1. Introduction There are multiple kinds of risks which can adversely affect people’s wealth or wellbeing, and therefore people usually buy insurance to reduce their losses. Unfortunately, in practice, the losses caused by some kind of risks are so great that the insurer is unable to insure it, and therefore exclusion clauses were frequently written in the insurance contract because of the limited capacity. For example, in car insurance, the insurer’s liabilities are often exempted if the losses result from wars or earthquakes.

Hence, it is necessary to study insurance problems in the presence of background risk.

The authors in [1] firstly studied the optimal insurance demand in the presence of background risk. Then insurance contracts with background risks have been studied in various cases. But most of the studies have focused on the demand of the insurance when there is background risk; for example, see the survey paper of [2] which discussed the proportional coinsurance policy under independent and dependent background risk.

The authors in [3] studied the efficient insurance contract when there is background risk. The optimal insurance contract is proved to contain a deductible if the background risk is independent with the insurable risk. Under given conditions of the utility function, the existence of the background risk will reduce the deductible. If the background risk increases with the insurable risk, and the insured is prudent, the optimal insurance contract entails a disappearing deductible. Efficient insurance contracts with independent background risk were also studied in [4–8]. The cases of positively correlated background risk are discussed in [9, 10]. The authors in [11] examined the qualitative properties of efficient insurance policy under the assumption of “stochastic increasingness” between the background risk and the insurable risk.

The authors in [8] proposed a cumulative distribution function (CDF) based method to derive the explicit form of the optimal reinsurance contract for an insurer when there is background risk. But the objective of [8] is maximizing the insured’s survival probability. In this paper, we will derive the explicit form of the optimal insurance policy which maximizes the insured’s expected utility. We first prove that when the background risk is discrete, the optimal solution should be contingent upon the realized values of the background risk without any extra assumptions on the dependence between the insurable risk and the background risk, or the distributions of them, or the utility function. And then we give the explicit form of the optimal solution and prove the uniqueness of the solution.

The paper is organized as follows: Section 2 introduces the model. Section 3 proves that when the background risk is discrete, the optimal insurance contract should be contingent upon the realized values of the background risk, and gives the explicit form of the optimal solution.

2. The Model Let (Ω,F,P) be a probability space and X, Y be two random variables defined on this space. We assume X is a continuous random variable and Y is a discrete random variable, X:Ω→C, and Y:Ω→D, where C and D are the sets of consequences of X and Y, respectively. We assume C=[0,x¯], x¯>0, and D is a countable subset of {0}∪R+.

Let us assume an agent faces two kinds of random losses X and Y. The insurance market provides insurance only for loss X. In the absence of background risk, the insurance policy or indemnity function is a deterministic function I(x) defined on C denoting the indemnity that is paid by the insurer if the observed loss is x. The indemnity function is usually assumed to satisfy 0≤I(x)≤x, which ensures the indemnity is nonnegative and does not exceed the occurred insurable loss. When there is background risk Y, the optimal insurance policy may be affected by Y, so we denote the insurance policy by IY(X). We denote the premium paid by the insured by π. The reasonable premium should satisfy 0≤π≤x¯.

Assume the insured’s initial wealth is W0. Facing a potential loss X and background loss Y, if the insured buys the contract, he is endowed with the random wealth W=W0-π-Y-X+IY(X). In this paper, we assume the insured has von Neumann-Morgenstern utility function u, and his preference over random wealth is represented by E[u(W)]. We assume the utility function u:R→R is increasing, strictly concave, and twice differentiable. The insured is supposed to choose a premium π and an insurance indemnity function IY(·) so that they maximize his/her expected utility on final wealth, i.e., (1)maxπ,I EuW0-π-Y-X+IYX.

We also assume the insurer is risk-neutral. From [12, 13], if the cost of offering the insurance is proportional to the expected value of the indemnity E[IY(X)], then, for the insurer, he/she requires (2)1+ρEIYX≤π.The expectation E[IY(X)] is the actuarial value of the insurance policy. For the risk-neutral insurer, (1+ρ)E[IY(X)] is the minimum price of the indemnity IY(X) to participate in the business.

Then the optimal design of the insurance contract can be modelled by the optimization problem (3)(3)maxπ,I EuW0-π-Y-X+IYX,1+ρEIYX≤π.

The model of the optimal index insurance contract in [14] can be modelled by the following optimization problem (4):(4)maxI EuW0-1-θπ-Y+IX,γEIX=π.

The two problems (3) and (4) have similar structure, but in model (4), X is not a loss term of the insured, so the solving methods and results are completely different.

3. The Solution of the Optimization Problem with Background Risk Assume the probability distribution of Y to be (5)PY=yi=pi, ∀yi∈D,with ∑yi∈Dpi=1.

The conditional distribution of X given Y is F(x∣Y=y). Then the objective function in (3) can be written as(6)EuW0-π-Y-X+IYX=EEuW0-π-Y-X+IYX∣Y=∑yi∈Dpi∫0x¯uW0-π-yi-x+IyixdFx∣Y=yi.

To be able to apply the Lagrange technique in the following, we write the constraint condition as(7)1+ρEIYX=1+ρEEIYX∣Y=1+ρ∑yi∈Dpi∫0x¯IyixdFx∣Y=yi≤π.

Lemma 1. For given π, and each fixed yi∈D, the solution of optimization problem(8)max0≤Iyix≤x∫0x¯uW0-π-yi-x+IyixdFx∣Y=yi,1+ρ∫0x¯IyixdFx∣Y=yi≤π.is(9)Iyix=x, with 1+ρEIyiX∣Y=yi=π,or(10)Iyix=x-dyi+, where dyi=W0-π-yi-u′-1λyi1+ρ and 1+ρEIyiX∣Y=yi=π.

Proof. For given π and fixed yi∈D, we solve the optimization problem (8). The Lagrangian of (8) is (11)LIyi,λyi=∫0x¯uW0-π-yi-x+IyixdFx∣Y=yi-λyi1+ρ∫0x¯IyixdFx∣Y=yi-π.

Let (12)HIyi,λyi=uW0-π-yi-x+Iyix-λyi1+ρIyix.

Since u is concave, H is concave about Iyi(·).

For x such that ∂H/∂Iyi(Iyi,λyi)Iyi=x=u′(W0-π-yi)-λyi(1+ρ)>0, we should let (13)Iyix=x.

For x such that ∂H/∂Iyi(Iyi,λyi)Iyi=0=u′(W0-π-yi-x)-λyi(1+ρ)<0, i.e., x<W0-π-yi-(u′)-1(λyi(1+ρ)) since u′ is nonincreasing, we should let (14)Iyix=0. Otherwise, (15)∂H∂IyiIyi,λyi=u′W0-π-yi-x+Iyix-λyi1+ρ=0, i.e., (16)Iyix=x+π+yi-W0+u′-1λyi1+ρ.

Since u is concave, u′ is nonincreasing. Therefore u′(W0-π-yi)-λyi(1+ρ)>0, and u′(W0-π-yi-x)-λyi(1+ρ)<0 cannot be satisfied simultaneously. Hence with the constraint condition, the optimal solution for (8) is (17)Iyix=x, if 1+ρEIyiX∣Y=yi=π, or (18)Iyix=x-dyi+, where dyi=W0-π-yi-u′-1λyi1+ρ and 1+ρEIyiX∣Y=yi=π.We get the result.

For the case of (10), dyi is determined by π uniquely; we define a function dyi(π):π→dyi. For the case of (9), we let dyi(π)≡0.

Lemma 2. For the case of (10), the first- and the second-order derivatives of dyi(π) for each yi∈D are(19)dyi′π=11+ρ1Fdyiπ∣Y=yi-1<0,dyi′′π=-11+ρF′dyiπ∣Y=yidyi′πFdyiπ∣Y=yi-12≥0.

Proof. By the constraint conditions in (10), we have (20)1+ρ∫dyiπx¯x-dyiπdFx∣Y=yi=π,and differentiating about π, we get (19).

Remark 3. For the case of (9), the first- and the second-order derivatives of dyi(π) for each yi∈D are(21)dyi′π=0,dyi′′π=0.

Lemma 4. There exists a unique solution of the following optimal problem without constraint:(22)max0≤π≤x¯∑yi∈Dpi∫0x¯uW0-π-yi-x+x-dyiπ+dFx∣Y=yi.

Proof. We denote the objective function in (22) by V(π), i.e., (23)Vπ=∑yi∈Dpi∫0x¯uW0-π-yi-x+x-dyiπ+dFx∣Y=yi. The second derivative of V(·), (24)V′′π=∑yi∈Dpi∫0x¯u′′W0-π-yi-x+x-dyiπ+-1-Ix>dyiπdyi′π2+u′W0-π-yi-x+x-dyiπ+-Ix>dyiπdyi′′πdFx∣Y=yi. Since u′′<0, u′>0, and dyi′′≥0, ∂2V/∂π2(π)<0, that is, V(·) is strictly concave. The concave maximization problem (22) can be solved by minimizing the convex objective function -V(π). By convex analysis ([15], Corollaire 3.20), and strict convexity of -V(π), there exists a unique π that attains the minimum of -V(π).

Theorem 5. The optimal solution π for the optimization problem (3) is the unique solution of the optimal problem without constraint (22), and the optimal solution IY(X) for the optimization problem (3) is(25)IYX=∑yi∈DIY=yiIyiX,where yi∈D are the consequences of Y, and for each fixed yi, Iyi(X) is the optimal solution for the optimization problem (8).

Proof. Since u is strictly concave, if the solution for problem (3) as well as (8) exists, it is unique.

From (6) and (7), the insured’s optimization problem (3) can be written as(26)maxπ,I∑yi∈Dpi∫0x¯uW0-π-yi-x+IyixdFx∣Y=yi,1+ρ∑yi∈Dpi∫0x¯IyixdFx∣Y=yi≤π.

Let π~,I~Y(X) be the solution of (3).

For each fixed yi∈D, solve the optimization problem(27)max0≤I~yix≤x∫0x¯uW0-π~-yi-x+I~yixdFx∣Y=yi,1+ρ∫0x¯I~yixdFx∣Y=yi≤π~.

Then (28)EuW0-π~-Y-X+I~YX≤∑yi∈Dpimax0≤I~yix≤x∫0x¯uW0-π~-yi-x+I~yixdFx∣Y=yi.

Since π~,I~Y(X) are the solution of (3), (29)EuW0-π~-Y-X+I~YX≥∑yi∈Dpimax0≤I~yix≤x∫0x¯uW0-π~-yi-x+I~yixdFx∣Y=yi. Hence (30)I~YX=IY=yiI~yiX,

From Lemma 1, the solutions of the optimization problems (27) for yi∈D are (9) (with dyi(π~)≡0) or (10) (with dyi(π~)>0) satisfying (31)1+ρ∫dyiπ~x¯x-dyiπ~dFx∣Y=yi=π~. Therefore (32)∑yi∈Dpimax0≤I~yix≤x∫0x¯uW0-π~-yi-x+I~yixdFx∣Y=yi=∑yi∈Dpi∫0x¯uW0-π~-yi-x+x-dyiπ~+dFx∣Y=yi≤∑yi∈Dpi∫0x¯uW0-π-yi-x+x-dyiπ+dFx∣Y=yiwhere the inequality is because of (22), and the equal sign “=” holds if and only if π=π~; otherwise “<” holds and it will contradict the assumption that π~,I~Y(X) is the solution of (3).