3.1. TVaR distortion

In this subsection, we consider the case where the insurer chooses TVaR as the distortion risk measure with $g_{i}(t)=\min\{1,t/\alpha\}$ , $\alpha\in(0,1)$ . In this case,

\begin{align*} \phi'(d) &=\min\left\{1,\frac{\bar{F}_X(d)}{\alpha}\right\}-\min\left\{1,\frac{1-p}{\alpha}\bar{F}_X(d)\right\}-(1+\theta)(p+(1-p)\gamma)\bar{F}_X(d)\\[5pt] &\quad+\frac{(1-p)\gamma}{1-\gamma} \int_d^{\infty} \frac{1}{\alpha}\mathbf{1}_{\left\{(1-p)\bar{F}_X\left(\frac{x-\gamma d}{1-\gamma}\right)\leq \alpha\right\}}(x)f_X\left(\frac{x-\gamma d}{1-\gamma}\right)\textrm{d} x.\end{align*}

By a change-of-variable, the above expression becomes

\begin{align*} \phi'(d) &=\min\left\{1,\frac{\bar{F}_X(d)}{\alpha}\right\}-\min\left\{1,\frac{1-p}{\alpha}\bar{F}_X(d)\right\}-(1+\theta)(p+(1-p)\gamma)\bar{F}_X(d)\\[5pt] &\quad+\frac{(1-p)\gamma}{\alpha} \int_d^{\infty} \mathbf{1}_{\left\{\bar{F}_X(x)\leq \frac{\alpha}{1-p}\right\}}(x)f_X(x)\textrm{d} x.\end{align*}

For fixed performance probability p, safety loading $\theta$ , and confidence level $\alpha$ , the optimal deductible can be derived explicitly and are summarized in the following theorem.

Figure 1. Flowchart illustration of Theorem 3.8.

The optimal deductibles $d^*$ depicted in Theorem 3.8 are schematically represented by the flowchart in Figure 1. The findings of Theorem 3.8 can be understood as follows:

• • When $\bar{F}_X(0)\leq \alpha$ , the probability that the aggregate loss is zero is higher than $1-\alpha$ . If the reinsurance premium is reasonable from the insurer’s perspective (i.e., $1/\alpha\geq 1+\theta$ ), he will buy reinsurance and indeed full reinsurance is optimal for him; otherwise, paying a high premium to cover the risk that is less likely to occur is irrational.

• • When $\bar{F}_X(0)> \alpha$ , the situations become more complicated, and multiple factors determine the optimal deductible level:

1. – (a) If $p\geq 1-\alpha/\bar{F}_X(0)$ , the default is unlikely to occur. The relations between $\kappa$ , $\alpha$ , and p affect the choice of optimal deductible level: if $\alpha< \kappa< \bar{F}_X(0)$ , the loading $\theta$ is subject to a range, indicating that the premium is charged within a range. It is optimal for the insurer to pay a premium in exchange for a stop-loss contract with deductible $d^*=\bar{F}^{-1}_X(\kappa)$ ; when $\bar{F}_X(0)\leq \kappa$ , the premium is charged at a lower loading level, so buying full reinsurance contract is optimal for the insurer; if $\kappa=\alpha$ , we know that $1/\alpha=1+\theta$ , displaying an equality relationship between confidence level and loading. From the insurer’s perspective, any deductible level equal to or greater than $\bar{F}^{-1}_X(\alpha)$ is optimal; otherwise, when $\kappa< \alpha$ , we have $1/\alpha< 1+\theta$ . As for the insurer, it is optimal not to buy any reinsurance because $1/\alpha< 1+\theta$ somehow means that the premium is too expensive.

2. – (b) If $p< 1-\alpha/\bar{F}_X(0)$ , there is a higher chance that the default might happen. The relations between $\alpha$ , p, $\kappa$ , and $\nu$ influence the choice of optimal deductible level: if $\alpha< \kappa\leq \alpha/(1-p)$ , the premium is charged within a range, then it is optimal for the insurer to buy a stop-loss contract with deductible $d^*=\bar{F}^{-1}_X(\kappa)$ ; if $\alpha/(1-p)< \kappa$ , we further define $\nu$ , satisfying $\nu> \kappa$ , which affects the choice of optimal deductible level. If $\nu< \bar{F}_X(0)$ , the premium is charged in a lower range, and it is still optimal for the insurer to buy a stop-loss contract with deductible $d^*=\bar{F}^{-1}_X(\nu)$ . If $\nu\geq \bar{F}_X(0)$ , the premium is charged at an even lower rate, and it is optimal to buy full reinsurance; if $\kappa=\alpha$ , similar arguments apply that buying a stop-loss treaty with deductible equal to or greater than $\bar{F}_X^{-1}(\alpha)$ is optimal to the insurer; otherwise if $\kappa\leq \alpha$ , given the tendency of default and the exceedingly expensive premium, it is not optimal for an insurer to buy any reinsurance.

When $\gamma=0$ , we arrive at a particular case where the reinsurer compensates nothing if the default occurs. In this case, Theorem 3.8 reduces to the following:

The case where $p=1$ corresponds to the situation that the reinsurer will never default. Theorem 3.8 then reduces to the following:

Case 2(a) and 2(c) of Corollary 3.11 echo the classical findings in Theorem 3.1 of Cai and Tan (Reference Cai and Tan2007). Our results enumerate all possible situations that might occur and articulate the optimal strategies the insurer should adopt.

3.2. Gini’s distortion

As for the Gini’s distortion principle, $g'_{i}(t)=(1+r)-2rt$ , for $r\in(0,1)$ , which leads to

\begin{align*} \phi'(d)&=(1+r)\bar{F}_X(d)-r\left(\bar{F}_X(d)\right)^2-(1-p)(1+r)\bar{F}_X(d)+r(1-p)^2\left(\bar{F}_X(d)\right)^2\\[5pt] &\quad-(1+\theta)(p+(1-p)\gamma)\bar{F}_X(d)+(1-p)(1+r)\gamma\int_d^{\infty} f_X(x)\textrm{d} x\\[5pt] &\quad-2\gamma r (1-p)^2\int_d^{\infty} \bar{F}_X(x)f_X(x) \textrm{d} x.\end{align*}

We can calculate the two integrals above as

\begin{equation*} \int_d^{\infty} f_X(x)\textrm{d} x = \bar{F}_X(d)\quad\text{and}\quad \int_d^{\infty} \bar{F}_X(x)f_X(x)\textrm{d} x = \frac{1}{2}\left(\bar{F}_X(d)\right)^2.\end{equation*}

As a result,

\begin{equation*} \phi'(d)=(r-\theta)\left(p+(1-p)\gamma\right)\bar{F}_X(d)-r\left(1-(1-p)^2(1-\gamma) \right)\left(\bar{F}_X(d)\right)^2.\end{equation*}

For fixed performance probability p, safety loading $\theta$ , and Gini coefficient r, the optimal deductible can then be found, and the result is summarized in the following theorem.

Proof. We know that

\begin{equation*} \phi'(d)=\bar{F}_X(d)\left((r-\theta)(p+(1-p)\gamma)-r(1-(1-p)^2(1-\gamma)\bar{F}_X(d)\right). \end{equation*}

If $r\leq \theta$ , it is easy to see $\phi'(d)\leq 0$ . As $\phi(d)$ is decreasing on $\mathbb{R}_+$ , we have $d^*=\infty$ . Otherwise, if $r> \theta$ , we have

\begin{equation*} \phi'(d)=\bar{F}_X(d)r(1-(1-p)^2(1-\gamma))(\zeta-\bar{F}_X(d)). \end{equation*}

Letting $\phi'(d)=0$ , $d=\bar{F}^{-1}_X(\zeta)$ is a potential optimal deductible level. If $0< \zeta< \bar{F}_X(0)$ , it is clear that $d^*=\bar{F}^{-1}_X(\zeta)$ . However, if $\zeta\geq \bar{F}_X(0)$ , $\phi'(d)\geq 0$ on $\overline{\mathbb{R}}_+$ , so $\phi(d)$ is increasing, and hence $d^*=0$ .

3.3. PH transform distortion

For the PH transform, we have $g_i(t)=t^k$ , $k\in (0, 1]$ and $g_{i}'(t)=kt^{k-1}$ . In this case,

\begin{align*} \phi'(d)&=\left(\bar{F}_X(d)\right)^k-(1-p)^k\left(\bar{F}_X(d)\right)^k-(1+\theta)(p+(1-p)\gamma)\bar{F}_X(d)\\[5pt] &\quad+\frac{(1-p)\gamma}{1-\gamma}\int_{d}^{\infty} k(1-p)^{k-1}\left(\bar{F}_X\left(\frac{x-\gamma d}{1-\gamma}\right)\right)^{k-1} f_X\left(\frac{x-\gamma d}{1-\gamma}\right)\textrm{d} x\\[5pt] &=\left(\bar{F}_X(d)\right)^k-(1-p)^k\left(\bar{F}_X(d)\right)^k-(1+\theta)(p+(1-p)\gamma)\bar{F}_X(d)\\[5pt] &\quad+(1-p)\gamma \int_{d}^{\infty} k(1-p)^{k-1}\left(\bar{F}_X(x)\right)^{k-1} f_X(x)\textrm{d} x,\end{align*}

in which the integral in the last line can be calculated as

\begin{align*} \int_{d}^{\infty} k(1-p)^{k-1}\left(\bar{F}_X(x)\right)^{k-1} f_X(x)\textrm{d} x &= -(1-p)^{k-1} \int_{d}^{\infty}k\left(\bar{F}_X(x)\right)^{k-1}\textrm{d} \bar{F}_X(x)\\[5pt] &= -(1-p)^{k-1} \left(\bar{F}_X(x)\right)^{k}\Big |_{d}^{\infty}\\[5pt] &=(1-p)^{k-1} \left(\bar{F}_X(d)\right)^k.\end{align*}

Consequently, we obtain a more concise expression of $\phi'(d)$ :

\begin{align*} \phi'(d)=&\left[1-(1-\gamma)(1-p)^k\right]\left(\bar{F}_X(d)\right)^k-(1+\theta)(p+(1-p)\gamma)\bar{F}_X(d).\end{align*}

For fixed performance level p, safety loading $\theta$ , and PH index k, the optimal deductible is derived in the next theorem.

Proof. We have

\begin{align*} \phi'(d)&=\bar{F}_X(d)\left[\left(1-(1-\gamma)(1-p)^k\right)\left(\bar{F}_X(d)\right)^{k-1}-(1+\theta)(p+(1-p)\gamma)\right]\\[5pt] &= \bar{F}_X(d)\left(1-(1-\gamma)(1-p)^k\right)\left(\left(\bar{F}_X(d)\right)^{k-1}-\eta\right).\end{align*}

If $\eta> (\bar{F}_X(0))^{k-1}$ , when $0\leq d< \bar{F}^{-1}_X(\eta^{1/(k-1)})$ , $\phi'(d)< 0$ , and $\phi(d)$ is decreasing; when $d> \bar{F}^{-1}_X(\eta^{1/(k-1)})$ , $\phi'(d)\geq 0$ , so $\phi(d)$ is increasing. Therefore, the optimal deductible equals $d^*=\bar{F}^{-1}_X(\eta^{1/(k-1)})$ . Otherwise, if $\eta \leq (\bar{F}_X(0))^{k-1}$ , we have $\phi'(d)\geq 0$ , so $\phi(d)$ is increasing on $\mathbb{R}_+$ . It then follows that $d^*=0$ .

As per Corollary 3.15, given a fixed loading $\theta$ , the two parties always agree with a stop-loss treaty or full reinsurance treaty under the setting of PH transform distortion measure.

4.1. TVaR-TVaR case

This part considers the Bowley solution under the setting where both the insurer and the reinsurer employ the TVaR measures. As the insurer is usually more risk averse than the reinsurer, we assume that the respective confidence levels satisfy $1>\beta>\alpha> 0$ .

As in Theorem 3.8, when $\kappa=\alpha$ , that is, $1/\alpha=1+\theta$ , the insurer is indifferent between $d^*_1$ , $d^*_2$ as long as both of them are greater than $\bar{F}^{-1}_X(\alpha)$ . However, if the reinsurer increases the loading $\theta$ , it becomes more expensive for the insurer to transfer the risk. In the limiting case as $\theta\rightarrow \infty$ , we obviously have $d^*\rightarrow \infty$ so that the insurer will not reinsure his risk. In the general framework, the reinsurer usually takes the lead and has the advantage over the insurer. The reinsurer will encourage insurers to buy contracts that would offer better coverage. Even though the insurer is indifferent between two deductibles as long as they yield the same minimum risk level, the reinsurer will choose to provide a contract with a lower deductible. In the following analysis of the reinsurer’s problem, we assume whenever $\kappa=\alpha$ in Theorem 3.8, the optimal deductible $d^*$ equals $\bar{F}^{-1}_X(\alpha)$ .

In the sequel, we write $\kappa_\theta$ , $\nu_{\theta}$ instead of $\kappa$ , $\nu$ to indicate that they are functions of $\theta$ . Regarding Theorem 3.8, we discuss solutions to the reinsurer’s problem (2.5) as follows:

1. (a) For $p\geq 1-\alpha/\bar{F}_X(0)$ , we find the optimal $\theta^*$ by examining relationships between $\kappa_{\theta}$ , $\alpha$ and p.

1. (i) If $\alpha\leq \kappa_{\theta}< \bar{F}_X(0)$ , then $d^*=\bar{F}^{-1}_X(\kappa_{\theta})$ . We now have

   (4.4) \begin{equation}\begin{split} \psi'(\theta) &=(p+(1-p)\gamma)\int_{\bar{F}^{-1}_X(\kappa_{\theta})}^{\infty} \bar{F}_X(x)\textrm{d} x+(1+\theta-c){(p+(1-p)\gamma)}\frac{\kappa_{\theta} \kappa'_{\theta}}{f_X(\bar{F}^{-1}_X(\kappa_{\theta}))}\\[5pt] &\quad -\frac{p\kappa'_{\theta}}{\beta f_X(\bar{F}^{-1}_X(\kappa_{\theta}))}\int_0^{\infty} f_X(x+\bar{F}_X^{-1}(\kappa_{\theta}))\mathbf{1}_{\left\{\bar{F}_{I_{d^*}(X,Y)}(x)\leq \beta\right\}}(x)\textrm{d} x\\[5pt] &\quad -\frac{(1-p)\kappa'_{\theta}}{\beta f_X(\bar{F}^{-1}_X(\kappa_{\theta}))}\int_0^{\infty}f_X\left(\frac{x}{\gamma}+\bar{F}_X^{-1}(\kappa_{\theta})\right)\mathbf{1}_{\left\{\bar{F}_{I_{d^*}(X,Y)}(x)\leq \beta\right\}}(x)\textrm{d} x.\end{split}\end{equation}

Denote the solution set of $\psi'(\theta)=0$ by $\mathcal{A}^*\;:\!=\;\{\theta_1, \theta_2, \cdots\}$ . Define

\begin{equation*}\iota=(1/\bar{F}_X(0)-(1-p)(1-\gamma)/\alpha)/(p+(1-p)\gamma)-1.\end{equation*}

• • If $\iota> 0$ , we let $\mathcal{A}_{1,\iota}^*=\left\{\theta\in\mathcal{A}^*\;:\;\iota < \theta\leq 1/\alpha-1\right\}$ denote the admissible solution set;

• • otherwise if $\iota\leq 0$ , we define $\mathcal{A}_{2,\iota}^*=\left\{\theta\in\mathcal{A^*}\;:\; 0\leq \theta\leq 1/\alpha-1\right\}$ .

1. (ii) If $\bar{F}_X(0)\leq \kappa$ , then $d^*=0$ . In this case, $\psi(\theta)$ has the expression of (4.3).

2. (iii) Otherwise if $\kappa_{\theta}< \alpha$ , we have $d^*=\infty$ . In this case, $\psi(\theta)=0$ .

1. (b) For $0< p< 1-\alpha/\bar{F}_X(0)$ , we consider the following cases:

1. (i) if $\alpha\leq \kappa_{\theta}\leq \alpha/(1-p)$ , then $d^*=\bar{F}_X^{-1}(\kappa_\theta)$ . In this case, $\psi'(\theta)$ is again given by (4.4). As in Case 2(a)(i) above, we denote the solution set of the equation $\psi'(\theta)=0$ by $\mathcal{A}^*\;:\!=\;\{\theta_1, \theta_2, \cdots\}$ . Define

   \begin{equation*}\varphi=(1-p)\gamma/(\alpha(p+(1-p)\gamma))-1.\end{equation*}

• • If $\varphi> 0$ , we let $\mathcal{A}_{1,\varphi}^*=\left\{\theta\in\mathcal{A^*}\;:\; \varphi \leq\theta\leq 1/\alpha-1\right\}$ denote the admissible solution set;

• • otherwise if $\varphi\leq 0$ , we define $\mathcal{A}_{2,\varphi}^*=\left\{\theta\in\mathcal{A^*}\;:\; 0\leq \theta\leq 1/\alpha-1\right\}$ .

1. (ii) If $\kappa_{\theta}> \alpha/(1-p)$ and $\alpha/(1-p)< \nu_{\theta}< \bar{F}_X(0)$ , then $d^*=\bar{F}_X^{-1}(\nu_{\theta})$ . The derivative function of $\psi(\theta)$ now equals

   (4.5) \begin{equation}\begin{split} \psi'(\theta) &=(p+(1-p)\gamma)\int_{\bar{F}^{-1}_X(\nu_{\theta})}^{\infty} \bar{F}_X(x)\textrm{d} x+(1+\theta-c){(p+(1-p)\gamma)}\frac{\nu_{\theta} \nu'_{\theta}}{f_X(\bar{F}^{-1}_X(\nu_{\theta}))}\\[5pt] &\quad -\frac{p\nu'_{\theta}}{\beta f_X(\bar{F}^{-1}_X(\nu_{\theta}))}\int_0^{\infty} f_X(x+\bar{F}_X^{-1}(\nu_{\theta}))\mathbf{1}_{\left\{\bar{F}_{I_{d^*}(X,Y)}(x)\leq \beta\right\}}(x)\textrm{d} x\\[5pt] &\quad -\frac{(1-p)\nu'_{\theta}}{\beta f_X(\bar{F}^{-1}_X(\nu_{\theta}))}\int_0^{\infty}f_X\left(\frac{x}{\gamma}+\bar{F}_X^{-1}(\nu_{\theta})\right) \mathbf{1}_{\left\{\bar{F}_{I_{d^*}(X,Y)}(x)\leq \beta\right\}}(x)\textrm{d} x.\end{split}\end{equation}

As before, $\mathcal{A^*}=\{\theta_1, \theta_2, \cdots\}$ denotes the solution set of $\psi'(\theta)=0$ . Further, we let

\begin{equation*}\xi=\gamma/(\bar{F}_X(0)(p+(1-p)\gamma))-1.\end{equation*}

• • If $\xi> 0$ , we let $\mathcal{A}_{1,\xi,\varphi}^*=\left\{\theta\in\mathcal{A^*}\;:\; \xi< \theta< \varphi\right\}$ denote the admissible solution set;

• • if $\xi\leq 0$ and $\varphi> 0$ , we define $\mathcal{A}_{2,\xi,\varphi}^*=\left\{\theta\in\mathcal{A^*}\;:\; 0\leq \theta< \varphi\right\}$ ;

• • otherwise if $\varphi\leq 0$ , we define $\mathcal{A}_{3,\xi,\varphi}^*=\{0\}$ , which is a singleton.

Moreover, if $\bar{F}_X(0)\leq \nu_{\theta}$ , we have $d^*=0$ . The expression of $\psi(\theta)$ is given by (4.3).

(iii) If $\kappa_{\theta}< \alpha$ , we have $d^*=\infty$ . In this case, $\psi(\theta)=0$ .

The various cases described in Theorem 4.2 are schematically shown in Figure 2. The takeaways of Theorem 4.2 are as follows:

• • When $\bar{F}_X(0)\leq \alpha$ and the reinsurance premium is set within a reasonable range (i.e., $0\leq \theta\leq 1/\alpha-1$ ), the optimal contract is full reinsurance, corresponding to $d^*=0$ . For the profit-seeking purpose, the reinsurer would charge the premium as much as possible. Thus, the optimal safety loading would be $\theta^*=1/\alpha-1$ , and the complete Bowley solution is given by $(\theta^*,d^*(\theta^*))=(1/\alpha-1, 0)$ .

• • When $\bar{F}_X(0)> \alpha$ , multiple parameters determine the Bowley solution. The comparisons regarding $\kappa_{\theta}$ associated with a certain parameter select local optimal value for $\theta$ . A comparison among local solutions that achieve maximum for the reinsurer’s objective determines the global maximum $\theta^*$ . The pair $(\theta^*, d^*(\theta^*))$ denotes the Bowley solution.

  1. – (a1) If $p\geq 1-\alpha/\bar{F}_X(0)$ , and $\kappa_{\theta}\geq \alpha$ , we introduce a parameter $\iota$ . By comparing $\kappa_{\theta}$ and $\bar{F}_X(0)$ , there would be three local solutions parameterized by $\iota$ , that is, $\theta^*_{1,\iota}$ , $\theta^*_{2,\iota}$ , and $\theta^*_{\iota}$ . The final step is to select the global solution that yields the maximum objective value from the set $\{\theta^*_{1,\iota}, \theta^*_{2,\iota}, \theta^*_{\iota}\}$ .

  2. – (a2) If $p\geq 1-\alpha/\bar{F}_X(0)$ , and $\kappa_{\theta}\geq \alpha$ , it is not optimal for the insurer to enter the reinsurance contract, that is, $d^*=\infty$ . However, the Bowley solution can still be characterized as $(\theta^*, \infty)$ as long as $\theta^*> 1/\alpha-1$ .

  3. – (b1) If $0< p< 1-\alpha/\bar{F}_X(0)$ and $\kappa_{\theta}\geq \alpha$ , we introduce parameters $\varphi$ and $\xi$ to find six local solutions. The global solution is selected from local solutions. The discussion is similar to that in (a1), and we omit it here.

  4. – (b2) If $0< p< 1-\alpha/\bar{F}_X(0)$ and $\kappa_{\theta}> \alpha$ , the Bowley solution is given by $(\theta^*, \infty)$ as long as $\theta^*> 1/\alpha-1$ . The discussion is omitted as it is similar to (a2).

Figure 2. Flowchart illustration of Theorem 4.2 for finite $d^*$ only.

4.2. Gini-TVaR case

When $r\leq \theta$ , it is optimal for an insurer not to enter into any reinsurance arrangement ( $d^*=\infty$ ). In this case, we discover that $\psi(\theta)=0$ . Besides, the finite optimal deductible exists when $r> \theta$ . We write $\zeta_{\theta}$ instead of $\zeta$ to emphasize the dependence on $\theta$ . If $0< \zeta_{\theta}< \bar{F}_X(0)$ , then $d^*=\bar{F}_X^{-1}(\zeta_{\theta})$ . The derivative function $\psi'(\theta)$ is given by

(4.6) \begin{equation}\begin{split}\psi'(\theta) &=(p+(1-p)\gamma)\int_{\bar{F}^{-1}_X(\zeta_{\theta})}^{\infty} \bar{F}_X(x)\textrm{d} x+(1+\theta-c){(p+(1-p)\gamma)}\frac{\zeta_{\theta} \zeta'_{\theta}}{f_X(\bar{F}^{-1}_X(\zeta_{\theta}))}\\[5pt] &\quad -\frac{p\zeta'_{\theta}}{\beta f_X(\bar{F}^{-1}_X(\zeta_{\theta}))}\int_0^{\infty} f_X(x+\bar{F}_X^{-1}(\zeta_{\theta}))\mathbf{1}_{\left\{\bar{F}_{I_{d^*}(X,Y)}(x)\leq \beta\right\}}(x)\textrm{d} x\\[5pt] &\quad -\frac{(1-p)\zeta'_{\theta}}{\beta f_X(\bar{F}^{-1}_X(\zeta_{\theta}))}\int_0^{\infty}f_X\left(\frac{x}{\gamma}+\bar{F}_X^{-1}(\zeta_{\theta})\right) \mathbf{1}_{\left\{\bar{F}_{I_{d^*}(X,Y)}(x)\leq \beta\right\}}(x)\textrm{d} x.\end{split}\end{equation}

Let $\mathcal{B^*}\;:\!=\;\{\theta_1, \theta_2,\cdots\}$ be the solution set of the equation $\psi'(\theta)=0$ . Denote

\begin{equation*} \varrho=r(1-\bar{F}_X(0)(1-(1-p)^2(1-\gamma))/(p+(1-p)\gamma)).\end{equation*}

• • If $\varrho> 0$ , we let $\mathcal{B}_{1,\varrho}^*=\left\{\theta\in\mathcal{B^*}\;:\;\varrho < \theta< r\right\}$ denote the admissible solution set;

• • otherwise if $\varrho\leq 0$ , we define $\mathcal{B}_{2,\varrho}^*=\left\{\theta\in\mathcal{B^*}\;:\; 0\leq \theta< r\right\}$ .

Moreover, if $\zeta_{\theta}\geq \bar{F}_X(0)$ , then $d^*=0$ . In this case, $\psi(\theta)$ is given by (4.3), and it is increasing in $\theta$ .

4.3. PH-TVaR case

To emphasize the dependence on $\theta$ , we write $\eta_{\theta}$ in the sequel. If $\eta_{\theta}> (\bar{F}_X(0))^{k-1}$ , then $d^*=\bar{F}^{-1}_X(\eta_{\theta}^{1/(k-1)})$ and we have

(4.7) \begin{equation}\begin{split} \psi'(\theta) &=(p+(1-p)\gamma)\int_{\bar{F}^{-1}_X(\eta_{\theta})}^{\infty} \bar{F}_X(x)\textrm{d} x+(1+\theta-c){(p+(1-p)\gamma)}\frac{\eta_{\theta} \eta'_{\theta}}{f_X(\bar{F}^{-1}_X(\eta_{\theta}))}\\[5pt] &\quad -\frac{p\eta'_{\theta}}{\beta f_X(\bar{F}^{-1}_X(\eta_{\theta}))}\int_0^{\infty} f_X(x+\bar{F}_X^{-1}(\eta_{\theta}))\mathbf{1}_{\left\{\bar{F}_{I_{d^*}(X,Y)}(x)\leq \beta\right\}}(x)\textrm{d} x\\[5pt] &\quad -\frac{(1-p)\eta'_{\theta}}{\beta f_X(\bar{F}^{-1}_X(\eta_{\theta}))}\int_0^{\infty}f_X\left(\frac{x}{\gamma}+\bar{F}_X^{-1}(\eta_{\theta})\right) \mathbf{1}_{\left\{\bar{F}_{I_{d^*}(X,Y)}(x)\leq \beta\right\}}(x)\textrm{d} x.\end{split}\end{equation}

We denote by $\mathcal{C}^*\;:\!=\;\{\theta_1, \theta_2,\cdots\}$ the solution set of $\psi'(\theta)=0$ , and set

\begin{equation*} \varsigma=(\bar{F}_X(0))^{k-1}(1-(1-\gamma)(1-p)^k)/(p+(1-p)\gamma)-1.\end{equation*}

• • If $\varsigma> 0$ , we let $\mathcal{C}_{1,\varsigma}^*=\left\{\theta\in\mathcal{C^*}\;:\;\varsigma < \theta< \infty\right\}$ denote the admissible solution set;

• • otherwise if $\varsigma\leq 0$ , we define $\mathcal{C}_{2,\varsigma}^*=\left\{\theta\in\mathcal{C^*}\;:\; 0\leq \theta< \infty\right\}$ .

Moreover, if $\eta_{\theta}\leq (\bar{F}_X(0))^{k-1}$ , then $d^*=0$ . In this case, $\psi(\theta)$ is expressed by (4.3) and increases in $\theta$ .

5.1. Sensitivity analysis on the insurer’s optimal deductible level

In this subsection, we visually exhibit curves of optimal deductible $d^*$ and investigate some quantitative properties of the resulting results for the insurer’s problem. By varying one parameter at a time, we analyze the effect of performance probability p, recovery rate $\gamma$ , and safety loading $\theta$ on optimal deductible $d^*$ from the insurer’s side.

We start with an example where the TVaR measure with $\alpha=0.05$ is adopted. Besides, we take $\theta=0.1$ and $\gamma=0.3$ . We plot $d^*$ as a function of p in Figure 3. The two strategies, full insurance and the stop-loss contract, are denoted in red and blue color, respectively. In this example, $\bar{F}_X(0)=0.7> \alpha$ , so only Case 2 of Theorem 3.8 is illustrated. The figure is separated into two regions by the rightmost red dashed vertical line located at $p=1-\alpha/\bar{F}_X(0)\approx 0.929$ . For $p\in(0, 0.929)$ , subcases are treated separately, see the left black dashed vertical line at $p\approx 0.838$ corresponding to $\kappa=\alpha/(1-p)$ . The highest point corresponds to $p=0.838$ and $d^*=315.1498$ , which means that the maximum optimal deductible value that the insurer would set is $315.1498$ when the default probability is 0.162.

Figure 3. An illustration of $d^*$ for the TVaR measure with $\theta=0.1$ , $\gamma=0.3$ and $\alpha =0.05$ .

Figure 4. Effect of p on $\kappa$ and $\nu$ for the TVaR measure with $\theta=0.1$ , $\gamma=0.3$ and $\alpha = 0.05$ .

Figure 5. Effect of p and $\gamma$ on $d^*$ with $\theta=0.1$ .

Figure 6. Effect of p and $\theta$ on $d^*$ with $\gamma=0.3$ .

In Figure 3, the five labels $\mathcal{R}_1,\ \mathcal{R}_2,\ \mathcal{R}_3,\ \mathcal{R}_4$ , and $\mathcal{R}_5$ denote the following regions:

\begin{align*} \mathcal{R}_1 &=\{p\in(0,1]\;:\; 0< p< 1-\alpha/\bar{F}_X(0),\ \kappa> \alpha/(1-p),\ \nu\geq \bar{F}_X(0)\};\\[5pt] \mathcal{R}_2 &=\{p\in(0,1]\;:\; 0< p< 1-\alpha/\bar{F}_X(0),\ \kappa> \alpha/(1-p),\ \nu< \bar{F}_X(0)\};\\[5pt] \mathcal{R}_3 &=\{p\in(0,1]\;:\; 0< p< 1-\alpha/\bar{F}_X(0),\ \alpha< \kappa\leq \alpha/(1-p)\};\\[5pt] \mathcal{R}_4 &=\{p\in(0,1]\;:\; p\geq1-\alpha/\bar{F}_X(0),\ \alpha< \kappa< \bar{F}_X(0)\};\\[5pt] \mathcal{R}_5 &=\{p\in(0,1]\;:\; p\geq1-\alpha/\bar{F}_X(0),\ \kappa\geq\bar{F}_X(0)\}.\end{align*}

We further provide Figure 4 to address the relationship among strategies and shaded regions, which are determined by the effect of p on $\kappa$ and $\nu$ . Combining Theorem 3.8, we connect Figures 3 and 4 as follows:

• • There are three subcases when $p\in (0, 0.929)$ :

1. – for $p\in \mathcal{R}_1$ , corresponding to the second part of Case (2b)(ii), reinsurance is cheap, and buying full insurance benefits the insurer most though the reinsurer may default;

2. – for $p\in \mathcal{R}_2$ , corresponding to the first part of Case (2b)(ii), buying some reinsurance is desirable when the probability of full reimbursement is small;

3. – for $p\in \mathcal{R}_3$ , corresponding to Case (2b)(i), $d^*$ is contingent on the value of $\kappa$ and decreasing in p.

• • There are two subcases when $p\in[0.929, 1]$ , representing an even higher possibility of full recovery:

1. – for $p\in \mathcal{R}_4$ , corresponding to Case (2a)(i), the insurer continues to decrease his own retained loss;

2. – for $p\in \mathcal{R}_5$ , corresponding to Case (2a)(ii), the insurer finds it optimal to buy full insurance even if the premium is expensive.

Figure 5 shows how the value of $d^*$ is affected by the interaction between the performance probability p and the recovery rate $\gamma$ for a fixed safety loading. When $\gamma$ increases, the insurer pays more attention to the loading compared to the default risk. It agrees with the intuition that the optimal deductible decreases in general.

Figure 7. Effect of p and $\alpha$ on $d^*$ for the TVaR measure with $\theta=0.1$ and $\gamma=0.3$ .

In Figure 6, we examine the effects of the performance probability p and safety loading $\theta$ on $d^*$ for the three distortion functions, subject to a fixed recovery rate $\gamma$ . As $\theta$ increases, the optimal deductible increases. This makes sense because the more expensive the contract premium is, the more reluctant the insurer would be to purchase a reinsurance contract. The insurer would rather retain more risk by raising the deductible level. In the extreme case, if $\theta\rightarrow \infty$ , we have $d^*\rightarrow \infty$ . The insurer is discouraged from buying any reinsurance contract.

Moreover, for the TVaR measure, we show how the performance probability p and $\alpha$ affect $d^*$ with fixed $\theta=0.1$ and $\gamma=0.3$ in Figure 7. When default risk exists, a higher $\alpha$ reflects a less prudent attitude of the insurer, so the value of $d^*$ decreases. The insurer would be willing to transfer more risk to the reinsurer.

Two special scenarios are examined. When $p=1$ , which corresponds to the no-default case, the values of the optimal level of deductibles for three distortion risk measures are plotted in Figure 8. As $\theta$ increases, the optimal strategies switch from full insurance to a stop-loss treaty with increasing deductibles. It makes sense that when $\theta$ is small, the premium is cheap and the insurer would transfer more risk. When $\theta$ gets larger, the premium becomes more expensive, which discourages reinsurance. As we expect, even if full repayment is available, $d^*\rightarrow \infty$ as $\theta\rightarrow \infty$ , so the insurer will not reinsure his risk.

Figure 8. Effect of $\theta$ on $d^*$ when $p=1$ .

When $\gamma=0$ so that there is no compensation in case the reinsurer defaults, we pay attention to the effect of p and $\theta$ on $d^*$ for three distortion risk measures. The results are presented in Figure 9. When the loading is low and the full performance probability is high, the insurer opts for full insurance. Otherwise, a stop-loss reinsurance contract is optimal. As the loading $\theta$ increases, the optimal deductible increases, reflecting that the insurer is discouraged from buying reinsurance.

Figure 9. Effect of $\theta$ and p on $d^*$ when $\gamma=0$ .

5.2. Scenario analysis on the Bowley solution

In the following, we adopt a discretization method on $\theta\in[0,\infty)$ , provide some examples to illustrate the reinsurer’s problem (2.5), and present the Bowley solutions.

It is noted that the Bowley solutions are given by the same strategy $(\theta^*, d^*(\theta^*))=(19, 1410.22)$ under different parameter settings described above. This is due to the assumption in Section 4.1 on the insurer’s indifferent attitude when $1+\theta=1/\alpha$ . In this case, the insurer would buy a stop-loss treaty with $\bar{F}^{-1}_X(\alpha)$ being the deductible. As the reinsurer takes the lead and she is profit seeking, the reinsurer opts for selecting a loading which as large as possible.