An Optimal Multi-layer Reinsurance Policy under Conditional Tail Expectation

TL;DR

This paper develops a method to construct optimal multi-layer reinsurance policies under the conditional tail expectation risk measure, extending traditional stop-loss policies and demonstrating their practical application through simulations.

Contribution

It introduces a systematic extension of optimal stop-loss reinsurance to multi-layer policies and provides estimation methods for unknown parameters.

Findings

Multi-layer reinsurance policies are optimal under CTE risk measure.

The proposed policies outperform traditional single-layer policies.

Simulation studies confirm practical applicability and parameter estimation methods.

Abstract

A usual reinsurance policy for insurance companies admits one or two layers of the payment deductions. Under optimal criterion of minimizing the conditional tail expectation (CTE) risk measure of the insurer's total risk, this article generalized an optimal stop-loss reinsurance policy to an optimal multi-layer reinsurance policy. To achieve such optimal multi-layer reinsurance policy, this article starts from a given optimal stop-loss reinsurance policy $f(\cdot).$ In the first step, it cuts down an interval $[0,\infty)$ into two intervals $[0,M_1)$ and $[M_1,\infty).$ By shifting the origin of Cartesian coordinate system to $(M_{1},f(M_{1})),$ and showing that under the $CTE$ criteria $f(x)I_{[0, M_1)}(x)+(f(M_1)+f(x-M_1))I_{[M_1,\infty)}(x)$ is, again, an optimal policy. This extension procedure can be repeated to obtain an optimal k-layer reinsurance policy. Finally, unknown parameters of the optimal multi-layer reinsurance policy are estimated using some additional appropriate criteria. Three simulation-based studies have been conducted to demonstrate: ({\bf 1}) The practical applications of our findings and ({\bf 2}) How one may employ other appropriate criteria to estimate unknown parameters of an optimal multi-layer contract. The multi-layer reinsurance policy, similar to the original stop-loss reinsurance policy is optimal, in a same sense. Moreover it has some other optimal criteria which the original policy does not have. Under optimal criterion of minimizing general translative and monotone risk measure $\rho(\cdot)$ of {\it either} the insurer's total risk {\it or} both the insurer's and the reinsurer's total risks, this article (in its discussion) also extends a given optimal reinsurance contract $f(\cdot)$ to a multi-layer and continuous reinsurance policy.

Equations

$$T_h(X)$$

$$\mathcal{C}$$

$$f_k(X)$$

$$\displaystyle X_{R}^{opt}=\left\{{\begin{array}[]{*{20}{c}}0&{X<{d_{\alpha}}}\\ {X-{d_{\alpha}}}&{{d_{\alpha}}\leq X<{M_{1}}}\\ {{M_{1}}-{d_{\alpha}}}&{{M_{1}}\leq X<{M_{1}}+{d_{\alpha}}}\\ {X-2{d_{\alpha}}}&{{M_{1}}+{d_{\alpha}}\leq X<{M_{2}}}\\ \vdots&{}\hfil\\ {{M_{k}}-k{d_{\alpha}}}&{{M_{k}}\leq X<{M_{k}}+{d_{\alpha}}}\\ {X-k{d_{\alpha}}}&{{M_{k}}+{d_{\alpha}}\leq X}\end{array}}\right.$$

$$F_{X_R^{opt}}(t)$$

$$M_{X_R^{opt}}(t)=$$

$${\int }_{M_2^{\ast }}^{M_3^{\ast }}{e}^{t(x+(M_1^{\ast }-M_0^{\ast })-M_2^{\ast })}d{F}_X(x)$$

$$M_{X-X_R^{opt}}(t)$$

$$E(X_R^{opt})$$

$$Q_h=\omega Var(h(X))+(1-\omega )Var(X-h(X)),$$

$$\underset{h\in \mathcal{C}}{\mathrm{argmin}}{Q}_h$$

$$Q_I=[\omega \frac{{(1-2\omega -{\beta }_1)}^2}{{(1-{\beta }_1)}^2}+(1-\omega )\frac{{(2\omega )}^2}{{(1-{\beta }_1)}^2}]Var(X).$$

$$\displaystyle{X_{R}^{2-layer}}(X)=\left\{\begin{array}[]{l}0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,X<{d_{\alpha}}\\ X-{d_{\alpha}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{d_{\alpha}}\leq X<{M_{1}}\\ {M_{1}}-{d_{\alpha}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{M_{1}}\leq X<{M_{1}}+{d_{\alpha}}\\ X-2{d_{\alpha}}\,\,\,\,\,\,\,{M_{1}}+{d_{\alpha}}\leq X<M_{2}\\ M_{2}-2{d_{\alpha}}\,\,\,\,\,\,\,{M_{2}}\leq X<M_{2}+{d_{\alpha}}\\ X-3{d_{\alpha}}\,\,\,\,\,\,\,{M_{2}}+{d_{\alpha}}\leq X\\ \end{array}\right.$$

$$U_h=\omega E(\exp (-\beta (h(X))))+(1-\omega )E(\exp (-\beta (X-h(X)))).$$

$$g_{X_R^{(i)}|\Theta }(y^{(i)})$$

$$f_{X_R^{(1)},\cdots ,X_R^{(n)}}(y^{(1)},\cdots ,y^{(n)}|\Theta )$$

$$\pi (\theta ,M_0^{\ast },\cdots ,M_{m-2}^{\ast }|y^{(1)},\cdots ,y^{(n)})=$$

$$\frac{f_{X_R^{(1)},\cdots ,X_R^{(n)}}(y^{(1)},\cdots ,y^{(n)}|\theta ,M_0^{\ast },\cdots ,M_{k-2}^{\ast })\pi (\theta ,M_0^{\ast },\cdots ,M_{k-2}^{\ast })}{{\int }_{{{\mathcal{M}}^{\ast }}_{k-2}}\cdots {\int }_{\Theta }{f}_{X_R^{(1)},\cdots ,X_R^{(n)}}(y^{(1)},\cdots ,y^{(n)}|\theta ,M_0^{\ast },\cdots ,M_{(}^{\ast }k-2))\pi (\theta ,M_0^{\ast },\cdots ,M_{k-2}^{\ast })d\theta d{M}_0^{\ast },\cdots ,d{M}_{k-2}^{\ast }}.$$

$${{\hat{M}}^{\ast }}_i$$

$$\displaystyle{X_{R}^{1-layer}}=\left\{\begin{array}[]{l}0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,X<{M_{0}}\\ X-{M_{0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{M_{0}}\leq X<{M_{1}}\\ {M_{1}}-{M_{0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{M_{1}}\leq X<{M_{2}}\\ X+({M_{1}}-{M_{0}})-{M_{2}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{M_{2}}\leq X\end{array}\right..$$

$$g(X)$$

$$\rho (X-g(X)+{\pi }_g^X)$$

$$g(X)$$

$${\omega }^{\ast }{\rho }_1(X-g(X)+{\pi }_g^X)+(1-{\omega }^{\ast }){\rho }_2(g(X)-{\pi }_g^X)$$

Full text

**An Optimal Multi-layer Reinsurance Policy under Conditional Tail Expectation

**Amir T. Payandeh Najafabadi 111Corresponding author: amirtpayandeh@sbu.ac.ir; Phone no. +98-21-29903011; Fax no. +98-21-22431649 & Ali Panahi Bazaz

Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, 1983963113, Tehran, Iran.

Abstract

A usual reinsurance policy for insurance companies admits one or two layers of the payment deductions. Under optimal criterion of minimizing the conditional tail expectation (CTE) risk measure of the insurer’s total risk, this article generalized an optimal stop-loss reinsurance policy to an optimal multi-layer reinsurance policy. To achieve such optimal multi-layer reinsurance policy, this article starts from a given optimal stop-loss reinsurance policy $f(\cdot).$ In the first step, it cuts down an interval $[0,\infty)$ into two intervals $[0,M_{1})$ and $[M_{1},\infty).$ By shifting the origin of Cartesian coordinate system to $(M_{1},f(M_{1})),$ and showing that under the $CTE$ criteria $f(x)I_{[0,M_{1})}(x)+(f(M_{1})+f(x-M_{1}))I_{[M_{1},\infty)}(x)$ is, again, an optimal policy. This extension procedure can be repeated to obtain an optimal k-layer reinsurance policy. Finally, unknown parameters of the optimal multi-layer reinsurance policy are estimated using some additional appropriate criteria. Three simulation-based studies have been conducted to demonstrate: (1) The practical applications of our findings and (2) How one may employ other appropriate criteria to estimate unknown parameters of an optimal multi-layer contract. The multi-layer reinsurance policy, similar to the original stop-loss reinsurance policy is optimal, in a same sense. Moreover it has some other optimal criteria which the original policy does not have. Under optimal criterion of minimizing general translative and monotone risk measure $\rho(\cdot)$ of either the insurer’s total risk or both the insurer’s and the reinsurer’s total risks, this article (in its discussion) also extends a given optimal reinsurance contract $f(\cdot)$ to a multi-layer and continuous reinsurance policy.

Keywords: Reinsurance policy; Stop-loss reinsurance; Translative and monotone risk measures; Optimization; Conditional tail expectation (CTE); Bayesian method.

AMS 2010 subject classifications: 97M30, 97K80, 62F15

1 Introduction

Designing an optimal reinsurance policy, in some sense, is one of the most attractive aspects in actuarial science. Reinsurance is a form of an insurance contract, that reinsurer accepts to pay a portion of an insurer’s risk by receiving a reinsurance premium. Therefore, both reinsurance and insurance companies try to design an optimal reinsurance policy to improve their ability to managing their risks under a certain criteria, e.g., increasing their surplus/wealth of company, decreasing the ruin probability, etc.

Several authors considered the problem of designing an optimal reinsurance policy under a certain optimal criteria. Surprisingly, in the most of studies the stop-loss reinsurance policy (or some its modification) established as an optimal policy. For instance, Borch (1960) proved that, under the variance retained risk optimal criteria and in the class of reinsurance policies with an equal reinsurance premium, the stop-loss reinsurance minimizes such variance. Under Borch (1960)’s class of reinsurance policies , Hesselager (1990) showed that the stop-loss reinsurance is an optimal policy which provides the smallest Lundberg’s upper bound for the ruin probability. Optimality of the one-layer stop-loss contract under minimizes the ruin probability criteria and several premium principles has been established by Kaluszka (2005). Passalacqua (2007) studied impacts of multi-layer stop-loss reinsurance contract on the valuation of risk capital (assessed under the Solvency II framework) for credit insurance. Cai et al. (2008) showed that the one-layer stop-loss contract is optimal whenever either both the ceded and the retained loss functions are increasing or the retained loss function is increasing and left-continuous. Kaluszka & Okolewski (2008) established the one-layer stop-loss contract is an optimal contract under the maximization of the expected utility, the stability and the survival probability of the cedent. Tan et al. (2011) and Chi & Tan (2011) showed that under the expectation premium principle assumption and the Conditional tail expectation (CTE) minimization criteria the stop-loss reinsurance contract is optimal. Porth et al. (2013) employed an empirical reinsurance model (introduced by Weng, 2009) to show that, under the standard deviation premium principle and consistency with market practice, a one-layer stop-loss reinsurance contract is optimal. In a situation that both the ceded and the retained loss functions are constrained to be increasing and under the variance premium principle assumption, Chi (2012) showed that one-layer stop-loss reinsurance is always optimal over both the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR) criteria. Ouyang & Li (2010) constructed a multi-layer reinsurance policy to achieve sustainable development of an agricultural insurance policy in the sense of adverse selection and mortal hazard problems. In 2012, Dedu generalized the stop-loss reinsurance to a multi-layer reinsurance policy. In the first step, she considered a certain class of multi-layer reinsurance policies with some unknown parameters. An optimal reinsurance policy, in such class, have been obtained by estimating unknown parameters such that the VaR and the CTE of the insurer’s total risk have been minimized. Chi (2012) showed that under minimizes the risk-adjusted value of an insurer’s liability and the VaR (or the CVaR) criteria the two-layer reinsurance contract is optimal under the Dutch premium principle assumption. Cortes et al. (2013) considered a multi-layer reinsurance contract consisting of a fixed number of layers. Then, they determined an optimal multi-layer contract such that for a given expected return the associated risk value is minimized. Chi & Tan (2013) established that a one-layer stop-loss contract is always optimal over both the VaR and the CVaR criteria and the prescribed premium principles. Cai & Weng (2014) showed under risk margin associated with an expectile risk measure criteria a two-layer reinsurance contract minimizes the liability of an insurer for a general class of reinsurance premium principles. Panahi Bazaz & Payandeh Najafabadi (2015) estimated parameters of a one-layer reinsurance policy such that a convex combination of the CTE of both the insurer’s and reinsurer’s random risks are minimized. Optimality of the stop-loss contract under distortion risk measures and premiums has been established by Assa (2015). Zhuang et al. (2016) showed that in a situation that the premium budget is not sufficiently high enough, under the CVaR optimality criteria, the optimal reinsurance policy will change from the stop-loss contract to a one-layer stop-loss. Payandeh Najafabadi & Panahi Bazaz (2016) considered a co-reinsurance contract which is a combination of several reinsurance contracts. Using a Bayesian approach parameters of co-reinsurance contract have been estimated.

In order to exclude the moral hazard, an appropriate reinsurance contract has to assign increasing functions to both insurer and reinsurer portions. On the other hand, reported claims in insurance industry have the property that higher claim size is less frequent with more severe probability of loss. Whereas lower claim sizes are more frequent with less severe probability of loss. Unfortunately, the stop-loss reinsurance contract despite several well-known properties does not consider these two important facts.

This article considers minimizing the $CTE$ risk measure of the insurer’s total risk as an optimal criterion to design an optimal reinsurance contract. Then, it introduces an algorithm which generalized a given optimal stop-loss policy to a multi-layer optimal reinsurance policy. To achieve such optimal multi-layer reinsurance policy, this article starts from a given optimal stop-loss reinsurance policy $f(\cdot).$ In the first step, it cuts down an interval $[0,\infty)$ into two intervals $[0,M_{1})$ and $[M_{1},\infty).$ By shifting the origin of Cartesian coordinate system to $(M_{1},f(M_{1})),$ and showing that under the $CTE$ criteria $f(x)I_{[0,M_{1})}(x)+(f(M_{1})+f(x-M_{1}))I_{[M_{1},\infty)}(x)$ is, again, an optimal policy. This extension procedure can be repeated to obtain an optimal k-layer reinsurance policy. Finally, unknown parameters of the multi-layer reinsurance policy are estimated using some additional appropriate criteria. Practical application of our findings have been shown through a simulation study. The multi-layer reinsurance policy, similar to the original stop-loss reinsurance policy is optimal, in a same sense. Moreover, it has some other optimal criteria which the original policy does not have. Under optimal criterion of minimizing a general translative and monotone risk measure $\rho(\cdot)$ of either the insurer’s total risk or both the insurer’s and the reinsurer’s total risks, this article (in its discussion) also extends an optimal reinsurance contract $f(\cdot)$ to an optimal multi-layer and continuous reinsurance policy.

This article is organized as the following. Section 2 collects some elements that play vital roles in the rest of this article. Moreover, Section 2 represents an algorithm that extends a given optimal stop-loss reinsurance policy to an optimal multi-layer policy. Section 3 describes three simulation-based studies illustrating the practical application of our results. Parameters of the optimal multi-layer contract, for each simulation study, have been estimated using an additional appropriate criteria. In Discussion results of this article (from two different senses) extends an optimal reinsurance contract $f(\cdot),$ under a general translative and monotone risk measure $\rho(\cdot),$ to an optimal multi-layer and continuous reinsurance policy.

2 Preliminary

Suppose continuous and nonnegative random variable $X$ stands for the aggregate claim initially assumed by an insurer. In addition, suppose that random claim $X$ with a cumulative distribution function $F_{X}(t)$ and a survival function $\bar{F}_{X}(t),$ and a density function $f_{X}$ defines on the probability space $(\Omega,\mathcal{F},P),$ where $\Omega=[0,\infty)$ and $\mathcal{F}$ is the Borel $\sigma$-field on $\Omega.$ Now, let $X_{I}$ and $X_{R},$ (or $X_{R}=h(X)$) respectively, stand for the insurer’s and the reinsurer’s risk portions from random claim $X,$ such that $X=X_{I}+X_{R}$ and $0\leq X_{I}~{}\&~{}X_{R}=h(X)\leq X.$ Under this presentation, the total risk of the insurance company can be restated as

[TABLE]

where $h(\cdot)$ is a functional form of a reinsurance contract and $\pi_{h}^{X}$ stands for a reinsurance premium.

Now, we collect some elements that play vital roles in the rest of this article.

Definition 1**.**

Risk measure $\rho(\cdot)$ is called translative and monotone if and only if $\rho(X+c)=\rho(X)+c$ and $\rho(X)\leq\rho(Y)$ whenever, $P(X\leq Y)=1$ and $c\in{\mathbb{R}}.$

In the sense of above definition a wide class of risk measures, such as coherent, spectral, distortion, Quantile-based, and Wang, are translative and monotone risk measures, see Denuit et al. (2006) for other possible classes of translative and monotone risk measures.

Consider the following class of reinsurance policies.

[TABLE]

where $\pi_{h}^{X}$ stands for the reinsurance premium under a reinsurance contract $h(\cdot).$

Suppose $f(\cdot)$ in class of reinsurer contracts $\mathcal{C},$ given by (2), minimizes given translative and monotone risk measure $\rho(\cdot)$ of the total risk of insurance company, i.e., $\displaystyle f(X)\equiv\operatornamewithlimits{argmin}_{h\in\mathcal{C}}\rho\left(T_{h}(X)\right).$ Now one may cut down an interval $[0,\infty)$ into two intervals $[0,M_{1})$ and $[M_{1},\infty)$ and shift the origin of Cartesian coordinate system to $(M_{1},f(M_{1})),$ see Figure 1(a) for an illustration. Again, in the new Cartesian coordinate system, the shifted reinsurance contract $f(\cdot)$ is an optimal contract and, in the old Cartesian coordinate system, the reinsurance contract $g(x)=f(x)I_{[0,M_{1})}(x)+(f(M_{1})+f(x-M_{1}))I_{[M_{1},\infty)}(x)$ is an appropriate contract. Since $f(\cdot)$ is an optimal contract, optimality of $g(\cdot)$ arrives by showing that $\rho(T_{g}(X))\equiv\rho(T_{f}(X)).$ Unfortunately proof of such identity is not available for general translative and monotone risk measures. Hopefully, Tan et al. (2011, Theorem 3.1) showed that under the $CTE$ criteria as far as $g(\cdot)\in\mathcal{C}$ and $0\leq g(x)\leq f^{*}(x)=\max\{x-d_{\alpha},0\},$ for a given $\alpha\in(0,1)$ and all $x\geq 0,$ any contract $g(\cdot)$ is again optimal, i.e., $\rho(T_{g}(X))\equiv\rho(T_{f^{*}}(X)).$ Using such seminal result, we can conclude that under the $CTE$ minimization criteria, the new contract $g(x)=f^{*}(x)I_{[0,M_{1})}(x)+(f^{*}(M_{1})+f^{*}(x-M_{1}))I_{[M_{1},\infty)}(x)$ is optimal. Again cutting down an interval $[M_{1},\infty)$ into two intervals $[M_{1},M_{2})$ and $[M_{2},\infty)$ and shifting the origin of Cartesian coordinate system to $(M_{2},f^{*}(M_{2}-M_{1})),$ we can obtain new contract $f^{*}(x)I_{[0,M_{1})}(x)+(f^{*}(M_{1})+f^{*}(x-M_{1}))I_{[M_{1},M_{2})}(x)+(f^{*}(M_{2})+f^{*}(x-M_{2}))I_{[M_{2},\infty)}(x)$ which Tan et al. (2011, Theorem 3.1) warranties its optimality. Several implementation of the above procedure leads to an optimal multi-layer reinsurance contract, under the $CTE$ minimization criteria. The following algorithm provides such multi-layer contract.

Algorithm 1**.**

Suppose $X_{R}$ stands for the reinsurer’s risk portion from random claim $X.$ The following steps design a multi-layer reinsurance policy which minimizes the $CTE$ of the insurer’s total risk.

Step 1)

A multi-layer reinsurance policy is obtained by the following iterative algorithm:

Part 1)

For $k\geq 2;$ Cut down an interval $[M_{k},\infty)$ into two intervals $[M_{k},M_{k+1})$ and $[M_{k+1},\infty)$ and define the reinsurer’s risk portion by

[TABLE]

where $f_{0}(X)=f(X)=\max\{X-d_{\alpha},0\};$

Part 2)

Go to Step 2 if a given stop criteria is met, otherwise set $k=k+1$ and go to Part (1)

Step 2)

Part 1)

The reinsurer’s risk portion under the k-layer reinsurance policy is $X_{R}=f(X)I_{[0,M_{1})}(X)+\sum_{j=1}^{k-1}f_{j}(X)I_{[M_{j},M_{j+1})}(X)+\left[f_{k-1}(M_{k})+f(X-M_{k})\right]I_{[M_{k},\infty)}(X).$

Part 2)

Now estimate unknown parameters by some additional appropriate criteria (or estimation methods) along the fact that the fact that $E(\max\{X-d_{\alpha},0\})=E(X_{R}).$

Closeness to an appropriate criteria (such as an optimal ruin probability) can be considered, in advance, as a stopping criteria in the above algorithm.

Algorithm (1) designs an optimal multi-layer reinsurance policy which the insurer’s and the reinsurer’s portion of both companies are increasing functions in the initial insurer claim $X.$ Moreover it provides a sharing system that its higher layer works appropriately for large reported claim size.

Application of Algorithm (1) leads to the following optimal k-layer reinsurance policy.

[TABLE]

Figure 1(b) illustrate optimal multi-layer reinsurance policy (11).

For the sake of simplicity, hereafter now, we set $M_{0}^{*}:=d_{\alpha},$ $M_{1}^{*}:=M_{1},$ $M_{2}^{*}:=M_{1}+d_{\alpha}$ and so on.

The cumulative distribution function for optimal k-layer reinsurance policy (11) can be restated as

[TABLE]

The following provides the moment generating function for the reinsurer’s risk portion from random claim $X,$ under optimal k-layer reinsurance policy (11).

Proposition 1**.**

Suppose $X_{R}$ stands for the reinsurer’s risk portion from random claim $X,$ under an optimal k-layer reinsurance policy which minimizes the $CTE$ of the insurer’s total risk. Then, the moment generating function for the reinsurer’s risk portion $X_{R}^{opt}$ under an optimal k-layer reinsurance policy.

[TABLE]

where $\sum_{j=a}^{b}c_{j}=0$ whenever $b<a.$

Proof. Observe that the moment generating function of $X_{R}^{opt},$ given by Equation (11) can be calculated as follows

[TABLE]

The odd terms can be evaluated directly. The following calculation represents that how one cab evaluate other terms.

[TABLE]

The desired proof arrives by a straightforward calculation. $\square$

Similar to Proposition (1), one may show that under the optimal k-layer reinsurance contract, the moment generating function for the insurer’s risk portion, $X_{I}=X-X_{R}^{opt},$ from random claim $X,$ is

[TABLE]

where $\sum_{j=a}^{b}c_{j}=0$ whenever $b<a.$

Using Proposition (1) the expectation of the reinsurer’s risk portion $X_{R}^{opt},$ under an optimal k-layer reinsurance can be evaluated as

[TABLE]

The next section conducts several simulation-based studies, to show “how one can employ some other appropriate criteria to fully determine an optimal k-layer reinsurance contract”.

3 Simulation Study

This section provides four numerical examples to show how the above findings, along with some other additional appropriate criteria , can be applied in practice. These examples consider a given multi-layer reinsurance policies which arrives by an extension of the optimal stop-loss reinsurance policy. Unknown parameters of each multi-layer reinsurance policy are estimated using an additional appropriate criteria.

Borch (1960) showed that, under the variance retained risk optimal criteria, in class of reinsurance contracts $\mathcal{C},$ given by Equation (2), the stop-loss reinsurance is optimal. The following shows that the proportional reinsurance contract minimizes a convex combination of variance of the insurer’s and the reinsurer’s risk portions from random claim $X.$

Proposition 2**.**

Suppose $X_{R}=h(X)$ and $X_{I}=X-h(X)$, respectively, stand for the reinsurer’s and the insurer’s risk portions from random claim $X.$ Then, in class of reinsurance contracts $\mathcal{C},$ given by Equation (2), proportional contract $h^{*}(X)=\frac{1}{{1+\omega}}X$ minimizes the following convex combination of variance of $X_{R}=h(X)$ and $X_{I}=X-h(X)$

[TABLE]

where $\omega\in[0,1].$

Proof. The above convex combination of two variances can be restated as

[TABLE]

Therefore, one may conclude that the above convex combination is minimal whenever $(X-h(X))$ and $(1-2\omega)X-h(X)]$ are linearly dependent. Choosing $(1-2\omega)X-h(X)=\beta_{0}+\beta_{1}(X-h(X))$ leads to $h(X)=(1-2\omega-\beta_{1})X/(1-\beta_{1})-\beta_{0}/(1-\beta_{1}).$ The fact that $0\leq h(X)\leq X$ implies that $\beta_{0}=0.$ Now by substituting back $h(X)=(1-2\omega-\beta_{1})X/(1-\beta_{1})$ in the above convex combination, we have

[TABLE]

Minimizing this expression, with respect to $\beta_{1},$ leads to desired result. $\square$

Proposition (2) shows that the proportional reinsurance the contract minimizes a convex combination of variance of $X_{R}$ and $X-X_{R}.$ The following example considers this observation as an appropriate criteria to estimate unknown parameters of an optimal 2-layer contract.

Example 1**.**

Suppose that random claim $X$ has been distributed according to one of the distributions given in the first column of Table 1. Moreover suppose that the optimal multi-layer contract has 2 layers and restated as

[TABLE]

For the sake of simplicity, we set $M_{1}=d_{\alpha}+d_{1}$ and $M_{2}=2d_{\alpha}+d_{1}+d_{2}.$ Now $M_{0}$ has been estimated such that $E(X_{R})=E(\max\{X-d_{\alpha},0\}).$ Other two parameters $d_{1}$ and $d_{2}$ have been estimated such that the square distance $\left[Q_{{X_{R}^{2-layer}}}-Q_{h^{*}}\right]^{2}$ is minimized, where $Q_{h}$ and $h^{*}$ are given in Proposition (2).

Table 1 shows estimation for unknown parameters of the above optimal 2-layer ${X_{R}^{2-layer}}.$

Table 1: Estimation for unknown parameters of the optimal 2-layer contract under variance optimal criteria, whenever $\omega=0.2$ and $\alpha=0.1$.

Random claim distribution $d_{\alpha}$ $M_{1}$ $M_{2}$ $E(h^{SL}(X))=E(h^{2-layer}(X))$ $CTE_{h^{SL}}=CTE_{h^{2-layer}}$ $Q_{h^{*}}$ $Q_{h^{SL}}$ $Q_{h^{2-layer}}$

Exp(10) 23.0259 24.4258 48.4516 $1$ 10.423 16.667 52.948 46.1586

Exp(8) 18.4206 26.4986 45.9192 $0.4498$ 8.14 6.6707 33.8867 29.5415

Exp(4) 9.2103 13.2103 18.1928 $0.4099$ 4.0743 1.6692 8.4717 7.3853

Weibull(1,2) 1.5174 4.1396 6.657 $0.028$ 0.2865 0.0358 0.1639 0.1338

Weibull(3,2) 4.5523 12.7469 18.2992 $0.02135$ 1.2235 0.322 1.475 1.204

$Q_{h}$ and $h^{*}$ are given in Proposition (2), $h^{SL}(X)=\max\{X-d_{\alpha},0\}$ and $h^{2-layer}(X)=X_{R}^{2-layer}(X).$

The last three columns of Table 1 show the convex combination of variance of $X_{R}=h(X)$ and $X_{I}=X-h(X)$ for optimal stop-loss, optimal 2-layer and proportional (given by Proposition, 2) contracts, respectively. As one may observe that, under the optimal 2-layer contract such convex combination of variances, compare to optimal stop-loss, has been improved. We conjecture that by increasing number of layer such convex combination of variances will be improved.

Under criteria of maximizing of the expected utility, one may either determine an optimal reinsurance contract (see Kaluszka & Okolewski, 2008, for more details) or estimate unknown parameters of an optimal reinsurance contract (see Dickson, 2005 §9.2, for more details).

The following example considers criteria of maximizing of a convex combination of the expected exponential utility of $X_{R}$ and $X-X_{R}$ as an additional appropriate criteria to estimate unknown parameters of a 2-layer optimal reinsurance contract.

Example 2**.**

Suppose that random claim $X$ has been distributed according to one of the distributions given in the first column of Table 2. Moreover consider the optimal 2-layer contract given in Example (1).

Similar to Example (1), for the sake of simplicity, we set $M_{1}=d_{\alpha}+d_{1}$ and $M_{2}=2d_{\alpha}+d_{1}+d_{2}.$ Now $M_{0}$ has been estimated such that $E(X_{R})=E(\max\{X-d_{\alpha},0\}).$ Other two parameters $d_{1}$ and $d_{2}$ are estimated such that the following convex combination of the expected exponential utilities of $X_{R}$ and $X-X_{R}$ has been minimized.

[TABLE]

where we set $\omega=0.2$ and $\beta_{1}=\beta_{2}=1.$

Table 2 shows estimation for unknown parameters of the optimal 2-layer ${X_{R}^{2-layer}}.$

Table 2: Estimation for unknown parameters of the optimal 2-layer contract under minimization $U_{h}$ as an optimal criteria, whenever $\omega=0.2$ and $\alpha=0.1$.

Random claim distribution $d_{\alpha}$ $M_{1}$ $M_{2}$ $E(h^{SL}(X))=E(h^{2-layer}(X))$ $CTE_{h^{SL}}=CTE_{h^{2-layer}}$ $U_{h^{SL}}$ $U_{h^{2-layer}}$

Exp(10) 23.0259 24.4259 48.4518 $1$ 10.423 $0.9312$ $0.9163$

Exp(8) 18.4206 31.4132 51.1488 $0.4498$ 8.14 $0.6412$ $0.5629$

Exp(4) 9.2103 13.2103 23.4037 $0.4099$ 4.0743 $0.8449$ $0.2000$

Weibull(1,2) 1.5174 4.1396 6.657 $0.028$ 0.2865 $0.5629$ $0.4593$

Weibull(3,2) 4.5523 12.7469 18.2992 $0.02135$ 1.2235 $0.3069$ $0.1465$

$Q_{h}$ and $h^{*}$ are given by Equation (13), $h^{SL}(X)=\max\{X-d_{\alpha},0\}$ and $h^{2-layer}(X)=X_{R}^{2-layer}(X).$

The last two columns of Table 2 show the convex combination of expected exponential utility of $X_{R}=h(X)$ and $X_{I}=X-h(X)$ for the optimal stop-loss and the optimal 2-layer contracts, respectively. As one may observe, under the optimal 2-layer contract such convex combination of utilities, compare to optimal stop-loss contract, is improved.

The Bayesian method under name of the credibility method is well-known in various areas of the actuarial sciences. For instance see: Whitney (1918) and Payandeh Najafabdi et al. (2015) for its application in the experience rating system; Bailey (1950), Payandeh Najafabdi (2010), and Payandeh Najafabdi et al. (2012) for its application in evaluating insurance premium; Hesselager & witting (1998) and England & Verral (2002) for its application in the IBNR claims reserving system; and see Makov et al. (1996), Makov (2001), and Hossack et al. (1999) for its general applications in actuarial science.

Now we employ the Bayesian estimation method as an appropriate method to estimate unknown parameters of an optimal multi-layer reinsurance contract.

To derive any Bayes estimator for $M^{*}_{0},\cdots,M^{*}_{m-2},$ based upon i.i.d. random claim $X^{(1)},\cdots,X^{(n)}.$ One has to consider initial values for $M^{*}_{0},\cdots,M^{*}_{m-2}.$ Then, using such initial values, he/she can define i.i.d reinsurer’s random claim $X^{(1)}_{R},\cdots,X^{(n)}_{R}.$ Now, using information given by $X^{(1)}_{R},\cdots,X^{(n)}_{R}$ accompanied with prior information on parameters $M^{*}_{0},\cdots,M^{*}_{m-2}$ and other unknown parameters, the Bayes estimator for parameters $M^{*}_{0},\cdots,M^{*}_{m-2},$ say $\hat{M}^{*}_{0},\cdots,\hat{M}^{*}_{m-2}$, under an appropriate loss function can be obtained. Certainly, such Bayes estimator may be, iteratively, improved by using $\hat{M}^{*}_{0},\cdots,\hat{M}^{*}_{m-2}$ as a new initial estimator for $M^{*}_{0},\cdots,M^{*}_{m-2}.$ And determining $X^{(1)}_{R},\cdots,X^{(n)}_{R},$ and finally reevaluating the Bayes estimator $\hat{M}^{*}_{0},\cdots,\hat{M}^{*}_{m-2},$ again.

Suppose $X^{(1)},\cdots,X^{(n)},$ given parameter $\theta,$ are i.i.d. random claims with a common density function $f_{X}$ and a distribution function $F_{X}.$ Moreover, suppose that $m^{*}_{0},\cdots,m^{*}_{k-2}$ stand for the initial values for $M^{*}_{0},\cdots,M^{*}_{k-2}$. Using a straightforward calculation, the density function for random variable $X_{R}^{(i)},$ for $i=1,\cdots,n,$ given parameters $\Theta:=(\theta,M^{*}_{0},\cdots,M^{*}_{k-2})$ at observed value $y^{(i)},$ is equal to

[TABLE]

Using the fact that random variables $X^{(1)}_{R},\cdots,X^{(n)}_{R}$ are i.i.d. Therefore, the joint density function for $X^{(1)}_{R},\cdots,X^{(n)}_{R},$ given parameters $\Theta:=(\theta,M^{*}_{0},\cdots,M^{*}_{k-2})$ can be restated as

[TABLE]

where $n_{0}:=\#(y^{(i)}=0),$ $n_{1}:=\#(0<y^{(i)}<(M^{*}_{1}-M^{*}_{0})),$ $n_{2}:=\#(y^{(i)}=(M^{*}_{1}-M^{*}_{0})),\cdots,n_{k-2}:=\#(\sum\limits_{i=1}^{k/2-2}{({M^{*}_{2i+1}}-{M^{*}_{2i}})}<y^{(i)})$.

Assuming $\pi{(\theta,M^{*}_{0},...,M^{*}_{k-2})}$ is the prior distribution for vector ${(\theta,M^{*}_{0},\cdots,M^{*}_{m-2})}$, joint posterior distribution for vector $\Theta:=(\theta,M^{*}_{0},\cdots,M^{*}_{k-2})$ is

[TABLE]

Using the above joint posterior distribution, the Bayes estimator for each $M^{*}_{0},...,M^{*}_{k-2}$ under the square error loss function, is

[TABLE]

for $i=0,\cdots,k-2.$

Now as an application of the above findings, we consider the following example.

Example 3**.**

Suppose that random claim $X$ has been distributed according to one of the distributions given in the first column of Table 3. Moreover, suppose that the optimal multi-layer contract has 1 layer and restated as

[TABLE]

For the sake of simplicity, we set $d_{0}=M_{0},$ $d_{1}=M_{1}-M_{0},$ and $d_{2}=M_{2}-M_{1}.$ Now, suppose that the prior distributions of the unknown parameters $d_{0},$ $d_{1},$ and $d_{2}$ are independent and given in the second, third, and fourth columns of Table 3, respectively.

To construct a Bayes estimator for unknown parameters, we employed $d_{0}=0.20,$ $d_{1}=0.15,$ and $d_{2}=0.02$ as initial values.

Table 3: Mean (standard deviation) of Bayes estimator for$d_{0}$, $d_{1}$ and $d_{2}$ based upon 100 sample size and 100 iterations, whenever $\alpha=0.1$.

Claim Distribution Prior distribution Prior distribution Prior distribution Mean (variance) Mean (variance) Mean (variance) $E(h^{SL}(X))=$

$CTE_{h^{SL}}=$

for $d_{0}$ for $d_{1}$ for $d_{2}$ of estimated $d_{0}$ of estimated $d_{1}$ of estimated $d_{2}$ $E(h^{1-layer}(X))$ $h^{SL}(X)$ $CTE_{h^{1-layer}}$

EXP(1) EXP(1) EXP(1) EXP(1) 0.0599 0.4474 0.0643 0.1 $(X-2.3026)_{+}$ 1.01

(4.795$\times 10^{-16}$) (4.439$\times 10^{-14}$) (8.458$\times 10^{-7}$)

EXP(4) Gamma(2,3) Gamma(3,2) Gamma(2,2) 0.0526 0.6575 0.0638 0.4 $(X-9.2103)_{+}$ 4.0743

(3.823$\times 10^{-18}$) (9.003$\times 10^{-13}$) (1.093$\times 10^{-5}$)

Weibull(1,2) Gamma(2,2) Gamma(3,2) Gamma(2,3) 0.0746 0.6575 0.0798 $0.028249$ $(X-1.5174)_{+}$ 0.2865

(1.661$\times 10^{-17}$) (4.393$\times 10^{-15}$) (3.542$\times 10^{-6}$)

$h^{SL}(X)=\max\{X-d_{\alpha},0\}$ and $h^{1-layer}(X)=X_{R}^{1-layer}(X).$

The three last columns of Table 3 represent the mean and the standard deviation, respectively, of the Bayes estimator for $d_{0}$, $d_{1}$ and $d_{2}$, which generates 100 random numbers from a given distribution. This estimators were derived using Equation (14) when the mean of 100 iterations of the Bayes estimator for $d_{0}$, $d_{1}$ and $d_{2}$ was used as an estimator for $d_{0}$, $d_{1}$ and $d_{2}.$

The small variance of these estimators shows that the estimation method is an appropriate method to use with the different samples.

4 Conclusion and suggestions

This article generalizes the stop-loss reinsurance policy to a new continuous multi-layer reinsurance policy which minimizes the conditional tail expectation (CTE) risk measure of the insurer’s total risk. Unknown parameters of the new optimal multi-layer reinsurance policy can be estimated using other additional appropriate criteria. Therefore, the new multi-layer reinsurance policy not only similar to the original stop-loss reinsurance policy is optimal, in a same sense, but also it has some other appropriate criteria which the original stop-loss policy does not have. Estimation method of this article can be generalized to the other appropriate criteria such as the ruin probability (Fang & Qu, 2014), percentile matching estimating method (Teugels & Sundt, 2004), etc.

The following two propositions are generalized result of this article under the general translative and monotone risk measure $\rho(\cdot)$.

The following suppose that under minimization criteria of a translative and monotone risk measure $\rho(\cdot)$ of the insurer’s total risk reinsurance contract $f(\cdot)$ is optimal. Then, it provides a multi-layer reinsurance contract which its corresponding risk measure coincides with the insurer’s total risk under contract $f(\cdot),$ see Figure 2(a) for an illustration.

Proposition 3**.**

Suppose $\rho(\cdot)$ is a translative and monotone risk measure. Moreover, suppose that $f(\cdot)$ in the class of reinsurance strategies $\mathcal{C}$ minimizes risk measure of the total risk of insurance company. Then, reinsurance $g(\cdot)$ also minimizes the risk measure of total risk of insurance company.

[TABLE]

where $M_{1},M_{2},\cdots,M_{k}$ are unknown parameters of the new optimal reinsurance and $M_{1}^{*},M_{2}^{*},\cdots,M_{k}^{*}$ have to be evaluated using equation $f(M_{2}^{*})=M_{2}-M_{1}+f(M_{1})$ and $f(M_{i}^{*})=M_{i}-M_{i-1}^{*}+f(M_{i-1}^{*})$ for $i=3,\cdots,k.$

Proof. Since $\rho(\cdot)$ is a translative risk measure, one may write that

[TABLE]

The above inequality arrives from the fact that $\rho(\cdot)$ is a monotone risk measure and $X-g(X)\leq X-f(X)$ with probability

1. Now using the fact that $\rho(X-f(X))=\mathop{\min}\limits_{h\in\mathcal{C}}\rho(X-h(X)+\pi_{h}^{X})$ we conclude that the above inequality has to be changed to an equality. $\square$

Now we provide an optimal multi-layer reinsurance contract, for a situation that the optimal reinsurance $f(\cdot)$ arrives by minimizing a convex combination of two translative and monotone risk measures $\rho_{1}(\cdot)$ and $\rho_{2}(\cdot)$ of the insurer’s total risk, $X_{R}=h(X),$ and the reinsurer’s total risk $X_{I}=X-h(X),$ i.e., $f(X)=\operatornamewithlimits{argmin}_{h\in\mathcal{C}}\{\omega\rho_{1}(X-h(X)+\pi_{h}^{X})+(1-\omega)\rho_{2}(h(X)-\pi_{h}^{X})\},$ where $\omega\in[0,1],$ see Figure 2(b) for an illustration.

As an example for such optimal reinsurance $f(\cdot),$ under such the convex combination of two distortion risk measures, see Assa (2015).

Proposition 4**.**

Suppose $\rho_{1}(\cdot)$ and $\rho_{2}(\cdot)$ are two translative and monotone risk measures. Moreover, suppose that $f(\cdot)$ in class of reinsurance strategies $\mathcal{C}$ minimizes a convex combination of two risk measures $\rho_{1}(\cdot)$ and $\rho_{2}(\cdot),$ i.e., $f(X)=\operatornamewithlimits{argmin}_{h\in\mathcal{C}}\{\omega\rho_{1}(X-h(X)+\pi_{h}^{X})+(1-\omega)\rho_{2}(h(X)-\pi_{h}^{X})\},$ where $\omega\in[0,1].$ Then, for $\omega^{*}\in(0,a_{\min}/(a_{\min}+a_{\max})),$ the following k-layer reinsurance $g(\cdot)$ also minimizes such the convex combination of two risk measures $\rho_{1}(\cdot)$ and $\rho_{2}(\cdot)$.

[TABLE]

where $M_{1},M_{2},\cdots,M_{k}$ are unknown parameters of the new optimal reinsurance and $M_{1}^{*},M_{2}^{*},\cdots,M_{k}^{*}$ have to be evaluated using equation $f(M_{2}^{*})=M_{2}-M_{2}+f(M_{1}),$ $f(M_{2j-1}^{*})=M_{2j-1}^{*}-M_{2j-1}+f(M_{2(j-1)}^{*}),$ $f(M^{*}_{2j})=f(M^{*}_{2(j-1)})+M_{2j}-M_{2j-1},$ for $j=2,\cdots,k,$ $a_{\min}:=\operatornamewithlimits{min}_{x\in A}\{|2f(x)-x|\},$ $a_{\max}:=\operatornamewithlimits{max}_{x\in A}\{|2f(x)-x|\}$ and $A:=[M^{1},M^{*}_{2})\cup_{j=2}^{k}[M^{*}_{2j-1},M^{*}_{2j}].$

Proof. Set $\pi_{g}^{*}:=\omega^{*}\pi_{g}^{X}-(1-\omega^{*})\pi_{g}^{X}.$ Since $\rho_{1}(\cdot)$ and $\rho_{2}(\cdot)$ are a translative risk measures, one may write that

[TABLE]

The last inequality arrives from the fact that $\omega^{*}\in[0,a_{\min}/(a_{\min}+a_{\max}).$ Now using the fact that $\omega^{*}\rho_{1}(X-f(X)+\pi_{f}^{X})+(1-\omega^{*})\rho_{2}(f(X)-\pi_{f}^{X})=\mathop{\min}\limits_{h\in\mathcal{C}}\left\{\omega^{*}\rho_{1}(X-h(X)+\pi_{h}^{X})+(1-\omega^{*})\rho_{2}(h(X)-\pi_{h}^{X})\right\},$ we conclude that the k-layer reinsurance $g(\cdot)$ also minimizes such the convex combination. $\square$

5 Acknowledgements

The second author also would like thanks support of the Central Insurance of the Islamic Republic of Iran.