3.1. Solution to the optimization problem of the insurer

In this subsection, we solve the optimization problem of the insurer for each admissible safety loading $\theta_R\in[0,\theta^{\max}]$ . We face two challenges in solving the optimization problem $(\mathcal{P}_I)$ : the insurer cannot invest in $S_2$ , hence, the optimization problem $(\mathcal{P}_I)$ is constrained, and it has a fixed-term investment in the put option.

The procedure of solving the optimization problem of the insurer $(\mathcal{P}_I)$ is as follows. First, we consider a family of auxiliary financial markets as per Cvitanić and Karatzas (Reference Cvitanić and Karatzas1992). In each auxiliary market, we can solve an unconstrained optimization problem with the generalized martingale method introduced by Desmettre and Seifried (Reference Desmettre and Seifried2016), that is, we find the optimal investment strategy and the optimal reinsurance strategy. Second, we find the optimal auxiliary market for which the solution to the unconstrained optimization problem coincides with the solution to the constrained optimization problem.

Auxiliary Market. The financial market consisting of $S_0$ , $S_1$ , and $S_2$ is referred to as the basic financial market $\mathcal{M}$ . Next, we introduce the concept of an auxiliary market as per Cvitanić and Karatzas (Reference Cvitanić and Karatzas1992). Consider a convex set $K=\mathbb{R}\times\{0\}$ . Its support function is given by $\delta:\mathbb{R}^2\to\mathbb{R}\cup\{+\infty\}$ with

\begin{align*} \delta(x)\,:\!=\,-\inf_{y\in K}\left(x^\top y\right)=-\inf_{y_1\in\mathbb{R}}(x_1y_1)=\begin{cases} 0, & \text{if }x_1=0,\\ +\infty, &\text{otherwise}. \end{cases} \end{align*}

The barrier cone of K is defined by:

\begin{align*} \tilde{K}\,:\!=\,\left\{x\in\mathbb{R}^2|\delta(x)<+\infty\right\}=\left\{x\in\mathbb{R}^2|x_1=0\right\}=\{0\}\times\mathbb{R}. \end{align*}

For $x\in\tilde{K}$ , we have $\delta(x)=0$ . Let the class of $\mathbb{R}^2$ -valued dual processes be given by:

\begin{align*} \mathcal{D}\,:\!=\,\bigg\{\lambda=(\lambda(t))_{t\in[0,T]}\text{ prog. measurable}\bigg|\mathbb{E}\bigg[\int_0^T\lVert\lambda(t)\rVert^2dt\bigg]<\infty,\;\mathbb{E}\bigg[\int_0^T\delta(\lambda(t))dt\bigg]<\infty\bigg\}. \end{align*}

It holds for $\lambda\in\mathcal{D}$ that $\lambda(t)\in\tilde{K}$ $\mathbb{Q}$ -a.s for all $t\in[0,T]$ , that is, $\lambda_1(t)=0$ $\mathbb{Q}$ -a.s. for all $t\in[0,T]$ .

Let $\lambda\in\mathcal{D}$ . In the auxiliary market $\mathcal{M}_\lambda$ the dynamics of assets is given by:

\begin{align*} dS_0^\lambda(t)=&S_0^\lambda(t)(r+\delta(\lambda(t)))dt=S_0^\lambda(t)rdt,\\[4pt] dS_1^\lambda(t)=&S_1^\lambda(t)\left[(\mu_1+\lambda_1(t)+\delta(\lambda(t)))dt+\sigma_1dW_1(t)\right]=S_1^\lambda(t)(\mu_1dt+\sigma_1dW_1(t)),\\[4pt] dS_2^\lambda(t)=&S_2^\lambda(t)\left[(\mu_2+\lambda_2(t)+\delta(\lambda(t)))dt+\sigma_2\left(\rho dW_1(t)+\sqrt{1-\rho^2}dW_2(t)\right)\right]\\[4pt] =&S_2^\lambda(t)\left[(\mu_2+\lambda_2(t))dt+\sigma_2\left(\rho dW_1(t)+\sqrt{1-\rho^2}dW_2(t)\right)\right], \end{align*}

since $\delta(\lambda(t))=0$ and $\lambda_1(t)=0$ . We denote the market price of risk and the discount factor (also called pricing kernel) in the auxiliary market by:

\begin{align*} \gamma_\lambda(t)\,:\!=\,\gamma+\sigma^{-1}\lambda(t)\,\,\text{and}\,\, \widetilde{Z}_\lambda(t)\,:\!=\,\exp\bigg(-rt-\frac{1}{2}\int_0^t\lVert\gamma_\lambda(s)\rVert^2ds-\int_0^t\gamma_\lambda(s)^\top dW(s)\bigg). \end{align*}

To apply the generalized martingale method introduced by Desmettre and Seifried (Reference Desmettre and Seifried2016), we only need the stochastic payoff of the fixed-term security and its price at time 0, which is not necessarily a price based on the risk-neutral valuation. Therefore, we are only interested in P(T) and P(0). P(T) is a random variable that is $\mathcal{F}_T$ -measurable. By risk-neutral valuation, $P(T) = (G_T-V^{v_I,\pi_B}(T))^+$ and $P(0) = \mathbb{E}\left[{\widetilde{Z}(T)(G_T-V^{v_I,\pi_B}(T))^+}\right]$ . Note that in general $P(0) = \mathbb{E}[\widetilde{Z}(T)P(T)] \neq \mathbb{E}[\widetilde{Z}_\lambda(T)P(T)]$ .

The wealth process of the insurer $V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda$ in $\mathcal{M}_\lambda$ is given by:

(3.1) \begin{align} dV^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(t) & =(1-\pi_{I,1}(t)-\pi_{I,2}(t))V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(t)\frac{dS_0^\lambda(t)}{S_0^\lambda(t)}\\ &\quad +\pi_{I,1}(t)V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(t)\frac{dS_1^\lambda(t)}{S_1^\lambda(t)}+\pi_{I,2}(t)V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(t)\frac{dS_2^\lambda(t)}{S_2^\lambda(t)}\nonumber\end{align}

(3.2) \begin{align} & =(1-\pi_{I,1}(t)-\pi_{I,2}(t))V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(t)\frac{dS_0(t)}{S_0(t)}\\ &\quad +\pi_{I,1}(t)V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(t)\frac{dS_1(t)}{S_1(t)}+\pi_{I,2}(t)V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(t)\frac{dS_2(t)}{S_2(t)}\nonumber\\ &\quad +\underbrace{V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(t)\pi_I(t)^\top\lambda(t)dt}_{\text{additional term}}, \nonumber\\ V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(0)&=v_I-\xi_I(1+\theta_R)P(0)=v_{I,0}(\xi_I,\theta_R). \nonumber \end{align}

The unconstrained optimization problem of the insurer in $\mathcal{M}_\lambda$ is given by:

\begin{align*} \qquad\qquad\qquad\qquad\qquad\qquad\sup_{(\pi_I,\xi_I)\in\Lambda_I^\lambda}\mathbb{E}\left[U_I\left(V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(T)+\xi_I P(T)\right)\right], \qquad\qquad\qquad\qquad\qquad (\mathcal{P}_I^\lambda) \end{align*}

where

\begin{align*} \Lambda_I^\lambda\,:\!=\,\{(\pi_I,\xi_I)|\;&\pi_I\text{ self-financing, }V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(t)\geq0\;\mathbb{Q}\text{-a.s. }\forall t\in[0,T],\\ &\xi_I\in[0,\xi^{\max}]\text{ and }\mathbb{E}\left[U_I\left(V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(T)+\xi_I P(T)\right)^-\right]<\infty\}. \end{align*}

Note that $(\mathcal{P}_I^\lambda)$ does not have the investment strategy constraint as in $(\mathcal{P}_I)$ , but for the optimal $\lambda^{\ast}$ , which has to be found, the solution to the unconstrained problem will satisfy the constraint $\pi_2 = 0$ . We denote the solution to $(\mathcal{P}_I^\lambda)$ by:

\begin{equation*} \left(\pi_\lambda^\ast,\xi_\lambda^\ast\right)\in\underset{(\pi_I,\xi_I)\in\Lambda_I^\lambda}{\text{arg sup }}\mathbb{E}\left[U_I\left(V^{v_{I,0}(\xi_I,\theta_R),\pi_I}_\lambda(T)+\xi_I P(T)\right)\right]. \end{equation*}

Random utility function. As in Desmettre and Seifried (Reference Desmettre and Seifried2016), we define the random utility function by:

\begin{align*} \hat{U}_I(x)\,:\!=\,U_I(x+\xi_I P(T)) \end{align*}

for $x\in [0,\infty)$ , where $\xi_I\in[0,\xi^{\text{max}}]$ . The utility function $\hat{U}_I$ is random, since P(T) is a random variable. Hence, $\hat{U}_I:[0,+\infty)\to[U_I(\xi_I P(T)),+\infty)$ and $\hat{U}_I$ is continuously differentiable, strictly increasing and strictly concave. Therefore, it holds $\hat{U}^{\prime}_I:[0,+\infty)\to(0,U^{\prime}_I(\xi_I P(T))]$ and

\begin{align*} \hat{U}^{\prime}_I(x)=U^{\prime}_I(x+\xi_I P(T)). \end{align*}

We denote the inverse function of $\hat{U}^{\prime}_I$ by $\hat{I}_I:(0,+\infty)\to[0,+\infty)$ . For $y\in(0,U^{\prime}_I(\xi_I P(T))]$ , it is given by $I_I(y)-\xi_I P(T)$ , where $I_I$ denotes the inverse of $U^{\prime}_I$ . For $y>U^{\prime}_I(\xi_I P(T))$ , we set $\hat{I}(y)\,:\!=\,0$ . Hence, the random inverse function $\hat{I}_I$ is bijective on $(0,U^{\prime}_I(\xi_I P(T))]$ .

Proposition 3.1 (Optimal solution to $(\mathcal{P}_I^\lambda)$ ). Assume that for all $y\in(0,\infty)$

\begin{align*} \mathbb{E}\left[\widetilde{Z}_\lambda(T)I_I(y\widetilde{Z}_\lambda(T))\right]<\infty\text{ and }\mathbb{E}\left[U_I\left(I_I\left(y\widetilde{Z}_\lambda(T)\right)\right)\right]<\infty \end{align*}

holds. Then, there exists a solution $\left(\pi_\lambda^\ast,\xi_\lambda^\ast\right)$ to $(\mathcal{P}_I^\lambda)$ , where

\begin{align*} \xi_\lambda^\ast\in\underset{\xi_I\in[0,\xi^{\text{max}}]}{\text{arg max }}\nu(\xi_I). \end{align*}

The function $\nu$ is given by:

\begin{align*} \nu(\xi_I)\,:\!=\,\mathbb{E}\left[U_I\left(\max\left\{I_I(y^\ast(\xi_I)\widetilde{Z}_\lambda(T)),\xi_I P(T)\right\}\right)\right], \end{align*}

where the Lagrange multiplier $y^\ast\,:\!=\, y^\ast(\xi_I)$ is given by the budget constraint:

\begin{align*} \mathbb{E}[\widetilde{Z}_\lambda(T)\hat{I}_I(y^\ast\widetilde{Z}_\lambda(T))]=v_I-\xi_I(1+\theta_R)P(0). \end{align*}

The optimal terminal wealth $V^\ast_\lambda(T)\,:\!=\,V^{v_{I,0}^\lambda(\xi_{\lambda}^\ast,\theta_R),\pi_\lambda^\ast}_\lambda(T)$ is given by:

\begin{align*} V^\ast_\lambda(T) = \hat{I}_I\left(y^\ast\left(\xi_\lambda^\ast\right)\widetilde{Z}_\lambda(T)\right)=\max\{I_I\left(y^\ast\left(\xi_\lambda^\ast\right)\widetilde{Z}_\lambda(T)\right)-\xi_\lambda^\ast P(T),0\} \end{align*}

and the optimal wealth process $V_\lambda^\ast$ is given by:

\begin{align*} V_\lambda^\ast(t)=\widetilde{Z}_\lambda(t)^{-1}\mathbb{E}[\widetilde{Z}_\lambda(T)V_\lambda^\ast(T)|\mathcal{F}_t] \quad \text{for}\quad t \in [0, T]. \end{align*}

If $\hat{I}_I$ and $\frac{d\hat{I}_I(y)}{dy}$ are polynomially boundedFootnote ¹⁰ at 0 and $\infty$ , the optimal $\pi_\lambda^\ast$ is given by:

(3.3) \begin{align} \pi_\lambda^\ast(t)V_\lambda^\ast(t)=-\left(\sigma^\top\right)^{-1}\gamma_\lambda\widetilde{Z}_\lambda(t)^{-1}\mathbb{E}\bigg[\widetilde{Z}_\lambda(T)y^\ast\left(\xi_\lambda^\ast\right)\widetilde{Z}_\lambda(T)\frac{d\hat{I}}{dy}\left(y^\ast(\xi_I^\ast)\widetilde{Z}_\lambda(T)\right)\bigg|\mathcal{F}_t\bigg] \end{align}

$\mathbb{Q}$ -a.s. for all $t \in [0, T]$ .

The assumption of the finiteness of expectations in Proposition 3.1 is standard (cf. Cvitanić and Karatzas, Reference Cvitanić and Karatzas1992; Desmettre and Seifried, Reference Desmettre and Seifried2016) and ensures that the optimal Lagrange multiplier $y^\ast(\xi_I) > 0$ (given an arbitrary but fixed $\xi_I$ ) in the corresponding problem exists.

Hence, by Proposition 3.1, we know that under some conditions there exists the optimal solution $\left(\pi_\lambda^\ast,\xi_\lambda^\ast\right)$ to the unconstrained optimization problem of the insurer $(\mathcal{P}_I^\lambda)$ . In the next proposition, we show how the solutions of the optimization problem of the insurer $(\mathcal{P}_I)$ and the unconstrained optimization problem of the insurer $(\mathcal{P}_I^\lambda)$ are linked.

Remark to Proposition 3.2. In the case of a deterministic utility function, $\lambda^\ast$ can be found by a minimization criterion, which is given in Example 15.1 of Cvitanić and Karatzas (Reference Cvitanić and Karatzas1992). As we have a random utility function, we cannot apply this minimization criterion, and the search for the optimal $\lambda^\ast$ is more involved. In Section 4, we have a concrete power utility function for the insurer and we find the explicit form of $\lambda^\ast$ that satisfies Proposition 3.2.

3.2. Solution to the optimization problem of the reinsurer

We now know the optimal responses of the insurer to each possible safety loading $\theta_R\in[0,\theta^{\max}]$ . We write $\left(\xi_I^\ast(\theta_R),\pi_I^\ast\left(t|\theta_R,\xi^\ast_I(\theta_R)\right)\right)$ to emphasize the dependence of the insurer’s optimal strategy on $\theta_R\in[0,\theta^{\max}]$ . To derive the Stackelberg equilibrium, we need to solve the optimization problem $\big(\mathcal{P}_R^{\pi_R,\theta_R}\big)$ of the reinsurer. In contrast to $(\mathcal{P}_I)$ , we face only one challenge in $\big(\mathcal{P}_R^{\pi_R,\theta_R}\big)$ : the reinsurer has a fixed short position in the put option.

In comparison to the insurer, the reinsurer has a larger investment universe and can hedge the put option. Thus, we adapt the approach proposed by Korn and Trautmann (Reference Korn and Trautmann1999) based on replicating strategies. In this approach, we work mainly with trading strategies instead of the relative portfolio processes. A trading strategy is an $\mathbb{R}^{3}$ -valued progressively measurable stochastic process $\varphi(t) = (\varphi_0(t), \varphi_1(t), \varphi_2(t))',\,t \in [0, T],$ such that $\mathbb{Q}$ -a.s.:

\begin{equation*} \int \limits_{0}^{T}|\varphi_0(t)|\,ds <+\infty \quad \text{and}\quad \int \limits_{0}^{T}\varphi_i^2(t)\,ds <+\infty \,\,, i \in \{1,2\}. \end{equation*}

We consider only self-financing trading strategies, which means that they satisfy

\begin{equation*} \underbrace{\varphi_0(t) S_0(t) + \varphi_1(t) S_1(t) + \varphi_2(t) S_2(t)}_{\,=\!:\, V(\varphi, t)} = V(\varphi, 0) + \int \limits_{0}^{t}\varphi_0(s)\,d S_0(s) + \int \limits_{0}^{t}\varphi_1(s)\,d S_1(s) + \int \limits_{0}^{t}\varphi_2(s)\,d S_2(s), \end{equation*}

where $V(\varphi, t)$ is the portfolio value and $V(\varphi, 0)$ is the initial wealth of the decision-maker. We denote the trading strategy of the reinsurer by $\varphi_R$ . If $\varphi_R$ is a self-financing strategy with $V_R(\varphi_R, t) > 0$ $\mathbb{Q}$ -a.s. $\forall t \in [0, T]$ , it is linked to the relative portfolio process $\pi_R$ via the formula:

(3.4) \begin{align} \pi_{R,i}(t)=\frac{\varphi_{R,i}(t)\cdot S_i(t)}{V_R(\varphi_R, t)},\;\forall t \in [0, T],i\in\{0,1,2\}, \end{align}

with $V_R(\varphi_R, t) = V^{v_{R,0}(\xi_I^\ast(\theta_R),\theta_R),\pi_R}_R(t)$ $\mathbb{Q}$ -a.s. for all $t \in [0, T]$ .

We solve the optimization problem of the reinsurer $\big(\mathcal{P}_R^{\pi_R,\theta_R}\big)$ as follows. First, we transform the optimization problem $\big(\mathcal{P}_R^{\pi_R,\theta_R}\big)$ with respect to (w.r.t.) the portfolio process $\pi_R$ and $\theta_R$ into an optimization problem $\big(\mathcal{P}_R^{\varphi_R,\theta_R}\big)$ w.r.t. the trading strategy $\varphi_R$ and $\theta_R$ . Second, we use the idea of Desmettre and Seifried (Reference Desmettre and Seifried2016) and solve the optimization problem $\big(\mathcal{P}_R^{\varphi_R,\theta_R}\big)$ given a fixed $\theta_R\in[0,\theta^{\max}]$ . For this, we transform the optimization problem $\big(\mathcal{P}_R^{\varphi_R,\theta_R}\big)$ into the optimization problem $(\mathcal{P}_R^{\varphi_R|\theta_R})$ w.r.t. the trading strategy $\varphi_R$ given fixed $\theta_R$ and afterward transform the optimization problem $\big(\mathcal{P}_R^{\varphi_R,\theta_R}\big)\big)$ into the optimization problem $(\mathcal{P}_R^{\varphi_R|\theta_R,\xi(t)=-\xi_I^\ast(\theta_R)})$ w.r.t. the trading strategy $\varphi_R$ given fixed $\theta_R$ and a fixed position $\xi(t)=-\xi_I^\ast(\theta_R)$ in the put option P. Then, we solve the optimization problem $(\mathcal{P}_R^{\varphi_R|\theta_R,\xi(t)=-\xi_I^\ast(\theta_R)})$ by applying the idea of Korn and Trautmann (Reference Korn and Trautmann1999) and use (3.4) to calculate $\pi_R^\ast(\cdot|\theta_R)$ from the optimal trading strategy $\varphi_R^\ast(\cdot|\theta_R)$ . The optimization problems $(\mathcal{P}_R^{\varphi_R|\theta_R})$ and $(\mathcal{P}_R^{\varphi_R|\theta_R,\xi(t)=-\xi_I^\ast(\theta_R)})$ are stated in Appendix A in the proof of Proposition 3.3 to keep this subsection clearer. Lastly, we solve the optimization problem $\big(\mathcal{P}_R^{\pi_R,\theta_R}\big)$ w.r.t. $\theta_R$ for the given optimal portfolio process $\pi_R^\ast(\cdot|\theta_R)$ .

Therefore, we first transform the optimization problem $\big(\mathcal{P}_R^{\pi_R,\theta_R}\big)$ w.r.t. the portfolio process $\pi_R$ into the optimization problem $\big(\mathcal{P}_R^{\varphi_R,\theta_R}\big)$ w.r.t. the trading strategy $\varphi_R$ . We use the notation $V^{v_{I,0}(\xi_I,\theta_R),\varphi_R}_R$ for the wealth process of the reinsurer w.r.t. the trading strategy $\varphi_R$ . Hence, the optimization problem of the reinsurer w.r.t. the trading strategy $\varphi_R$ is given by:

\begin{align*} \quad\qquad\qquad\qquad\qquad\sup_{(\varphi_R,\theta_R)\in\Lambda_R^{\varphi_R}}\mathbb{E}\left[U_R\left(V^{v_{R,0}(\xi_I^\ast(\theta_R),\theta_R),\varphi_R}_R(T)-\xi_I^\ast(\theta_R)P(T)\right)\right], \qquad\qquad\qquad\qquad(\mathcal{P}_R^{\varphi_R,\theta_R}) \end{align*}

where $\Lambda_R^{\varphi_R}$ is the set of all admissible trading strategies and safety loadings of the reinsurer:

\begin{align*} & \Lambda_R^{\varphi_R}\,:\!=\,\left\{(\varphi_R,\theta_R)|\;\varphi_R\text{ self-financing, }V^{v_{R,0}(\xi_I^\ast(\theta_R),\theta_R),\varphi_R}_R(t)\geq0\;\mathbb{Q}\text{-a.s. }\forall t\in[0,T],\right.\\ &\qquad \qquad\! \left. \theta_R\in[0,\theta^{\max}]\text{ and }\mathbb{E}\left[U_R(V^{v_{R,0}(\xi_I^\ast(\theta_R),\theta_R),\varphi_R}_R(T)-\xi_I^\ast(\theta_R)P(T))^-\right]<\infty\right\}. \end{align*}

Let $\theta_R\in[0,\theta^{\max}]$ be arbitrary but fixed. Our aim is to solve the optimization problem:

\begin{align*} \qquad\qquad\qquad\qquad\qquad\sup_{\varphi_R:(\varphi_R,\theta_R)\in\Lambda_R^{\varphi_R}}\mathbb{E}\left[U_R\left(V^{v_{R,0}(\xi_I^\ast(\theta_R),\theta_R),\varphi_R}_R(T)-\xi_I^\ast(\theta_R)P(T)\right)\right], \qquad\qquad\qquad (\mathcal{P}_R^{\varphi_R|\theta_R}) \end{align*}

where $\theta_R$ in $(\mathcal{P}_R^{\varphi_R|\theta_R})$ emphasizes that $\theta_R$ is not a control variable there. To find the optimal trading strategy $\varphi_R^\ast$ that solves $(\mathcal{P}_R^{\varphi_R|\theta_R})$ , we adjust Theorem 5.1 mentioned in Remark 5.2 in Korn and Trautmann (Reference Korn and Trautmann1999).

Proposition 3.3 (Optimal solution to $(\mathcal{P}_R^{\varphi_R|\theta_R})$ . Assume that it holds for all $y\in(0,\infty)$

\begin{align*} \mathbb{E}[\widetilde{Z}(T)I_R(y\widetilde{Z}(T))]<\infty, \end{align*}

where $I_R$ is the inverse function of $U^{\prime}_R$ .

1. (a) There exists an optimal trading strategy $\varphi_R^\ast$ for the optimization problem $(\mathcal{P}_R^{\varphi_R|\theta_R})$ . The optimal total terminal wealth in the optimization problem $(\mathcal{P}_R^{\varphi_R|\theta_R})$ is given by:

   \begin{align*} V^{v_{R,0}(\xi_I^\ast(\theta_R),\theta_R),\varphi_R^\ast}_R(T)-\xi_I^\ast(\theta_R)P(T)=I_R\left(y^\ast_R(\theta_R)\widetilde{Z}(T)\right), \end{align*}
   where $y^\ast_R\equiv y^\ast_R(\theta_R)$ is the Lagrange multiplier determined by:
   (3.5) \begin{align} \mathbb{E}\left[\widetilde{Z}(T)I_R\left(y^\ast_R\widetilde{Z}(T)\right)\right]=v_{R}+\xi_I^\ast(\theta_R)\theta_RP(0). \end{align}

2. (b) Let $\zeta_R^\ast$ be the optimal trading strategy of the stock optimization problem, where the reinsurer only invests in the assets $S_0$ , $S_1$ , and $S_2$ (i.e., no investment in options). Then, the optimal trading strategy $\varphi_R^\ast$ to the optimization problem $(\mathcal{P}_R^{\varphi_R|\theta_R})$ is given by:

   \begin{align*} \varphi_{R,0}^\ast(t)&=\frac{V^{v_{R,0}(\xi_I^\ast(\theta_R),\theta_R),\varphi_R^\ast}(t)-\sum_{i=1}^2\varphi^\ast_{R,i}(t)S_i(t)}{S_0(t)},\quad \varphi_{R,1}^\ast(t)=\zeta_{R,1}^\ast(t);\\ \varphi_{R,2}^\ast(t)&=\zeta_{R,2}^\ast(t) + \xi_I^\ast(\theta_R) \frac{\pi^{CM}V^{v_I,\pi_B}(t)(\Phi(d_+)-1)}{S_2(t)}, \end{align*}
   where
   \begin{align*} d_+\,:\!=\,d_+(t,V^{v_I,\pi_B}(t))\,:\!=\,\frac{\ln\Big(\frac{V^{v_I,\pi_B}(t)}{G_T}\Big)+\big(r+\frac{1}{2}\left(\sigma_2\pi^{CM}\right)^2\big)(T-t)}{\pi^{CM}\sigma_2\sqrt{T-t}}. \end{align*}

Remark to Proposition 3.3. The assumption of the finiteness of expectations in Proposition 3.3 is standard (cf. Korn and Trautmann, Reference Korn and Trautmann1999) and ensures the existence of the optimal Lagrange multiplier $y^\ast_R > 0$ .

Proposition 3.3 yields the the optimal terminal value $V^{v_{R,0}(\xi_I^\ast(\theta_R),\theta_R),\varphi_R^\ast}_R(T)$ and the optimal trading strategy $\varphi_R^\ast$ in the reinsurer’s optimization problem $\big(\mathcal{P}_R^{\varphi_R,\theta_R}\big)$ for an arbitrary but fixed $\theta_R$ . The next proposition determines the optimal safety loading.

Proposition 3.4 (Optimal safety loading). Let $\varphi^\ast_R(\cdot|\theta_R)$ be the optimal trading strategy for $\big(\mathcal{P}_R^{\varphi_R,\theta_R}\big)$ given an arbitrary but fixed $\theta_R\in[0,\theta^{\text{max}}]$ . Then, the optimal $\theta_R^\ast$ is given by:

\begin{align*} \theta_R^\ast=\underset{\theta_R\in[0,\theta^{\max}]}{\text{arg max }}\mathbb{E}\left[U_R\left(V^{v_{R,0}(\xi_I^\ast(\theta_R),\theta_R),\varphi^\ast_R}_R(T)-\xi_I^\ast(\theta_R)P(T)\right)\right]. \end{align*}

3.3. Stackelberg equilibrium

Remarks to Proposition 3.5.

1. 1. (3.6) ensures that $(SEC_2)$ is fulfilled, that is, if there exist several best responses of the insurer to the reinsurer’s optimal action, then the insurer chooses among them the best response that is also best from the reinsurer’s perspective.

2. 2. We get analytical representations of the optimal relative portfolio processes $\pi_R^\ast$ and $\pi_I^\ast$ , which depend on $\theta_R^\ast$ and $\xi_I^\ast\left(\theta_R^\ast\right)$ . For common utility functions, such as power, HARA, or logarithmic, the optimal reinsurance amount $\xi_I^\ast\left(\theta_R^\ast\right)$ , $\xi_I^\ast\left(\theta_R^\ast\right) = \bar{\xi}$ , but the optimal safety loading $\theta_R^\ast$ has to be numerically computed by solving a nonlinear equation.

5.1. Parameter selection

In this part, we discuss the selection of the parameters summarized in Table 1. For the majority of parameters, we choose the same values as in Escobar-Anel et al. (Reference Escobar-Anel, Havrylenko, Kschonnek and Zagst2022), where they are calibrated to the German market in the period from January 1, 2003, to June 8, 2020.

Table 1. Parameters for the numerical analysis.

The risk-free rate is modeled by the Euro OverNight Index Average (EONIA) daily data. For calibrating the parameters of $S_1$ , we use the TecDAX daily data and for calibrating the parameters of $S_2$ , we use the DAX daily data.

In this way, we model the following situation: the insurer invests in bonds and prefers the technological sector to a broad stock portfolio. One reason could be that the insurer’s asset manager has special knowledge in the technology sector and/or believes that the TecDAX has a better performance than the DAX. In contrast, the reinsurer considers the insurer’s technology-focused portfolio too risky to be reinsured. Therefore, the reinsurer offers reinsurance only on a mixed portfolio consisting of bonds and the broad market index DAX.

The relative proportion $\pi^{CM}$ of the constant-mix portfolio is selected such that it equals the optimal initial proportion of the insurer in the risky asset without reinsurance.

In the German Life Insurance Market, the capital guarantee for the representative client is usually less than or equal to 100% of the representative client’s initial endowment. For example, ERGO offers the equity-linked insurance product “ERGO Rente Garantie” where the guarantee lies between 80% and 100%.Footnote ¹¹ In contrast, Allianz offers a guarantee between 60% and 90%.Footnote ¹² Hence, we assume that the representative client has a 100% capital guarantee of the initial capital and investigate in the sensitivity analysis a $G_T$ varying from 60 to 110% of the initial contribution of the client.

For convenience, we set the insurer’s initial wealth $v_I$ to 100. It is natural to assume that the reinsurer is a larger company with more initial capital. If reinsurance is offered on the whole company level, then the initial wealth of the reinsurer should be higher than the initial wealth of the insurer, as stated in Chen and Shen (Reference Chen and Shen2018). Since we consider reinsurance within a single insurance product, we assume that the initial product-related capital of the reinsurer coincides with the initial wealth of the insurer, that is, $v_R=v_I=100$ .

In Chen and Shen (Reference Chen and Shen2018), the authors assume that the reinsurer and insurer have the same risk aversion. In contrast, Bai et al. (Reference Bai, Zhou, Xiao, Gao and Zhong2022) assume that the insurer is more risk-averse than the reinsurer. Thus, we first consider the situation where both parties have the same risk aversion. Afterward, we explore the situations where the parties have different risk aversion. In the base case, we choose $b_I=b_R=-9$ , which is consistent with Escobar-Anel et al. (Reference Escobar-Anel, Havrylenko, Kschonnek and Zagst2022).

For the maximal level of safety loading, we choose 50%. It was chosen in line with Chen and Shen (Reference Chen and Shen2018) and Chen and Shen (Reference Chen and Shen2019) who chose an upper bound on the safety loading of the reinsurer of 45%.

We do not allow that the insurer can speculate with the reinsurance by going short or buying too much of it. Since the underlying of the put option is not the portfolio of the insurer but a correlated portfolio, we allow that $\xi^{\max}=\bar{\xi}=1.5$ .

5.2. Stackelberg equilibrium

First, we calculate the Stackelberg equilibrium in the base case of parametrization and its sensitivity. It is given by:

\begin{align*} \pi_R^\ast(0)=&{(31.67\%,-16.42\%)^\top},\quad \theta_R^\ast = 20.86\%,\\ \pi_I^\ast(0)=&(31.69\%,0\%)^\top,\quad \quad \quad \xi_I^\ast\left(\theta_R^\ast\right)=1.5. \end{align*}

The optimal reinsurance strategy of the insurer is to buy the maximal amount of put options (i.e., $\xi_I^\ast\left(\theta_R^\ast\right)=\xi^{\max}$ ), and the reinsurer sets the maximal price for the reinsurance such that the insurer still buys reinsurance, which is about 20% higher than the fair price of the put option.

Next, we describe the results of our sensitivity of the Stackelberg equilibrium w.r.t. the change of risk aversion of each party, the interest rate, the time horizon, and the capital guarantee. The corresponding plots can be found online in Section 2 of Supplementary Materials.

Varying relative risk aversion (RRA) coefficients, we find that the higher the RRA of a party, the less the party invests in or speculates with the risky assets. Interestingly, the values of the reinsurance contract parameters in the equilibrium do not depend on the individual RRA coefficient of each party. This is consistent with Corollary 4.2, from which we know that the equilibrium safety loading $\theta_R^\ast$ depends on the pricing kernel in the original as well as the auxiliary markets and the put-option parameters. Hence, RRA coefficients do not influence $\theta_R^\ast$ , if the benchmark strategy $\pi_B$ is independent of them. Figure 2.1 in Supplementary Materials corresponds to such a choice of $\pi_B$ . However, the insurer’s RRA coefficient can have an indirect impact on the reinsurer’s choice of $\theta_R^\ast$ through $\pi_B$ , because the benchmark portfolio influences the fair price of the put option. If the risky asset investment share $\pi^{CM}$ in $\pi_B$ decreases with increasing $b_I$ , then the equilibrium safety loading increases. The remaining components of the equilibrium have the same trends as indicated in Figure 2.1. The amount of reinsurance $\xi_I^\ast\left(\theta_R^\ast\right)$ bought by the insurer in equilibrium equals $\bar{\xi}$ for any considered RRA coefficient, as expected from (4.2) and $(SEC_2)$ .

Varying the interest rate $r\in\{-2\%,-1\%,0\%,1\%,2\%\}$ , we find that the higher r, the higher the optimal safety loading $\theta_R^\ast$ , since the fair price of the put option P(0) in the basic market $\mathcal{M}$ decreases faster than the fair price of the put option $\mathbb{E}[\widetilde{Z}_{\lambda^\ast}(T)P(T)]$ in the auxiliary market $\mathcal{M}_{\lambda^\ast}$ (see (4.5)). Consistent with Corollary 4.2, the interest rate has no influence on the reinsurance strategy of the insurer. The higher the interest rate, the less the insurer invests in the risky asset. The interest rate has no notable influence on the reinsurer’s optimal investment portfolio. The corresponding plots can be found in Figure 2.2 in Supplementary Materials.

For increasing time horizon $T\in\{1,5,10,15,20\}$ , the equilibrium safety loading increases. In contrast, the time horizon has no influence on the optimal reinsurance strategy of the insurer and no notable influence on the optimal investment strategy of the reinsurer and the insurer. The corresponding plots can be found in in Figure 2.3 in Supplementary Materials.

Finally, we explore the sensitivity of the Stackelberg equilibrium w.r.t. the client guarantee $G_T\in\{0.6\cdot v_I,0.7\cdot v_I,0.8\cdot v_I,\dots, 1.1\cdot v_I\}$ , which is also the strike of the put option that models the reinsurance contract. We observe that the optimal safety loading of the reinsurer is decreasing in $G_T$ . This follows from the fact that the higher the guarantee, the more expensive the put option, which is why the maximal additional safety loading at which the insurer may still be willing to buy reinsurance decreases. However, $G_T$ has no influence on the reinsurance amount that is purchased by the insurer in the equilibrium, that is, $\xi_I = \xi^{\max} = 1.5$ . This stems from the definition of the Stackelberg equilibrium, where, in the case of several equivalent optimal responses to the leader’s action, the follower chooses the response that is also best for the leader’s perspective. When $G_T$ increases, the reinsurer’s investment in $S_2$ decreases due to the hedge of its put position. There is no notable influence of $G_T$ on the optimal investment strategy of the insurer. Figure 2.4 in Supplementary Materials contains the corresponding plots.

5.3. Impact of deviating from Stackelberg equilibrium

Recall from (4.2) that if the reinsurer chooses the safety loading below the critical value $({\mathbb{E}[\widetilde{Z}_{\lambda^\ast}(T)P(T)]-P(0)})/{P(0)}$ , then the best action of the insurer is to buy the maximal amount of reinsurance. However, if the reinsurer chooses $\theta_R$ equal to that critical value, then the insurer’s best response is to buy any reinsurance amount from the interval $[0,\bar{\xi}]$ . Condition $(SEC_2)$ in the definition of the Stackelberg equilibrium implies that the reinsurer chooses the critical value for safety loading and has an optimistic view on the insurer’s response. In other words, the reinsurer “hopes” that the insurer selects $\xi_I^\ast\left(\theta_R^\ast\right)=\bar{\xi}$ . However, in practice, the insurer could also buy no reinsurance or any amount in between, because its set of best responses is $[0,\bar{\xi}]$ . Therefore, we study next the effects of a deviation of the reinsurer from the optimal safety loading $\theta_R^\ast$ on the reinsurer and the insurer. In the following, we denote $\xi_I^\ast\left(\theta_R^\ast\right)$ only by $\xi_I^\ast$ to simplify notation.

The reinsurer should offer safety loadings equal to or below $\theta_R^\ast$ , as otherwise the reinsurance price is too high for the insurer to buy it. We denote a discounted safety loading by $\theta_R(\alpha)=\alpha\cdot\theta_R^\ast$ with $\alpha\in[0,1]$ . If the reinsurance premium is higher than the fair price of the put option (i.e., $\alpha>0$ ), then the reinsurer always gains from offering reinsurance with a discounted safety loading in comparison with not offering reinsurance. In the Stackelberg equilibrium, the reinsurer has the highest gain in its utility from offering reinsurance but is uncertain about the amount of reinsurance the insurer will buy. To get rid of this uncertainty, the reinsurer can offer reinsurance with a safety loading $\theta_R(\alpha)$ with $\alpha\in(0,1)$ . Note that a deviation from the optimal safety loading means that the parties are not in the theoretical Stackelberg equilibrium anymore. However, both of them are incentivized to do so. The insurer’s expected utility is strictly larger than it is in the Stackelberg equilibrium as well as in the case of not buying reinsurance. The reinsurer’s expected utility for a discounted safety loading is strictly larger than it is in the case of not selling reinsurance. See Section 3 of Supplementary Materials for details.

After the reinsurer has decided to grant a discount on the safety loading to the insurer, we consider the disadvantage/advantage to the reinsurer and the insurer in comparison with the Stackelberg equilibrium. We study this using the concept of the wealth-equivalent utility change (WEUC) of a party, that is, insurer and reinsurer. In the calculation of WEUC, the expected utility of the party in the Stackelberg equilibrium (or the Stackelberg equilibrium with a discount) is compared as a reference action combination to an alternative action combination. The alternative action combination refers to a specific choice of the safety loading as well as the investment strategy by the reinsurer and the amount of reinsurance as well as the investment strategy by the insurer. WEUC equals the relative change of the party’s initial wealth that is required to bring the party in the alternative action combination to the same expected utility as in case of the reference combination of actions. We define it in a way such that our definition is consistent with the definition of the wealth-equivalent utility loss in Larsen and Munk (Reference Larsen and Munk2012).

To be specific, we denote a reference combination of actions by $\left(\tilde{\pi}_R\left(\cdot|\tilde{\theta}_R,\tilde{\xi}_I\right),\tilde{\theta}_R,\tilde{\pi}_I\left(\cdot|\tilde{\theta}_R,\tilde{\xi}_I\right),\tilde{\xi}_I\right)$ and an alternative combination of actions by $\left(\hat{\pi}_R\left(\cdot|\hat{\theta}_R,\hat{\xi}_I\right),\hat{\theta}_R,\hat{\pi}_I\left(\cdot|\hat{\theta}_R,\hat{\xi}_I\right),\hat{\xi}_I\right)$ .

Then the WEUC of the reinsurer is denoted by:

\begin{equation*} WEUC_R\left(\tilde{\theta}_R,\tilde{\xi}_I,\hat{\theta}_R,\hat{\xi}_I\right)\,:\!=\,WEUC_R\left(\left(\tilde{\pi}_R\left(\cdot|\tilde{\theta}_R,\tilde{\xi}_I\right),\tilde{\theta}_R\right), \left(\hat{\pi}_R\left(\cdot|\hat{\theta}_R,\hat{\xi}_I\right),\hat{\theta}_R\right)\right) \end{equation*}

and satisfies the relation:

\begin{equation*} \mathbb{E}\left[U_R\left(V_R^{v^{WEUC}_{R,0}\left(\hat{\xi}_I,\hat{\theta}_R\right),\hat{\pi}_R}(T)-\hat{\xi}_I P(T)\right)\right]= \mathbb{E}\left[U_R\left(V_R^{v_{R,0}(\tilde{\xi}_I,\tilde{\theta}_R),\tilde{\pi}_R}(T)-\tilde{\xi}_I P(T)\right)\right], \end{equation*}

where

\begin{equation*} v^{WEUC}_{R,0}\left(\hat{\xi}_I,\hat{\theta}_R\right)=v_R\cdot\left(1+WEUC_R\left(\tilde{\theta}_R,\tilde{\xi}_I,\hat{\theta}_R,\hat{\xi}_I\right)\right)+\hat{\xi}_I\left(1+\hat{\theta}_R\right)P(0). \end{equation*}

Analogously, the WEUC of the insurer is denoted by:

\begin{equation*} WEUC_I\left(\tilde{\theta}_R,\tilde{\xi}_I,\hat{\theta}_R,\hat{\xi}_I\right)\,:\!=\,WEUC_I\left(\left(\tilde{\pi}_I\left(\cdot|\tilde{\theta}_R,\tilde{\xi}_I\right),\tilde{\xi}_I\right), \left(\hat{\pi}_I\left(\cdot|\hat{\theta}_R,\hat{\xi}_I\right),\hat{\xi}_I\right)\right) \end{equation*}

and satisfies the relation:

\begin{align*} \mathbb{E}\left[U_I\left(V_I^{v^{WEUC}_{I,0}\left(\hat{\xi}_I,\hat{\theta}_R\right),\hat{\pi}_I}(T)+\hat{\xi}_I P(T)\right)\right]&= \mathbb{E}\left[U_I\left(V_I^{v_{I,0}(\tilde{\xi}_I,\tilde{\theta}_R),\tilde{\pi}_I}(T)+\tilde{\xi}_I P(T)\right)\right], \end{align*}

where

\begin{align*} v^{WEUC}_{I,0}\left(\hat{\xi}_I,\hat{\theta}_R\right)&=v_I\cdot\left(1+WEUC_I\left(\tilde{\theta}_R,\tilde{\xi}_I,\hat{\theta}_R,\hat{\xi}_I\right)\right)-\hat{\xi}_I\left(1+\hat{\theta}_R\right)P(0). \end{align*}

The WEUC has an intuitive interpretation. If this quantity is positive, then the reference combination of actions is better for the party of interest than the alternative combination of actions. In this case, the WEUC indicates by which proportion the party has to increase its initial capital in case of the alternative combination of actions so that it has the same expected utility as in the case of the reference combination of actions. If the WEUC is negative, then the reference combination of actions is worse for the considered party than the alternative combination of actions. In this case, the WEUC indicates by which proportion the party can decrease its initial capital in case of the alternative combination of actions so that it has the same expected utility as in the case of the reference combination of actions.

Figure 1 shows the loss of the reinsurer and the gain of the insurer in their utilities, when the Stackelberg equilibrium is compared to an alternative situation with all actions being the same as in the Stackelberg equilibrium but with a discounted safety loading. The gain of the insurer is of the same amount as the loss of the reinsurer. Moreover, further investigations reveal that it does not depend on the risk aversions of the parties. Hence, the insurer does not benefit more from deviating from the Stackelberg equilibrium as the reinsurer loses. For example, a discount of 5% from $\theta_R^\ast$ would increase the WEUC of the reinsurer by 6bp of the initial capital.

Figure 1. Comparison of WEUC: Stackelberg equilibrium (reference) and the same combination of actions but with a discounted safety loading (alternative).

In the next step, the reinsurer needs to choose a discount level $\alpha$ such that its WEUC is not too “painful”. In the following, we present some ways to choose $\alpha$ . One way would be for the reinsurer to choose an acceptable upper bound on its WEUC (see the blue line in Figure 1) and to exchange uncertainty for a fixed defined WEUC. For example, if the reinsurer wants its WEUC to be at most 25bp, then it can choose $\alpha= 79.27\%$ (see turquoise line in Figure 1).

The other ways of choosing $\alpha$ can be related to product profitability. It can be measured by calculating the probability that the reinsurer’s total terminal wealth is below its initial wealth:

\begin{align*} \mathbb{Q}(\alpha)\,:\!=\,\mathbb{Q}\left(V_R^{v_{R,0}(\theta_R(\alpha),\xi_I^\ast),\pi_R^\ast(\cdot|\theta_R(\alpha),\xi_I^\ast)}(T)-\xi_I^\ast P(T)< v_R\right). \end{align*}

This measure shows the product profitability of the reinsurance for the reinsurer and is shown in Figure 2. The reinsurer can choose $\alpha$ in several different ways, which are based on:

• a tolerance level for the increase of the loss probability. For example, if the reinsurer is not willing to increase its loss probability by more than 0.01% (compared to the loss probability of $\mathbb{Q}(1)=0.4413\%$ in the case of the Stackelberg equilibrium), then the reinsurer can choose $\alpha=86.73\%$ (see green line in Figure 2);

• the maximal acceptable probability of loss. For example, if the reinsurer can tolerate at most $0.5\%$ probability of loss, it can choose $\alpha=20.74\%$ (see purple line in Figure 2).

The reinsurance company can also use other criteria for choosing $\alpha$ , for example, other risk measures, as the standard deviation, or different performance measures, as the adjusted Sharpe ratio.

Figure 2. Probability of loss at time T for reinsurer.

In the insurance industry, the analysis of the product profitability is a common procedure, which is also easier to communicate. Therefore, we choose as an example $\alpha=86.73\%$ . In this case, the reinsurer limits the increase of the probability of loss to $0.0001$ .

In the remaining analysis, we study how the insurer benefits from buying reinsurance with a safety loading of $\theta_R(86.73\%)$ (reference) instead of buying no reinsurance (alternative).

First, we determine the benefit of the insurer if it follows a dynamic portfolio strategy with reinsurance $(\pi_I^\ast(\cdot|\theta_R(86.73\%),\xi_I^\ast),\xi_I^\ast)$ (reference) instead of a dynamic portfolio strategy without reinsurance $(\pi_I^\ast(\cdot|0,0),0)$ (alternative), the latter being its Merton strategy. Therefore, we calculate $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)$ and consider how the relative risk aversion and the time horizon influence this value. The result is shown in Figure 3(a). Since $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)>0$ , the insurer needs less initial capital if it follows the dynamic portfolio strategy with reinsurance instead of the dynamic portfolio strategy without reinsurance. This means that the insurer can decrease its product costs by buying reinsurance. The more risk-averse the insurer, the lower $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)$ , and the longer the investment period, the higher $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)$ . If the insurer’s relative risk aversion is given by $RRA_I= 5$ and the insurer offers a long-term equity-linked insurance product with $T= 20$ , then $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)=60bp$ . If the insurer gets more risk-averse, then $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)$ decreases strongly. For $RRA_I= 15$ and $T= 20$ , we have $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)= 10bp$ .

Figure 3. Impact of relative risk aversion of insurer and investment horizon on $WEUC_I$ .

Lastly, we consider the benefit of the insurer if it follows the dynamic investment strategy with reinsurance $(\pi_I^\ast(\cdot|\theta_R(86.73\%),\xi_I^\ast),\xi_I^\ast)$ (reference strategy) instead of the constant-mix strategy without reinsurance $((15\%,0),0)$ (alternative strategy). We consider this constant-mix strategy without reinsurance, since it approximates the long-term investment strategy of an average life insurer due to Gründl and Gal (Reference Gründl and Gal2017). Therefore, we calculate $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)$ and analyze the impact of the relative risk aversion and investment horizon; see Figure 3(b). Analogous to the analysis before, the insurer needs less initial capital if it follows the dynamic strategy with reinsurance instead of the constant-mix strategy without reinsurance. The more risk-averse the insurer, the lower $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)$ , and the longer the investment period, the higher $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)$ . If the insurer’s relative risk aversion is given by $RRA_I= 5$ and the insurer offers a long-term equity-linked insurance product with $T= 20$ , then $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)=7275bp$ . If the insurer gets more risk-averse, then $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)$ decreases strongly. For $RRA_I= 15$ and $T= 20$ , we have $WEUC_I(\theta_R(86.73\%),\xi_I^\ast,0,0)= 287bp$ .