Optimal quota-share and stop-loss reinsurance from the perspectives of insurer and reinsurer

Abstract

Reinsurance plays a vital role in the insurance activities. The insurer and the reinsurer, which have conflicting interests, compose the two parties of a reinsurance contract. In this paper, we extend the results achieved by Tan et al. (N Am Actuar J 13(4):459–482, 2009) to the case in which the perspectives of both the insurer and the reinsurer are considered. We study the optimal quota-share and stop-loss reinsurance models by minimizing the convex combination of the VaR risk measures of the insurer’s cost and the reinsurer’s cost. Furthermore, as many as 16 reinsurance premium principles are investigated. The results show that optimal quota-share and stop-loss reinsurance may or may not exist depending on the chosen principles. Moreover, we establish the sufficient and necessary conditions for the existence of the nontrivial optimal reinsurance.

Appendix

Proof of Theorem 3.1

(1) If the reinsurance premium $\pi \left( \cdot \right)$ satisfies $\pi \left( {bX} \right) = b\pi \left( X \right)$ for any $0 \le b \le 1$, then it follows from (3.1) that the loss function for quota-share reinsurance is given by

$$\begin{aligned} L\left( b \right) = \lambda S_X^{ - 1}\left( \alpha \right) + b\left( {\lambda \pi \left( X \right) - \left( {2\lambda - 1} \right) S_X^{ - 1}\left( \alpha \right) } \right) . \end{aligned}$$

(7.1)

Note that $L\left( b \right)$ is linear in b. Therefore,

(a) If $\lambda \in [0, \frac{1}{2}]$, then ${\lambda \pi \left( X \right) - \left( {2\lambda - 1} \right) S_X^{ - 1}\left( \alpha \right) }>0$, hence the optimal quota-share coefficient ${b^ * } = 0$.

(b) If $\lambda \in (\frac{1}{2},1]$, the sign of ${\lambda \pi \left( X \right) - \left( {2\lambda - 1} \right) S_X^{ - 1}\left( \alpha \right) }$ depends on the relative magnitude between $\lambda \pi \left( X \right)$ and $\left( {2\lambda - 1} \right) S_X^{ - 1}\left( \alpha \right)$, then the optimal quota-share coefficient ${b^ * }$ is determined as in (3.3).

(2) If $\pi \left( {bX} \right)$ is strictly convex in b for $0 \le b \le 1$, then it follows from (3.1) that

$$\begin{aligned} L'\left( b \right) = \left( {1-2\lambda } \right) S_X^{ - 1}\left( \alpha \right) + \lambda \pi _{b} '\left( {bX} \right) , \qquad L''\left( b \right) = \lambda \pi _{b} ''\left( {bX} \right) > 0. \end{aligned}$$

Therefore, the loss function L(b) is also strictly convex in b. Hence the nontrivial optimal quota-share reinsurance exists if and only if that L(b) attains its global minimum value at some $b^{*}\in (0,1)$, i.e., there exists a constant ${b^ * } \in \left( {0,1} \right)$ such that $L'(b)|_{b = b^{*} }=\lambda \pi _{b}'( b^{*}X ) + ( 1 - 2\lambda )S_{X}^{ - 1}( \alpha ) = 0.$ $\square$

Proof of Proposition 3.1

Note that for premium principles (P1)–(P11),

$$\begin{aligned} \pi (0)=0, \quad \pi \left( {bX} \right) = b\pi \left( X \right) . \end{aligned}$$

(7.2)

Then it follows from Theorem 3.1 (1) the optimal quota-share reinsurance is trivial and the optimal quota-share coefficient is determined as in Theorem 3.1 (1). $\square$

Proof of Proposition 3.2

(a) P12 (Variance principle): The reinsurance premium $\pi \left( {bX} \right) = bE\left( X \right) + {b^2}\beta D\left( X \right)$ is strictly convex in b, and the loss function $L\left( b \right)$ is also strictly convex in b. Therefore, it follows from Theorem 3.1 (b) that the optimal quota-share reinsurance is nontrivial if and only if there exists a constant ${b^ * } \in \left( {0,1} \right)$, such that

$$\begin{aligned} L'(b^{*})= & {} \lambda \pi '\left( {{b^ * }X} \right) + \left( {1 - 2\lambda } \right) S_X^{ - 1}\left( \alpha \right) \\= & {} \lambda \left[ {E\left( X \right) + 2{b^ * }\beta D\left( X \right) } \right] + \left( {1 - 2\lambda } \right) S_X^{ - 1}\left( \alpha \right) = 0. \end{aligned}$$

This is equivalent to $L'(0)<0$ and $L'(1)>0$, i.e. $\lambda E\left( X \right)< \left( {2\lambda - 1} \right) S_X^{ - 1}\left( \alpha \right) < \lambda E\left( X \right) + 2\lambda \beta D\left( X \right) .$

(b)–(e) We omit the proofs of (b)–(e) because the proofs are similar to (a). $\square$

Proof of Theorem 4.1

(1) If the loss function L(d) is concave on $[0, S_X^{ - 1}( \alpha )]$, then L(d) is increasing or decreasing or first increasing and then decreasing on $\left[ {0,S_X^{ - 1}\left( \alpha \right) } \right]$. And since the loss function $L\left( d \right)$ is decreasing in d on $\left( {S_X^{ - 1}\left( \alpha \right) ,\infty } \right)$, then the loss function $L\left( d \right)$ attains its minimum value at either $d=0$ or $d\rightarrow \infty$, which implies that the optimal stop-loss reinsurance is trivial.

Moreover, if L(d) is increasing or first increasing and then decreasing on $\left[ {0,S_X^{ - 1}\left( \alpha \right) } \right]$, then the optimal stop-loss constant ${d^ * }$ depends on the relative magnitude between $\lambda S_X^{ - 1}\left( \alpha \right)$ and $\lambda \pi \left( X \right) + \left( {1 - \lambda } \right) S_X^{ - 1}\left( \alpha \right)$. The optimal stop-loss retentions are indicated below

$$\begin{aligned} {d^ * } = {\left\{ \begin{array}{ll} 0, &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) > \lambda \pi \left( X \right) + \left( {1 - \lambda } \right) S_X^{ - 1}\left( \alpha \right) , \\ 0 \ or \ \infty , &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) = \lambda \pi \left( X \right) + \left( {1 - \lambda } \right) S_X^{ - 1}\left( \alpha \right) ,\\ \infty ,&{}\quad \lambda S_X^{ - 1}\left( \alpha \right) < \lambda \pi \left( X \right) + \left( {1 - \lambda } \right) S_X^{ - 1}\left( \alpha \right) . \end{array}\right. } \end{aligned}$$

(7.3)

If L(d) is decreasing on $\left[ {0,S_X^{ - 1}\left( \alpha \right) } \right]$, then the optimal stop-loss retention ${d^ * }=\infty$ because of the continuity of the loss function.

(2) If the loss function $L\left( d \right)$ is decreasing in d on $\left[ {0,{d_0}} \right]$ while increasing on $\left[ {{d_0},S_X^{ - 1}\left( \alpha \right) } \right]$, (4.7) ensures that the loss function $L\left( d \right)$ attains its global minimum at ${d_0}$, then the optimal stop-loss reinsurance is nontrivial; On the contrary, while the optimal stop-loss reinsurance is nontrivial, then ${d_0}$ is the global minimum for $L\left( d \right)$, hence (4.7) holds. $\square$

Proof of Proposition 4.1

The results for these premium principles can be verified by resorting to Theorem 4.1 (2).

(a) P1 (Expectation principle): Under expectation principle, the loss function and its derivatives on $[0, S_{X}^{-1}(\alpha )]$ are given by

$$\begin{aligned} L\left( d \right)= & {} \left( {2\lambda - 1} \right) d + \lambda \left( {1 + \beta } \right) \int _d^\infty {{S_X}\left( x \right) dx} + \left( {1 - \lambda } \right) S_X^{ - 1}\left( \alpha \right) ,\quad \end{aligned}$$

(7.4)

$$\begin{aligned} L'\left( d \right)= & {} 2\lambda - 1 - \lambda \left( {1 + \beta } \right) {S_X}\left( d \right) , \end{aligned}$$

(7.5)

$$\begin{aligned} L''\left( d \right)= & {} - \lambda \left( {1 + \beta } \right) {{S'}_X}\left( d \right) > 0. \end{aligned}$$

(7.6)

Therefore, the first derivative $L'\left( d \right)$ is strictly increasing in d on $[0, S_{X}^{-1}(\alpha )]$. Note that

$$\begin{aligned} L'\left( 0 \right) = 2\lambda - 1 - \lambda \left( {1 + \beta } \right) {S_X}\left( 0 \right) ,\qquad L'\left( {S_X^{ - 1}\left( \alpha \right) } \right) = 2\lambda - 1 - \lambda \alpha \left( {1 + \beta } \right) . \end{aligned}$$

(1) If $L'\left( 0 \right) > 0$, that is $\frac{1}{{2 - \left( {1 + \beta } \right) {S_X}\left( 0 \right) }}< \lambda < 1$. For any $d \in \left[ {0,S_{X}^{-1}(\alpha ) } \right]$, $L'\left( d \right) > 0$, then $L\left( d \right)$ is increasing in d, therefore the optimal stop-loss retention ${d^ * }$ depends on the relative magnitude between $L\left( 0 \right)$ and $\lambda S_X^{ - 1}\left( \alpha \right)$, i.e.

$$\begin{aligned} {d^ * } = {\left\{ \begin{array}{ll} 0, &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) > L\left( 0 \right) ,\\ 0 \ or \ \infty , &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) = L\left( 0 \right) ,\\ \infty , &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) < L\left( 0 \right) . \end{array}\right. } \end{aligned}$$

In this case, the optimal reinsurance is trivial.

(2) If $L'\left( {S_X^{ - 1}\left( \alpha \right) } \right) < 0$, that is $0< \lambda < \frac{1}{{2 - \alpha \left( {1 + \beta } \right) }}$, then $L\left( d \right)$ is decreasing in d, therefore the optimal stop-loss retention is ${d^ * = \infty }$. The optimal reinsurance is also trivial in this case.

(3) If $L'\left( 0 \right) < 0$ and $L'\left( {S_X^{ - 1}\left( \alpha \right) } \right) > 0$, that is

$$\begin{aligned} \frac{1}{{2 - \alpha \left( {1 + \beta } \right) }}< \lambda < \frac{1}{{2 - \left( {1 + \beta } \right) {S_X}\left( 0 \right) }}. \end{aligned}$$

Then there exists a positive constant ${d_0}$ such that the loss function $L\left( d \right)$ is decreasing in d on $\left[ {0,{d_0}} \right]$ while increasing on $\left[ {{d_0},S_X^{ - 1}\left( \alpha \right) } \right]$, where ${d_0}$ satisfies

$$\begin{aligned} L'\left( {{d_0}} \right) = 2\lambda - 1 - \lambda \left( {1 + \beta } \right) {S_X}\left( {{d_0}} \right) = 0. \end{aligned}$$

Thus the optimal stop-loss retention ${d^ * }$ depends on the relative magnitude between $L\left( {d_0} \right)$ and $\lambda S_X^{ - 1}\left( \alpha \right)$, that is

$$\begin{aligned} {d^ * } = {\left\{ \begin{array}{ll} {d_0}, &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) > L\left( {d_0} \right) ,\\ {d_0} \ or \ \infty , &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) = L\left( {d_0} \right) ,\\ \infty , &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) < L\left( {d_0} \right) . \end{array}\right. } \end{aligned}$$

Therefore, the optimal stop-loss reinsurance is nontrivial if and only if

$$\begin{aligned} \frac{1}{{2 - \alpha \left( {1 + \beta } \right) }}< \lambda < \frac{1}{{2 - \left( {1 + \beta } \right) {S_X}\left( 0 \right) }} \end{aligned}$$

(7.7)

and

$$\begin{aligned} \lambda S_X^{ - 1}\left( \alpha \right) > \left( {2\lambda - 1} \right) {d_0} + \lambda ( 1 + \beta )\int _{d_{0}}^\infty {S_X}( x )dx + \left( {1 - \lambda } \right) S_X^{ - 1}\left( \alpha \right) . \end{aligned}$$

(7.8)

(b)-(d) We omit the proofs of (b)-(d), because they are similar to (a).

(e) P8 (Standard deviation principle): Under standard deviation principle, the loss function and its first and second derivatives are given by

$$\begin{aligned} L\left( d \right)= & {} \left( {2\lambda - 1} \right) d + \lambda E{\left[ {X - d} \right] _ + } + \lambda \beta \sqrt{D{{\left[ {X - d} \right] }_ + }} + \left( {1 - \lambda } \right) S_X^{ - 1}\left( \alpha \right) ,\nonumber \\ \end{aligned}$$

(7.9)

$$\begin{aligned} L'\left( d \right)= & {} \lambda - 1 + \lambda \left( {1 - {S_X}\left( d \right) } \right) \left( {1 - \beta \frac{{E{{\left[ {X - d} \right] }_ + }}}{{\sqrt{D{{\left[ {X - d} \right] }_ + }} }}} \right) , \end{aligned}$$

(7.10)

$$\begin{aligned} L''\left( d \right)= & {} - \lambda {{S'}_X}\left( d \right) \left( {1 - \beta \frac{{E{{\left[ {X - d} \right] }_ + }}}{{\sqrt{D{{\left[ {X - d} \right] }_ + }} }}} \right) \nonumber \\&+\,\beta \lambda \left( {1 - {S_X}\left( d \right) } \right) \frac{{{S_X}\left( d \right) D{{\left[ {X - d} \right] }_ + } - \left( {1 - {S_X}\left( d \right) } \right) {E^2}{{\left[ {X - d} \right] }_ + }}}{{{{\left( {D{{\left[ {X - d} \right] }_ + }} \right) }^{{3/2}}}}}.\nonumber \\ \end{aligned}$$

(7.11)

Let

$$\begin{aligned} {g_1}\left( d \right) = 1 - \beta \frac{{E{{\left[ {X - d} \right] }_ + }}}{{\sqrt{D{{\left[ {X - d} \right] }_ + }} }}, \end{aligned}$$

(7.12)

then

$$\begin{aligned} {{g'}_1}\left( d \right) = \beta \frac{{{S_X}\left( d \right) D{{\left[ {X - d} \right] }_ + } - \left( {1 - {S_X}\left( d \right) } \right) {E^2}{{\left[ {X - d} \right] }_ + }}}{\left( D{\left[ {X - d} \right] }_ {+}\right) ^{3/2}}. \end{aligned}$$

(7.13)

Let

$$\begin{aligned} q\left( d \right) = {S_X}\left( d \right) D\left[ {X - d} \right] _ {+} - \left( {1 - {S_X}\left( d \right) } \right) {E^2}\left[ {X - d} \right] _ +, \end{aligned}$$

(7.14)

then

$$\begin{aligned} q'\left( d \right) = {{S'}_X}\left( d \right) E\left[ {X - d} \right] _ + ^2 < 0. \end{aligned}$$

(7.15)

Note that $\lim \limits _{d\rightarrow \infty } q(d)= 0$. Therefore $q\left( d \right) > 0$ and ${{g}_1}\left( d \right)$ is strictly increasing in d. If condition $\beta E\left[ X \right] < \sqrt{D\left[ X \right] }$ holds, then $L''\left( d \right) >0$ and $L'\left( d \right)$ is increasing in d. Note that

$$\begin{aligned} L'\left( 0 \right)= & {} \lambda - 1 + \lambda \left( {1 - {S_X}\left( 0 \right) } \right) \left( {1 - \beta \frac{{E\left[ X \right] }}{{\sqrt{D\left[ X \right] } }}} \right) ,\\ L'\left( {S_X^{ - 1}\left( \alpha \right) } \right)= & {} \lambda - 1 + \lambda \left( {1 - \alpha } \right) \left( {1 - \beta \frac{{E{{\left[ {X - S_X^{ - 1}\left( \alpha \right) } \right] }_ + }}}{{\sqrt{D{{\left[ {X - S_X^{ - 1}\left( \alpha \right) } \right] }_ + }} }}} \right) . \end{aligned}$$

(1) If $L'\left( 0 \right) > 0$, that is $\lambda _g^1< \lambda < 1$, then $L\left( d \right)$ is increasing in d on $[0, S_{X}^{-1}(\alpha )]$. Therefore the optimal stop-loss constant ${d^ * }$ depends on the relative magnitude between $L\left( 0 \right)$ and $\lambda S_X^{ - 1}\left( \alpha \right)$, i.e.

$$\begin{aligned} {d^ * } = {\left\{ \begin{array}{ll} 0, &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) > L\left( 0 \right) ,\\ 0 \ or \ \infty , &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) = L\left( 0 \right) ,\\ \infty , &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) < L\left( 0 \right) , \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \lambda _g^1 = \frac{1}{{1 + \left( {1 - {S_X}\left( 0 \right) } \right) \left( {1 - \beta \frac{{E\left[ X \right] }}{{\sqrt{D\left[ X \right] } }}} \right) }}. \end{aligned}$$

In this case, the optimal reinsurance is trivial.

(2) If $L'\left( {S_X^{ - 1}\left( \alpha \right) } \right) < 0$, that is $0< \lambda < \lambda _g^2$, then $L\left( d \right)$ is decreasing in d, therefore the optimal stop-loss retention ${d^ * = \infty }$, where

$$\begin{aligned} \lambda _g^2 = \frac{1}{{1 + \left( {1 - \alpha } \right) \left( {1 - \beta \frac{{E{{\left[ {X - S_X^{ - 1}\left( \alpha \right) } \right] }_ + }}}{{\sqrt{D{{\left[ {X - S_X^{ - 1}\left( \alpha \right) } \right] }_ + }} }}} \right) }}. \end{aligned}$$

In this case, the optimal reinsurance is also trivial.

(3) If $L'\left( 0 \right) < 0$ and $L'\left( {S_X^{ - 1}\left( \alpha \right) } \right) > 0$, that is $\lambda _g^2< \lambda < \lambda _g^1$. Then there exists a positive constant ${d_0}$ such that the loss function $L\left( d \right)$ is decreasing in d on $\left[ {0,{d_0}} \right]$ while increasing on $\left[ {{d_0},S_X^{ - 1}\left( \alpha \right) } \right]$, where ${d_0}$ satisfies

$$\begin{aligned} L'\left( d_0 \right) = \lambda - 1 + \lambda \left( {1 - {S_X}\left( d_0 \right) } \right) \left( {1 - \beta \frac{{E\left[ X-d_{0} \right] _{+}}}{{\sqrt{D\left[ X -d_{0}\right] _{+}} }}} \right) = 0. \end{aligned}$$

Thus the optimal stop-loss retention ${d^ * }$ depends on the relative magnitude between $L\left( {d_0} \right)$ and $\lambda S_X^{ - 1}\left( \alpha \right)$, that is

$$\begin{aligned} {d^ * } = {\left\{ \begin{array}{ll} {d_0}, &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) > L\left( {d_0} \right) ,\\ {d_0} \ or \ \infty , &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) = L\left( {d_0} \right) ,\\ \infty , &{}\quad \lambda S_X^{ - 1}\left( \alpha \right) < L\left( {d_0} \right) . \end{array}\right. } \end{aligned}$$

Therefore, under condition $\beta E\left[ X \right] < \sqrt{D\left[ X \right] }$, the optimal stop-loss reinsurance is nontrivial if and only if $\lambda _g^2< \lambda < \lambda _g^1$ and $\lambda S_X^{ - 1}\left( \alpha \right) > L\left( {d_0} \right)$.

(f) P10 (Generalized percentile principle): We omit the proof of (f), because it is similar to (a).

(g)–(i): We omit the proofs of (g)–(i), because they are similar to (e). $\square$