2. Model setup

2.1. A proportional insurance contract

An individual endowed with initial wealth $w_0$ faces an insurable risk X, interpreted as a loss. The risk X, defined on a probability space $(\Omega, \mathcal{G},\mathbb{P})$ , is a non-negative, bounded random variable with an essential infimum of 0 and an essential supremum of $M\gt 0$ .Footnote ² To reduce her risk exposure, she purchases a proportional insurance contract with indemnity $I_\theta(x)\,:\!=\,\theta x$ for some proportion $\theta\in[0,1]$ and insurance premium P, ceding partial risk to an insurer. Note that the admissible insurance premium must be non-negative and cannot exceed M. Thus, the proportional insurance contract is completely determined by the pair $(\theta,P)\in[0,1]\times[0,M]$ . The individual is risk averse and is endowed with a utility function u satisfying $u'(\cdot)\gt 0$ and $u''(\cdot)\lt 0$ on the domain $[w_0-2M,w_0]$ . A rational condition for purchasing this contract is that the individual’s expected utility is enhanced, that is,

\begin{align*} \mathbb{E}[u(w_{0}-X+I_\theta(X)-P)] \geqslant\mathbb{E}[u(w_{0}-X)].\end{align*}

Following Raviv (Reference Raviv1979), we assume that the insurer is risk averse and endowed with bounded initial wealth $W_1$ and a utility function v satisfying $v'(\cdot) \gt 0 $ and $v''(\cdot) \lt 0$ on the relevant domain, where $W_1$ is affected by the fluctuation of his existing business and may be random. The insurer will not offer this contract unless his welfare is weakly improved, that is,

\begin{align*} \mathbb{E}[v(W_{1}+P-(1+\tau)I_\theta(X))] \geqslant \mathbb{E}[v(W_{1})],\end{align*}

where $\tau\geqslant 0$ is the deadweight cost rate. The factor $\tau$ is used to include the overhead costs of providing insurance by the insurer, such as marketing and administration costs.

For any proportion $\theta\in[0,1]$ , these two rationality conditions are equivalent to

(2.1) \begin{eqnarray}P_{-}(\theta) \leqslant P \leqslant P_{+}(\theta),\end{eqnarray}

where $P_{-}(\theta)$ and $P_{+}(\theta)$ are the solutions to the following equations

(2.2) \begin{eqnarray} \mathbb{E}[v(W_{1}+P-(1+\tau)I_\theta(X))] =\mathbb{E}[v(W_{1})],\, P\geqslant 0,\end{eqnarray}

and

(2.3) \begin{eqnarray}\mathbb{E}[u(w_{0}-X+I_\theta(X)-P)] = \mathbb{E}[u(w_{0}-X)],\, P\geqslant 0,\end{eqnarray}

respectively. In other words, the proportional insurance contract is acceptable only if the insurance premium is less than the maximum amount the insured is willing to pay and exceeds the minimum level required by the insurer. Thus, compared with the status quo (i.e., $(\theta, P)=(0,0)$ ), Equation (2.1) guarantees that positive insurance is demanded only if both insured and insurer will (weakly) benefit from the insurance transaction. Note that $P_{-}(\theta)$ and $P_{+}(\theta)$ are unique due to the strict increasingness of u and v.

Clearly, the final insurance contract depends heavily on the negotiation between the insured and the insurer. In the literature, Nash bargaining is widely used to model the negotiation between two parties. Thus, we will give a brief introduction of Nash bargaining in the next subsection.

2.2. Asymmetric Nash bargaining

A two-person Nash bargaining solution was first introduced by Nash (Reference Nash1950) and then extended by Kalai (Reference Kalai1977) to the asymmetric case. Let $\mathbf{S}$ be the set of all feasible bargaining outcome vectors for the two parties and is a compact convex subset of $\mathbb{R}^2$ . A bargaining problem is composed of a pair $(\mathbf{a_0},\mathbf{S})$ , where the 2-dimensional vector $\mathbf{a_0}\in \mathbf{S}$ represents the status quo before bargaining (“disagreement point”), and there exists at least one point $\mathbf{x}=(x_1,x_2)^T\in\mathbf{S}$ such that $x_1\gt a_{01}$ and $x_2\gt a_{02}$ . Let $\mathcal{B}$ denote the collection of all 2-person bargaining problems. We use a map $\mu: \mathcal{B}\mapsto \mathbb{R}^2$ to characterize the bargaining process. According to Kalai (Reference Kalai1977), the negotiation is referred to as asymmetric Nash bargaining if it satisfies the following four axioms:

• • Feasibility: $\mu(\mathbf{a_0},\mathbf{S})\in \mathbf{S}$ and $\mu(\mathbf{a_0},\mathbf{S})\gt \mathbf{a_0}$ , where $\mu(\mathbf{a_0},\mathbf{S})\gt \mathbf{a_0}$ means a strict inequality in both components.

• • Invariance under change of scale of utilities: If $G: \mathbb{R}^2\mapsto\mathbb{R}^2$ is such that $G (\mathbf{a}) = (c_1a_1 + b_1, c_2a_2 + b_2)^T$ where $c_i \gt 0$ , then $G(\mu(\mathbf{a_0},\mathbf{S})) = \mu (G (\mathbf{a_0}), G (\mathbf{S}))$ .

• • Independence of irrelevant alternatives: For all two bargaining problems $(\mathbf{a_0},\mathbf{S})$ and $(\mathbf{a_0},\mathbf{U})$ such that $\mathbf{S}\subset \mathbf{U}$ and $\mu(\mathbf{a_0},\mathbf{U})\in\mathbf{S}$ , it holds that $\mu (\mathbf{a_0},\mathbf{S}) = \mu (\mathbf{a_0},\mathbf{U})$ .

• • Pareto optimality: If $\mu (\mathbf{a_0},\mathbf{S})=(z_1, z_2)^T$ and $y_i\geqslant z_i$ for $i=1,2$ , then either $\mathbf{y}\notin\mathbf{S}$ or $y_i=z_i$ for $i=1,2$ .

Kalai (Reference Kalai1977) shows that $\mu$ satisfies the above four axioms if and only if there exists a $\delta\in(0,1)$ such that $\mu(\mathbf{a_0},\mathbf{S})$ is the unique point in $\mathbf{S}$ satisfying

\begin{eqnarray*}\mu(\mathbf{a_0},\mathbf{S})= \arg\max_{\mathbf{x}\in \mathbf{S}}\, (x_1-a_{01})^{1-\delta}(x_2-a_{02})^\delta.\end{eqnarray*}

This solution is referred to as the asymmetric Nash bargaining solution. Here, $\delta \in (0,1)$ represents the bargaining power of the second person relative to the first person. This means that the second person has more power in the negotiation if the value of $\delta$ becomes larger (see, e.g., Kalai Reference Kalai1977). Feasibility is an attractive property, as it implies that the asymmetric Nash bargaining solution strictly exceeds $\mathbf{a_0}$ in every component. This strict inequality does not hold true for Bowley or competitive equilibria in the context of distortion risk measures, as shown by Boonen and Ghossoub (Reference Boonen and Ghossoub2023).

3. Nash insurance bargaining

In this paper, we analyze the insurance negotiation between a risk-averse insured and a risk-averse insurer using the asymmetric Nash bargaining framework. More specifically, the status quo is characterized by

\begin{eqnarray*}\mathbf{a_0}=\binom{\mathbb{E}[u(w_0-X)]}{\mathbb{E}[v(W_1)]}.\end{eqnarray*}

Let the feasible set of insurance contracts be given by

\begin{eqnarray*}\mathcal F=\left\{(\theta,P)\,:\,\theta\in[0,1], P_{-}(\theta)\leqslant P\leqslant P_{+}(\theta)\right\}.\end{eqnarray*}

Then, the set of feasible bargaining outcome vectors under proportional insurance can be given by

\begin{eqnarray*}\mathbf{S}=\left\{\binom{\mathbb{E}[u(w_0-X+I_\theta(X)-P)]}{\mathbb{E}[v(W_1+P-(1+\tau)I_\theta(X))]}\,:\, (\theta,P)\in\mathcal F \right\}.\end{eqnarray*}

Since $(0,0)\in\mathcal{F}$ , it follows that $\mathbf{a_0}\in \mathbf{S}$ .

Let $\mathcal{PO}\subset\mathcal F$ be the class of all $(\theta,P)$ such that there is no $(\theta',P')\in\mathcal F$ satisfying

(3.1) \begin{align} \mathbb{E}[u(w_{0}-X+I_{\theta'}(X)-P')] &\geqslant \mathbb{E}[u(w_{0}-X+I_{\theta}(X)-P)],\end{align}

(3.2) \begin{align} \mathbb{E}[v(W_{1}+P'-(1+\tau)I_{\theta'}(X))] &\geqslant \mathbb{E}[v(W_{1}+P-(1+\tau)I_{\theta}(X))], \end{align}

with at least one inequality being strict. Denoting the frontier of $\mathbf{S}$ by $\partial \mathbf{S}$ , we have

\begin{equation*}\partial \mathbf{S}=\left\{ \binom{\mathbb{E}[u(w_0-X+I_\theta(X)-P)]}{\mathbb{E}[v(W_1+P-(1+\tau)I_\theta(X))]}\,:\, (\theta,P)\in \mathcal{PO}\right\} . \end{equation*}

Next, we analyze whether the status quo $\mathbf{a_0}$ belongs to the frontier of $\mathbf{S}$ .

Proposition 1. Under the assumption of proportional insurance, the status quo is Pareto optimal if and only if

(3.3) \begin{equation}\frac{\mathbb{E}[u'(w_{0}-X)X]}{\mathbb{E}[u'(w_{0}-X)]} \leqslant (1+\tau)\frac{\mathbb{E}[v'(W_1)X]}{\mathbb{E}[v'(W_1)]}.\end{equation}

From the above proposition, we know that no insurance will be purchased if the insured’s marginal welfare improvement $\frac{\mathbb{E}[u'(w_{0}-X)X]}{\mathbb{E}[u'(w_{0}-X)]}$ is less than the marginal cost from the insurer $(1+\tau)\frac{\mathbb{E}[v'(W_1)X]}{\mathbb{E}[v'(W_1)]}$ . In particular, when $\tau$ is sufficiently high, insurance becomes unattractive. Furthermore, when $W_1$ is independent of X, some interesting phenomena can be observed:

• • The right-hand side of Equation (3.3) degenerates to $(1+\tau)\mathbb{E}[X]$ such that this necessary and sufficient condition is no longer affected by the degree of the risk aversion of the insurer. Notably, Proposition 3 in Braun and Muermann (Reference Braun and Muermann2004) also states that when the insurer is risk neutral, no insurance will be purchased if this condition is satisfied.

• • Note that $u'(w_{0}-X)$ and X are comonotonic, and thus the left-hand side of Equation (3.3) is larger than $\mathbb{E}[X]$ by the Hardy-Littlewood inequality (Hardy et al., Reference Hardy, Littlewood and Pólya1952). This means that if $\tau=0$ , the status quo is Pareto optimal only if X is almost surely deterministic.

Further, if $W_1$ is stochastically increasing in X (denoted by $W_1\uparrow_{st}X$ ),Footnote ³ we have

\begin{align*}\mathbb{E}[v'(W_1)X]/\mathbb{E}[v'(W_1)]\leqslant \mathbb{E}[X]\end{align*}

such that Equation (3.3) fails to be satisfied for a relatively small $\tau$ . This is intuitive: if the insurer can hedge existing business by underwriting new risks, he will provide insurance when the deadweight cost is low. For the opposite case of $W_1\downarrow_{st}X$ , it is harder to evaluate because the right-hand side of Equation (3.3) also exceeds $\mathbb{E}[X]$ .

To ensure the feasibility of Nash bargaining, we impose the following assumption, which requires the deadweight cost rate $\tau$ to be sufficiently small.

In addition to the Pareto inefficiency of the status quo, Nash bargaining problems also require the set $\mathbf{S}$ to be compact and convex. This is shown in the following proposition.

From the above proposition, Nash bargaining is feasible if Assumption 1 is satisfied. The asymmetric Nash bargaining solution solves the following optimization problem:

(3.4) \begin{eqnarray} &&\max _{\theta \in [0,1]\atop P_{-}(\theta) \leqslant P \leqslant P_{+}(\theta)} \left\{\mathbb{E}[v(W_{1}+P-(1+\tau)I_\theta(X))]- \mathbb{E}[v(W_{1})]\right\}^{\delta} \nonumber\\&&\qquad\qquad\qquad\quad \times\left\{\mathbb{E}[u(w_{0}-X +I_{\theta}(X)-P)] -\mathbb{E}[u(w_{0}-X)]\right\}^{1-\delta}\end{eqnarray}

for some $\delta\in(0,1)$ , where $\delta$ represents the bargaining power of the insurer. Obviously, the optimization objective function is continuous in $\theta$ and P and equals to zero if P is equal to either $P_{-}(\theta)$ or $P_{+}(\theta)$ . Thus, the Nash bargaining solutions $(\theta^*,P^*)$ must exist and satisfy

(3.5) \begin{eqnarray} \theta^*\in\Theta\,:\!=\, \{\theta \in (0,1] \,:\, P_+(\theta) \gt P_-(\theta) \}\quad\text{and}\quad P_{-}(\theta^*)\lt P^*\lt P_{+}(\theta^*)\end{eqnarray}

due to Assumption 1. The set $\Theta$ plays an important role in deriving optimal solutions, and we provide an alternative characterization in the following proposition.

Proposition 3. Under Assumption 1,

\[\Theta=\left\{\begin{array}{l@{\quad}l}(0,1],&\text{if}\, P_+(1)\gt P_{-}(1),\\(0,\theta_0),&\text{otherwise}\end{array}\right.\]

for some $\theta_0\in(0,1]$ .

Example 1. Let

\begin{align*}w_0=20, u(w) =-w^{-1},\, v(w) =w^{0.6},\end{align*}

and $W_1=100$ almost surely. Moreover, assume that the loss X has the cumulative distribution function

\begin{align*}F_X(x)=\frac{5}{6}+\frac{4}{7}\int_0^x\frac{10^3}{(y+10)^4}dy,\,x\in[0,10].\end{align*}

Clearly, $\mathbb{P}(X=0)=\frac{5}{6}$ and $M=10$ . Assumption 1 is satisfied whenever $\tau\lt 0.817$ , since $\frac{\mathbb{E}[u'(w_{0}-X)X]}{\mathbb{E}[u'(w_{0}-X)]\mathbb{E}[X]}=1.817$ .

Figure 1. The functions $ P_+(\theta)$ and $P_-(\theta)$ corresponding to Example 1.

Based on the above assumptions, it is easy to calculate $ P_+(\theta)$ and $P_-(\theta)$ numerically. For $\tau=0.4$ , we display these two functions in Figure 1. The set $\Theta$ can be determined for different values of the cost rate $\tau$ . More specifically, when $\tau=0.3$ , we have $\theta_0=1$ , and thus $\Theta=(0, 1]$ . The supremum of $\Theta$ changes to approximately $0.773$ if the value of $\tau$ is set to be $0.4$ (see Figure 1). If $\tau$ further increases to $0.5$ , then $\theta_0$ decreases to approximately $0.540 $ . A similar set $\Theta$ is obtained when the insurer’s risk attitude is changed from power utility function to

\begin{align*}v(w) =1-e^{-aw}\end{align*}

for some positive a. The outcomes of $\theta_0$ in those experiments are presented in Table 1.

Table 1. The supremum of $\Theta$ , given by $\theta_0$ , when the insurer’s risk preference follows an exponential utility function. Here, the value $-\infty$ corresponds to the cases that Assumption 1 is violated.

Now, we can solve the optimization problem (3.4) and obtain the Nash bargaining solutions in the following proposition.

Proposition 4. Under Assumption 1, the optimal proportional insurance contract $(\theta^*,P^*)$ that solves Problem (3.4) is unique. It satisfies the first-order condition

(3.6) \begin{align}\left\{\begin{array}{ll}\!(1-\delta) \times\dfrac{\mathbb{E}[u'(w_{0}-X +I_{\theta^*}(X)-P^*)]}{\mathbb{E}[u(w_{0}-X +I_{\theta^*}(X)-P^*)] -\mathbb{E}[u(w_{0}-X)]}&=\delta\times\dfrac{\mathbb{E}[v'(W_{1}+P^*-(1+\tau)I_{\theta^*}(X))]}{\mathbb{E}[v(W_{1}+P^*-(1+\tau)I_{\theta^*}(X))]- \mathbb{E}[v(W_{1})]},\\&\\\qquad\qquad\qquad\dfrac{\mathbb{E}[u'(w_{0}-X +I_{\theta^*}(X)-P^*)X]}{\mathbb{E}[u'(w_{0}-X +I_{\theta^*}(X)-P^*)]}&=(1+\tau)\dfrac{\mathbb{E}[v'(W_{1}+P^*-(1+\tau)I_{\theta^*}(X))X]}{\mathbb{E}[v'(W_{1}+P^*-(1+\tau)I_{\theta^*}(X))]},\end{array}\right.\end{align}

if

(3.7) \begin{equation}P_{+}(1)\leqslant P_{-}(1)\quad\text{or}\quad\mathbb{E}[X]\lt (1+\tau)\frac{\mathbb{E}\left[v'(W_1+\hat{P}-(1+\tau)X)X\right]}{\mathbb{E}\left[v'(W_1+\hat{P}-(1+\tau)X)\right]},\end{equation}

and equals to $(1, \hat{P})$ otherwise, where $\hat{P}$ is a solution to the following equation

(3.8) \begin{equation}\frac{(1-\delta)u'(w_{0}-P)}{u(w_{0}-P) -\mathbb{E}[u(w_{0}-X)]}=\frac{\delta\mathbb{E}[v'(W_{1}+P-(1+\tau)X)]}{\mathbb{E}[v(W_{1}+P-(1+\tau)X)]- \mathbb{E}[v(W_{1})]},\,P\geqslant 0.\end{equation}

From the above proposition, we can see that full insurance is unlikely to be optimal, except for very extreme cases such as that of a quite small $\tau$ and the insurer’s risk exposure $W_1-(1+\tau)X$ being stochastically increasing in X. Especially when $W_1\downarrow_{st}X$ , which includes the independent case between $W_1$ and X, we have

\begin{align*}\frac{\mathbb{E}\left[v'(W_1+\hat{P}-(1+\tau)X)X\right]}{\mathbb{E}\left[v'(W_1+\hat{P}-(1+\tau)X)\right]}\gt \mathbb{E}[X]\end{align*}

such that condition (3.7) is satisfied even if the deadweight cost rate $\tau$ is zero, then partial insurance is optimal. Moreover, the optimal insurance solution depends not only on the degree of risk aversion of both parties but also on the bargaining power $\delta$ .

4. Comparative statics analysis

In this section, we will carry out comparative statics analysis to investigate the effect of some interesting factors on the Nash bargaining solution.

First, we investigate the effect of the degree of the insured’s risk aversion on the optimal rate and insurance premium of proportional insurance. In the literature, the insured’s degree of risk aversion is often measured by Arrow-Pratt coefficient of absolute risk aversion

\begin{eqnarray*}\mathcal{A}_u(w)=-\frac{u''(w)}{u'(w)}.\end{eqnarray*}

The insured’s risk preference is called to exhibit constant absolute risk aversion (CARA) if $\mathcal{A}_u(w)$ is a constant function. It is equivalent to that the insured has an exponential utility function with $\mathcal{A}_u(w)=a$ for some positive $a$ .

To proceed, we consider a very special case. If there are no deadweight costs and risk preferences of the two agents in the contract exhibit CARA, then the optimal proportion has a closed-form expression and is independent of the bargaining power.

The result in the above proposition can also be verified through Equation (3.6). Under these strict assumptions, we can see that the optimal proportion $\theta^*$ does not depend on the bargaining power $\delta$ and increases in $\lambda_u$ . In other words, as the insured becomes more risk averse, more insurance coverage is demanded. We claim that this effect can also be held for general cases, while the optimal parameters $\theta^*$ and $P^*$ may not have closed-form expressions. Another insured’s risk attitude with increasing and concave utility function $\tilde{u}$ is called to be more risk averse than the insured’s with utility function u if $\mathcal{A}_{\tilde{u}}(w)\geqslant \mathcal{A}_u(w)$ for any w. Equivalently, there exists a twice differentiable increasing concave function $\kappa$ such that $\tilde{u}(w)=\kappa(u(w))$ ; see Proposition 2 in Gollier (Reference Gollier2001). Furthermore, the utility function u is said to exhibit a DARA risk preference if $\mathcal{A}_u(w)$ is decreasing.

The above proposition can be explained as follows: As the insured becomes more risk averse, she is willing to pay more insurance premium to mitigate the risk and transfer more risk to the insurer. Notably, a similar finding is obtained by Proposition 2 in Chi et al. (Reference Chi, Hu, Zhao and Zheng2024), which assumes a risk-neutral insurer. The above proposition extends the result to the case of a risk-averse insurer.

Next, we analyze the effects on the welfare improvement and the Nash bargaining solution by the change of the insurance market structure.

The above proposition indicates that the insurer will receive a larger reward from the contract negotiation as he becomes more powerful. However, this comes at the cost of a reduction in the insured’s welfare improvement. As more reward is asked by the insurer, the insured has to pay more extra cost to cede the risk. Equivalently, the insured’s initial wealth is relatively reduced. Under the DARA assumption of the insured’s risk preference, the insured becomes more risk averse and would like to cede more risk. In other words, the increase in the insurer’s bargaining power leads to more insurance demand. Thus, this proposition can be used to explain the phenomenon of the overinsurance preference observed in practice without incorporating behavioral elements (e.g., Braun and Muermann Reference Braun and Muermann2004). It is necessary to point out that a similar finding is also obtained by Proposition 5 in Chi et al. (Reference Chi, Hu, Zhao and Zheng2024) under the assumption of a risk-neutral insurer. However, we consider a risk-averse insurer with random initial wealth satisfying $-W_1\uparrow_{hr}X$ instead.

Finally, we assume that $W_1=w_1$ almost surely and attempt to analyze the effect on the optimal insurance coverage by the change of the insurer’s initial wealth $w_1$ .

When the insurer’s initial wealth increases, he will become less risk averse under the DARA assumption of his risk preference and charge less insurance premium. As the insurance becomes less costly, the insured would cede more risk, as stated in the above proposition.

5. Numerical examples

In this section, we provide examples to illustrate the theoretical results from the previous two sections. Specifically, we adopt the setting of Example 1 as our benchmark assumptions. That is,

\begin{align*}w_0=20, u(w)=-w^{-1},\quad v(w) =w^{0.6}\quad \text{and}\quad\tau=0.4,\end{align*}

and $W_1=100$ almost surely.

We show the insurance contract corresponding to the asymmetric Nash bargaining solution in Figure 2. In this figure, we display the optimal insurance contract $(\theta, P)$ under the asymmetric Nash bargaining for $\tau=0.3$ and $\tau=0.4$ . The contracts are shown as functions of the bargaining power of the insurer $\delta$ . Consistent with Proposition 7, the numerical results show that both the coinsurance rate $\theta$ and the premium P strictly increase with $\delta$ . Thus, greater insurance coverage is induced when the insurer has more bargaining power. In addition, we note that the result in Proposition 5 may not hold if the assumptions on $\tau=0$ and the CARA risk preferences of the insured and the insurer are relaxed. The difference between the cases $\tau=0.3$ and $\tau=0.4$ can be described as follows. When the deadweight cost rate $\tau$ is lower, insurance becomes more attractive, as reflected by a larger coinsurance rate of approximately 0.48 for $\tau=0.3$ , compared to approximately 0.36 when $\tau=0.4$ . As the insurance coverage gets larger, the corresponding insurance premium also becomes higher.

Figure 2. The optimal parameter pair $(\theta, P)$ in the asymmetric Nash bargaining solution as a function of the insurer’s bargaining power $\delta\in(0,1)$ , with $\tau=0.3$ (left) and $\tau=0.4$ (right).

Next, we present a sensitivity analysis to illustrate the comparative statics. For related problems involving different parameter choices for $\delta$ or minor adjustments to the relative risk-aversion parameters in u or v, we find that the pattern of the optimal insurance contract $(\theta, P)$ is roughly consistent with Figure 2. That is, both $\theta$ and P increase with the bargaining power, albeit only slightly. We now present four additional examples, focusing on the following sensitivities: (1) the impact of the insured’s risk aversion, (2) the use of a different class of utility functions, (3) the effect of the insurer’s initial wealth, and (4) the effect of background risk. Throughout the following examples, we fix $\tau=0.4$ and $w_0=20$ .

First, we examine the effect of the insured’s risk-aversion parameter. Let $W_1=100$ a.s., $v(w)=w^{0.6}$ , and let the bargaining power be $\delta=0.5$ . The utility function u of the insured is given by

(5.1) \begin{equation} u(w)=\left\{\begin{array}{ll}\frac{w^{1-\gamma}}{1-\gamma}&\text{ if }\gamma\neq 1,\\ \ln(w),&\text{ if }\gamma= 1,\end{array}\right. \end{equation}

which is the constant relative risk aversion (CRRA) utility function with coefficient $\gamma\gt 0$ . Thus, the utility function of the insurer $v(w)=w^{0.6}$ belongs to the same class, with parameter $\gamma=0.4$ . In Figure 3, we show the optimal insurance contract as a function of the risk-aversion parameter $\gamma$ of the insured. We observe that as the insured becomes more risk averse, the insurance coverage increases, and the insurance premium rises disproportionately.

Figure 3. The optimal parameter pair $(\theta, P)$ in the asymmetric Nash bargaining solution as a function of the risk-aversion parameter $\gamma$ of the insured, with $\delta=0.5$ and $\tau=0.4$ .

Second, we present a case in which the insurance coverage no longer increases in the bargaining power. We assume that the utility function of the insured is given as

(5.2) \begin{equation}u(w)=w-\frac{\beta}{2} w^2.\end{equation}

Here, $\beta=0.05$ controls the degree of risk aversion. We can easily verify that this utility function is increasing for wealth levels within the relevant range. However, the Arrow-Pratt coefficient of absolute risk aversion $\mathcal A_u$ is increasing in w and the DARA assumption is not met. The optimal insurance contract is displayed in Figure 4. We find that the optimal insurance coverage $\theta$ is indeed decreasing in the bargaining power, while the insurance premium is increasing. Thus, if the insurer gets more bargaining power, he will charge a higher premium for less coverage. This represents a clear advantage for the insurer.

Figure 4. The optimal parameter pair $(\theta, P)$ in the asymmetric Nash bargaining solution as a function of the bargaining power $\delta\in(0,1)$ , where $\tau=0.4$ , $v(w)=w^{0.6}$ , and the utility function u is as in Equation (5.2) with $\beta=0.05$ .

Third, we study the effect of the initial wealth of the insurer. Let the utility function of the insured u be given as in Equation (5.1) with coefficient $\gamma=2$ . For two different utility functions v of the insurer, we show the optimal insurance contracts as a function of the initial wealth of the insurer in Figure 5. Proposition 8 shows that if v is DARA and u CARA, then under Assumption 1, the optimal rate of proportional insurance $\theta^*$ increases in the insurer’s initial wealth $w_1$ . This pattern is also evident in Figure 5(a), where both the insured and the insurer have DARA utility functions. However, Figure 5(b) shows that this pattern does not hold when v is not a DARA utility function but instead takes the functional form in Equation (5.2) with $\beta=0.01$ , which results in an increasing Arrow-Pratt coefficient of absolute risk aversion $\mathcal A_v$ for relevant wealth levels.

Figure 5. The optimal parameter pair $(\theta, P)$ in the asymmetric Nash bargaining solution as a function of the initial wealth of the insurer $w_1$ , with $\tau=0.4$ and $\delta=0.5$ . Here, we use u as in Equation (5.1) with $\gamma=2$ .

Fourth, and finally, we study the effect of background risk of the insurer. Recall that in our baseline example, we let the utility functions of the insured u and the insurer v be given as in Equation (5.1) with $\gamma=2$ for the insured and $\gamma=0.4$ for the insurer. We model background risk via the following assumption:

(5.3) \begin{equation}W_1=100-k X,\end{equation}

Here, $k\in\mathbb{R}$ measures the extent of background risk. If $k\geqslant 0$ , then $-W_1\uparrow_{hr}X$ . We show the optimal insurance contract as a function of $k\in[\!-\!2,2]$ in Figure 6. We interpret a larger value of k as indicating greater background risk. Additionally, $k\geqslant0$ implies that new insurance risk is not attractive for the insurer due to a lack of diversification opportunities. From Figure 6, we observe that larger background risk leads to less coverage and a lower corresponding insurance premium. Additionally, we do not observe any discontinuity at $k=0$ . Thus, a small change from positive to negative dependence, or vice versa, has little impact on the optimal insurance contract

Figure 6. The optimal parameter pair $(\theta, P)$ in the asymmetric Nash bargaining solution as a function of the weight k that measures background risk (see Equation (5.3)), where $\tau=0.4$ , $\delta=0.5$ , $u(w)=-w^{-1}$ , and $v(w)=w^{0.6}$ .

6. Concluding remarks

In this paper, we study optimal insurance design under asymmetric Nash bargaining, assuming that both the insured and the insurer are risk averse and allowing the insurer’s initial wealth to be random. To simplify the analysis, ensure the feasibility of asymmetric Nash bargaining, and facilitate comparative statics, we focus on proportional insurance contracts. After obtaining a necessary and sufficient condition for Pareto optimality of the status quo, we derive the optimal Nash bargaining solutions when this condition is not satisfied. We show that when the insurer’s initial wealth is negatively dependent with the insurable risk, full insurance cannot be optimal, even if the deadweight cost is set to zero. We further find that insurance coverage increases with the insured’s risk aversion or the insurer’s bargaining power if the insurer’s initial wealth decreases in the insurable risk in the sense of the reversed hazard rate order and both parties exhibit DARA risk preferences. In particular, when the insured has a CARA utility function, greater insurance coverage is induced by higher initial wealth of the insurer.

We acknowledge that our analysis relies heavily on the assumption of proportional insurance. It would be interesting to revisit this problem using other types of insurance contracts. However, stop-loss insurance, which is widely used in the actuarial science literature, may not be a suitable choice. This is because the corresponding set of bargaining outcomes may not be convex, which prevents the direct application of the representation theorem for asymmetric Nash bargaining in Kalai (Reference Kalai1977). On the other hand, this paper analyzes the effect of changes in the insurer’s constant initial wealth on the optimal insurance coverage under the strict assumption that the insured has a CARA utility function. It would be valuable to investigate whether this comparative statics result remains valid when the CARA assumption on the insured’s risk preference is relaxed. We leave these questions to future research.