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This paper studies an optimal reinsurance and investment problem for a loss-averse insurer. The insurer ’ s goal is to choose the optimal strategy to maximize the expected S-shaped utility from the terminal wealth. The surplus process of the insurer is assumed to follow a classical Cramér-Lundberg (C-L) model and the insurer is allowed to purchase excess-of-loss reinsurance. Moreover, the insurer can invest in a risk-free asset and a risky asset. The dynamic problem is transformed into an equivalent static optimization problem via martingale approach and then we derive the optimal strategy in closed- form. Finally, we present some numerical simulation to illustrate the effects of market parameters on the optimal terminal wealth and the optimal strategy, and explain some economic phenomena from these results.

Recently, optimal reinsurance and investment problems for insurers have attracted increasing attention from academics and industries. By purchasing reinsurance and investing in the financial market, insurance companies reduce their exposure risk and gain profits from investment. There are many literatures in this field. Browne [

Generally, the above mentioned researches assume that investors are rational and risk averse. However, rationality hypothesis cannot be suit due to the investors’ psychological effects. Based on the experiment and relative results, Kahneman and Tversky [

After Kahneman and Tversky [

To the best of our knowledge, there is few work incorporating loss aversion into the optimal reinsurance and investment problem. This paper adopts the S-shaped utility function to describe the insurer preference, and the insurer is allowed to invest in a risk-free asset and a risky asset. Moreover, the insurer can purchase excess-of-loss reinsurance, which is more practical in reality. Typically, three types of risk models are commonly considered in reinsurance and investment problems, the Cramér-Lundberg model (see Zeng et al. [

This paper is related to Guo [

The main contribution of this paper is as follows: 1) the optimal reinsurance and investment strategy with loss aversion is studied and the closed-form expression of the optimal strategy is derived; 2) we define a quasi-pricing kernel and construct a martingale process to solve the problem. We find that the optimal terminal wealth is piecewise function. In good states of market, the optimal wealths is of the same form with the smooth CRRA utility function case, on the contrary the optimal wealth approaches 0 in bad states of market. Similarly, the optimal investment and reinsurance strategy are also divided into two cases respectively. When the market deteriorates, the insurer will stop investing in the risky asset and purchasing reinvestment strategy.

The rest of this paper is organized as follows: The financial market and insurance model are described in Section 2. In Section 3, we establish the optimal reinsurance-investment problem, and the optimal strategy is derived by using Lagrangian duality and martingale method. Section 4 presents numerical illustrations to demonstrate our results. Section 5 concludes the paper and provides further discussion.

We impose the following standard assumptions: the insurer can trade in the financial market and in the insurance market continuously over time, no transaction costs or taxes are involved in trading. Let ( Ω , F , { F t ,0 ≤ t ≤ T } , P ) be a filtered, complete probability space satisfying the usual conditions, in which T > 0 is a finite time horizon. All stochastic processes introduced below are assumed to be adapted processes in this space.

Assume that an insurer’s basic surplus process is described by the classical Cramér-Lundberg (C-L) model: without reinsurance and investment, the insurer’s surplus U is given by:

d U ( t ) = c d t − d ∑ i − 1 N ( t ) Z i , U ( 0 ) > 0 , (1)

where U ( 0 ) is the initial surplus, c > 0 is the premium rate, { N ( t ) ,0 ≤ t ≤ T } is a homogeneous Poisson process with intensity λ > 0 , Z i represents the size of the ith claim and the claim sizes Z 1 , Z 2 , ⋯ are assumed to be independent and identically distributed (i.i.d.), non-negative random variables with common distribution F having finite first-order moment μ ∞ , second-order moment

σ ∞ 2 , and F ( z ) = P r { Z ≤ z } . Consequently, ∑ i = 1 N ( t ) Z i is a compound Poisson

process representing the cumulative amount of claims in time interval [ 0, t ] .

According to Gu et al. [

measure N ( d z , d t ) defined on Ω × R + × [ 0, T ] , therefore ∑ i = 1 N ( t ) Z i of discrete

time can be converted into continuous time. N ( d z , d t ) represents the number of insurance claims of size ( z , z + d z ) within the time period ( t , t + d t ) , and Z ( t ) stands for the claim at time t ∈ [ 0, T ] . We assume that the premium rate c is calculated according to the expected value principle, i.e., c = ( 1 + η ) λ μ ∞ , and we denote the υ ( d z ) = λ d F ( z ) , i.e.,

c = ( 1 + η ) ∫ 0 ∞ z υ ( d z ) ,

where η > 0 is the insurer’s relative safety loading, υ is a lévy measure such that ∫ 0 ∞ z υ ( d z ) < ∞ , υ ( d z ) represents the expected number of insurance claims of size ( z , z + d z ) within a unit time interval, and denotes the compensated measure of N ˜ ( d z , d t ) by N ˜ ( d z , d t ) = N ( d z , d t ) − υ ( d z ) d t . Putting it all together, the insurer’s surplus U without reinsurance is governed by

d U ( t ) = c d t − d ∑ i N ( t ) Z i = ( 1 + η ) ∫ 0 ∞ z υ ( d z ) d t − ∫ 0 ∞ z N ( d z , d t ) .

In this paper, the insurer can purchase a reinsurance strategy with retained claim { l ( t ) ,0 ≤ t ≤ T } , with the only restriction 0 ≤ l ( t ) ≤ Z ( t ) when the claim equals Z ( t ) at time t ∈ [ 0, T ] . Note that the reinsurer covers the excess loss Z ( t ) − l ( t ) . We will look for a reinsurance strategy given in feedback form by l ( t ) = l ( Z ( t ) , t ) , in which we slightly abuse notation by using l on both sides of this equation.

Here, we assume that reinsurance is not inexpensive, i.e., the safety loading of the reinsurer θ is greater than the safety loading of the insurer η . Assuming we use the expected value principle again for the reinsurer, the reinsurance premium rate calculates as

( 1 + θ ) ∫ 0 ∞ ( z − l ( t ) ) υ ( d z ) .

Under the retention l ( t ) , the dynamics of the surplus process is governed by

d U ( t ) = c d t − ( 1 + θ ) ∫ 0 ∞ ( z − l ( t ) ) υ ( d z ) d t − ∫ 0 ∞ l ( t ) N ( d z , d t ) = ∫ 0 ∞ [ ( η − θ ) z + θ l ( t ) ] υ ( d z ) d t − ∫ 0 ∞ l ( t ) N ˜ ( d z , d t ) .

Remark 2.1. According to the expected value principle, c = ( 1 + η ) λ μ ∞ = ( 1 + η ) ∫ 0 ∞ z υ ( d z ) , therefore ∫ 0 ∞ z υ ( d z ) = λ μ ∞ .

Assume that the financial market consists of one risk-free asset and one risky asset. The price process of the risk-free asset price solves

d S 0 ( t ) = r S 0 ( t ) d t ,

in which we assume the risk-free interest rate r > 0 is constant, and the price process of the risky asset is described by the geometric Brownian motion with dynamics

d S ( t ) S ( t ) = μ s d t + σ s d W ( t ) ,

in which μ s > r , σ s > 0 , and { W ( t ) , t ≥ 0 } is a standard Brownian motion, independent of the N ( d z , d t ) .

Therefore, the market price of risk process can be defined by

λ s = μ s − r σ s ,

and λ s is bounded.

During the time horizon [ 0, T ] , the insurer is allowed to dynamically purchase excess-of-loss reinsurance and invest in the financial market. Let π ( t ) be the total amount of money invested in the risky asset at time t. An rein-surance-investment strategy is described by ϕ ( t ) = ( π ( t ) , l ( t ) ) , and the amount invested in the risk-free asset S 0 ( t ) is X ( t ) − π ( t ) , where X ( t ) is the wealth process associated with the strategy. Then X ( t ) is a solution to the following stochastic differential equation (SDE):

d X ( t ) = ( X ( t ) − π ( t ) ) d S 0 ( t ) S 0 ( t ) + π ( t ) d S ( t ) S ( t ) + d U ( t ) = [ r X ( t ) + ( μ s − r ) π ( t ) + ∫ 0 ∞ ( ( η − θ ) z + θ l ( t ) ) υ ( d z ) ] d t + π ( t ) σ s d W ( t ) − ∫ 0 ∞ l ( t ) N ˜ ( d z , d t ) , X ( 0 ) = x 0 . (2)

Kahneman and Tversky [

Based on the experiments and relative results, Kahneman and Tversky proposed a utility function, which is defined over gains and losses relative to the reference point ξ as follows:

U ( x ) = ( α ( x − ξ ) γ 1 , x > ξ , − β ( ξ − x ) γ 2 , x ≤ ξ , (3)

α > 0 and β > 0 are required to ensure that U ( x ) is an increasing function, β > α holds for loss aversion, γ 1 and γ 2 are the curvature parameters for gains and losses, and 0 < γ 1 < 1 , 0 < γ 2 < 1 for the convex-concave shape (

Definition 3.1. A strategy ϕ ( t ) = ( π ( t ) , l ( t ) ) , t ∈ [ 0 , T ] is called admissible if it satisfies the following conditions:

1) ϕ ( t ) is { F t ,0 ≤ t ≤ T } -progressively measurable;

2) Z ( t ) ≥ 0 , 0 ≤ l ( t ) ≤ Z ( t ) , and t ∈ [ 0 , T ] ;

3) For all ( X ( t ) , t ) ∈ ℝ × [ 0, T ] , the stochastic differential Equation (2) has a unique solution.

Note that ϕ ( t ) is the admissible strategy and Φ is the admissible space. Following utility maximization criterion, the problem of choosing an optimal portfolio can be formulated as follows:

max ϕ ( ⋅ ) E [ U ( X ( T ) ) ]

s . t . ( ( X ( t ) , ϕ ( t ) ) satisfies ( 2 ) , ϕ ( t ) ∈ Φ . (4)

In order to facilitate the solution of this problem, markets are assumed to be complete, which implies the existence unique state pricing kernel. Since the S-shaped utility is convex-concave, the stochastic optimal control approach can not be feasible. In this case, martingale approach proposed by Cox and Huang [

H ( t ) = exp { − r t − 1 2 λ s 2 t − λ s W ( t ) + ∫ 0 t ∫ 0 ∞ ln ( 1 + θ ) N ( d z , d s ) − ∫ 0 t ∫ 0 ∞ θ υ ( d z ) d s } , (5)

and construct a martingale process, see Proposition 3.1.

Proposition 3.1. If H ( t ) is defined by (5) for [ 0, T ] , H ( t ) X ( t ) − ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 t H ( s ) d s is a martingale.

Proof. Consider a lévy-type stochastic integral of the form

Y i ( t ) = Y i ( 0 ) + Y c i ( t ) + ∫ 0 t ∫ A J i ( s , x ) N ( d s , d x ) ,

where

Y c i ( t ) = ∫ 0 t G i ( s ) d s + ∫ 0 t F j i ( s ) d B j ( s ) ,

Itô formula for lévy-type stochastic integrals can be written as

f ( Y ( t ) ) − f ( Y ( 0 ) ) = ∫ 0 t ∂ i f ( Y ( s − ) ) d Y c i ( s ) + 1 2 ∫ 0 t ∂ i ∂ j f ( Y ( s − ) ) d [ Y c i , Y c j ] ( s ) + ∫ 0 t ∫ A [ f ( Y ( s − ) + J ( s , x ) ) − f ( Y ( s − ) ) ] N ( d s , d x ) , (6)

for each f ∈ C ( ℝ d ) , t > 0 , 0 ≤ i ≤ d .

For more information about Lévy processes, please see the Lévy Process and Stochastic Calculus [

Using the Itô formula for lévy-type stochastic integrals, we find that

d H ( t ) = H ( t ) ( − r d t − λ s d W ( t ) + ∫ 0 ∞ θ N ˜ ( d z , d t ) ) , (7)

and

d ( H ( t ) X ( t ) ) = H ( t ) d X ( t ) + X ( t ) d H ( t ) + d [ H , X ] ( t ) = H ( t ) [ r X ( t ) + ( μ s − r ) π ( t ) + ∫ 0 ∞ ( ( η − θ ) z + θ l ( t ) ) υ ( d z ) d t + π ( t ) σ s d W ( t ) − ∫ 0 ∞ l ( t ) N ˜ ( d z , d t ) ] + X ( t ) H ( t ) ( − r d t − λ s d W ( t ) + ∫ 0 ∞ θ N ˜ ( d z , d t ) ) + H ( t ) ( − λ s π ( t ) σ s d t − ∫ 0 ∞ θ l ( t ) N ( d z , d t ) )

= H ( t ) ( η − θ ) [ ∫ 0 ∞ z υ ( d z ) ] d t + H ( t ) ( π ( t ) σ s − λ s X ( t ) ) d W ( t ) + H ( t ) [ X ( t ) ∫ 0 ∞ θ − ∫ 0 ∞ ( θ + 1 ) l ( t ) ] N ˜ ( d z , d t ) , (8)

Therefore, H ( t ) X ( t ) − ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 t H ( s ) d s is a martingale. □

Now, the dynamic maximization problem (4) can be converted into the following equivalent static optimization problem with constraint:

max X ( T ) E [ U ( X ( T ) ) ]

s . t . ( E [ H ( T ) X ( T ) − ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 t H ( s ) d s ] ≤ x 0 , X ( T ) ≥ 0. (9)

Theorem 3.1. The optimal terminal wealth for the loss-averse member in the dynamic problem (4) is

X ∗ , y ∗ ( T ) = ( ξ + ( α γ 1 y * H ( T ) ) 1 1 − γ 1 , H ( T ) < H ¯ , 0 , H ( T ) ≥ H ¯ , (10)

where H ¯ satisfies f ( H ¯ ) = 0 with

f ( x ) = 1 − γ 1 γ 1 y * x ( α γ 1 y * x ) 1 1 − γ 1 − y * x ξ + β ξ γ 2 ,

y * > 0 is a Lagrange multiplier, satisfies

E [ H ( T ) X ∗ , y ∗ ( T ) − ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 T H ( s ) d s ] = x 0 .

Proof. First we define the Lagrangian function of problem (9) as follows:

L ( X ( T ) , y ) = E [ U ( X ( T ) ) ] − y E [ H ( T ) X ( T ) ] + y E [ ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 t H ( s ) d s ] + y x 0 , (11)

where y is the Lagrangian multiplier. According to lagrange dual theory, we can get the solution of the optimal X ∗ , y ( T ) with fixed parameter y, and then figure out the optimal parameters y * . When KKT condition is satisfied, the optimal solution of the original problem and the dual problem is equal.

Hence, the equivalent problem of the original problem (9) can be written as:

( min y > 0 max X ( T ) L ( X ( T ) , y ) , X ( T ) ≥ 0. (12)

When we find the optimal X ∗ , y ( T ) with fixed parameter y, we can only focus on the part of X ( T ) in (11) and ignore irrelevant items that only influence the values of the Lagrangian multiplier. In this case, the problem (12) turns into the following problem:

( max X ( T ) { E [ U ( X ( T ) ) ] − y E [ H ( T ) X ( T ) ] } , X ( T ) ≥ 0. (13)

Denote U 1 ( X ) = α ( X − ξ ) γ 1 , and U 2 ( X ) = − β ( ξ − X ) γ 2 . If X > ξ , the utility function U 1 ( X ) is concave and we denote another Lagrangian function L 2 = U ( X ) − y H ( T ) X + ζ X where ζ is Lagrange multiplier. The maximum X 1 ∗ , y satisfies the KKT conditions:

( U ′ ( X ) − y H ( T ) + ζ = 0 , X 1 * ≥ 0 , ζ X 1 ∗ , y = 0 , ζ ≥ 0. (14)

Solving constraint (14), we obtain

X 1 ∗ , y = ξ + ( α γ 1 y H ( T ) ) 1 1 − γ 1 .

If X ≤ ξ , the U 2 ( X ) is convex, and the Weirestrass theorem implies that maximum X 2 ∗ , y must lie on the boundaries X 2 ∗ , y = 0 or X 2 ∗ , y = ξ .

In order to know whether X 1 ∗ , y or X 2 ∗ , y is the global maximum, we denote

f ( H ( T ) ) = U ( X 1 ∗ , y ) − y H ( T ) X 1 ∗ , y − [ U ( X 2 ∗ , y ) − y H ( T ) X 2 ∗ , y ] ,

if f ( H ( T ) ) > 0 , then X ∗ , y ( T ) = X 1 * , otherwise X ∗ , y ( T ) = X 2 * .

Comparing X 1 ∗ , y with X 2 ∗ , y = ξ , we find

f ( H ( T ) ) = α ( α γ 1 y H ( T ) ) γ 1 1 − γ 1 − y H ( T ) ( α γ 1 y H ( T ) ) 1 1 − γ 1 = [ α − y H ( T ) α γ 1 y H ( T ) ] ( α γ 1 y H ( T ) ) γ 1 1 − γ 1 = α ( 1 − γ 1 ) ( α γ 1 y H ( T ) ) γ 1 1 − γ 1 > 0 ,

so X 2 ∗ , y = ξ is never the optimal level of wealth.

Comparing X 1 ∗ , y with X 2 ∗ , y = 0 , we find

f ( H ( T ) ) = α ( α γ 1 y H ( T ) ) γ 1 1 − γ 1 − y H ( T ) ξ − y H ( T ) ( α γ 1 y H ( T ) ) 1 1 − γ 1 + β ξ γ 2 = 1 − γ 1 γ 1 y H ( T ) ( α γ 1 y H ( T ) ) 1 1 − γ 1 − y H ( T ) ξ + β ξ γ 2 ,

when H ( T ) ≤ β y ξ γ 2 − 1 , − y H ( T ) ξ + β ξ γ 2 > 0 , so f ( H ( T ) ) > 0 . Moreover, lim H ( T ) → ∞ f ( H ( T ) ) = − ∞ , f ′ ( H ( T ) ) < 0 , hence f ( H ( T ) ) has an unique root in the interval ( β y ξ γ 2 − 1 , + ∞ ) which we denote the root of f ( H ( T ) ) by H ¯ .

Summarizing the above analysis, we obtain

f ( H ( T ) ) ≤ 0, H ( T ) ≥ H ¯ ,

f ( H ( T ) ) > 0 , H ( T ) < H ¯ .

Hence the global optimizer of problem (13) can be written as

X ∗ , y ( T ) = ( ξ + ( α γ 1 y H ( T ) ) 1 1 − γ 1 , H ( T ) < H ¯ , 0 , H ( T ) ≥ H ¯ .

So far, we have got the optimal X ∗ , y ( T ) with fixed y, and next we will begin to figure out the optimal y * to solve the problem

min y > 0 L ( X ∗ , y ( T ) , y ) ,

According to KKT condition as follows:

y * E [ H ( T ) X ∗ , y ( T ) − ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 T H ( s ) d s ] − y * x 0 = 0. (15)

When H ( T ) < H ¯ , we can get the equation

y * E [ H ( T ) ( ξ + ( α γ 1 y * H ( T ) ) 1 1 − γ 1 ) − ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 T H ( s ) d s ] − y * x 0 = 0 ,

so the optimal y * satisfies

y ∗ 1 γ 1 − 1 = x 0 − E [ H ( T ) ξ ] + ( η − θ ) ∫ 0 ∞ z υ ( d z ) E [ ∫ 0 T H ( s ) d s ] E [ ( α γ 1 ) 1 1 − γ 1 H ( T ) γ 1 γ 1 − 1 ] . (16)

Substituting y * into X ∗ , y ( T ) , we get the optimal X ∗ , y ∗ ( T ) as follows:

X ∗ , y ∗ ( T ) = ( ξ + ( α γ 1 y * H ( T ) ) 1 1 − γ 1 , H ( T ) < H ¯ , 0 , H ( T ) ≥ H ¯ .

Let X ( T ) represent another possible optimal solution satisfying the static budget equation,

E [ U ( X ∗ , y ∗ ( T ) ) ] − E [ U ( X ( T ) ) ] = E [ U ( X ∗ , y ∗ ( T ) ) ] − E [ U ( X ( T ) ) ] − y * X ( 0 ) − y * E [ ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 T H ( s ) d s ] + y * X ( 0 ) + y * E [ ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 T H ( s ) d s ]

≥ E [ U ( X ∗ , y ∗ ( T ) ) ] − E [ U ( X ( T ) ) ] − y * E [ H ( T ) X ∗ , y ∗ ( T ) ] + y * E [ H ( T ) X ( T ) ] ≥ 0.

According to constraint (9) and (15), the first inequality follows from the fact that the static budget equation holds with equality for X ∗ , y ∗ ( T ) and with inequality for X ( T ) , that is,

y * X ( 0 ) + y * E [ ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 T H ( s ) d s ] = y * E [ H ( T ) X ∗ , y ∗ ( T ) ] , y * X ( 0 ) + y * E [ ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 T H ( s ) d s ] ≥ y * E [ H ( T ) X ( T ) ] .

The second inequality holds because X ∗ , y ∗ ( T ) is the optimal solution for problem (13). As such X ∗ , y ∗ ( T ) is the optimal solution of the static problem. □

From the Proposition 3.1, we find that the optimal terminal wealth for the

loss-averse insurer is discontinuous and achieves either ξ + ( α γ 1 y * H ( T ) ) 1 1 − γ 1 or 0.

H ¯ means the breakpoint of the economic states. H ( T ) < H ¯ stands for a good economic states, at this time the insurer gains from participating in the financial

market, ( α γ 1 y * H ( T ) ) 1 1 − γ 1 . As economic conditions deteriorate, the terminal

wealth drops to 0. When ξ = 0 , due to the X ( T ) is no less than 0, the utility function (3) degenerates to the CRRA types U ( X ( T ) ) = α X ( T ) γ 1 , in this case,

H ¯ = + ∞ and the optimal terminal wealth equals to ( α γ 1 y * H ( T ) ) 1 1 − γ 1 . Similar

results can be seen from Guan and Liang [

Remark 3.1. When H ( T ) < H ¯ , substituting (16) into X ∗ , y ( T ) , we obtain

X ∗ , y ∗ ( T ) = ξ + [ x 0 − E ( H ( T ) ξ ) + ( η − θ ) ∫ 0 ∞ z υ ( d z ) E ( ∫ 0 T H ( s ) d s ) ] × H ( T ) 1 γ 1 − 1 E [ H ( T ) γ 1 γ 1 − 1 ] . (17)

Next Lemma 3.1 and Lemma 3.2 will compute it.

If X ( t ) is martingale, s.t.

E [ X ( t ) | F s ] = X ( s ) , s < t ,

and Proposition 3.1 shows that H ( t ) X ( t ) − ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 t H ( s ) d s is a martingale, therefore

E [ ( H ( T ) X ∗ , y ∗ ( T ) − ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 T H ( s ) d s ) | F t ] = H ( t ) X ∗ , y ∗ ( t ) − ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 t H ( s ) d s ,

and hence

H ( t ) X ∗ , y ∗ ( t ) = E [ H ( T ) X ∗ , y ∗ ( T ) | F t ] − ( η − θ ) ∫ 0 ∞ z υ ( d z ) E [ ( ∫ 0 t H ( s ) d s + ∫ t T H ( s ) d s ) | F t ] + ( η − θ ) ∫ 0 ∞ z υ ( d z ) ∫ 0 t H ( s ) d s = E [ H ( T ) X * , y * ( T ) | F t ] − ( η − θ ) ∫ 0 ∞ z υ ( d z ) E [ ∫ t T H ( s ) d s | F t ] . (18)

When H ( T ) < H ¯ , substituting X ∗ , y ∗ ( T ) in Equation (17) into H ( t ) X ∗ , y ∗ ( t ) , (18) can be rewritten as

H ( t ) X ∗ , y ∗ ( t ) = [ x 0 − E ( H ( T ) ξ ) + ( η − θ ) ∫ 0 ∞ z υ ( d z ) E ( ∫ 0 T H ( s ) d s ) ] E [ H ( T ) γ 1 γ 1 − 1 | F t ] E [ H ( T ) γ 1 γ 1 − 1 ] + E [ ξ H ( T ) | F t ] − ( η − θ ) ∫ 0 ∞ z υ ( d z ) E [ ∫ t T H ( s ) d s | F t ] . (19)

When H ( T ) ≥ H ¯ , X ∗ , y ( T ) = 0 , the H ( t ) X ∗ , y ∗ ( t ) can be rewritten as

H ( t ) X ∗ , y ∗ ( t ) = − ( η − θ ) ∫ 0 ∞ z υ ( d z ) E [ ∫ t T H ( s ) d s | F t ] . (20)

Lemma 3.1. Duing to the martingale property and the conditional Fubini theorem, we obtain

E [ ξ H ( T ) | F t ] = ξ H ( t ) e − r ( T − t ) , E [ ∫ t T H ( s ) d s | F t ] = 1 r H ( t ) ( 1 − e − r ( T − t ) ) .

Proof. H ( t ) can be written as

H ( t ) = exp ( − ∫ 0 t r d s ) K ( t ) = e − r t K ( t ) , (21)

where K ( t ) is defined by

K ( t ) = exp { − 1 2 λ s 2 t − λ s W ( t ) + ∫ 0 t ∫ 0 ∞ ln ( 1 + θ ) N ( d z , d s ) − ∫ 0 t ∫ 0 ∞ θ υ ( d z ) d s } ,

using the Itô formula for lévy-type stochastic integrals, we find that

d K ( t ) = K ( t ) ( − λ s d W ( t ) + ∫ 0 ∞ θ z N ˜ ( d z , d t ) ) ,

therefore K ( t ) is a martingale, and we obtain

E [ H ( T ) ] = E [ e − r T K ( T ) ] = e − r T E [ K ( T ) ] = e − r T E [ K ( 0 ) ] = e − r T .

Similarly, we have

E [ ξ H ( T ) | F t ] = ξ E [ e − r T K ( T ) | F t ] = ξ e − r T E [ K ( T ) | F t ] = ξ e − r T K ( t ) = ξ H ( t ) e − r ( T − t ) .

Using the conditional Fubini theorem, the order of integral and expectation can be exchanged, so we obtain

E ( ∫ 0 T H ( s ) d s ) = ∫ 0 T E [ H ( s ) ] d s = ∫ 0 T E [ e − r s K ( s ) ] d s = ∫ 0 T e − r s d s = 1 r ( 1 − e − r T ) ,

and

E [ ∫ t T H ( s ) d s | F t ] = ∫ t T E [ H ( s ) | F t ] d s = ∫ t T H ( t ) e − r ( s − t ) d s = H ( t ) ∫ t T e − r ( s − t ) d s = 1 r H ( t ) ( 1 − e − r ( T − t ) ) .

□

Lemma 3.2.

E [ H ( T ) γ 1 γ 1 − 1 | F t ] E [ H ( T ) γ 1 γ 1 − 1 ] = M ( t ) ,

where

M ( t ) = exp { − γ 1 γ 1 − 1 ∫ 0 t λ s d W ( s ) − 1 2 ( γ 1 γ 1 − 1 ) 2 ∫ 0 t λ s 2 d s + γ 1 γ 1 − 1 ∫ 0 t ∫ 0 ∞ ln ( 1 + θ ) N ( d z , d s ) + ∫ 0 t ∫ 0 ∞ ( 1 − ( 1 + θ ) γ 1 γ 1 − 1 ) υ ( d z ) d s } . (22)

Proof. We substitute H ( T ) = e − r T K ( T ) into the E [ H ( T ) γ 1 γ 1 − 1 | F t ] E [ H ( T ) γ 1 γ 1 − 1 ] , then we

obtain

E [ H ( T ) γ 1 γ 1 − 1 | F t ] E [ H ( T ) γ 1 γ 1 − 1 ] = E [ e − γ 1 γ 1 − 1 r T K ( T ) γ 1 γ 1 − 1 | F t ] E [ e − γ 1 γ 1 − 1 r T K ( T ) γ 1 γ 1 − 1 ] ,

Using the Itô formula for lévy-type stochastic integrals, we know that

K ( t ) γ 1 γ 1 − 1 is not a martingale, so we introduce an exponential martingale

M ( t ) = exp { − γ 1 γ 1 − 1 λ s W ( t ) − 1 2 ( γ 1 γ 1 − 1 ) 2 λ s 2 t + γ 1 γ 1 − 1 ∫ 0 t ∫ 0 ∞ ln ( 1 + θ ) N ( d z , d s ) + ∫ 0 t ∫ 0 ∞ ( 1 − ( 1 + θ ) γ 1 γ 1 − 1 ) υ ( d z ) d s } , (23)

and the differential form of M ( t ) as follows:

d M ( t ) M ( t ) = − γ 1 γ 1 − 1 λ s d W ( t ) + ∫ 0 ∞ ( 1 − ( 1 + θ ) γ 1 γ 1 − 1 ) N ˜ ( d z , d s ) , (24)

then denote

N ( t ) = exp { − γ 1 γ 1 − 1 r t + 1 2 γ 1 ( γ 1 − 1 ) 2 λ s 2 t + ∫ 0 t ∫ 0 ∞ [ − γ 1 γ 1 − 1 θ − 1 + ( 1 + θ ) γ 1 γ 1 − 1 υ ( d z ) d s ] } ,

so H ( t ) γ 1 γ 1 − 1 can be written as

H ( t ) γ 1 γ 1 − 1 = M ( t ) N ( t ) .

Therefore,

E [ H ( T ) γ 1 γ 1 − 1 | F t ] E [ H ( T ) γ 1 γ 1 − 1 ] = E [ M ( T ) N ( T ) | F t ] E [ M ( T ) N ( T ) ] = N ( T ) E [ M ( T ) | F t ] N ( T ) E [ M ( T ) ] = M ( t ) M ( 0 ) = M ( t ) .

□

Theorem 3.2. 1) When H ( T ) < H ¯ , the optimal portfolio π * ( t ) and l * ( t ) are

π * ( t ) = μ s − r σ s 2 1 1 − γ 1 [ X ∗ , y ∗ ( t ) − ξ e − r ( T − t ) + η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) ] , l * ( t ) = θ − 1 + ( θ + 1 ) γ 1 γ 1 − 1 θ + 1 [ X ∗ , y ∗ ( t ) − ξ e − r ( T − t ) + η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) ] ∧ z . (25)

The optimal wealth at time t is given by

X ∗ , y ∗ ( t ) = ξ e − r ( T − t ) − η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) + [ x 0 − ξ e − r T + η − θ r ( 1 − e − r T ) ∫ 0 ∞ z υ ( d z ) ] exp { r t + 1 1 − γ 1 λ s W ( t ) + 1 2 1 − 2 γ 1 ( γ 1 − 1 ) 2 λ s 2 t − 1 1 − γ 1 ∫ 0 t ∫ 0 ∞ ln ( 1 + θ ) N ( d z , d s ) + ∫ 0 t ∫ 0 ∞ ( θ + 1 − ( θ + 1 ) γ 1 γ 1 − 1 ) υ ( d z ) d s } . (26)

2) When H ( T ) ≥ H ¯ , the optimal portfolio π * ( t ) and l * ( t ) are given, respectively, by

π * ( t ) = 0, l * ( t ) = 0. (27)

The optimal wealth at time t is given by

X ∗ , y ∗ ( t ) = − η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) . (28)

Proof. 1) According to Lemma 3.1 and Lemma 3.2 in the case of H ( T ) < H ¯ , H ( t ) X ∗ , y ∗ ( t ) in the (19) can be written as

H ( t ) X ∗ , y ∗ ( t ) = [ x 0 − ξ e − r T + η − θ r ∫ 0 ∞ z υ ( d z ) ( 1 − e − r T ) ] M ( t ) + ξ H ( t ) e − r ( T − t ) − η − θ r ( 1 − e − r ( T − t ) ) H ( t ) ∫ 0 ∞ z υ ( d z ) , (29)

so the optimal wealth at time t is given by

X ∗ , y ∗ ( t ) = ξ e − r ( T − t ) − η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) + [ x 0 − ξ e − r T + η − θ r ( 1 − e − r T ) ∫ 0 ∞ z υ ( d z ) ] M ( t ) H ( t ) = ξ e − r ( T − t ) − η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) + [ x 0 − ξ e − r T + η − θ r ( 1 − e − r T ) ∫ 0 ∞ z υ ( d z ) ] exp { r t + 1 1 − γ 1 λ s W ( t ) + 1 2 1 − 2 γ 1 ( γ 1 − 1 ) 2 λ s 2 t − 1 1 − γ 1 ∫ 0 t ∫ 0 ∞ ln ( 1 + θ ) N ( d z , d s ) + ∫ 0 t ∫ 0 ∞ ( θ + 1 − ( θ + 1 ) γ 1 γ 1 − 1 ) υ ( d z ) d s } .

Taking differential on both sides of (29)

d [ H ( t ) X ∗ , y ∗ ( t ) ] = [ x 0 − ξ e − r T + η − θ r ( 1 − e − r T ) ∫ 0 ∞ z υ ( d z ) ] d M ( t ) + M ( t ) d ( x 0 − ξ e − r T + η − θ r ( 1 − e − r T ) ∫ 0 ∞ z υ ( d z ) ) + ( ξ e − r ( T − t ) − η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) ) d H ( t ) + H ( t ) d ( ξ e − r ( T − t ) − η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) ) .

Note that the coefficient of M ( t ) in (29) can be replaced by

x 0 − ξ e − r T + η − θ r ( 1 − e − r T ) ∫ 0 ∞ z υ ( d z ) = [ X ∗ , y ∗ ( t ) − ξ e − r ( T − t ) + η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) ] H ( t ) M ( t ) . (30)

According to (7) (24) and (30) ,

d [ H ( t ) X ∗ , y ∗ ( t ) ] = ( ⋅ ) d t + H ( t ) λ s [ γ 1 1 − γ 1 X ∗ , y ∗ ( t ) − 1 1 − γ 1 ( ξ e − r ( T − t ) − η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) ) ] d W ( t ) + H ( t ) [ X ∗ , y ∗ ( t ) ∫ 0 ∞ ( 1 − ( θ + 1 ) γ 1 γ 1 − 1 ) + ( ξ e − r ( T − t ) − η − θ r ∫ 0 ∞ z υ ( d z ) ) ( ∫ 0 ∞ ( θ − 1 + ( θ + 1 ) γ 1 γ 1 − 1 ) ) ] N ˜ ( d z , d t ) . (31)

We are only interested in diffusion part, so Comparing (31) with (8), we obtain

π * ( t ) = μ s − r σ s 2 1 1 − γ 1 [ X ∗ , y ∗ ( t ) − ξ e − r ( T − t ) + η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) ] , l * ( t ) = θ − 1 + ( θ + 1 ) γ 1 γ 1 − 1 θ + 1 [ X ∗ , y ∗ ( t ) − ξ e − r ( T − t ) + η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) ] ∧ z .

2) Similarly, when H ( T ) ≥ H ¯ , the H ( t ) X ∗ , y ∗ in the (20) can be written as

H ( t ) X ∗ , y ∗ ( t ) = − η − θ r ∫ 0 ∞ z υ ( d z ) H ( t ) ( 1 − e − r ( T − t ) ) , (32)

so the optimal wealth at time t is given by

X ∗ , y ∗ ( t ) = − η − θ r ( 1 − e − r ( T − t ) ) ∫ 0 ∞ z υ ( d z ) .

Taking differential on both sides of Equation (32)

d [ H ( t ) X ∗ , y ∗ ( t ) ] = ( ⋅ ) d t + X ∗ , y ∗ ( t ) H ( t ) ( − λ s d W ( t ) + ∫ 0 ∞ θ N ˜ ( d z , d t ) ) ,

and comparing it with (8), we obtain

π * ( t ) = 0 , l * ( t ) = 0.

□

Remark 3.2. If we put the (26) into equation (25), we find that

π * ( t ) = μ s − r σ s 2 1 1 − γ 1 [ x 0 − ξ e − r T + η − θ r ( 1 − e − r T ) ∫ 0 ∞ z υ ( d z ) ] × exp { r t + 1 1 − γ 1 λ s W ( t ) + 1 2 1 − 2 γ 1 ( γ 1 − 1 ) 2 λ s 2 t − 1 1 − γ 1 ∫ 0 t ∫ 0 ∞ ln ( 1 + θ ) N ( d z , d s ) + ∫ 0 t ∫ 0 ∞ ( θ + 1 − ( θ + 1 ) γ 1 γ 1 − 1 ) υ ( d z ) d s } .

Similarly,

l * ( t ) = θ − 1 + ( θ + 1 ) γ 1 γ 1 − 1 θ + 1 [ x 0 − ξ e − r T + η − θ r ( 1 − e − r T ) ∫ 0 ∞ z υ ( d z ) ] × exp { r t + 1 1 − γ 1 λ s W ( t ) + 1 2 1 − 2 γ 1 ( γ 1 − 1 ) 2 λ s 2 t − 1 1 − γ 1 ∫ 0 t ∫ 0 ∞ ln ( 1 + θ ) N ( d z , d s ) + ∫ 0 t ∫ 0 ∞ ( θ + 1 − ( θ + 1 ) γ 1 γ 1 − 1 ) υ ( d z ) d s } .

As the insurance has more initial wealth, as measured by x 0 , the π * and l * also increase linearly. If the appreciation rate μ s of risky asset increases, the amount invested in the risky asset obviously increases. Furthermore, as the insurance market becomes more volatile, as measured by σ s , the amount invested in the risky asset decreases nonlinearly. Also, it makes sense that μ s and σ s have no effect on the excess-of-loss reinsurance.

Remark 3.3. Theorem 3.2 shows that, π * ( t ) = 0 and l * ( t ) = 0 in the case of H ( T ) ≥ H ¯ . That is to say, when the market deteriorate, the insurer will stop investing in the risky asset and purchasing reinvestment strategy.

Remark 3.4. Optimal y * in the Equation (16) can be rewritten as

y * = ( x 0 − ξ e − r T + η − θ r ( 1 − e − r T ) ∫ 0 ∞ z υ ( d z ) ) γ 1 − 1 ( α γ 1 ) − 1 N ( T ) γ 1 − 1 . (33)

Therefore, the optimal terminal wealth in Theorem 3.1 can be rewritten as

X ∗ , y ∗ ( T ) = ( ξ + [ x 0 − ξ e − r T + η − θ r ( 1 − e − r T ) ∫ 0 ∞ z υ ( d z ) ] H ( T ) 1 γ 1 − 1 N ( T ) , H ( T ) < H ¯ , 0 , H ( T ) ≥ H ¯ ,

(34)

where

N ( T ) = exp { − γ 1 γ 1 − 1 r T + 1 2 γ 1 ( γ 1 − 1 ) 2 λ s 2 T + ( − γ 1 γ 1 − 1 θ − 1 + ( 1 + θ ) γ 1 γ 1 − 1 ) T ∫ 0 ∞ υ ( d z ) } ,

H ( T ) 1 γ 1 − 1 N ( T ) = exp { r T + 1 − 2 γ 1 ( 1 − γ 1 ) 2 λ s T + 1 1 − γ 1 λ s W ( T ) − 1 1 − γ 1 ∫ 0 T ∫ 0 ∞ ln ( 1 + θ ) N ( d z , d s ) + ( 1 + θ − ( 1 + θ ) γ 1 γ 1 − 1 ) T ∫ 0 ∞ υ ( d z ) } .

In this section, we present several numerical examples to illustrate our results in the previous section. Thus throughout this section, unless otherwise stated, the values parameters are taken as:

α = 3 ; β = 4 ; ξ = 4 ; γ 1 = 0.2 ; γ 2 = 0.15 ; r = 0.05 ; λ = 6 η = 0.5 ; θ = 0.6 ; μ s = 0.1 ; σ s = 0.3 ; x 0 = 4 ; T = 4.

In addition, we assume that the claim size Z i follows the exponential

distribution and the claim size density function is f ( z ) = 6 e − 6 z , hence μ ∞ = 1 6

and acording to the remark 2.1, ∫ 0 ∞ z υ ( d z ) = λ μ ∞ = 1 . Subsequently, we analyze the effects of parameters on the optimal terminal wealth and the strategy. For convenience but without loss of generality, we focus on the case at time t = 0 .

In this paper, Proposition 3.1 and

Note that the breakpoint H ¯ is of importance in Theorem 3.1 and Theorem 3.2, and hence we next discuss the sensitivity of the strategy with H ¯ . We know

y * is the optimal lagrange multiplier and H ¯ is the root of f ( H ( T ) ) . There is a negative relationship between y * and H ¯ .

y * = ( x 0 − ξ e − r T + η − θ r ∫ 0 ∞ z υ ( d z ) ( 1 − e − r T ) ) γ 1 − 1 ( α γ 1 ) − 1 N ( T ) γ 1 − 1 , (35)

f ( H ( T ) ) = 1 − γ 1 γ 1 y * H ( T ) ( α γ 1 y * H ( T ) ) 1 1 − γ 1 − y * H ( T ) ξ + β ξ γ 2 , (36)

Moreover,

From

negative effect on the optimal investment weight π ( t ) X ( t ) . We know the coefficient of relative risk aversion of the utility u ( x ) equals − u ″ ( x ) u ′ ( x ) , which

means that 1 − γ 1 stands for the coefficient of relative risk aversion. In other words, 1 − γ 1 decreases with the increasing of γ 1 , and the insurer becomes more gain-aversion. Therefore, increasing γ 1 leads to a increase in the optimal

terminal wealth X ( T ) , a decrease in the breakpoint H ¯ , as is shown in

In this paper, we consider the optimal investment and reinsurance strategy for insurer with loss aversion. The insurer aims to maximize the expected utility of terminal wealth and the wealth is allowed to invest in a risk-free asset and a risky asset. Furthermore, the insurer can purchase excess-of-loss reinsurance. Since the S-shaped utility is convex-concave, the stochastic programming method is not suitable, and we obtain a close-form solution for the optimal strategy by using martingale method. We find that the optimal terminal wealth is piecewise function. In good states of market, the optimal wealths is of the same form with the smooth CRRA utility function case, on the contrary the optimal wealth approaches 0 in bad states of market. Similarly, the optimal investment and reinsurance strategy are also divided into two cases respectively. When the market deteriorate, the insurer will stop investing in the risky asset and purchasing reinvestment strategy. Finally, we present some numerical examples to show the effects of model parameters on the optimal terminal wealth and the optimal strategy.

Based on our current work, various directions may be followed in the future research. 1) the price process of the risky asset now is described by the GBM, further we can add the diffusion term to model, or try to use CEV model or Heston model; 2) notice that the interest rate used in this paper is a fixed constant, so we can introduce the stochastic interest rate process, such as Vasicek model or Ornstein-Uhlenbeck model; 3) derivatives, such as option, can be added to the paper and purchased by the insurer, thus we can research the effect of derivatives on investment strategies and control risks; 4) the reference point in the S-shape utility function can be dynamics by introducing inflation, which will change with time and correlate with inflation factors. All the future research directions will make the problem more comprehensive and complex, but also more relevant to reality in insurers daily business.

The authors are very grateful to the referees for helpful comments and suggestion. This research was supported by grants from the National Natural Science Foundation of China (11771329, 11871052, 11301376).

The authors declare no conflicts of interest regarding the publication of this paper.

Sun, Q.Y., Rong, X.M. and Zhao, H. (2019) Optimal Excess- of-Loss Reinsurance and Investment Problem for Insurers with Loss Aversion. Theoretical Economics Letters, 9, 1129-1151. https://doi.org/10.4236/tel.2019.94073