OPTIMAL REINSURANCE AND INVESTMENT STRATEGY WITH TWO PIECE UTILITY FUNCTION

. This paper studies optimal reinsurance and investment strategies that maximize expected utility of the terminal wealth for an insurer in a sto- chastic market. The insurer’s preference is represented by a two-piece utility function which can be regarded as a generalization of traditional concave utility functions. We employ martingale approach and convex optimization method to transform the dynamic maximization problem into an equivalent static op- timization problem. By solving the optimization problem, we derive explicit expressions of the optimal reinsurance and investment strategy and the optimal wealth process.


1.
Introduction. Study on continuous-time optimal control problem related to insurance risk management has predominantly centered around expected utility maximization (EUM) for decades. Traditional assumption of EUM is that decision makers are rational and risk averse when facing uncertainty. However, this assumption has been challenged by many researchers, for example, the Allais paradox (see Allais, 1953), the equity premium puzzle (see Mehra and Prescott, 1985) and so on.
A number of alternative preference measures have been proposed to overcome the drawbacks of EUM, such as Lopes' SP/A model, cumulative prospect theory (CPT, see Tversky 1979, 1992) and disappointment theory (DT, see Bell 1985, Loomes andSugden 1986). CPT has three significant features: existence of preference point, S-shaped utility function and probability distortion. More and more researchers incorporate these new preference measures into optimization problems and pricing principles. The early publications are limited to the single period setting, see, for example, Benarti and Thaler (1995), Lopes and Oden (1999), Shefrin and Statman (2000) and Bernard and Ghossoub (2010). Berkelaar et al. (2004) considers the dynamic portfolio selection problem under a two-piece power utility function with loss aversion, where investor takes up a risk-seeking attitude towards loss, and derive optimal investment strategy by employing a convex optimization technique. Jin and Zhou (2008) considers a continuous time portfolio selection model under CPT. They separate the optimization problem into gain part and loss part, use a Choquet integral formulation to deal with the probability distortion and develop a Choquet maximization and minimization technique to solve the problem. Mi and Zhang (2012) investigates an optimal portfolio selection problem assuming a two-piece utility function in an incomplete market.
The optimal reinsurance and investment strategy is an important research topic in insurance and actuarial science. Under EUM setting, the optimal strategy has attracted considerable interest recently. Browne (1995) uses a Brownian motion with drift to model the risk process of the insurer and obtains the corresponding optimal investment strategy. Yang and Zhang (2005) considers the portfolio selection problem under a jump diffusion risk process model. In order to deal with changes in the market environment, Zhang and Siu (2012) investigates the optimal proportional reinsurance and investment strategy under a Markovian regime switching economy. Other related papers include Plum (2000, 2003), Schmidli (2001), Liu and Yang (2004), Xu and Yao (2008), Yao et al. (2010). There is also burgeoning research interest in actuarial science under alternative preference measures mentioned above. Tasnakas and Desli (2003) derives a premium principle called generalized expected utility premium principle based on the rank dependent expected utility (RDEU). Sung et al. (2012) studies the optimal insurance policy using a behavioral principle. Chueng et al. (2015aChueng et al. ( , 2015b investigate the premium principle and optimal insurance using disappointment theory. He et al. (2015) studies the optimal insurance design using RDEU.
To the best of our knowledge, there has been limited publications incorporating non-concave preferences into the optimal strategy selection for an insurer. Guo (2014) considers the optimal investment problem for an insurer with loss aversion. In this paper, we derive the optimal strategy with a general utility function, a special case of our model can be reduced to that in Guo (2014), and we incorporate the proportional reinsurance for an insurer in our model. The decision maker's preference is represented by a two-piece utility function with a reference point. When the utility function is convex, we set a lower bound for the wealth. If the utility takes a concave preference, the lower bound can be ignored. Comparing to results in the relative literatures, the result in this paper is more general. We do not assume a specified function form for the utility function in this paper, the specified utility functions are given as examples for illustration. For some utility functions, for example, a two-piece power utility or exponential utility, we are able to obtain corresponding analogue results as those in Browne (1995) and Guo (2014). We give a uniform expression for concave utility function where the result is always coincident. But in some special two-piece concave utility case, the result is different and relies on the corresponding parameters of the positive part and negative part of utility function. In concave-convex utility function case, the result is seriously affected by the lower bound. In this paper, we apply traditional martingale technique, which is widely used in mathematical finance, to work out the closed form of the optimal strategy and the optimal wealth process.
The rest of this paper is arranged as follows. Section 2 describes the economy for the insurance company. The maximization problem of investment and reinsurance is presented in Section 3. The explicit expression of the optimal strategy and the optimal wealth process are obtained in Section 4. In Section 5, we present some examples for illustration purpose.
2. The market. In this section, We define a continuous-time financial market on a finite-time horizon T := [0, T ], where T < ∞ is the terminal time of decision making process. The uncertainty of the economy is represented by a filtered probability space (Ω, F, F, P), where F := {F(t)|t ∈ T } is the collection of information until time t and P is a real-world probability measure. All the processes defined below are presumed to be adapted to F. We denote by E[·] the expectation under P.
The financial market consists of a risk-free asset and a risky asset which can be traded continuously on the time horizon T . The price process of the risky-free asset B := {B(t)} t≥0 evolves according to where r(t) denotes the risk-free interest rate for borrowing and is assumed to be deterministic and uniformly bounded. The price process of risky asset S := {S(t)|t ∈ T } is governed by a geometric Brownian motion where b(t) and σ(t) denote the appreciation rate and the volatility of the asset at time t respectively and satisfy b(t) > r(t), σ(t) > 0; b(t) and σ(t) are assumed to be deterministic and uniformly bounded; W := {W (t)|t ∈ T } is a standard one-dimensional Brownian motion on (Ω, F, F, P).
The surplus process of an insurer, U := {U (t)|t ∈ T }, is assumed to be the classical Cramér-Lunderberg model, namely where c(t) > 0 is the premium rate at time t; L := {L(t) = is an i.i.d. sequence of non-negative random variables, N := {N (t)|t ∈ T } is a poisson process with intensity λ(t) > 0 and represents the number of claims up to time t. Here N is assumed to be independent of Y i and E(Y i ) = µ 1 < ∞, E(Y 2 i ) = µ 2 < ∞. According to the expected premium principle, we set where t ∈ [0, T ] and η > 0 represents the safety loading. Thus the surplus process {U (t)|t ∈ T } is governed by the following equation It is well-known that the surplus process above can be approximated by following Brownian motion with drifted (see Grandll (1991)) where W 0 := {W 0 (t)|t ∈ T } is a standard one-dimensional Brownian motion on (Ω, F, F, P). Moreover, we assume that W 0 and W are stochastically independent. Thus the filtration {F(t)|t ∈ T } can be regarded as augmentation of the filtration {F W (t),W0(t) t |t ∈ T } that generated by (W, W 0 ). The insurer is allowed to invest its surplus in the financial market and purchase reinsurance to control its risk. In this paper, we restrict our attention to the proportional reinsurance and let p(t) denote the reinsurance proportion, that is 1 − p(t) portion of the insurance risk is divided to the reinsurer business. Here p(t) is restricted to be non-negative and p(t) > 1 means taking new reinsurance business from insurance market.
The insurer's objective is to choose an F−adapted process π(t), representing the amount invested in the risky stock, and an F−adapted process p(t), the proportional reinsurance process, so as to maximize the expected utility of terminal wealth at time T . The reinsurance and investment strategy is a two-dimensional stochastic The set of all admissible strategies is denoted by Π.
The wealth process X(t), associated with an admissible strategy u, takes the following form where x 0 denotes the initial wealth; θ represents the safety loading of the reinsurer and, in general, θ ≥ η, otherwise there are arbitrage opportunities. For simplicity, we restrict our analysis to the cheap reinsurance case: θ = η, that is, the reinsurance company uses the same safety loading as the cedent. Thereafter, X(t) is of the following form where λ(t), b(t), r(t), σ(t) are assumed to be deterministic and uniformly bounded on [0, T ].
3. The insurer's maximization problem. In this section, we introduce the insurer's utility function and propose the maximization problem. It is well known that one of the conventional assumptions in the theory of optimal reinsurance-investment strategy is that the utility function is a smooth, concave and increasing function over terminal wealth X(T ). In this paper, we consider a two-piece utility function, that is, the insurer is assumed to be an investor with following preference where U 1 (·) : + → , is strictly increasing, concave and twice differentiable with U 1 (+∞) = 0; U 2 (·) : − → , is strictly increasing, twice differentiable and U 1 (0) = U 2 (0). We consider two cases in this paper: the first case is that U 2 (·) is a concave function, the second case is that U 2 (·) is a convex function. The utility in (1.2) can be viewed as a generalization of the common concave utility and the utility with loss aversion. When U 1 (·) = U 2 (·) are both concave functions, the two-piece utility U (·) reduces to the common concave utility, that means, the insurer shows an invariable risk-averse attitude towards gain and loss.
In the case of convex preference for loss part and U 1 (0+) = U 2 (0−) = ∞, originating from the classical CPT theory in Tversky and Kehneman (1992), the utility U (·) is an S−shaped function, that is, the insurer is an investor with loss aversion. Moreover, if U 2 (·) is always steeper than U 1 (·), that reflects that the investor is more sensitive to losses than gains. Due to the convexity of U 2 (·), the agent is risk-averse in the gain domain and risk-seeking in the loss domain. We will consider this special case in Section 5.
Given the terminal wealth X(T ) ∈ F(T ), the objective function for the insurer is given by where k represents the reference point which divides the utility into two parts; I is an indicator function.
Following the expected utility maximization criterion, the problem of choosing an optimal reinsurance-investment strategy for an insurer is formulated as where L denotes the lower bound of X(t). Usually the value of L is equal to zero, indicating that the insurance company is not bankrupt throughout the investment period [0, T ].
4. The optimal strategy choice. In this section we derive the optimal terminal wealth and optimal proportional reinsurance and investment strategy. The steps to achieve the goal can be listed as follows: we reduce the original problem (1.3) to a static optimization problem which is subject to a linear constraint and then apply martingale technique to solve the static problem and derive the optimal terminal wealth. Finally, the optimal proportional reinsurance and investment strategy u * can be obtained. Define (1.4) We have two similar propositions as those in Guo (2014).
Applying Itô formula, we obtain where H(t), X(t) is the quadratic covariation process of H(t) and X(t). This shows that H(t)X(t) can be represented as an Itô integral with respect to twodimensional Brownian motion W 0 (t), W (t) . Therefore H(t)X(t) is a martingale under P.
Proposition 2. Let the initial wealth x 0 be given, then for any F(T ) random variable ξ with a lower bound L satisfying there exists an admissible strategy u such that Proof. Let us define a martingale From martingale representative theorem (Karatzas and Shreve 1991), there exist two progressive measurable processes ϕ : such that Compare dW 0 (t)−term and dW (t)−term with those in (1.5), let t = T , we have which imply that The admissibility of u can be obtained from the corresponding results in Guo (2014).
According to the above propositions, any F(t) random variable ξ, satisfying E H(T )ξ = x 0 , can be financed via trading an admissible strategy u. Thus the dynamic maximization problem (1.3) can be transformed into following static opti- (1.6) After a simple change, the above problem (1.6) turns to be where ξ := ξ − k, x 0 := x 0 − kE H(T ) and L := L − k. We only consider the case k = 0 in the following. We apply a divide formulation to split problem (1.6) into two parts Positive Part Problem. A problem with parameters (A, x + ): where x + ≥ x 0 , x + ≥ 0 and A ∈ F(T ) is given.
If the lower bound L > 0, problem (1.7) turns to be a single positive problem (1.8) with additional constraint on the value of L. If the lower bound L ≤ 0, we need to consider both problems (1.8) and (1.9), and make comparisons to obtain the optimal wealth. We solve problem (1.8) and (1.9) respectively in the following. Proof. We first solve the point-wise maximization problem where L ∨ 0 := max{L, 0}. Due to the concavity of utility function U 1 (·) and the restriction that ξ + ≥ L ∨ 0 , the maximizer ξ * + to problem (1.10) is given by where U 1 (·) denotes the derivative of U 1 (·); (U 1 ) −1 (·) denotes the inverse of U 1 (·) and (U 1 ) + (L ∨ 0) denotes the right-hand derivative of U 1 (·) at point L ∨ 0. We then show that ξ * + is the candidate optimal wealth for problem (1.8). Suppose ξ + represents any possible optimal solution satisfying the static budget equation in (1.8), then we have which manifests the optimality of ξ * + . Finally we verify that, for any x + ≥ x 0 and x + > 0, there exists a unique y > 0 satisfying the budget constraint in (1.8). Since U 1 (·) is strictly increasing, concave, twice differentiable, and defined on + , its derivative has a strictly decreasing, continuous inverse (U 1 ) −1 (·) : + −→ + . Hence, for any y 1 > y 2 > 0, and due to the non-negative of H(T ), we have By the dominated convergence theorem and monotone convergence theorem, it is easy to see that E ϕ(y) is continuous with limit Thus there exists a unique y such that E ϕ(y) = x + , and we denote it by y * . In conclusion we have  (1) If U 2 (·) is a strictly convex utility function, L ∈ (−∞, 0) and then the optimal wealth of the insurer in negative part problem (1.9) is given by is a strictly concave utility function, L ∈ (−∞, 0) and then the optimal wealth of the insurer in negative part problem (1.9) is given by where y * is the unique solution of (1.11) and define

Proof. (1) Consider the point-wise maximization problem
It is easy to see that φ (x) is increasing with respect to x, thus the optimal maximizer ξ * − for problem (1.11) is located at one of the boundaries ξ * − = 0, ξ * − = L. Comparing the value of φ(x) at ξ * − = 0 and ξ * − = L, we obtain when yH(T )L ≤ U 2 (L) − U 2 (0), the optimal value is ξ * − = L and when yH(T )L > U 2 (L) − U 2 (0), the optimal value is ξ * − = 0. Therefore, the optimal terminal wealth is given by The optimality of ξ * − and uniqueness of y * can be obtained by a similar proof as that in Proposition 3.
(2) Similar to Proposition 3, the result can be obtained.
It is worthy noting that when U 2 (·) is a strictly convex function, the existence of lower bound L is essential otherwise the problem will be ill-posed, because the prospect value will be unbounded. The condition is to ensure the existence of optimal strategy, otherwise there is no strategy satisfying the static budget equation in (1.9).
The following theorem presents the optimal solution to (1.6).
The optimal terminal wealth is given by where y * , H T (y * ) is the unique solution of E H(T )X * (T ) = x 0 and (1.12) where y * , H T (y * ) ∈ and on {yx ≤ (U 1 ) + (0)} we obtain that meanwhile, ψ(x) := U 1 (0) + yxL is decreasing with respect to x. Therefore, for a fixed y, g H(T ), y is decreasing with respect to H(T ). Similarly, for a fixed H(T ), g H(T ), y is also decreasing with respect to y. Thus, for a fixed y, there exists a H(T ), denoted by H T (y), such that when H(T ) ≤ H T (y), and when H(T ) > H T (y), Thus, the candidate optimal terminal wealth is given by where y * is the unique solution of Similar to Proposition 3, the optimality of X * (T ) and uniqueness of y * can be obtained .
(2) Compare and observe that h H(T ), y : and observe that Thus and h H(T ), y is decreasing with respect to H(T ). Similarly, for a fixed H(T ), h H(T ), y is also decreasing with respect to y.
1 . Assume (U 2 ) − (0) ≥ (U 1 ) + (0) and let H T (y) be any value on Thus h H T (y), y = 0, and the candidate optimal terminal wealth is given by Consider a special case U 1 (·) = U 2 (·) = U (·), we obtain the candidate optimal h H T (y), y = 0. The candidate optimal terminal wealth is given by where y * is the unique solution of E H(T )X * (T ) = x 0 . Similar to Proposition 3, the optimality of X * (T ) and uniqueness of y * can be obtained.
We derive the optimal proportional reinsurance and investment strategy for an insurer under two-piece utility (1.2).
It is easy to see that, for all cases in Theorem 4.1, C(t, H(t)) is always nonincreasing with respect to H(t). Thus we have that for an insurer whose wealth process evolves according to (1.1), with any two-piece utility function defined by (1.2) and under the expected value premium principle, the optimal proportional reinsurance strategy p(t) is non-negative. 5. Some examples. In this section, we consider some common utility functions.
Example 5.1. A particular case of utility (1.2) is the two-piece power utility function, see Tversky and Kehneman (1992), which is represented by: where 0 < γ 1 , γ 2 ≤ 1 are curvature parameters, α, β > 0 and β stands for the loss aversion coefficients of the investor. This utility is widely used models that consider the loss aversion, for example, in Berkelaar (2004) and Guo (2014). Let L be the lower bound of X(T ), satisfying L < k and E H(T )L ≤ x 0 , where x 0 represents the initial wealth. The utility function of the insurer is given bỹ The optimal terminal wealth is given bỹ where y * , H T (y * ) is the unique solution of The optimal wealth process is given by The optimal reinsurance-investment strategy is given by The optimal wealth process is given by The optimal wealth process is given by