Abstract
In this paper, we consider a robust optimal investment-reinsurance problem with a default risk. The ambiguity-averse insurer (AAI) may carry out transactions on a risk-free asset, a stock, and a defaultable corporate bond. The stock’s price is described by a jump-diffusion process, and both the jump intensity and the distribution of jump amplitude are uncertain, i.e., the jump is ambiguous. The AAI’s surplus process is assumed to follow an approximate diffusion process. In particular, the reinsurance premium is calculated according to the generalized mean-variance premium principle, and the reinsurance type has to follow a self-reinsurance function. In performing dynamic programming, both the predefault case and the postdefault case are analyzed, and the optimal strategies and the corresponding value functions are derived under the worst-case scenario. Moreover, we give a detailed proof of the verification theorem and give some special cases and numerical examples to illustrate our theoretical results.
1. Introduction
Investment is the most common way for the insurer to cope with the fierce competition in the insurance market and get higher returns, including risk-free investment (bank), risky investment (stock), and bond investment. The insurer can also transfer their risks by buying reinsurance. Therefore, the optimal investment-reinsurance problem of insurers has received extensive attention in the field of insurance and stochastic control. Browne [1] originally optimized the exponential utility of terminal wealth in order to obtain an optimal investment strategy for the insurer. Since then, a large number of works have been done concerning this topic (see Yang and Zhang [2], Bai and Guo [3], Huang et al. [4], Zhang et al. [5], Zeng et al. [6], Peng et al. [7], etc.).
For the stock’s price process, some scholars have paid attention to the jump risk, such as Yu et al. [8] and Zhang et al. [9]. This is because in the face of serious events (natural disasters and serious large-scale diseases), the stock price may jump to a new level. Therefore, it is not suitable for the stock price to be described by a geometric Brownian motion (GBM) with the constant appreciation rate and volatility. Moreover, the optimal investment-reinsurance problem under the jump-diffusion model has drawn much attention, for example, Li et al. [10], Cao [11], Liang et al. [12], Lin et al. [13], Yang and Zhang [2], and Zhao et al. [14].
Recently, the investors/insurers have an increasing interest in the default risk of corporate bonds with high yield. Default risk (credit risk) refers to the risk that the security issuer will not be able to repay the principal and interest at the maturity of the security, which makes the investor suffer losses. Therefore, it is the purpose of investors to reduce credit risk and obtain higher returns. In fact, several scholars have addressed the portfolio optimization on corporate bonds in the last several decades. Bielecki and Jang [15] studied an optimal allocation problem associated with defaultable bond, and their goal was to maximize the expected utility of the terminal wealth. Bo et al. [16, 17] considered an investment-consumption problem for an investor who can invest in a defaultable market. For more results about default risk, see Capponi and Figueroa-López [18], Zhu et al. [19], Zhao et al. [20], and Deng et al. [21].
For the insurer, reinsurance is an important method to balance their risk and obtain higher profit via an optimal reinsurance strategy. The reinsurance includes the reinsurance premium and reinsurance type. In most of the above results, the reinsurance premium principle is calculated according to the expected value principle or variance principle, such as Liang and Bayraktar [22] and Sun et al. [23]. In previous conclusions, two types of reinsurance policies are most commonly studied in the literature. One policy is the proportional (quota-share) reinsurance (see Zhou et al. [24] and Shen and Zeng [25]), and the other policy is the excess-of-loss reinsurance (see [10, 14]). Recently, Zhang et al. [26] studied the optimal investment-insurance for insurers with the generalized mean-variance principle. In their article, a generalized mean-variance principle included two special cases: the expected value principle and the variance principle. The reinsurance policy was considered a self-reinsurance function, which included the proportional reinsurance and the excess-of-loss reinsurance.
Apart from the investment-reinsurance, recent advances are on the ambiguity aversion, the uncertainty associated with the model, and the risk aversion. In reality, it is a notorious fact that the return of risky assets is difficult to estimate accurately. Therefore, investors may consider some alternative models that are close to the estimated model to deal with portfolio selection in case of ambiguity. Anderson et al. [27] introduced the concept of ambiguity aversion and formulated a robust control problem for investors. Uppal and Wang [28] extended the results of Anderson et al. [27] under the model uncertainty robustness framework with different levels of ambiguity. For investors, Maenhout [29, 30] derived the closed-form solutions to robust optimal strategies by innovating a “homothetic robustness” framework. A lot of descendent researches of Maenhout [29, 30] concentrated on the influences of ambiguity in the field of finance or insurance, and the representative publications are Zhang and Siu [31], Yi et al. [32], Flor and Larsen [33], Pun and Wong [34], Zeng et al. [35], Zheng et al. [26], Zhang et al. [36], etc.
Moreover, the jump risk, especially that associated with disaster events, is more difficult to estimate accurately. So many scholars have paid attention to the ambiguity of jump risks. For the sake of explanation, the distribution of jump amplitude is assumed to be known and is restricted to be identical under the reference model’s measure and the alternative measure , but the jump intensity is uncertain. This topic was studied by some scholars, for example, Branger and Larsen [37], Zeng et al. [6], Sun et al. [23], and Li et al. [10]. But in most instances, the distribution of jump amplitude is unknown. Jin et al. [38] considered the dynamic portfolio choice problem with ambiguous jump risks in a multidimensional jump-diffusion framework. In their results, both the jump amplitude distribution and the jump intensity were assumed to be uncertain.
For these reasons, we choose a jump-diffusion process to describe the price of the stock and consider the robust model to find an optimal strategy in this paper. Moreover, the AAI is allowed to buy reinsurance and allocate his/her wealth among a risk-free asset, a stock, and a default corporate bond. According to Zhang et al. [5], we assume that the reinsurance premium is calculated about the generalized mean-variance principle, which is more general than that reported by Sun et al. [23] and Li et al. [10]. Specifically, in our model, the stock’s price changes dramatically, while the parameters of the underlying jump processes are difficult to estimate accurately. Therefore, we assume that the stock’s jump amplitude distribution and the jump intensity are uncertain, which are different from those reported by Sun et al. [23] and Li et al. [10]. The surplus of an AAI is described by an approximate diffusion process. In light of the principle of dynamic programming, the corresponding Hamilton–Jacobi–Bellman (HJB) equations are deduced for both the postdefault case and the predefault case. Using the variable change and variable separation techniques, we obtain the optimal reinsurance and investment strategies and the corresponding value functions. Our goal is to maximize the expected utility of terminal wealth under the worst-case scenario according to the max-min expected utility. Finally, we exemplify our deductions by some special cases and numerical cases, which verify our theoretical results.
Here, we arrange the remaining part of this paper as follows: Section 2 formulates the robust investment-reinsurance optimization regarding the default risk under the jump-diffusion model. In Section 3, we derive the closed-form expressions for the optimal strategies and the corresponding value functions under the predefault case and postdefault case, respectively. Section 4 provides a proof of the verification theorem. Section 5 provides some special cases. Numerical examples of our results are demonstrated in Section 6. Finally, conclusions are given in Section 7.
2. Model Formulation
In this article, we consider a complete probability space . Let be the right-continuous, -complete filtration generated by two standard Brownian motions and , a Poisson process , and two families of random variables and . We assume that , , and are mutually independent. Let be the enlarged filtration of and , i.e., , where the filtration is generated by a default process . We assume that every -martingale is also a -martingale. The probability measure is the real-world probability measure, and Q is the risk-neutral measure.
2.1. The Financial Market
In this section, we consider a financial market consisting of three types of securities: a risk-free asset, a stock, and a defaultable corporate bond. The price process of the risk-free asset under the measure is described bywhere is the risk-free interest rate. The price process of a stock is described by a jump-diffusion process:where is the expected instantaneous rate of return of the stock; σ is a positive constant; is a homogeneous Poisson process with intensity , representing the number of a stock price’s jumps during the time interval ; and is the ith jump amplitude of the stock’s price, and are i.i.d. random variables with the common distribution function , the first moment , and the second moment . We assume that to ensure that the stock’s price remains positive. Generally, the expect return of the stock is larger than the risk-free interest rate, so we assume that .
Next, we consider that is a Poisson random measure on and is the compensator measure of . That is, , and thenwhere . By the definition of , for any , is a -martingale.
Next, we consider the price process of the defaultable corporate bond by the intensity-based approach. Let τ be the time of default and τ represent the first jump time of a Poisson process with constant intensity under P. A default indicator process is defined as for each , and the value of the corporate bond is assumed to be zero after default. Let be the filtration generated by the default process and augmented in the usual way. By definition, τ is naturally an -stopping time and a -stopping time. Furthermore, the martingale default process is thus given by , which is a -martingale. By Girsanov’s theorem in Bielecki and Jang [15], under the chosen risk-neutral measure Q, the arrival intensity of default is given by . We denote that is the default risk premium. We assume that there exists a defaultable zero coupon bond with a maturity date , and the insurer can recover a fraction of the market value of the defaultable bond just prior to default. Now, for the positive interest rate r, the price dynamics of the defaultable bond under P is (see Deng et al. [21])where represents the risk-neutral credit spread and denotes the loss rate of the bond when a default occurs.
2.2. Dynamics of Surplus Process
The insurer’s surplus process is described by a jump-diffusion risk model:where c is the premium rate and is a constant. represents the aggregate claim amount up to time t, where is a homogeneous Poisson process with intensity , and the individual claim sizes independent of , are i.i.d. positive random variables with the common distribution function , the first moment , and the second moment .
In addition, the insurance premium rate c under the expected value principle is given by , where is the relative safety loading of the insurer. Suppose the insurer wants to reduce his/her risk by purchasing reinsurance. If there is a claim at time t, a proportion (self-reinsurance function, see Schmidli [39]) is paid by the insurer, and the rest is paid by the reinsurer. is the premium rate of the reinsurer and is calculated according to the generalized mean-variance principle [5]. So, the premium rate of the insurer iswhere and is the relative safety loading of the reinsurer.
According to Grandell [40], the surplus process can be approximated by the following diffusion process:
While the self-reinsurance function may take various forms, Zhang et al. [5] supposed thatwhere . Next, we will only consider reinsurance strategies given by (8). In this case, since is uniquely characterized by the parameter a, we also rewrite as to emphasize the dependence on a and call a as the insurer’s reinsurance strategy. That is, . At any time t, with a larger a, the insurer reduces expenses on reinsurance and pays a larger proportion of each claim by himself/herself. Specially, when , , that is, the insurer pays all of the claims by himself/herself; when , he/she transfers all of the claims to the reinsurer according to Chen et al. [41]. Then, the surplus process of the insurer under the retention at time t is given by
Remark 1. If , then becomes a proportional reinsurance type (see also Zhou et al. [24] and Zheng et al. [26]); if , then becomes an excess-of-loss reinsurance type, the reinsurance premium under the mean principle (see also Zhao et al. [14] and Li et al. [10]). Therefore, proportional reinsurance and excess-of-loss reinsurance are special cases of (8).
2.3. The Wealth Process
In this section, we assume that the insurer is allowed to invest all his/her surplus in the financial market defined above. The insurer’s trading strategy is , where is the total amount of wealth invested in the risky asset (a stock) at time t, is the amount of wealth invested in the defaultable corporate bond, and a is the insurer’s reinsurance strategy. The remainder amount is invested in the risk-free asset. We assume that the corporate bond is not traded after default, and the investment horizon is , where . The reserve process subjected to this choice is denoted by . Thus, the wealth process can be presented as follows:
Suppose that the insurer has an exponential utility function defined by
For an admissible control and an initial value , we define the objective function as
We denote the set of all admissible strategies by . Then, we have the following definition for the set of admissible strategies.
Definition 1. A trading strategy is said to be admissible if(i) is -progressively measurable(ii)(iii), the stochastic differential equation (10) has a pathwise unique solution with , where is the chosen measure to describe the worst case and will be shown laterHowever, the AAI wants to guard himself/herself against worst-case scenarios. We assume that the knowledge of the AAI about ambiguity is described by probability , namely, the reference probability (or model). But, he/she is skeptical about this reference model and hopes to consider alternative models, which are defined as a class of probability measures equivalent to (Anderson et al. [27] and Zeng et al. [6]). At first, we define a process such that(1) is -measurable, for each (2) and a.s. (3) and , P-a.s.The alternative measures , and are positive stochastic processes. We write for the space of all such processes ϕ. Note that P is the probability measure associated with the reference model. For every , each probability measure has a Radon–Nikodym derivative:with respect to P, where the process is modelled by the stochastic differential equation (see Jin et al. [38])with , P-a.s. By the Itô differentiation rule, we getNote that and are positive stochastic processes, and satisfies the following relationship:The distribution of the stock’s jump amplitude and the jump intensity is ambiguous, so the density function is not equal under P and . Under the probability measure , the jump intensity and the density function of the stock’s price process are changed into and in the alternative model. That is, , and is a compensated Poisson random measure. For the default indicator process the intensity of the jumps becomes , and the jump size is always equal to 1, so the jump size distribution is identical under P and . For three standard Brownian motions, according to Girsanov’s theorem, Thus, the wealth process under becomes thatTo simplify further analysis, we define the following functions:Then, the dynamics of the wealth process under isNext, we assume that the insurer determines a robust portfolio strategy which is the best choice in some worst-case models as Anderson et al. [42]. The insurer penalizes any deviation from this reference model and the penalty increases with this deviation. Then, we use relative entropy to measure the deviation between the reference measure P and an alternative measure . The increase in relative entropy from t to is shown bywith . The increase is caused by three diffusion components and two jump components.
On the basis of Branger and Larsen [37], which allows the insurer’s ambiguity aversion with respect to the diffusion risk and jump risk to differ from each other, we can modify problem (12) and define the value function aswhere is calculated under the alternative measure , the initial values of the processes are given by , andThe five terms in (22) are scaled by , , , , and , which are state-dependent. We follow Maenhout [29] and setwhere , is the ambiguity aversion coefficient with respect to three diffusion risks and two jump risks. The larger the values of are, the less a given deviation from the reference model is penalized, the less the faith of the insurer in the reference model, and the more the worst-case model will deviate from the reference model.
3. The Main Result
In this section, the goal is to find the optimal allocation pair under the worst-case scenario. According to the dynamic programming principle, the HJB equation can be derived as (Anderson et al. [42])with the boundary condition , where is the infinitesimal generator of (21) under and is defined bywhere represent the value function’s partial derivatives with respect to the corresponding variables. We split equation (24) into two cases: the postdefault case () and the predefault case (), and denote V as follows:
According to (26) and (21), the HJB equation (24) transforms into the following two forms:
In the following two sections, we derive the optimal reinsurance and investment strategies and corresponding value functions in the postdefault case and predefault case, respectively.
3.1. The Postdefault Case
In this section, we will concentrate on the postdefault case, that is, HJB equation (27) for , and we conjecture that the value function has the following form:where is a deterministic function, with . We get
Substituting these partial derivatives and into equation (27), we get
Fixing π and a and maximizing over ϕ yield the following first-order condition for the minimum point (there is no ambiguity about the default jump risk after default):
Noting that (formula (16)), we have
Lemma 1. For any , the equation has a unique positive solution .
Proof. Suppose , then we getwhich implies that is a decreasing function w.r.t. . Furthermore, we have . Also, we can find that if , we have . Therefore, equation (34) has a unique positive root .
Lemma 2. For the functions , , defined above, we have the following relationship:
Substituting into (31), according to the first-order condition and Lemmas 1 and 2, we can obtain the maximum point given by
Substituting and into (31), we get the equation
Therefore, we can derive the following theorem.
Theorem 1 (postdefault strategy). The robust optimal reinsurance and investment strategies for the period after default are given as
Furthermore, the postdefault value function is given bywhere
3.2. The Predefault Case
In this section, we will concentrate on the predefault case, that is, HJB equation (28) for , and we conjecture that the value function has the following form:where is a deterministic function, with . We get
Substituting these partial derivatives, , and into equation (28), we get
Fixing π and maximizing over ϕ yield the following first-order condition for the minimum point :
Noting that , we have
Substituting into (44), according to the first-order condition, we can obtain the maximum point given by
Substituting into equation (45), we have and this equation has a unique positive root by the following lemma.
Lemma 3. Let , then has a unique positive root .
Proof. It is similar to the proof of Proposition 4.2 in the study of Sun et al. [23]. So we omit it.
According to (47) and (49), we obtain and . Substituting and into (44), we get the equationNote that ; in order to get the expression for with the boundary condition , we try the following form of :Therefore, we can derive the following theorem.
Theorem 2 (predefault strategy). The robust optimal reinsurance and investment strategies for the period before default are given as
Furthermore, the predefault value function is given bywhere , in which
Next, putting the predefault and postdefault cases together, we have the following solution to the HJB equation (24) associated with the value function :
Let us define the following processes which are candidate optimal strategies:
From the above expressions for , it is easy to verify that the expression with instead of is indeed a P-martingale, which ensures a well-defined .
In the next section, we shall show that the above stochastic control policies are indeed the optimal strategies and that the value function is unique.
4. Verification Theorem
In order to verify the candidate optimal strategies and are indeed optimal, and the candidate value function is (55), we give the verification theorem as follows.
Proposition 1. Let and denote the closure of . Suppose that there exist a function and a control such that(1), for all (2), for all (3)(4)For all : (5) and are uniformly integrable, where denotes the set of stopping times Then, and are an optimal control.
Proof. According to , choose , by the definition of , , and ; therefore, the Itô formula can be applied to :where , for , and and for short. Because the continuous function is bounded on the set , we obtain the following estimate with an appropriate constant :Due to (ii) of Definition 1, it follows thatBy the similar way as that in (59), we haveTherefore, taking expectations in (57) leads toIf we apply (61) to with and use property 1, we getLetting and using properties 4 and 5, we haveSince this holds for all , we deduce thatHence,Next, if we apply (61) to with and use property 2, we getLetting and using properties 4 and 5, we haveSince this holds for all , we deduce thatAccording to (65) and (68), we get . Finally, we apply (61) to as the above process. Then,For , we can obtain the following result by the similar method:Thus, is an optimal control strategy about (69)-(70), and .
Lemma 4. The following integral is finite:
Proof. Putting and into (71), with an appropriate constant , we havewhere the inequality is established because and are the deterministic and bound functions on .
Lemma 5. For the optimal control problem (21) with the exponential utility function (11), if is the solution of (24) with the boundary condition , then we have
Proof. Substituting into (19), we have the wealth process under :For the candidate value function (21), we getAt first, we prove . , and are deterministic continuous functions and are bounded on . Substituting (75) into (48), there are two constants and satisfying such thatwherewhere are deterministic continuous functions, which are bounded on , and we getApplying Lemma 5 in Zeng and Taksar [43], we know that , , and are martingales; then,Applying (78)–(81) with an appropriate constant satisfying , we haveConsequently, . Similarity, we can also proveThen, the formula holds. The first part of Lemma 4 is proved.
Letwhere is obviously bounded, then we getThe first inequality follows from Cauchy–Schwarz inequality. The last inequality follows from (71). Lemma 5 is proved.
Based on the discussion above, the main theorem is summarized as follows.
Theorem 3. For the robust optimal control problem (21) with the exponential utility function (11), is the solution of (24) with the boundary condition , is an optimal strategy, and then is the corresponding value function.
Proof. From Lemmas 1, 2, and 3 and Theorems 1 and 2, we can obtain properties 1–4 in Proposition 1. By Lemma 5, condition 5 in Proposition 1 also holds for . From Proposition 1, we can obtain the result of Theorem 3, is an optimal strategy, and is the corresponding value function.
5. Some Special Cases
In this section, we shall present some special cases of our results, such as the proportional reinsurance, the excess-of-loss reinsurance, and an ambiguity-neutral insurer of the insurance.
5.1. The Proportional Reinsurance
The parameter η satisfies , and the insurer purchases reinsurance in the form of the proportional reinsurance, that is, . The result is shown by corollary as follows.
Corollary 5.1. If , then the value function , and the optimal strategies and are given by
The optimal value function is as follows:whereand , in which
Proof. In this case, , and then the functions in our results are as follows:According to Theorems 3.3 and 3.5, we can obtain the result.
Remark 2. If , and , then becomes a proportional reinsurance type. There are similar studies, for example, Zhou et al. [24], Zheng et al. [26], and Wang et al. [44].
5.2. The Excess-of-Loss Reinsurance
The parameter satisfies , and the insurer purchases reinsurance in the form of the excess-of-loss reinsurance, that is, . The result is shown by corollary as follows.
Corollary 5.3. If , then the optimal value function , and the optimal strategies and are given by
The optimal value function is as follows:whereand , in which
Proof. In this case, , and then the functions in our results are as follows:According to Theorems 1 and 2, we can obtain the result.
Remark 3. If , then becomes an excess-of-loss reinsurance type. Without regard for the default bond, let , then the reinsurance strategy becomes the result of Li et al. [10].
5.3. Ambiguity-Neutral Insurer (ANI) Case
If the insurer is an ambiguity-neutral insurer, then the aversion ambiguity coefficient . In this case, for an admissible control and an initial value , the objective function is described by
The corresponding HJB equation is
From the value functionswe can get the following candidate optimal strategies by the same way:whereand , in which
Remark 4. If all of the ambiguity aversion coefficients equal 0, i.e., , our model reduces to an optimization problem for an ambiguity-neutral insurer (ANI). For the ANI, the optimization investment-reinsurance is researched by Cao [11], Yang and Zhang [2], etc.
6. Sensitivity Analysis
In this section, we will give several numerical examples to illustrate the influences of the parameters on the optimal strategies and the optimal value functions. Unless otherwise stated, the basic parameters are given in Table 1.
Some analyses of the optimal reinsurance strategy are shown in Figures 1–4. is the ambiguity aversion coefficient of the AAI. From Figure 1, it is found that affects the reinsurance strategy of the insurer. As increases, the insurer has lower risk exposure in the insurance market, so less amount of money will be paid to purchase reinsurance. η and are the relative safety loadings of the reinsurer. Figures 2 and 3 show the effects of η or on the insurer’s reinsurance strategy. As η or increases, the reinsurer pays more concern on his/her risk exposures and charges more for them. Consequently, the insurer decreases his/her demand for reinsurance and pays more claims by himself/herself. In Figure 4, we show the common effects of the safety loading and the ambiguity aversion coefficient on .
In Figures 5–8, we illustrate the impacts of the ambiguity aversion coefficients and on the stock strategy . From Figures 5 and 6, we find that the AAI will reduce the wealth invested on the stock, when there is a higher ambiguity aversion coefficient. From Figures 7 and 8, we find that as the time t increases, the AAI increases the stock investment amount. These figures show that the robust optimal strategies can effectively reduce the sensitivity of on the stock.
For the defaultable corporate bond, the value is assumed to be zero after default. In Figures 9–13, we show the numerical analysis of the defaultable corporate bond strategy before default. and are the ambiguity aversion coefficients. As increases, the insurer will reduce the money on the defaultable corporate bond, but does not affect the investment, as shown in Figure 9. In Figure 10, the insurer will invest more amount of his/her money, if the defaultable corporate bond with a higher premium induces a higher potential yield. Contrary to the default risk premium , the accession of ζ reduces the insurer’s investment on the defaultable bond, as shown in Figure 11. A higher loss rate ζ leads to a less recovery value, which implies a higher potential loss of the insurer. When , the reinsurance type is a proportional reinsurance, and when , the reinsurance type becomes an excess-of-loss reinsurance. We show that the proportion reinsurance is always below the excess-of-loss reinsurance at the same time in Figure 12. Figure 13 provides a full description of with respect to and ζ with two different reinsurance types, respectively.
(a)
(b)
In Figure 14, we illustrate the predefault value function and the postdefault value function with respect to the initial wealth about a proportional reinsurance and an excess-of-loss reinsurance, respectively. We can see that the value functions of the insurer increase as the initial wealth increases and the predefault value function is always greater than or equal to the postdefault value function.
(a)
(b)
7. Conclusion
In this paper, we consider a robust optimal reinsurance-investment problem of an insurer under the generalized mean-variance premium principle and a defaultable market. The insurer can trade in a risk-free asset, a stock, and a defaultable corporate bond. The surplus of the AAI is described by an approximate diffusion process. The stock’s price process is described by a jump-diffusion model. Using the dynamic programming approach, we study the predefault case and the postdefault case, respectively, and derive the optimal strategies and the corresponding value functions under the worst-case scenario. We give some sensitivity analysis to illustrate our theoretical results. In future research, we will consider some complex models, such as the robust optimal reinsurance-investment problem of stochastic differential games.
Data Availability
All data generated or analyzed during this study are included in this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (grant nos. 71701068, 11701175, and 11671132), the Natural Science Foundation of Hunan Province, China (no. 2018JJ3360), and the Scientific Research Fund of Hunan Provincial Education Department, China (nos. 19B343, 17C1001, and 17K057).