assume that the insurer is a closely-held corporation with risk aversion for reasonable utility analysis (see, Mayers and Smith [18], Loubergé and Watt [19]).

3. The Gain of Excess-of-Loss Reinsurance

In this section, we will show that the optimal excess-of-loss reinsurance policy is always better than proportional reinsurance policy. For the proportional reinsurance with retention level 𝑎_pr (i.e., 𝑌_𝑖(𝑎_pr)=𝑎_pr𝑌_𝑖, 0<𝑎_pr≤1), 𝜇𝑎_pr=𝐸𝑎_pr𝑌_𝑖=𝑎_pr𝜇_∞,𝜎²𝑎_pr=𝐸𝑎_pr𝑌_𝑖²=𝑎²_pr𝜎²_∞. (15) Then, the diffusion claim process (6) becomes d𝑅^(𝑎pr)(𝑡)=𝜃𝑎_pr𝜇_∞+𝜂−𝜃𝜇_∞d𝑡+𝑎_pr𝜎_∞d𝑊(𝑡). (16)

Lemma 4. Let 0<𝑎_pr≤1 be a (fixed) retention level in proportional reinsurance model satisfying the condition 𝜎²(𝑎)=𝐸𝑌_𝑖∧𝑎²=𝐸𝑎_pr𝑌_𝑖²=𝑎²_pr𝜎²_∞, (17) then 𝜇(𝑎)=𝐸𝑌_𝑖∧𝑎≥𝑎_pr𝐸𝑌_𝑖=𝑎_pr𝜇_∞. (18)

Proof. The proof of the lemma can be found in Bai and Guo [11].

Theorem 5. For all (𝑡,𝑥,𝑠)∈[0,𝑇]×𝑅×𝑅, there exists policy 𝛼∈Π, satisfying 𝑉^𝛼(𝑡,𝑥,𝑠)≥𝑉^𝑎∗pr(𝑡,𝑥,𝑠), (19) where 𝑉^𝛼(𝑡,𝑥,𝑠) is the value function for the excess-of-loss reinsurance model and 𝑉^𝑎∗pr(𝑡,𝑥,𝑠) is the optimal value function for the proportional reinsurance model.

Proof. Let (𝑎^∗_pr,𝜋^∗) be the optimal feedback retention level and investment policy for the proportional reinsurance model (see Gu et al. [15]). The dynamics of the resulting surplus process (11) becomes d𝑋(𝑡)=𝑟𝑋(𝑡)+𝜇−𝑟𝜋^∗(𝑡)+𝜃𝑎^∗_pr𝜇_∞+𝜂−𝜃𝜇_∞d𝑡+𝜎𝜋^∗(𝑡)𝑆(𝑡)^𝛽d𝐵(𝑡)+𝑎^∗_pr𝜎_∞d𝑊(𝑡). (20) Since 𝜎²(𝑎) is continuous function with respect to 𝑎 and 𝜎²(∞)=∫^∞₀𝑦²d𝐹(𝑦)=𝜎²_∞, we can choose a feedback control 𝛼=(𝜋,𝑎) in the excess-of-loss model in such a way that 𝜋=𝜋^∗ and 𝜎²(𝑎(𝑡))=∫₀^𝑎(𝑡)𝑦²d𝐹(𝑦)=(𝑎^∗_pr𝜎_∞)². From Lemma 4, we have 𝜇(𝑎(𝑡))≥𝑎^∗_pr𝜇_∞. Hence, with the same diffusion coefficient, the drift coefficient of excess-of-loss reinsurance model is bigger, which implies that 𝑉^𝛼(𝑡,𝑥,𝑠)≥𝑉^𝑎∗pr(𝑡,𝑥,𝑠).

Remark 6. From the proof of Theorem 5, we can know that the preference for excess-of-loss reinsurance does not depend on utility function. We can also see that the result is true under our objective function in the case of cheap reinsurance (similar to Asmussen et al. [8]). Note that the premium is calculated by means of the expected value principle in the model. However, other premium principles are used, for example, the variance principle (see, Waters [20], Hesselager [21]), there may be different from the result of Theorem 5. In general this is a complicated matter. We leave this problem as an area for future research.

From now on, we only consider the excess-of-loss reinsurance model.

4. Solution to the Model under Exponential Utility

Suppose now that the insurer has exponential utility 𝑢(𝑥)=𝜆₀−𝛾𝑚𝑒^−𝑚𝑥, (21) where 𝛾>0 and 𝑚>0. This utility has constant absolute risk aversion (CARA) parameter 𝑚. Such utility functions play a prominent role in insurance mathematics and actuarial practice, since they are the only utility functions under which the principle of “zero utility” gives a fair premium, that is, independent of the level of reserves of an insurer (see Gerber [22]).

We use the standard dynamic programming approach to solve the problem of maximizing expected exponential utility. We see that if the optimal value function 𝑉 and its partial derivatives 𝑉_𝑡, 𝑉_𝑥, 𝑉_𝑥𝑥, 𝑉_𝑠, 𝑉_𝑠𝑠, and 𝑉_𝑠𝑥 are continuous on [0,𝑇)×𝑅×𝑅, then 𝑉 satisfies the following Hamilton-Jacobi-Bellman (HJB) equation: sup_{𝜋∈𝑅,𝑎∈[0,𝑁]}𝒜^𝛼𝑉(𝑡,𝑥,𝑠)=0, (22) with boundary condition 𝑉(𝑇,𝑥,𝑠)=𝑢(𝑥), (23) where 𝒜^𝛼𝑉(𝑡,𝑥,𝑠)=𝑉_𝑡+𝑟𝑥+𝜇−𝑟𝜋+𝜃𝜇(𝑎)+𝜇_∞𝜂−𝜃𝑉_𝑥+𝜇𝑠𝑉_𝑠+12𝜋²𝜎²𝑠^2𝛽+𝜎²(𝑎)𝑉_𝑥𝑥+12𝜎²𝑠^2𝛽+2𝑉_𝑠𝑠+𝜋𝜎²𝑠^2𝛽+1𝑉_𝑥𝑠,𝑁=sup𝑦∶𝐹𝑦<1≤∞. (24) The following verification theorem is essential in solving the associated stochastic control problem.

Theorem 7. Let 𝑊∈𝐶^1,2 be concave solution to HJB equation (22) subject to the boundary condition (23). Then, the value function 𝑉 given by expression (13) coincides with 𝑊. That is, 𝑊(𝑡,𝑥,𝑠)=𝑉(𝑡,𝑥,𝑠). (25) Furthermore, let (𝜋^∗,𝑎^∗) be such that 𝒜^{(𝜋∗,𝑎∗)}𝑉(𝑡,𝑥,𝑠)=0 (26) for all (𝑡,𝑥,𝑠)∈[0,𝑇)×𝑅×𝑅. Then, the feedback (Markov) strategies 𝜋^∗𝑡,𝑋^∗(𝑡),𝑆(𝑡),𝑎^∗𝑡,𝑋^∗(𝑡),𝑆(𝑡) (27) are the optimal policies.

Proof. The proof of the verification theorem is standard (see chapter III in Fleming and Soner [23]).

Remark 8. In order to use the verification theorem in its basic form, a sufficient condition is that the following kind of expectation is finite: 𝐸^𝑡′_𝑡𝒜^{(𝜋∗,𝑎∗)}𝑊𝑢,𝑋^∗(𝑢),𝑆(𝑢)d𝑢<∞,∀𝑡^′≥𝑡, (28) so as to be able to use Dynkin’s formula and conclude that 𝑊(𝑡,𝑥,𝑠) is indeed 𝑉(𝑡,𝑥,𝑠). This point can be verified from the following conclusion of Theorem 9: 𝜋^∗(𝑡)=𝑂(1/𝑆(𝑡)^2𝛽) and 𝑎^∗(𝑡)=𝑂(1).

Theorem 9. When 𝑁𝑚>𝜃, the optimal value function is 𝑉(𝑡,𝑥,𝑠)=𝜆₀−𝛾𝑚exp−𝑚𝑥𝑒^{𝑟(𝑇−𝑡)}+𝐾₁(𝑡)+𝐿(𝑡)𝑠^−2𝛽, (29) where 𝐿(𝑡)=−𝜇−𝑟²4𝑟𝜎²1−𝑒^{−2𝑟(𝑇−𝑡)},𝐾₁(𝑡)=𝛽2𝛽+1𝜇−𝑟²)−4𝑟(𝑇−𝑡1−𝑒^{−2𝑟(𝑇−𝑡)}+2𝑟𝑚𝜇_∞𝜂−𝜃𝑒^{𝑟(𝑇−𝑡)}𝑟−^𝑇_𝑡−𝜃𝑚𝑒^{𝑟(𝑇−𝑧)𝜃𝑒−𝑟(𝑇−𝑧)}₀^/𝑚𝐹𝑦d𝑦+𝑚²𝑒^{2𝑟(𝑇−𝑧)𝜃𝑒−𝑟(𝑇−𝑧)}₀^/𝑚𝑦𝐹𝑦d𝑦d𝑧. (30) In this case, the optimal excess-of-loss reinsurance and investment policy is 𝜋^∗(𝑡)=2𝑟𝜇−𝑟+𝛽𝜇−𝑟²1−𝑒^{−2𝑟(𝑇−𝑡)}2𝑟𝜎²𝑠^2𝛽×𝑒^{−𝑟(𝑇−𝑡)}𝑚1=𝑂𝑆(𝑡)^2𝛽,𝑎^∗(𝑡)=𝜃𝑒^{−𝑟(𝑇−𝑡)}𝑚=𝑂(1). (31) When 𝑁𝑚≤𝜃, the optimal value function is 𝑉(𝑡,𝑥,𝑠)=𝜆₀−𝛾𝑚exp−𝑚𝑥𝑒^{𝑟(𝑇−𝑡)}+𝐾₁(𝑡)+𝐿(𝑡)𝑠^−2𝛽+𝑘,0≤𝑡<𝜆𝑇,₀−𝛾𝑚exp−𝑚𝑥𝑒^{𝑟(𝑇−𝑡)}+𝐾₂(𝑡)+𝐿(𝑡)𝑠^−2𝛽,𝑇≤𝑡<𝑇, (32) where 𝑇=𝑇+(ln(𝑁𝑚)−ln𝜃)/𝑟, 𝐾₂(𝑡)𝛽=−2𝛽+1𝜇−𝑟²)−4𝑟(𝑇−𝑡1−𝑒^{−2𝑟(𝑇−𝑡)}−2𝑟𝑚𝜇_∞𝜂𝑟𝑒^{𝑟(𝑇−𝑡)}−1−𝑚𝜎²_∞4𝑟1−𝑒^{2𝑟(𝑇−𝑡)}, (33) and 𝑘=𝐾₂(𝑇)−𝐾₁(𝑇). In this case, the corresponding optimal excess-of-loss reinsurance and investment policy is 𝜋^∗(𝑡),𝑎^∗(𝑡=𝜋)^∗(𝑡),𝜃𝑒^{−𝑟(𝑇−𝑡)}𝑚,0≤𝑡<𝜋𝑇,_∗(𝑡),𝑁,𝑇≤𝑡<𝑇, (34) where 𝜋^∗(𝑡)=((2𝑟(𝜇−𝑟)+𝛽(𝜇−𝑟)²(1−𝑒^{−2𝑟(𝑇−𝑡)}))/2𝑟𝜎²𝑠^2𝛽)(𝑒^{−𝑟(𝑇−𝑡)}/𝑚)=𝑂(1/𝑆(𝑡)^2𝛽).

Remark 10. From Theorem 9, we can see that when the total expected claims exceed the ratio of the reinsurer’s safety loading to the coefficient of risk aversion, that is, 𝑁>𝑚/𝜃, the optimal excess-of-loss portfolio retention is the ratio of discounted reinsurer’s safety loading to the coefficient of risk aversion. If 𝑁≤𝑚/𝜃, the optimal excess-of-loss policy is no reinsurance when 𝑡≥𝑇 and is also the ratio of discounted reinsurer’s safety loading to the coefficient of risk aversion when 𝑡≤𝑇.

Proof. Following the methods of Browne [1] or Liang et al. [24], we conjecture a solution of the form 𝑉(𝑡,𝑥,𝑠)=𝜆₀−𝛾𝑚exp−𝑚𝑥𝑒^{𝑟(𝑇−𝑡)}+𝐺(𝑡,𝑠), (35) where 𝐺(𝑡,𝑠) is a suitable function to be determined. And the boundary condition 𝑉(𝑇,𝑥,𝑠)=𝑢(𝑥) implies that 𝐺(𝑇,𝑠)=0. (36) Let 𝐺_𝑡, 𝐺_𝑠, and 𝐺_𝑠𝑠 be the partial derivatives of 𝐺(𝑡,𝑠). Note that 𝑉_𝑡=𝑉(𝑡,𝑥,𝑠)−𝜆₀𝑚𝑥𝑟𝑒^{𝑟(𝑇−𝑡)}+𝐺_𝑡,𝑉_𝑥=𝑉(𝑡,𝑥,𝑠)−𝜆₀−𝑚𝑒^{𝑟(𝑇−𝑡)},𝑉_𝑠=𝑉(𝑡,𝑥,𝑠)−𝜆₀𝐺_𝑠,𝑉_𝑥𝑥=𝑉(𝑡,𝑥,𝑠)−𝜆₀𝑚²𝑒^{2𝑟(𝑇−𝑡)},𝑉_𝑠𝑠=𝑉(𝑡,𝑥,𝑠)−𝜆₀𝐺²_𝑠+𝐺_𝑠𝑠,𝑉_𝑥𝑠=𝑉(𝑡,𝑥,𝑠)−𝜆₀−𝑚𝑒^{𝑟(𝑇−𝑡)}𝐺_𝑠. (37) Substituting (37) back into the HJB equation (22), since 𝑉(𝑡,𝑥,𝑠)−𝜆₀<0, we get 𝐺_𝑡−𝑚𝜇_∞𝜂−𝜃𝑒^{𝑟(𝑇−𝑡)}+𝜇𝑠𝐺_𝑠+12𝜎²𝑠^2𝛽+2𝐺²_𝑠+𝐺_𝑠𝑠+inf_𝜋∈𝑅−𝜇−𝑟𝜋−𝜋𝜎²𝑠^2𝛽+1𝐺_𝑠+12𝜎²𝜋²𝑠^2𝛽𝑚𝑒^{𝑟(𝑇−𝑡)}𝑚𝑒^{𝑟(𝑇−𝑡)}+inf_{𝑎∈[0,𝑁]}(𝑎)1−𝜇𝜃+2𝜎²(𝑎)𝑚𝑒^{𝑟(𝑇−𝑡)}𝑚𝑒^{𝑟(𝑇−𝑡)}=0. (38) Let 𝑓₁()=−𝜋,𝑡𝜇−𝑟𝜋−𝜋𝜎²𝑠^2𝛽+1𝐺_𝑠+12𝜎²𝜋²𝑠^2𝛽𝑚𝑒^{𝑟(𝑇−𝑡)}𝑚𝑒^{𝑟(𝑇−𝑡)},𝑓₂()=(𝑎)1𝑎,𝑡−𝜇𝜃+2𝜎²(𝑎)𝑚𝑒^{𝑟(𝑇−𝑡)}𝑚𝑒^{𝑟(𝑇−𝑡)}. (39) Differentiating 𝑓₁(𝜋,𝑡) with respect to 𝜋 yields the minimizer 𝜋^∗=𝜇−𝑟+𝜎²𝑠^2𝛽+1𝐺_𝑠𝜎²𝑠^2𝛽𝑒^{−𝑟(𝑇−𝑡)}𝑚, (40) and the value of 𝑓₁(𝜋,𝑡) at this minimizer is 𝑓₁𝜋^∗1,𝑡=−2𝜇−𝑟+𝜎²𝑠^2𝛽+1𝐺_𝑠²𝜎²𝑠^2𝛽. (41) Similarly, from the first order condition 𝜕𝑓₂(𝑎,𝑡)=𝜕𝑎−𝜃𝐹(𝑎)+𝑎𝐹(𝑎)𝑚𝑒^{𝑟(𝑇−𝑡)}𝑚𝑒^{𝑟(𝑇−𝑡)}=0, (42) we know that without restriction with respect to 𝑎, 𝜃𝑎=𝑚𝑒^{−𝑟(𝑇−𝑡)}, (43) which leads to 𝑓₂()=(𝑎)1𝑎,𝑡−𝜇𝜃+2𝜎²(𝑎)𝑚𝑒^{𝑟(𝑇−𝑡)}𝑚𝑒^{𝑟(𝑇−𝑡)}=−𝜃𝑚𝑒^{𝑟(𝑇−𝑡)𝜃𝑒−𝑟(𝑇−𝑡)}₀^/𝑚𝐹𝑦d𝑦+𝑚²𝑒^{2𝑟(𝑇−𝑡)𝜃𝑒−𝑟(𝑇−𝑡)}₀^/𝑚𝑦𝐹𝑦d𝑦. (44)
We need to discuss the two cases according to the value of 𝑎.
Case 1. When 𝑁𝑚>𝜃
if 𝑡<𝑇+(ln(𝑁𝑚)−ln𝜃)/𝑟, then 𝑎∈[0,𝑁). So, 𝜋^∗(𝑡),𝑎^∗(𝑡)=𝜇−𝑟+𝜎²𝑠^2𝛽+1𝐺_𝑠𝜎²𝑠^2𝛽⋅𝑒^{−𝑟(𝑇−𝑡)}𝑚,𝜃𝑒^{−𝑟(𝑇−𝑡)}𝑚 (45) coincides with the optimal policy. Since 𝑇<𝑇+(ln(𝑁𝑚)−ln𝜃)/𝑟, (𝜋^∗(𝑡),𝑎^∗(𝑡)) is optimal policy on [0,𝑇].
Up to now, we still need to solve 𝐺(𝑡,𝑠) to find 𝜋^∗(𝑡) and 𝑉(𝑡,𝑥,𝑠) in this case. Substituting (𝜋^∗(𝑡),𝑎^∗(𝑡)) (i.e., expression (45)) back to (38), we can get 𝐺_𝑡−𝑚𝜇_∞𝜂−𝜃𝑒^{𝑟(𝑇−𝑡)}+𝑟𝑠𝐺_𝑠+12𝜎²𝑠^2𝛽+2𝐺_𝑠𝑠−𝜇−𝑟²2𝜎²𝑠^2𝛽+𝑓₂𝑎^∗,𝑡=0. (46) We appeal to power transformation technique and variable change method to solve the problem.
Let 𝐺(𝑡,𝑠)=ℎ𝑡,𝑦,𝑦=𝑠^−2𝛽, (47) with boundary condition ℎ𝐺𝑇,𝑦=0,_𝑡=ℎ_𝑡,𝐺_𝑠=−2𝛽𝑠^−2𝛽−1ℎ_𝑦,𝐺_𝑠𝑠=2𝛽2𝛽+1𝑠^−2𝛽−2ℎ_𝑦+4𝛽²𝑠^−4𝛽−2ℎ_𝑦𝑦, (48) where ℎ_𝑡, ℎ_𝑦, and ℎ_𝑦𝑦 are partial derivatives of ℎ(𝑡,𝑦).
Putting the partial derivatives of 𝐺(𝑡,𝑠) into (46), we obtain ℎ_𝑡+𝜎²2𝛽+1−2𝑟𝑦𝛽ℎ_𝑦+2𝜎²𝛽²𝑦ℎ_𝑦𝑦−𝜇−𝑟²2𝜎²𝑦+𝑀₁(𝑡)=0, (49) where 𝑀₁(𝑡)=−𝑚𝜇_∞(𝜂−𝜃)𝑒^{𝑟(𝑇−𝑡)}+𝑓₂(𝑎^∗,𝑡). We try to find a solution of the above equation with the following form ℎ𝑡,𝑦=𝐾₁(𝑡)+𝐿₁(𝑡)𝑦, (50) with boundary condition 𝐾₁(𝑇)𝐿=0,₁(𝑇)ℎ=0,_𝑡=𝐾^′₁+𝐿^′₁𝑦,ℎ_𝑦=𝐿₁,ℎ_𝑦𝑦=0, (51)(52)(53) where 𝐾^′₁, 𝐿^′₁ are the derivatives of 𝐾₁, 𝐿₁, respectively. Putting (53) into (49), we derive 𝐾^′₁+𝜎²𝛽2𝛽+1𝐿₁+𝐿^′₁−2𝑟𝐿₁−𝜇−𝑟²2𝜎²𝑦+𝑀₁(𝑡)=0. (54) By matching coefficients, we have 𝐾^′₁+𝜎²𝛽2𝛽+1𝐿₁+𝑀₁(𝑡)𝐿=0,_′1−2𝑟𝐿₁−𝜇−𝑟²2𝜎²=0. (55) Taking into account the boundary condition, we have the solution of (55): 𝐿₁(𝑡)=−𝜇−𝑟²4𝑟𝜎²1−𝑒^{−2𝑟(𝑇−𝑡)},𝐾₁(𝑡)=^𝑇_𝑡𝛽2𝛽+1𝜎²𝐿₁(𝑧)+𝑀₁(𝑧𝛽)d𝑧=−2𝛽+1𝜇−𝑟²)−4𝑟(𝑇−𝑡1−𝑒^{−2𝑟(𝑇−𝑡)}−2𝑟𝑚𝜇_∞𝜂−𝜃𝑒^{𝑟(𝑇−𝑡)}−1𝑟+^𝑇_𝑡−𝜃𝑚𝑒^{𝑟(𝑇−𝑧)𝜃𝑒−𝑟(𝑇−𝑧)}₀^/𝑚𝐹𝑦d𝑦+𝑚²𝑒^{2𝑟(𝑇−𝑧)𝜃𝑒−𝑟(𝑇−𝑧)}₀^/𝑚𝑦𝐹𝑦d𝑦d𝑧. (56) Putting these parameters into 𝐺(𝑡,𝑠), we obtain 𝐺(𝑡,𝑠)=𝐾₁(𝑡)+𝐿₁(𝑡)𝑠^−2𝛽. (57) So, the optimal investment policy is 2𝑟𝜇−𝑟+𝛽𝜇−𝑟²1−𝑒^{−2𝑟(𝑇−𝑡)}2𝑟𝜎²𝑠^2𝛽⋅𝑒^{−𝑟(𝑇−𝑡)}𝑚, (58) and the corresponding value function has the form 𝑉(𝑡,𝑥,𝑠)=𝜆₀−𝛾𝑚exp−𝑚𝑥𝑒^{𝑟(𝑇−𝑡)}+𝐾₁(𝑡)+𝐿₁(𝑡)𝑠^−2𝛽, (59) where 𝐿₁(𝑡) and 𝐾₁(𝑡) are determined by (56), respectively.
Case 2. When 𝑁𝑚≤𝜃
If 𝑡<𝑇+(ln(𝑁𝑚)−ln𝜃)/𝑟 (noting that 𝑇+(ln(𝑁𝑚)−ln𝜃)/𝑟≤𝑇), we know that 𝑎∈[0,𝑁) from expression (43). Similar to Case 1, incorporating the constants of the calculations, we get the optimal value function 𝑉(𝑡,𝑥,𝑠)=𝜆₀−𝛾𝑚exp−𝑚𝑥𝑒^{𝑟(𝑇−𝑡)}+𝐾₁(𝑡)+𝐿₁(𝑡)𝑠^−2𝛽+𝑘, (60) where the constant 𝑘 will be determined from the following (70), and the optimal policies are 𝜋^∗(𝑡),𝑎^∗(𝑡)=𝜇−𝑟+𝜎²𝑠^2𝛽+1𝐺_𝑠𝜎²𝑠^2𝛽𝑒^{−𝑟(𝑇−𝑡)}𝑚,𝜃𝑒^{−𝑟(𝑇−𝑡)}𝑚, (61) where 𝐺(𝑡,𝑠)=𝐾₁(𝑡)+𝐿₁(𝑡)𝑠^−2𝛽+𝑘. If 𝑇+(ln(𝑁𝑚)−ln𝜃)/𝑟≤𝑡≤𝑇, then 𝑎≥𝑁. We get that the optimal retention level is 𝑎^∗(𝑡)=𝑁. In this case, 𝜇(𝑁)=𝜇_∞ and 𝜎²(𝑁)=𝜎²_∞. Putting the optimal policies 𝜋^∗(𝑡),𝑎^∗(𝑡)=𝜇−𝑟+𝜎²𝑠^2𝛽+1𝐺_𝑠𝜎²𝑠^2𝛽𝑒^{−𝑟(𝑇−𝑡)}𝑚,𝑁, (62) into (38), we obtain 𝐺_𝑡−𝑚𝜇_∞𝜂𝑒^{𝑟(𝑇−𝑡)}+𝑟𝑠𝐺_𝑠+12𝜎²𝑠^2𝛽+2𝐺_𝑠𝑠−12𝜇−𝑟²𝜎²𝑠^2𝛽+12𝜎²_∞𝑚²𝑒^{2𝑟(𝑇−𝑡)}=0. (63)
Again, we use the power transformation technique and variable change method to solve (63) with the boundary condition (21).
Similarly, let 𝐺(𝑡,𝑠)=ℎ(𝑡,𝑦), 𝑦=𝑠^−2𝛽, we have ℎ_𝑡+𝜎²2𝛽+1−2𝑟𝑦𝛽ℎ_𝑦+2𝜎²𝛽²𝑦ℎ_𝑦𝑦−𝜇−𝑟²2𝜎²𝑦−𝑚𝜇_∞𝜂𝑒^{𝑟(𝑇−𝑡)}+12𝜎²_∞𝑚²𝑒^{2𝑟(𝑇−𝑡)}=0. (64) And we try the following form and match coefficients, ℎ𝑡,𝑦=𝐾₂(𝑡)+𝐿₂(𝑡)𝑦. (65) Therefore, we get 𝐿₂(𝑡)=−𝜇−𝑟²4𝑟𝜎²1−𝑒^{−2𝑟(𝑇−𝑡)}, (66) which is the same as the expression 𝐿₁(𝑡), denoted by 𝐿(𝑡) for simplicity: 𝐾₂(𝑡)𝛽=−2𝛽+1𝜇−𝑟²)−4𝑟(𝑇−𝑡1−𝑒^{−2𝑟(𝑇−𝑡)}−2𝑟𝑚𝜇_∞𝜂𝑟𝑒^{𝑟(𝑇−𝑡)}−1−𝑚𝜎²_∞4𝑟1−𝑒^{2𝑟(𝑇−𝑡)},𝐺(𝑡,𝑠)=𝐾₂(𝑡)+𝐿₂(𝑡)𝑠^−2𝛽. (67) Let 𝑇=𝑇+(ln(𝑁𝑚)−ln𝜃)/𝑟. So, in this case, the optimal excess-of-loss reinsurance and investment policies are 𝜋^∗(𝑡),𝑎^∗(𝑡=𝜋)^∗(𝑡),𝜃𝑒^{−𝑟(𝑇−𝑡)}𝑚,0≤𝑡<𝜋𝑇,_∗(𝑡),𝑁,𝑇≤𝑡<𝑇, (68) where 𝜋^∗(𝑡)=((2𝑟(𝜇−𝑟)+𝛽(𝜇−𝑟)²(1−𝑒^{−2𝑟(𝑇−𝑡)}))/2𝑟𝜎²𝑠^2𝛽)⋅(𝑒^{−𝑟(𝑇−𝑡)}/𝑚). And the corresponding value function has the form 𝑉(𝑡,𝑥,𝑠)=𝜆₀−𝛾𝑚exp−𝑚𝑥𝑒^{𝑟(𝑇−𝑡)}+𝐾₁(𝑡)+𝐿(𝑡)𝑠^−2𝛽+𝑘,0≤𝑡<𝜆𝑇,₀−𝛾𝑚exp−𝑚𝑥𝑒^{𝑟(𝑇−𝑡)}+𝐾₂(𝑡)+𝐿(𝑡)𝑠^−2𝛽,𝑇≤𝑡<𝑇, (69) where choose 𝑘 in the way that 𝑉(𝑡,𝑥,𝑠) given by (69) is continuous at 𝑇; that is, 𝑘=𝐾₂𝑇−𝐾₁𝑇. (70) Thus, we complete the proof.

Remark 11. From Theorem 9, we can see that the optimal investment policy is independent of claim size distribution 𝐹 and the value of 𝑥 but is dependent on the value of the risk asset price 𝑠 and time 𝑡.

5. Numerical Examples of the Optimal Policies

In this section, to give some intuitive interpretation of optimal investment and reinsurance policies, we demonstrate numerical examples of two main claim sizes distributions—the exponential and uniform distributions. We set the riskless rate at 𝑟=0.04 per year, the mean excess returns at 𝜇−𝑟=0.04 per year, and the parameter 𝜎 in the expression of an annual standard volatility at 0.19. The estimates of parameters can be based on annual equity return on the stock price index. We refer the readers to Chacko and Viceira [13] and Schroder [14] and references therein.

Let the time horizon 𝑇=3 years be fixed. Because the optimal investment policy is independent of claim size distribution, we firstly give the graph of the optimal investment policies in Figure 1 with the risk aversion parameter 𝑚=0.1 and the elasticity parameter 𝛽=−1/3.

Figure 1 

The surface of 𝜋∗(𝑡,𝑠) with 𝑚=0.1, 𝜎=0.19, 𝑟=0.04, 𝜇=0.08, and 𝛽=−1/3.

From Figure 1, we can see that the effect of the risky asset price on the optimal investment policies 𝜋^∗(𝑡,𝑠) is relatively small. In practice, the optimal investment policy is comparatively more responsive to changes of the mean excess return and volatility of returns. We provide some reports concerning the sensitivity to these parameterizations in Figures 2 and 3.

Figure 2 

The effect of 𝜎 on 𝜋∗(2,5) with 𝛽=−1/3.

Figure 3 

The effect of 𝜇−𝑟 on 𝜋∗(2,5) with 𝑚=2.5.

Let time 𝑡=2 and the risky asset price 𝑠=5 are fixed. We consider values of 𝜎 between 0.1 and 0.4, 𝜇−𝑟=0.04, and 𝛽=−1/3 in Figure 2, and values of 𝜇−𝑟 between 0.04 and 0.2, 𝜎=0.19, and 𝑚=2.5 in Figure 3.

Figure 2 shows that the optimal investment policy decreases as the standard volatility increases. Moreover, a higher level 𝑚 yields a lower value of the optimal investment policy, which is the natural consequence since the larger value of 𝑚 means more risk aversion.

Figure 3 reports that the optimal investment policy increases as the excess return 𝜇−𝑟 increases. The result also shows that a higher elasticity parameter 𝛽 yields a larger value of the optimal investment policy. Especially, when 𝛽 attains its maximum 0, the model degenerates to a GBM model.

The following examples are about reinsurance policy.

Example 1. Assume that the claim size is a standard exponential distribution, 𝐹(𝑦)=𝑒^−𝑦, then 𝑁=+∞. In this case, 𝑁𝑚>𝜃 is always true. Thus, the optimal excess-of-loss reinsurance level 𝑎^∗(𝑡)=(𝜃/𝑚)𝑒^{−𝑟(𝑇−𝑡)} on [0,𝑇]. Figure 4 presents the optimal excess-of-loss reinsurance level for different 𝜃/𝑚=0.5,2.2, respectively.

Figure 4 

𝑌∼𝑒(1), the optimal excess-of-loss reinsurance level 𝑎∗(𝑡) with 𝜃/𝑚=0.5,2.2, respectively.

Example 2. Assume that the claim size is a uniform distribution on (0,2), then 𝑁=2. If 𝜃/𝑚=0.5, then 𝑁𝑚>𝜃. Thus, 𝑎^∗(𝑡)=(𝜃/𝑚)𝑒^{−𝑟(𝑇−𝑡)} on [0,𝑇]. If 𝜃/𝑚=2.2, then 𝑁𝑚<𝜃. Thus, 𝑎^∗(𝑡)=(𝜃/𝑚)𝑒^{−𝑟(𝑇−𝑡)} on [0,0.617) and 𝑎^∗(𝑡)=2 on [0.617,3]. Figure 5 presents the optimal excess-of-loss reinsurance level for different 𝜃/𝑚=0.5,2.2, respectively.

Figure 5 

𝑌∼𝑈(0,2), the optimal excess-of-loss reinsurance level 𝑎∗(𝑡) with 𝜃/𝑚=0.5,2.2, respectively.

From Figure 5, we can see that if 𝜃/𝑚=2.2, the optimal retention 𝑎^∗(𝑡) is a linear increasing function with respect to 𝑡 when 𝑡∈[0,0.617) and is flat for all 𝑡≥0.617. But, in Figure 4, the optimal retention 𝑎^∗(𝑡) is always a linear increasing function with respect to 𝑡 since the potential maximal value of the claim size 𝑌_𝑖 is infinity.

Acknowledgments

This research was supported by SJTU Special Funds for Cross Disciplines (Grant no. 10JCY11) and the National Natural Science Foundation of China (Grant no. 11101215).