2. Problem formulation

We confine ourselves to a one-period economy. Suppose there is a decision maker (DM, also called insured) who is faced with a non-negative, bounded random loss X whose support is [0,M] (i.e., the set of numbers which have non-zero probability densities). The cumulative distribution function and density function of X are given by F(x) and f(x), respectively. The DM would like to purchase an insurance contract $(I,\pi)$ where I is the indemnity function and $\pi(I)$ is the premium principle that is used by the insurer.

There exists counterparty risk in the sense that the insurer may fail to pay its promised indemnity as per the contract at the end of period. We assume that the coverage received by the DM is given by $I(X)\cdot Y$ , where Y is a random variable distributed over [0,1] and may be correlated with X. In this paper, we follow Cummins and Mahul (Reference Cummins and Mahul2003) and Bernard and Ludkovski (Reference Bernard and Ludkovski2012) by assuming that the default event is exogenous in the sense that it is not affected by the DM’s transactions. To avoid some trivial cases of the default event, we adopt the following assumption throughout the paper.

Assumption 1(ii) states that regardless of the DM’s loss, the probability of that the insurer recovers some partial liability (e.g., through selling its remaining assets) is positive. Under this setting, the end-of-period loss of the DM is

(2.1) \begin{equation}L(I, \pi)=X-I(X)\cdot Y+\pi(I).\end{equation}

In insurance or reinsurance, some policies provide the DM an incentive to misreport the loss. For example, a franchise deductible indemnity function, that is $I(x)=x\cdot \unicode{x1d7d9}_{[d,\infty)}(x)$ for some $d\ge0$ , may incentivize the DM to over-report the loss or create an increment in the loss; a truncated excess-of-loss indemnity function, that is $I(x)=(x-d_1)_+\cdot\unicode{x1d7d9}_{[0,d_2]}(x)$ for some $d_2\ge d_1\ge0$ , may incentivize the DM to under-report the loss if it exceeds $d_2$ . Such behavior is called ex post moral hazard and should be seriously treated. A popular way in the literature to handle this issue is to restrict the indemnity functions to the following class:

\begin{align*} \mathcal{I} &=\Big\{I\;:\;[0,M]\mapsto[0,M]\ \Big|\ I(0)=0, \; 0\le I(x_2)-I(x_1)\le x_2-x_1, \\ & \qquad \forall\ 0\le x_1\le x_2\le M\Big\}.\end{align*}

The indemnity functions belonging to the set $\mathcal{I}$ are said to satisfy the incentive-compatibility or no-sabotage condition (Huberman et al. Reference Huberman, Mayers and Smith1983; Carlier and Dana, Reference Carlier and Dana2003). The advantages of limiting the indemnity function to the class $\mathcal{I}$ are twofold. First, the indemnity function is always non-decreasing, that is $I'(x)\ge 0$ , and satisfies $0\le I(x)\le x$ . That means, the indemnity is increasing with respect to the loss and can never be negative nor exceed the loss x generated by the DM’s loss. Second, one unit increment of loss cannot be compensated by more than one unit of indemnity. For any $I\in\mathcal{I}$ , I(X) and $X-I(X)$ are comonotonic with respect to the loss X. In particular, if the insurable loss X increases, then both the indemnity I(X) and the retained loss $X-I(X)$ increase.

For the premium, we assume that it is based on the coverage $I(X)\cdot Y$ rather than the promised indemnity I(X). We further assume that the premium is determined as a function of the expected coverage:

(2.2) \begin{equation} \pi(I)=h(\mathbb{E}[I(X)\cdot Y]),\end{equation}

where $h(\!\cdot\!)$ is some differentiable function satisfying $h(0)=0$ and $h'(x)>1$ for $x\ge 0$ . A special case of h is $h(x)=(1+\theta)x$ with $\theta>0$ , which leads to the expectation premium principle.

We follow Chi and Tan (Reference Chi and Tan2021) and study the optimal insurance problem within a mean-variance framework. We focus on the following general problem with counterparty risk.

Problem 1 (Main problem)

\begin{equation*} \min_{I\in\mathcal{I}}\ \mathcal{M}\left(\mathbb{E}[L(I,\pi)], \textit{Var}(L(I,\pi))\right), \end{equation*}

where $\mathcal{M}(z_1,z_2)$ is increasing with respect to both $z_1$ and $z_2$ .

Problem 1 accommodates a wide range of mean-variance problems. For example, if let $\mathcal{M}(z_1,z_2)=z_1+\frac{\mathcal{B}}{2}z_2$ for some $\mathcal{B}\ge 0$ , the traditional mean-variance criterion is recovered, where $\mathcal{B}$ measures the DM’s aversion towards volatility.

We remark that the optimal indemnity function within the mean–variance framework for additive background risk model has been derived by Chi and Tan (Reference Chi and Tan2021) through a constructive approach with stochastic ordering techniques. The marginal indemnity function formulation, which is obtained through the calculus of variations, not only helps us to solve the multiplicative background risk (or counterparty risk) model but also provides an alternative way to solve the additive background risk model.

3. Optimal indemnity function

Solving Problem 1 usually takes a two-step procedure. In the first step, we fix the mean (i.e., the first argument of $\mathcal{M}(\cdot,\cdot)$ ) and minimize the variance (i.e., the second argument of $\mathcal{M}(\cdot,\cdot)$ ):

(3.1) \begin{equation} \min_{I\in\mathcal{I}}\ \text{Var}(L(I,\pi)),\quad \text{s.t.}\ \mathbb{E}[L(I,\pi)]=c,\end{equation}

where c is a constant. As problem (3.1) depends on c, we denote its solution by $I_c$ . In the second step, we search for the optimal c such that $\mathcal{M}(c,\text{Var}(L(I_c,\pi)))$ reaches its minimum. The second step is a one-dimensional problem, and can be solved using standard techniques. The first step plays a vital role as it identifies the optimal indemnity function with c as its parameter. In the sequel, we focus on this first step, and thus on the problem (3.1).

Let $\psi_1(x)=\mathbb{E}[Y|X=x]$ and $\psi_2(x)=\mathbb{E}[Y^2|X=x]$ . Under Assumption 1, it follows that $\psi_1(x)>0$ and $\psi_2(x)>0$ for all $x\in[0,M]$ . Apply the conditional variance formula to the objective function of (3.1) gives

\begin{align*}\text{Var}(L(I,\pi))&=\mathbb{E}\left[\text{Var}(L(I,\pi)\big| X)\right]+\text{Var}\left(\mathbb{E}[L(I,\pi)\big|X]\right) \\&=\mathbb{E}[I(X)^2\cdot \text{Var}(Y\big| X)]+\text{Var}(X-I(X)\cdot \mathbb{E}[Y\big| X]) \\&=\mathbb{E}[I(X)^2\cdot \psi_2(X)]-\mathbb{E}[I(X)^2\cdot \psi_1(X)^2]+\mathbb{E}\left[(X-I(X)\cdot \psi_1(X))^2\right] \\ & \quad -\left(\mathbb{E}[X-I(X)\cdot\psi_1(X)]\right)^2\\ &=\mathbb{E}[I(X)^2\cdot \psi_2(X)]-\mathbb{E}[I(X)^2\cdot \psi_1(X)^2]+\mathbb{E}[X^2] \\ & \quad -2\mathbb{E}[X\cdot I(X)\cdot \psi_1(X)] +\mathbb{E}[I(X)^2\cdot \psi_1(X)^2]-\mathbb{E}[X]^2 \\ & \quad +2\mathbb{E}[X]\cdot\mathbb{E}[I(X)\cdot\psi_1(X)]-\mathbb{E}[I(X)\cdot\psi_1(X)]^2\\&=\mathbb{E}[I(X)^2\cdot \psi_2(X)]-2\mathbb{E}[X\cdot I(X)\cdot \psi_1(X)]+\text{Var}(X) \\& \quad +2\mathbb{E}[X]\cdot\mathbb{E}[I(X)\cdot\psi_1(X)]-\mathbb{E}[I(X)\cdot\psi_1(X)]^2.\end{align*}

The constraint of (3.1) reduces to

\begin{equation*}\begin{aligned}&\mathbb{E}[X-I(X)\cdot Y+\pi]=\mathbb{E}[X]-\mathbb{E}[I(X)\cdot Y]+\pi=c \\\Longrightarrow\ &h(\mathbb{E}[I(X)\cdot Y])-\mathbb{E}[I(X)\cdot Y]=c-\mathbb{E}[X] \\\Longrightarrow\ &h(\mathbb{E}[I(X)\cdot \psi_1(X)])-\mathbb{E}[I(X)\cdot \psi_1(X)]=c-\mathbb{E}[X].\end{aligned}\end{equation*}

Since $\left(h(x)-x\right)'=h'(x)-1>0$ and $h(0)=0$ , for any $c\ge \mathbb{E}[X]$ , the equation $h(x)-x=c-\mathbb{E}[X]$ has only one solution, which is denoted as $x^{\ast}$ . Based on the above simplifications, problem (3.1) reduces to

(3.2) \begin{equation} \begin{aligned} \min_{I\in\mathcal{I}}&\ \mathbb{E}[I(X)^2\cdot\psi_2(X)]-2\mathbb{E}[X\cdot I(X)\cdot \psi_1(X)]\\ \text{s.t.}&\ \mathbb{E}[I(X)\cdot\psi_1(X)]=x^{\ast}. \end{aligned}\end{equation}

Solving problem (3.2) is equivalent to solving its Lagrangian dual problem:

(3.3) \begin{equation} \min_{I\in\mathcal{I}}\ \mathbb{E}[I(X)^2\cdot\psi_2(X)]-2\mathbb{E}[X\cdot I(X)\cdot \psi_1(X)]+\lambda\cdot \mathbb{E}[I(X)\cdot \psi_1(X)]\end{equation}

where $\lambda\in\mathbb{R}$ is the Lagrangian coefficient.

The following lemma characterizes the optimal indemnity function to problem (3.3). In this lemma, $\unicode{x1d7d9}_{A}(t)$ is defined as the indicator function: $\unicode{x1d7d9}_{A}(t)=1$ if $t\in A$ and $\unicode{x1d7d9}_{A}(t)=0$ otherwise.

Lemma 3.1 Let Assumption 1 hold, and

\begin{equation*} L(t;I^{\ast},\lambda)=\int_{t}^{M}\psi_2(x)\cdot\left(I^{\ast}(x)-\frac{\psi_1(x)}{\psi_2(x)}\left(x-\frac{\lambda}{2}\right)\right)dF_X(x). \end{equation*}

Then, $I^{\ast}(x)=\int_{0}^{x}\eta^{\ast}(t)dt$ is an optimal solution to problem (3.3) if and only if

\begin{equation*} \eta^{\ast}(t)=\unicode{x1d7d9}_{D_{\lambda}}(t)+\xi(t)\cdot\unicode{x1d7d9}_{E_{\lambda}}(t), \end{equation*}

where

\begin{equation*} D_{\lambda}=\left\{t\;:\; L(t;I^{\ast},\lambda)<0\right\},\quad E_{\lambda}=\left\{t\;:\; L(t;I^{\ast},\lambda)=0\right\}, \end{equation*}

and $\xi(t)\in[0,1]$ is such that $I^{\ast}\in\mathcal{I}$ .

Lemma 3.1 characterizes the optimal marginal indemnity functions. However, the optimal indemnity functions given by Lemma 3.1 are implicit since $I^{\ast}$ also appears in $L(t;I^{\ast},\lambda)$ . Nevertheless, Lemma 3.1 provides insights about the explicit structure of $I^{\ast}$ , which will be derived in detail in the rest of this section. We remark that in deriving the above result, the calculus of variations plays important role. In recent years, such technique has been widely applied to obtain an implicit characterization of the optimal indemnity function under other preference functionals. See, for example, Chi and Wei (Reference Chi and Wei2020) and Chi and Zhuang (Reference Chi and Zhuang2020).

For now, let $\lambda$ be fixed and

\begin{equation*}\phi_{\lambda}(x)=\frac{\psi_1(x)}{\psi_2(x)}\left(x-\frac{\lambda}{2}\right).\end{equation*}

It is easy to verify that $\phi_{\lambda}(x)$ is the element-wise minimizer to problem (3.3).Footnote ¹ The following assumption is needed to proceed.

Under Assumption 2, $\phi_{\lambda}(x)$ is also continuously differentiable over [0,M]. As such, the whole domain [0,M] could be partitioned as per the first order derivative of $\phi_{\lambda}(x)$ such that

(3.4) \begin{equation}[0,M]=\bigcup_{i=1}^mS_{i,j_i},\end{equation}

where

(3.5) \begin{equation}j_i=\left\{\begin{aligned}&1,&\quad& \text{if}\ \phi'_{\!\!\lambda}(x)\in(1,\infty), \\&2,&\quad& \text{if}\ \phi'_{\!\!\lambda}(x)\in[0,1], \\&3,&\quad& \text{if}\ \phi'_{\!\!\lambda}(x)\in(-\infty,0),\end{aligned}\right.\end{equation}

and m is the smallest positive integer required for such a partition. Throughout the rest of this paper, m is assumed to be finite. Under this partition rule, we have $j_{i+1}\neq j_i$ and $|j_{i+1}-j_i|=1$ for $i=1,2,\dots,m-1$ . Let $x_{i-1}=\inf\left\{x\;:\; x\in S_{i,j_i}\right\}$ for $i=1,2,\dots,m$ , we have $0=x_0\le x_1\le\cdots\le x_{m-1}\le x_m=M$ . Intuitively, $x_i=\sup\left\{x\;:\;x\in S_{i,j_i}\right\}$ for $i=1,2,\dots,m$ . These points $\{x_i\}_{i=1,2,\dots,m-1}$ are referred to as the change points (see also Chi and Tan, Reference Chi and Tan2021). Our goal is to obtain the explicit structure of the optimal indemnity function over each piece $S_{i,j_i}$ . For the ease of presentation, the following layer-type indemnity function is defined:

\begin{equation*}I_{(a,b]}(x)=(x-a)_+-(x-b)_+\quad\text{where}\quad 0\le a\le b\le M.\end{equation*}

The following theorem gives the optimal parametric indemnity function over each $S_{i,j_i}$ for $i=1,2,\dots,m$ .

An illustration of the optimal indemnity function $I^{\ast}$ is shown in Figure 1, where the red line denotes $I^{\ast}$ and the green dashed line denotes the element-wise minimizer $\phi_{\lambda}$ to problem (3.3). We can see that $\phi_{\lambda}$ does not satisfy the incentive compatibility condition, so the optimal indemnity function $I^{\ast}$ is obtained based on the slope of $\phi_{\lambda}$ . In Figure 1, there are two change points, that is $x_1$ and $x_2$ , and the domain of loss is partitioned into three pieces. We obtain the parametric form of the optimal indemnity function on each piece by using Theorem 3.1.

• Over $[0,x_1]$ , $\phi'_{\!\!\lambda}(x)<0$ , so $j_1=3$ , and

  \begin{equation*} I^{\ast}(x)=x-I_{(\gamma_{1,3},\gamma_{1,3}+x_1-I^{\ast}(x_1)]}(x). \end{equation*}

• Over $[x_1,x_2]$ , $0\le \phi'_{\!\!\lambda}(x)\le 1$ , so $j_2=2$ , and

  \begin{equation*} I^{\ast}(x)=\min\left\{\max\left\{\phi_{\lambda}(x),I^{\ast}(x_2)+x-x_2,I^{\ast}(x_{1})\right\},I^{\ast}(x_{1})+x-x_{1},I^{\ast}(x_2) \right\}. \end{equation*}

• Over $[x_2,M]$ , $\phi'_{\!\!\lambda}(x)>1$ , so $j_3=1$ , and

  \begin{equation*} I^{\ast}(x)=I^{\ast}(x_2)+(x-\gamma_{3,1})_+. \end{equation*}

Figure 1: An illustration of an indemnity function $I^{\ast}$ that solves problem (3.3), when $m=3$ , $j_1=3$ , $j_2=2$ and $j_3=1$ .

Theorem 3.1 shows the applicability of Lemma 3.1 in practice and gives the explicit structure of the optimal indemnity function. It reduces the dimension of the original optimization problem from $\infty$ to at most $2m-1$ (i.e., $I^{\ast}(x_i)$ for $i=1,2,\dots,m-1$ , $\gamma_{i,1}$ and $\gamma_{i,3}$ for $i=1,2,\dots,m$ , and $\lambda$ ).

We remark that the partition (3.4) varies with respect to the Lagrangian coefficient $\lambda$ , which is a key parameter pertaining to the premium $\pi$ . Therefore, if the number of change points (i.e., $m-1$ ) is large, numerically optimizing the parameters in Theorem 3.1 is still computationally expensive. To slightly simplify the computation, the following proposition is given.

Proposition 3.1 Let Assumption 1 hold. If $\lambda>0$ , then for problem (3.3) the optimal indemnity function over $[0,\frac{\lambda}{2}]$ is given by

\begin{equation*} I^{\ast}(x)=(x-d)_+ \end{equation*}

for some $d\in[0,\frac{\lambda}{2}]$ .

Proposition 3.1 shows that the optimal indemnity function is of the stop-loss form in a neighborhood of 0 if $\lambda>0$ . To derive the general solution, only the interval $[\frac{\lambda}{2},M]$ needs to be partitioned as per (3.4) and (3.5).

4. Two special dependence structures

In this section, we study two special cases of our problem, that is when the counterparty risk Y is independent of X and when Y is a decreasing function of X. In both special cases, we will derive a much simpler solution.

4.1. Y is independent of X

In this section, we study the indemnity function for the case where Y is independent of X. This happens when the DM’s loss does not affect the solvency status of the insurer. In such a case, $\frac{\psi_1(x)}{\psi_2(x)}=\frac{\mathbb{E}[Y|X=x]}{\mathbb{E}[Y^2|X=x]}=\frac{\mathbb{E}[Y]}{\mathbb{E}[Y^2]}$ is a constant. Then $\phi'_{\!\!\lambda}(x)=\frac{\mathbb{E}[Y]}{\mathbb{E}[Y^2]}>1$ for any $x\in[0,M]$ . This implies that $m=1$ and there is no change point. Applying Theorem 3.1 leads to the following corollary.

Corollary 4.1 If Y is independent of X, then the optimal indemnity function to Problem 1 is given by

\begin{equation*} I^{\ast}(x)=(x-d)_+ \end{equation*}

for some $d\ge 0$ .

The optimality of stop-loss indemnity function is also verified in the expected utility framework. For example, Cummins and Mahul (Reference Cummins and Mahul2003) show the optimality of a stop-loss function in a situation where the recovery rate can only take 1 or 0 and both the DM and insurer have the same belief about the default probability. Bernard and Ludkovski (Reference Bernard and Ludkovski2012) extend the result of Cummins and Mahul (Reference Cummins and Mahul2003) by considering a budget constraint, where the stop-loss function is proved to be optimal again. In the above-mentioned works, the recovery rate does not need to be independent of the insurable loss.

4.2. Y is a decreasing function of X

In the literature, the recovery rate Y is generally assumed to be negatively correlated with X (see Bernard and Ludkovski Reference Bernard and Ludkovski2012; Li and Li Reference Li and Li2018). This is intuitive, as a larger loss would make the insurer more likely to default, which results in a smaller recovery rate. In this section, we analyze a special case where Y is a decreasing function X, that is $Y=g(X)$ where g is a decreasing function.Footnote ²

To simplify our discussion, we focus on the following situation.

Assumption 3

1. (i) there exists an $x_1\in[0,M]$ such that $g(x)=1$ for all $x\in(0,x_1]$ , and $g(x)<1$ for all $x\in(x_1,M]$ ;

2. (ii) $g(M)>0$ .

Assumption 3(i) states that the insurer will not default when the DM’s loss is less than some threshold. In the rare case where the realized loss is the largest, the insurer may default, but is able to sell its remaining assets and recover part of the indemnity to the DM. This leads Assumption 3(ii) to hold. Assumption 3 is for instance related to the model of Cai et al. (Reference Cai, Lemieux and Liu2014), when the insurer sells an insurance contract to only one DM. In such case, default happens if and only if the insurable loss exceeds a certain threshold.

Note that $\frac{\psi_1(x)}{\psi_2(x)}=\frac{\mathbb{E}[g(X)|X=x]}{\mathbb{E}[g(X)^2|X=x]}=\frac{g(x)}{g(x)^2}=\frac{1}{g(x)}$ . Thus, under Assumption 3

\begin{equation*}\phi_{\lambda}(x)=\left\{\begin{aligned}&x-\frac{\lambda}{2},&\quad& x\in[0,x_1],\\[5pt] &\frac{x-\frac{\lambda}{2}}{g(x)},&\quad& x\in(x_1,M].\end{aligned}\right.\end{equation*}

For $x>\max\left\{x_1,\frac{\lambda}{2}\right\}$ , we have

\begin{equation*}\phi'_{\!\!\lambda}(x)=\left(\frac{x-\frac{\lambda}{2}}{g(x)}\right)\!{}^{^{^{'}}}=-\frac{g'(x)}{g(x)^2}\left(x-\frac{\lambda}{2}\right)+\frac{1}{g(x)}\ge\frac{1}{g(x)}>1.\end{equation*}

Depending on the value of $\lambda$ , we have the following two sub-cases.

• Case 1: $\lambda\le 2x_1$ In this case, we have $\phi'_{\!\!\lambda}(x)=1$ for $x\in[0,x_1]$ and $\phi'_{\!\!\lambda}(x)>1$ for $x\in(x_1,M]$ . Therefore, $x_1$ is the only change point. Applying Theorem 3.1 and Proposition 3.1 gives

  \begin{equation*} I^{\ast}(x)=(\min\left\{x,d_2\right\}-d_1)_++(x-d_3)_+, \end{equation*}
  where $0\le d_1\le\frac{\lambda}{2}\le d_2\le x_1\le d_2\le M$ .

• Case 2: $\lambda>2x_1$ In this case, as per Proposition 3.1, for any $x\in[0,\frac{\lambda}{2}]$ we have $I^{\ast}(x)=(x-d_1)_+$ for some $d_1\in[0,\frac{\lambda}{2}]$ . For $x\in(\frac{\lambda}{2},M]$ , applying Theorem 3.1 leads to $I^{\ast}(x)=I^{\ast}(\frac{\lambda}{2})+(x-d_2)_+$ for some $d_2\in[\frac{\lambda}{2},M]$ . In conclusion, the optimal indemnity function for this case is given by

  \begin{equation*} I^{\ast}(x)=(\min\left\{x,\frac{\lambda}{2}\right\}-d_1)_++(x-d_2)_+, \end{equation*}
  where $0\le d_1\le\frac{\lambda}{2}\le d_2$ .

Note that the solutions for Case 1 and 2 are of similar formats. The above discussions are summarized in the next corollary.

Corollary 4.2 If Y is a decreasing function of X, under Assumption 3, the optimal indemnity function to Problem 1 is given by

\begin{equation*} I^{\ast}(x)=(\min\left\{x,a_2\right\}-a_1)_++(x-a_3)_+, \end{equation*}

where $0\le a_1\le a_2\le a_3\le M$ .

5. Numerical illustrations

In this section, we present a numerical example illustrating the main result of this paper. For the ease of discussion, we assume that the DM uses the traditional mean–variance criterion: $\mathcal{M}(z_1,z_2)=z_1+\frac{\mathcal{B}}{2}z_2$ . The premium is determined by the expectation premium principle, that is $\pi(I)=(1+\theta)\mathbb{E}[I(X)Y]$ for some $\theta>0$ . Moreover, we assume the following structure of $Y|X=x$ to depict the dependence between the recovery rate and loss:

(5.1) \begin{equation} \left\{ \begin{aligned} &Y=1,&\quad&\text{with probability}\ p(x) \\ &Y\sim U(0,1),&\quad&\text{with probability}\ 1-p(x), \end{aligned} \right.\end{equation}

where x is the realization of X and U(0,1) denotes the uniform distribution over [0,1). In this case, the random variable $Y|Y<1$ is independent of X. Such structure is inspired by Bernard and Ludkovski (Reference Bernard and Ludkovski2012), who study the case where Y takes one out of two values: $Y\in\left\{y_0,1\right\}$ for some $y_0\in[0,1]$ . As Y is usually negatively correlated with X, $p(\!\cdot\!)$ is usually a decreasing function. Then, the larger the loss is, the smaller is the probability of full recovery. Moreover, for any increasing function g, it holds that

\begin{align*}\mathbb{E}[g(Y)|X=x]&=g(1)p(x)+(1-p(x))\int_{0}^{1}g(y)dy \\ & =\int_{0}^{1}g(y)dy+p(x)(g(1)-\int_{0}^{1}g(y)dy),\end{align*}

which is decreasing with respect to x. This implies that Y is stochastically decreasing with respect to X. For simplicity, we assume that p(x) is subject to exponential decay, that is $p(x)=e^{-ax}$ for some $a>0$ .

Under these assumptions, we can easily get

\begin{equation*}\psi_1(x)=\mathbb{E}[Y|X=x]=\frac{1}{2}+\frac{1}{2}p(x),\quad \psi_2(x)=\mathbb{E}[Y^2|X=x]=\frac{1}{3}+\frac{2}{3}p(x).\end{equation*}

Furthermore,

\begin{equation*}\left(\frac{\psi_1(x)}{\psi_2(x)}\right)\!{}^{^{'}}=\frac{3}{2}\cdot\frac{ae^{-ax}}{(1+2e^{-ax})^2}> 0.\end{equation*}

Therefore $\frac{\psi_1(x)}{\psi_2(x)}\ge\frac{\psi_1(0)}{\psi_2(0)}=1$ .

Based on the value of $\lambda$ , we have the following cases.

• If $\lambda>0$ , applying Proposition 3.1 leads to

  \begin{equation*} I^{\ast}(x)=(x-d_1)_+ \end{equation*}
  on $[0,\frac{\lambda}{2}]$ for some $d_1\in[0,\frac{\lambda}{2}]$ . When $x>\frac{\lambda}{2}$ ,
  \begin{equation*} \phi'_{\!\!\lambda}(x)=\left(\frac{\psi_1(x)}{\psi_2(x)}\right)\!{}^{^{'}}\left(x-\frac{\lambda}{2}\right)+\frac{\psi_1(x)}{\psi_2(x)}>1. \end{equation*}
  Applying Theorem 3.1 leads to
  \begin{equation*} I^{\ast}(x)=I^{\ast}\left(\frac{\lambda}{2}\right)+(x-d_2)_+ \end{equation*}
  for some $d_2\ge\frac{\lambda}{2}$ .

• If $\lambda\le 0$ , then since $\phi'_{\!\!\lambda}(x)>1$ on [0,M], applying Theorem 3.1 leads to

  \begin{equation*} I^{\ast}(x)=(x-d_3)_+ \end{equation*}
  over the whole domain, where $d_3\in[0,M]$ .

Summarizing the above discussions leads to the general solution

\begin{equation*}I^{\ast}(x)=I_{a_{1\kern-1pt},a_2,a_3}(x)\;:\!=\;\left(\min\left\{x,a_2\right\}-a_1\right)_++(x-a_3)_+,\end{equation*}

where $0\le a_1\le a_2\le a_3\le M$ . Note that if $a_1=a_2$ or $a_2=a_3$ , this solution reduces to a stop-loss function. An illustrative $I^{\ast}$ is given by Figure 2.

Figure 2: Illustrative $I^{\ast}$ .

Under the traditional mean-variance criterion, our goal is to minimize

\begin{equation*}\begin{aligned}&\mathcal{M}(\mathbb{E}[L(I,\pi)], \text{Var}(L(I,\pi))) \\[5pt] =\;&\mathbb{E}[X]+\theta\mathbb{E}[I(X)Y]+\frac{\mathcal{B}}{2}\left\{\mathbb{E}[(X-I(X)Y)^2]-\left(\mathbb{E}[X-I(X)Y]\right)^2\right\}\\[5pt] =\;&\mathbb{E}[X]+\theta\mathbb{E}[I(X)\psi_1(X)]+\frac{\mathcal{B}}{2}\left\{\text{Var}(X)-2\mathbb{E}[XI(X)\psi_1(X)]+\mathbb{E}[I(X)^2\psi_2(X)]\right.\\[5pt] &\left.+2\mathbb{E}[X]\mathbb{E}[I(X)\psi_1(X)]-\mathbb{E}[I(X)\psi_1(X)]^2\right\}.\end{aligned}\end{equation*}

With the optimal indemnity function $I_{a_{1\kern-1pt},a_2,a_3}$ , the above problem reduces to

(5.2) \begin{align}\min_{0\le a_1\le a_2\le a_3\le M}\ & \theta\mathbb{E}[I_{a_{1},a_2,a_3}(X)\psi_1(X)]+\frac{\mathcal{B}}{2}\left\{\mathbb{E}[I_{a_{1\kern-1pt},a_2,a_3}^2(X)\psi_2(X)]\right. \nonumber\\[5pt] & \left.+2\mathbb{E}[X]\mathbb{E}[I_{a_{1\kern-1pt},a_2,a_3}(X)\psi_1(X)] -2\mathbb{E}[XI_{a_{1\kern-1pt},a_2,a_3}(X)\psi_1(X)]\right. \nonumber \\[5pt] & \left.-\mathbb{E}[I_{a_{1\kern-1pt},a_2,a_3}(X)\psi_1(X)]^2\right\}.\end{align}

To optimize $a_{1\kern-1pt},a_2$ and $a_3$ , we use the “fmincon” or “patternsearch” function in MATLAB. Under different loss distributions and volatility aversion parameters, the optimal $a_{1\kern-1pt},a_2$ and $a_3$ are presented in Tables 1 and 2.

Interestingly, in most of the cases in Tables 1 and 2, we get either $a_1=a_2$ or $a_2=a_3$ . This implies that the optimal indemnity function for the considered model settings are of the stop-loss type. For the sensitivity of the retention point with respect to model parameters (e.g., the mean loss, $\mathcal{B}$ , $\theta$ and a), we have the following observations.

• The retention point increases with respect to the mean loss. Usually a larger mean loss results in a larger premium. The DM may choose to increase the retention point to keep the premium affordable.

• A larger volatility aversion level leads to a smaller retention point. This is as expected since a smaller retention point implies that the DM cedes more risk to the insurer so that its retained risk is subject to less uncertainty.

• A larger safety loading parameter leads to a larger retention point. Relatively speaking, a larger safety loading parameter corresponds to a smaller volatility aversion level by looking at problem (5.2). Then as per the second bullet point, a smaller volatility aversion level leads to a larger retention point. From the perspective of premium, larger safety loading increases the premium. So the DM increases the retention point to maintain the affordable premium.

• A larger a leads to a smaller retention point. It is straightforward that a larger a leads to a smaller expected recovery rate, then the DM tends to cede out more risk to reduce the uncertainty of its retained risk.

Table 1 The effects of the mean loss and $\mathcal{B}$ on $(a_{1\kern-1pt},a_2,a_3)$ for exponentially distributed loss, $\theta=0.01$ and $a=0.001$ .

Table 2 The effects of $\theta$ and a on $(a_{1\kern-1pt},a_2,a_3)$ for exponentially distributed loss with mean 500 and $\mathcal{B}=0.005$ .