Optimal reinsurance designs based on risk measures: a review

Abstract

Reinsurance is an effective way for an insurance company to control its risk. How to design an optimal reinsurance contract is not only a key topic in actuarial science, but also an interesting research question in mathematics and statistics. Optimal reinsurance design problems can be proposed from different perspectives. Risk measures as tools of quantitative risk management have been extensively used in insurance and finance. Optimal reinsurance designs based on risk measures have been widely studied in the literature of insurance and become an active research topic. Different research approaches have been developed and many interesting results have been obtained in this area. These approaches and results have potential applications in future research. In this article, we review the recent advances in optimal reinsurance designs based on risk measures in static models and discuss some interesting problems on this topic for future research.

Keywords:

• Value-at-risk
• conditional value-at-risk
• distortion risk measures
• layer reinsurance
• optimal reinsurance designs

1. Introduction

Reinsurance is one of the key tools for insurance companies to manage their risks. It can help insurance companies reduce their risk exposures, stabilise their profits over times and even help them underwrite more insurance policies. It is thus imperative for an insurance company to design an optimal reinsurance strategy.¹

The optimal reinsurance design is typically formulated as an optimisation problem. More explicitly, let X denote the loss that an insurer is facing over a fixed time period. We assume that X is a non-negative random variable on the probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ with cumulative distribution function (c.d.f.) $F_X\left(x\right)=\mathbb{P}\{X\le x\}$ and finite expectation $\mathbb{E}\left[X\right]<\mathrm{\infty }$. Roughly speaking, a reinsurance policy is to separate the loss X into $f\left(X\right)$ and $R_f\left(X\right)$ with $X=f\left(X\right)+R_f\left(X\right)$ so that $f\left(X\right)$, satisfying $0\le f\left(X\right)\le X$, is the loss ceded to a reinsurer, while $R_f\left(X\right)$ is the loss retained by the insurer (cedent). Therefore, $f\left(x\right)$ and $R_f\left(x\right)$ are usually called the insurer's ceded loss function and the retained loss function, respectively. Covering part of the loss X for the insurer, the reinsurer will be compensated by a payment in the form of reinsurance premium. The reinsurance premium is often calculated according to a premium principle. Denote by $\pi (\cdot )$ a reinsurance premium principle, then the reinsurance premium is $\pi \left(f\right(X\left)\right)$ and hence the insurer's total risk exposure $T_f\left(X\right)$ in the presence of reinsurance is given by $T_f\left(X\right)=R_f\left(X\right)+\pi \left(f\right(X\left)\right).$ An optimal reinsurance design is to find a ceded loss function $f^{\ast }$ so that it is optimal under an optimisation criterion. To build an optimal reinsurance model from the perspective of an insurer, we have to consider three indispensable factors:

1. a feasible set of ceded loss functions $f\left(x\right)$;

2. a premium principle $\pi (\cdot )$ for pricing the contract;

3. the insurer's optimisation objective on $T_f\left(X\right)$.

Notably, different assumptions may lead to different optimal reinsurance models. The study of these interesting models has attracted great attention from academics since the seminal work of Borch (1960).

To the best of our knowledge, Borch (1960) is the first to study the problem of this kind. His optimisation objective is to minimise the variance of the insurer's risk exposure, assuming that the reinsurance premium is calculated by the expected value principle and that the contract satisfies the principle of indemnity (that is, $0\le f\left(x\right)\le x$). He finds that the stop-loss reinsurance, which is the full reinsurance above a constant deductible, is optimal. Using a different optimisation criterion from Borch (1960)'s model, Arrow (1963) maximises the expected utility of the insurer's final wealth and obtains a similar result in favour of stop-loss reinsurance. Arrow's model is extended in many different directions. Just to name a few, Van Heerwaarden et al. (1989) and Chi and Lin (2014) generalise Arrow's result by assuming a quite general optimisation criterion that preserves the stop-loss order.² On the other hand, Arrow's model is extended by assuming other premium principles. More precisely, Raviv (1979), Deprez and Gerber (1985) and Young (1999) assume the principle of equivalent utility, convex and Gâteaux differentiable premium principle, and Wang's premium principle, respectively. For more references on the extensions of Arrow's model, we refer to Albrecher et al. (2017).

Since 1990s, risk measures such as value at risk (VaR) and conditional value at risk (CVaR) have been frequently used to quantify the risks in finance and insurance. Their properties have been extensively analysed. See, e.g., Artzner et al. (1999), Föllmer and Schied (2004), Dhaene et al. (2006), and Mao and Cai (2018). Due to prevalent uses of VaR and CVaR for quantifying the insurance risks and setting the regulatory capitals, VaR and CVaR based optimal reinsurance models have attracted great attention from academics. Mathematically, denote by $\rho (\cdot )$ the risk measure used to quantify the insurer's risk exposure, the optimal reinsurance problem can be formulated by (1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) where $\mathfrak{C}$ is the set of admissible ceded loss functions f. It is worth noting that risk measure based optimisation objectives are often quite different from ones under the expected utility framework. More specifically, the objective functions in (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) usually lack smoothness, and thus the classical approaches such as Lagrange multiplier approach and the stochastic control method may not work on such problems. In order to solve the minimisation problem (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ), new approaches have been developed and many new results have been obtained.

Cai and Tan (2007) first study the optimal reinsurance problem (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) for VaR and CVaR risk measures and the class of stop-loss ceded loss functions. Since their work, many new results and approaches on the optimal reinsurance problem (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) have been developed. To give a clear picture of the recent developments of this research under a static model, in this article we will review the main contributions to this topic developed over the past decade.

Further, to attract more academics to get involved in this topic, we will list some interesting and/or unsolved problems related to the topics for future studies.

2. VaR and CVaR based optimal reinsurance designs

In insurance practice, the regulators often calculate the regulatory capitals by adopting VaR and CVaR risk measures. For example, VaR is adopted by Solvency II and CVaR is used in Swiss Solvency Test. These two risk measures are formally defined as follows:

Definition 2.1

The VaR of a non-negative loss random variable Y at a confidence level $1-\alpha$ for $0<\alpha <1$ is defined as (2) ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(Y\right)=inf\{y\ge 0:\mathbb{P}\{Y>y\}\le \alpha \}.$(2) Based upon the definition of VaR, the CVaR of Y at a confidence level $1-\alpha$ is defined as (3) ${\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(Y\right)=\frac{1}{\alpha }{\int }_0^{\alpha }{\mathrm{V}\mathrm{a}\mathrm{R}}_s\left(Y\right)\mathrm{d}s.$(3)

Note that CVaR is also known as the ‘average value at risk (AVaR)’ or ‘expected shortfall (ES)’. VaR is more appropriately referred to as the quantile risk measure since ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(Y\right)$ is exactly a $(1-\alpha )$-quantile of random variable Y. It follows from the definition of ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)$ that (4) ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(Y\right)\le y⟺S_Y\left(y\right)\le \alpha ,$(4) where $S_Y\left(y\right)$ is the survival function of Y. Another important property associated with ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(Y\right)$ is that for any left-continuous and increasing³ function g, we have (see Theorem 1 in Dhaene et al., 2002) (5) ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(g\right(Y\left)\right)=g\left({\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\right(Y\left)\right).$(5) In the literature, there is another widely used risk measure called conditional tail expectation (CTE), which is defined as ${\mathrm{C}\mathrm{T}\mathrm{E}}_{\alpha }\left(Y\right)=\mathbb{E}\left[Y|Y>{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(Y\right)\right].$ In general, CTE is different from CVaR. However, for a continuous random variable Y or more generally, for a random variable Y satisfying $\mathbb{P}\{Y={\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(Y\left)\right\}=0$, it holds that ${\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(Y\right)={\mathrm{C}\mathrm{T}\mathrm{E}}_{\alpha }\left(Y\right)$.

One advantage of CVaR over CTE is that the former preserves the convex order but the latter fails. See Dhaene et al. (2006) for a detailed discussion on the relationship between CVaR and CTE.

To the best of our knowledge, VaR and CTE based optimal reinsurance models are first studied by Cai and Tan (2007), who assume

1. The reinsurer is risk neutral and the reinsurance premium principle is given by (6) $\pi (\cdot )=(1+\kappa )\mathbb{E}[\cdot ]$(6) for some safety loading coefficient $\kappa >0$;

2. The risk X has a one-to-one continuous distribution function on $(0,\mathrm{\infty })$ with a possible jump at 0, that is, ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)$ is the unique solution to the equation $\mathbb{P}\{X>x\}=\alpha$ for all $\alpha \in (0,S_X(0\left)\right)$;

3. The set of admissible ceded loss functions is given by ${\mathfrak{C}}_{sl}=\left\{(x-d{)}_{+}:d\ge 0\right\},$where $(x{)}_{+}=max\{x,0\}$.

Under the above assumptions, their optimal reinsurance models (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) are exactly minimisation problems with one decision variable. Their problems are reduced to find the optimal retention level $d^{\ast }$. The optimal retention of the stop-loss reinsurance is obtained in the following theorem.

Theorem 2.1

Theorems 2.1 and 3.1 in Cai & Tan, 2007

1. When the risk measure $\rho (\cdot )$ is adopted by ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(\cdot ),$ the optimal retention level $d^{\ast }>0$ exists if and only if $\begin{array}{rl}\alpha & <\frac{1}{1+\kappa }<S_X\left(0\right)\mathrm{a}\mathrm{n}\mathrm{d}\\ {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)& \ge {\mathrm{V}\mathrm{a}\mathrm{R}}_{1/(1+\kappa )}\left(X\right)\\ & +(1+\kappa )\mathbb{E}\left[\right(X-{\mathrm{V}\mathrm{a}\mathrm{R}}_{1/(1+\kappa )}\left(X\right){)}_{+}].\end{array}$Furthermore, if the optimal retention level $d^{\ast }$ exists, then $d^{\ast }={\mathrm{V}\mathrm{a}\mathrm{R}}_{1/(1+\kappa )}\left(X\right)$.

2. When $\rho (\cdot )$ is given by ${\mathrm{C}\mathrm{T}\mathrm{E}}_{\alpha }(\cdot ),$ the optimal retention level $\tilde{d}>0$ exists if and only if $0<\alpha \le 1/(1+\kappa )<S_X\left(0\right)$. When it exists, the optimal retention level $\tilde{d}$ satisfies $\tilde{d}={\mathrm{V}\mathrm{a}\mathrm{R}}_{1/(1+\kappa )}\left(X\right)\text{for}\alpha <\frac{1}{1+\kappa }$and $\tilde{d}\ge {\mathrm{V}\mathrm{a}\mathrm{R}}_{1/(1+\kappa )}\left(X\right)\text{for}\alpha =\frac{1}{1+\kappa }.$

In addition to the stop-loss reinsurance, many other types of reinsurance such as quota-share reinsurance and one-layer reinsurance are also widely used in reinsurance practice. Also, there are many other risk measures used in insurance and finance. Hence, it is easy to propose different optimal reinsurance models by considering other sets of admissible ceded loss functions and different risk measures and reinsurance premium principles. In fact, the model and the problem of Cai and Tan (2007) have been extended in different directions. Moreover, many new research approaches have been developed in order to solve more general models and problems than Cai and Tan (2007)'s. These extensions will be discussed in detail in the following sections.

3. The effect of the choice set on the optimal reinsurance strategy

3.1. Increasing convex ceded loss functions

The optimal reinsurance models in Cai and Tan (2007) are extended by Cai et al. (2008) to study a more general class of ceded loss functions. More explicitly, Cai et al. (2008) consider the choice set (7) $\begin{array}{rl}{\mathfrak{C}}_1& =\{0\le f(x)\le x:f(x)\\ & \text{ is an increasing convex function}\}.\end{array}$(7) Obviously, the ceded loss function of stop-loss reinsurance $f\left(x\right)=(x-d{)}_{+}$ for $d\ge 0$ belongs to ${\mathfrak{C}}_1$. In addition, it also includes the ceded loss function of quota-share reinsurance $f\left(x\right)=\theta x$ for $\theta \in [0,1]$.

Different to Cai and Tan (2007), the optimal reinsurance models (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) with ${\mathfrak{C}}_1$ become infinite-dimensional minimisation problems and are more difficult to be solved. VaR and CTE risk measures are not smooth, and thus the classical approaches such as Lagrange multiplier approach and the stochastic control method may not work for such problems. Since the ceded loss functions are assumed to be convex, Cai et al. (2008) make an attempt to solve the problems by using the tools from convex analysis. In fact, any $f\in {\mathfrak{C}}_1$ can be approximated by the following ceded loss functions $f_n\left(x\right)=\sum _{i=1}^n{c}_{n,i}(x-d_{n,i}{)}_{+},$ where $c_{n,i}>0$ and $d_{n,i}\ge 0$ are constants such that $0\le \sum _{i=1}^n{c}_{n,i}\le 1\mathrm{a}\mathrm{n}\mathrm{d}0\le d_{n,1}\le d_{n,2}\le \cdots \le d_{n,n}.$ They first derive optimal parameters $c_{n,i}^{\ast }$ and $d_{n,i}^{\ast }$ for a given type of ceded loss functions $f_n$. The technique for the optimal parameters $c_{n,i}^{\ast }$ and $d_{n,i}^{\ast }$ becomes easier because such optimisation problems are finite-dimensional. Following, they show that these solutions can converge to the globally optimal one. As a consequence, they show that the stop-loss reinsurance is optimal. See Theorems 3.1 and 4.1 in Cai et al. (2008) for more results.

Although this approximation approach is very constructive, their proofs are very lengthy and complicated. In fact, by the convex analysis, a concise expression of the increasing convex function can be obtained. More specifically, as pointed out by Chi and Tan (2011), any $f\in {\mathfrak{C}}_1$ can be expressed by $f\left(x\right)=c{\int }_{0-}^{\mathrm{\infty }}(x-t{)}_{+}\nu (\mathrm{d}t)$ for some $c\in [0,1]$ and probability measure $\nu (\cdot )$ defined on $[0,\mathrm{\infty }]$. Noting that VaR risk measure is translation invariant, Chi and Tan (2011) show $\begin{array}{rl}{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(T_f\right(X\left)\right)& ={\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)\\ & +c{\int }_{0-}^{\mathrm{\infty }}(1+\kappa )\mathbb{E}\left[\right(X-t{)}_{+}]\\ & -\left({\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\right(X)-t{)}_{+}\nu (\mathrm{d}t).\end{array}$ As a consequence, the analysis of the optimal reinsurance problem under VaR risk measure is simplified to finding optimal c and probability measure $\nu (\cdot )$, which is a very easy task. See Theorem 3.1 in Chi and Tan (2011) for more details.

On the other hand, Cheung (2010) simplifies the proofs of Cai et al. (2008) by introducing a geometric approach. More specifically, noting that $f\in {\mathfrak{C}}_1$ is an increasing continuous function, then a new reinsurance strategy can be constructed as $\tilde{f}\left(x\right)={\left(f\left({\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\right(X\left)\right)+f_{+}^{\prime }\left({\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\right(X\left)\right)(x-{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(X\left)\right)\right)}_{+}$ such that $f\left(x\right)\ge \tilde{f}\left(x\right)$ with an identity at $x={\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)$, where $f_{+}^{\prime }$ represents the right-derivative of f. Therefore, via (5(5) ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(g\right(Y\left)\right)=g\left({\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\right(Y\left)\right).$(5) ), it holds that $\begin{array}{rl}{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(T_f\right(X\left)\right)& ={\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)-f\left({\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\right(X\left)\right)\\ & +(1+\kappa )\mathbb{E}\left[f\right(X\left)\right]\ge {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(T_{\tilde{f}}\right(X\left)\right).\end{array}$ Notably, $\tilde{f}\left(x\right)$ is a special change-loss reinsurance policy, and hence the analysis of the problem under VaR risk measure can be simplified to deriving the optimal parameters of change-loss reinsurance, which is equivalent to solving an optimisation problem of two decision variables. Similar arguments can be applied to the case of CTE risk measure. Although this geometric approach introduced by Cheung (2010) is straightforward, its improved version becomes very powerful in solving the optimal reinsurance model (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) especially for distortion risk measures.

For any $f\left(x\right)\in {\mathfrak{C}}_1$, it is easy to see that $f_{+}^{\prime }\left(x\right)$ is increasing. In other words, the reinsurer has to pay a larger proportion of per unit of insurable risk for a larger loss. In practice, the reinsurer would not like to reinsure too much tail risk, and this convex assumption seems quite strict. On the other hand, the set ${\mathfrak{C}}_1$ excludes the one-layer reinsurance policy.

Definition 3.1

A layer $(a,b]$ of the risk X is given by (8) ${\mathcal{L}}_{(a,b]}\left(X\right)=min\left\{(X-a{)}_{+},b-a\right\},0\le a<b.$(8)

Therefore, in the next subsection we proceed to consider a larger set of admissible ceded loss functions, which includes the one-layer reinsurance. Note that the one-layer reinsurance is also called the limited stop-loss reinsurance.

3.2. Incentive compatibility

In this subsection, we remove the convex constraint on the ceded loss function. However, we have to take into consideration the moral hazard, which is a great concern in the contract design. To reduce ex post moral hazard, Huberman et al. (1983) suggest that the contract should be incentive-compatible. That is, both the insurer and the reinsurer should pay more for a large realisation of the loss. Mathematically, both $f\left(x\right)$ and $R_f\left(x\right)$ are increasing functions. Equivalently, $f\left(x\right)$ is increasing and Lipschitz-continuous, i.e., (9) $0\le f\left(x\right)-f\left(y\right)\le x-y,\mathrm{\forall }0\le y\le x.$(9) It is further equivalent to $f^{\prime }\left(x\right)\in [0,1]$ almost everywhere. Following the way of Huberman et al. (1983), this subsection assumes the set of admissible ceded loss functions by ${\mathfrak{C}}_2=\left\{0\le f\left(x\right)\le x:f\left(x\right)\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{s}\left(9\right)\right\}.$ As pointed out by Cai et al. (2008), the set ${\mathfrak{C}}_2$ is strictly larger than ${\mathfrak{C}}_1$.

VaR and CVaR based optimal reinsurance models (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) with ${\mathfrak{C}}_2$ are first studied in Chi and Tan (2011) by using the constructive approach, which is an improved version of the geometric approach introduced by Cheung (2010). More precisely, for any $f\in {\mathfrak{C}}_2$, with the help of the characteristics of the objective function and the reinsurance premium principle, they could construct an improved ceded loss function $f_{\mu }^{\ast }\left(x\right)\triangleq \left\{\begin{array}{l}(min\{x,{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(X\left)\right\}-d{)}_{+},\\ \mathbb{E}[min\{X,{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)\left\}\right]\ge \mu ;\\ min\{x,{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(X\left)\right\}+\theta (x-{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(X){)}_{+},\\ \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}\right.$ where $0\le d\le {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)$ and $0<\theta \le 1$ are determined by $\mathbb{E}\left[f_{\mu }^{\ast }\right(X\left)\right]=\mu =\mathbb{E}\left[f\right(X\left)\right]$. From the above equation, it is easy to see that all the new constructed reinsurance strategies have the same concise form, and hence the analysis is simplified to analysing a minimisation problem with a few decision variables. It can be easily solved and the optimal solution is listed in the following theorem.

Theorem 3.1

Theorem 3.2 in Chi & Tan, 2011

The optimal ceded loss function $f^{\ast }$ that solves the reinsurance model (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) under VaR risk measure and the set ${\mathfrak{C}}_2$ is given by $\begin{array}{r}{f}^{\ast }\left(x\right)=\left\{\begin{array}{l}{\mathcal{L}}_{\left({\mathrm{V}\mathrm{a}\mathrm{R}}_{1/(1+\kappa )}\right(X),{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(X\left)\right]}\left(x\right),\\ {\mathrm{V}\mathrm{a}\mathrm{R}}_{1/(1+\kappa )}\left(X\right)<{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right);\\ 0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}\right.\end{array}$

Another interesting type of reinsurance is the truncated stop-loss policy, which has the following form of the ceded loss function (10) $f\left(x\right)=(x-d{)}_{+}\mathbb{I}(x\le m),$(10) where $0\le d\le m$ and $\mathbb{I}\left(A\right)$ denotes the indicator function of an event A. Such a truncated stop-loss policy has been identified as an optimal solution in some optimal reinsurance problems studied by Kaluszka and Okolewski (2008) and Bernard and Tian (2009). Unfortunately, the ceded loss function in (10(10) $f\left(x\right)=(x-d{)}_{+}\mathbb{I}(x\le m),$(10) ) is excluded from ${\mathfrak{C}}_2$. The next subsection is devoted to analysing the optimal reinsurance design over a larger choice set.

3.3. Increasing retained loss functions

In Chi and Tan (2011), they consider another set of admissible ceded loss functions $\begin{array}{rl}{\mathfrak{C}}_3& =\{0\le f(x)\le x:R_f(x)\\ & \text{is an increasing left-continuous function}\}.\end{array}$ Mathematically, the set ${\mathfrak{C}}_3$ is strictly larger than ${\mathfrak{C}}_2$. From the economic point of view, the assumption of an increasing retained loss function $R_f\left(t\right)$ in the set ${\mathfrak{C}}_3$ will reduce the risk for the insurer to relax the audit of the claim. According to (5(5) ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(g\right(Y\left)\right)=g\left({\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\right(Y\left)\right).$(5) ), the technical assumption of increasing and left-continuous retained loss functions is used to obtain $R_f\left({\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\right(X\left)\right)={\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(R_f\right(X\left)\right).$ Consequently, the constructive approach is still applicable to solve the reinsurance model (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) under VaR risk measure and the set ${\mathfrak{C}}_3$.

Theorem 3.2

Theorem 3.3 in Chi & Tan, 2011

The optimal ceded loss function $f^{\ast }$ to the VaR based reinsurance model (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) with ${\mathfrak{C}}_3$ can be given by $f^{\ast }\left(x\right)=(x-\gamma )\mathbb{I}(x\le {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(X\left)\right),$ where $\vartheta \triangleq S_X\left({\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\right(X\left)\right)+\frac{1}{1+\kappa }and\gamma \triangleq {\mathrm{V}\mathrm{a}\mathrm{R}}_{\vartheta }\left(X\right).$

The above theorem manifests that under VaR risk measure, the optimal solution can be a truncated stop-loss reinsurance policy and hence the insurer would retain all the risk above ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)$. In fact, the left-continuous constraint on the retained loss functions can be further removed if the $F_X\left(x\right)$ is strictly increasing and continuous at a neighbourhood of ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)$. See Theorem 3.4 in Chi and Tan (2011) for a detailed discussion.

From the above three subsections, we can see that the optimal solutions are different over three choice sets under VaR risk measure. They can be stop-loss, one-layer and truncated stop-loss. However, for CVaR risk measure, the optimal solutions are very robust with respect to the change of the choice set. More specifically, for any ceded loss function satisfying the principle of indemnity $0\le f\left(x\right)\le x$, Van Heerwaarden et al. (1989) show that⁴ $X-(X-d{)}_{+}{\le }_{cx}X-f(X),$ where $d\ge 0$ is determined by $\mathbb{E}\left[\right(X-d{)}_{+}]=\mathbb{E}[f\left(X\right)]$. Noting that the reinsurance premium calculated by the expected value principle only depends upon the expected indemnity, the stop-loss reinsurance is always optimal over $\{{\mathfrak{C}}_i{\}}_{i=1}^3$ under CVaR risk measure.

Theorem 3.3

Theorem 4.1 in Chi & Tan, 2011

When $\rho (\cdot )={\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(\cdot ),$ the optimal solution over ${\mathfrak{C}}_i$ for i = 1, 2, 3 is given by $f^{\ast }\left(x\right)=\left\{\begin{array}{ll}(x-{\mathrm{V}\mathrm{a}\mathrm{R}}_{1/(1+\kappa )}(X){)}_{+},& \alpha <\frac{1}{1+\kappa };\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}\right.$

Under CTE risk measure, Tan et al. (2011) and Cheung, Sung, Yam, and Yung (2014) obtain a similar result in favour of stop-loss reinsurance. Notably, in this section the optimal reinsurance design is investigated under the expected value premium principle. In addition to the expected value principle, there are many other premium principles widely used in actuarial science. For example, Young (2004) lists as many as eleven premium principles. Therefore, it is very necessary to analyse the effect of reinsurance premium principle on the insurer's optimal reinsurance strategy.

4. Optimal reinsurance under different premium principles

Besides the expected value principle, VaR and CVaR based optimal reinsurance problems have been studied for other premium principles. To the best of our knowledge, Tan et al. (2009) is the first to extend the analysis in Cai and Tan (2007) to consider as many as seventeen reinsurance premium principles. However, their analysis is confined to study the optimal parameters of stop-loss reinsurance and quota-share reinsurance. Cheung (2010) instead assumes the choice set by ${\mathfrak{C}}_1$, and focuses on VaR risk measure and Wang's premium principle (11) ${\pi }_w\left(Y\right)={\int }_0^{\mathrm{\infty }}w\left(S_Y\right(y\left)\right)\mathrm{d}y,\mathrm{\forall }Y\in \chi$(11) for some increasing concave distortion function $w(\cdot )$ with $w\left(0\right)=0$ and $w\left(1\right)=1$, where χ is the collection of all non-negative random variables.

Theorem 4.1

Theorem 3 in Cheung, 2010

Assume that the reinsurance premium is calculated by Wang's principle (11(11) ${\pi }_w\left(Y\right)={\int }_0^{\mathrm{\infty }}w\left(S_Y\right(y\left)\right)\mathrm{d}y,\mathrm{\forall }Y\in \chi$(11) ). Over ${\mathfrak{C}}_1$, the optimal ceded loss function $f^{\ast }$ to the reinsurance model (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) under VaR risk measure is given by $f^{\ast }\left(x\right)=\left\{\begin{array}{ll}x,& {\pi }_w\left(X\right)<{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right);\\ cx\text{ for any}\\ \text{constant }c\in (0,1],& {\pi }_w\left(X\right)={\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right);\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$ for $0<\alpha <S_X\left(0\right)$.

Different to Cheung (2010), Chi (2012b) quantifies the insurance risk by using VaR as well as CVaR and analyses the optimal solution over a larger choice set ${\mathfrak{C}}_2$. Furthermore, in order to incorporate more premium principles into the analysis, he only assumes that the reinsurance premium principle $\pi (\cdot )$ satisfies the following three axioms:

1. Law invariance: $\pi \left(Y\right)$ depends only on the c.d.f. $F_Y\left(y\right)$;

2. Risk loading: $\pi \left(Y\right)\ge \mathbb{E}\left[Y\right]$ for $Y\in \chi$;

3. Preserving the convex order: For $Y,Z\in \chi$, $\pi \left(Y\right)\le \pi \left(Z\right)$, if $Y{\le }_{cx}Z$.

The first axiom is an implicit assumption in actuarial science, the second axiom is adopted to guarantee the safety of the reinsurer according to Strong Law of Large Numbers, and the third axiom is consistent with the utility framework of a risk-averse reinsurer. This premium principle is quite general in the sense that it includes ten of eleven widely used premium principles listed in Young (2004) except Esscher principle. More specifically, they are net, expected value, exponential, proportional hazards, principle of equivalent utility, Wang's, Swiss, Dutch, variance and standard deviation principles.

Instead of minimising the risk exposure of the insurer under VaR or CVaR risk measure, Chi (2012b) follows Asimit, Badescu, and Tsanakas (2013) to adopt the criterion of minimising the risk-adjusted value of the insurer's liability, which is often composed of two parts. The first part is the best estimate of the insurer's liability, which is usually set to be $\mathbb{E}\left[T_f\right(X\left)\right]$. The second part is based on the additional capital held against the unexpected loss $T_f\left(X\right)-\mathbb{E}\left[T_f\right(X\left)\right]$. If the unexpected loss is measured by the risk measure ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(\cdot )$ and the cost-of-capital rate is $\delta \in (0,1)$, then the risk-adjusted value of the insurer's liability can be represented by $\begin{array}{rl}{\rho }_v\left(T_f\right(X\left)\right)& =\mathbb{E}\left[T_f\right(X\left)\right]+\delta {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(T_f\left(X\right)-\mathbb{E}\left[T_f\right(X\left)\right]\right)\\ & =(1-\delta )\mathbb{E}\left[T_f\right(X\left)\right]+\delta {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(T_f\right(X\left)\right).\end{array}$ Similarly, if CVaR is used to quantify the unexpected loss, the corresponding risk-adjusted value is given by ${\rho }_{cv}\left(T_f\right(X\left)\right)=(1-\delta )\mathbb{E}\left[T_f\right(X\left)\right]+\delta {\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(T_f\right(X\left)\right).$ In particular, if $\delta =1$, this objective reduces to the CVaR of the insurer's risk exposure.

It is unclear whether the objective functions are smooth because of the general premium principle. Chi (2012b) resorts to the constructive approach and narrows the choice set by (12) $\begin{array}{rl}{\mathfrak{C}}_v& =\left\{{\mathcal{L}}_{(0,a]}\right(x)+{\mathcal{L}}_{(b,{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(X\left)\right]}(x\left)\mathrm{o}\mathrm{r}{\mathcal{L}}_{(0,c]}\right(x):\\ & 0\le a\le b\le {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)\le c\}\end{array}$(12) and (13) $\begin{array}{rl}{\mathfrak{C}}_{cv}& =\left\{{\mathcal{L}}_{(0,a]}\right(x)+{\mathcal{L}}_{(b,c]}(x):\\ & 0\le a\le b\le {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)\le c\}.\end{array}$(13)

Theorem 4.2

Theorems 3.1 and 3.2 in Chi, 2012b

Assume that the reinsurance premium principle satisfies the axioms of law invariance, risk loading and preserving the convex order. Under ${\rho }_v(\cdot )$, any admissible ceded loss function $f\in {\mathfrak{C}}_2$ to the reinsurance model (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) is dominated by an $\tilde{f}\in {\mathfrak{C}}_v$. Similarly, if the insurer's objective changes to minimise ${\rho }_{cv}\left(T_f\right(X\left)\right)$, the optimal ceded loss function can be in ${\mathfrak{C}}_{cv}$.

It should be emphasised that optimal solutions may not be unique and some possibly don't belong to ${\mathfrak{C}}_v$ or ${\mathfrak{C}}_{cv}$. From the above theorem, we can see that the optimal reinsurance can be in the form of one-layer or two-layer. The optimal reinsurance form can be further simplified when the premium principle is subject to an additional constraint, as illustrated in the following corollary.

Corollary 4.1

Corollary 3.3 in Chi, 2012b

Let ${\pi }_{\delta }\left(Y\right)\triangleq \pi \left(Y\right)-(1-\delta )\mathbb{E}\left[Y\right]$ for $Y\in \chi$. If the premium principle $\pi (\cdot )$ further satisfies the axiom of translation invariance, i.e., $\pi (Y+c)=\pi \left(Y\right)+c\text{for all }c\ge 0\text{ and all }Y\in \chi$ or ${\pi }_{\delta }(\cdot )$ preserves the first-order stochastic dominance (FOSD), i.e., ${\pi }_{\delta }\left(Y\right)\le {\pi }_{\delta }\left(Z\right),\text{if}{S}_Y\left(t\right)\le S_Z\left(t\right),\mathrm{\forall }t\ge 0,$ we have (14) $\begin{array}{rl}\underset{f\in {\mathfrak{C}}_2}{min}{\rho }_v\left(T_f\right(X\left)\right)& =\underset{f\in {\mathfrak{C}}_v^{\mathrm{\prime }}}{min}{\rho }_v\left(T_f\right(X\left)\right)\mathrm{a}\mathrm{n}\mathrm{d}\\ \underset{f\in {\mathfrak{C}}_2}{min}{\rho }_{cv}\left(T_f\right(X\left)\right)& =\underset{f\in {\mathfrak{C}}_{cv}^{\mathrm{\prime }}}{min}{\rho }_{cv}\left(T_f\right(X\left)\right),\end{array}$(14) where ${\mathfrak{C}}_v^{\mathrm{\prime }}=\left\{{\mathcal{L}}_{(a,{\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(X\left)\right]}\left(x\right):0\le a\le {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)\right\}\subset {\mathfrak{C}}_v$ and ${\mathfrak{C}}_{cv}^{\mathrm{\prime }}=\left\{{\mathcal{L}}_{(b,c]}\left(x\right):0\le b\le {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)\le c\right\}\subset {\mathfrak{C}}_{cv}.$

The additional constraint on the premium principle in the above corollary is rather weak and is satisfied by the widely used premium principles listed in Young (2004), except Esscher, Dutch and Swiss premium principles. In other words, for most of widely used premium principles, the optimal reinsurance is always in the form of one layer such that the analysis of the infinite-dimensional optimal reinsurance problem is reduced to solving a minimisation problem with less than two decision variables. It is not very challenging for a specific premium principle, as discussed in Chi (2012b). It is worth noting that the optimal reinsurance models with some special premium principles are also investigated in Chi (2012a), Cheung et al. (2012) and Chi and Tan (2013) for $\delta =1$, and they are also analysed by Tan and Weng (2014) via an empirical approach. These analyses are extended by Cai and Weng (2016) to assume the expectile risk measure.

It is important to note that the reinsurance premium calculated by the afore-mentioned premium principles is based only on the ceded loss. Bühlmann (1980), on the other hand, argues that the reinsurance market is incomplete so that the reinsurance premium depends not only on the ceded loss but also on the existing market environment. Therefore, he advocates an economic pricing of reinsurance contracts. Following the way of Bühlmann (1980), Chi et al. (2017) investigate the optimal reinsurance design under an economic premium principle ${\pi }_{\mathcal{M}}\left(Y\right)=\mathbb{E}\left[Y\mathcal{M}\right],\mathrm{\forall }Y\in \chi ,$ where $\mathcal{M}$ is a non-negative random variable with $\mathbb{E}\left[\mathcal{M}\right]=1$ and reflects the reinsurance environment and other risks the reinsurer is facing.

Theorem 4.3

Theorems 3.1 and 3.2 in Chi et al., 2017

Under the reinsurance premium principle ${\pi }_{\mathcal{M}}(\cdot )$, the optimal ceded loss function that solves the reinsurance model (1(1) $\underset{f\in \mathfrak{C}}{min}\rho \left(T_f\right(X\left)\right),$(1) ) with risk measure ${\rho }_v(\cdot )$ and the choice set ${\mathfrak{C}}_2$ has the derivative $\begin{array}{rl}{f_v^{\ast }}^{\prime }\left(t\right)& =\mathbb{E}\left[\mathcal{M}\mathbb{I}\right(X>t\left)\right]-(1-\delta )S_X\left(t\right)\\ & -\delta \mathbb{I}\left(S_X\right(t)>\alpha ).\end{array}$ Similarly, for the risk measure ${\rho }_{cv}(\cdot )$, the optimal solution becomes $f_{cv}^{\ast }\left(x\right)={\int }_0^x\psi \left(t\right)\mathrm{d}t$, where $\begin{array}{rl}\psi \left(t\right)& =\mathbb{E}\left[\mathcal{M}\mathbb{I}\right(X>t\left)\right]-(1-\delta )S_X\left(t\right)\\ & -\delta min\{1,S_X(t\left)/\alpha \right\}.\end{array}$

The above theorem manifests that the optimal reinsurance strategy depends heavily on the factor $\mathcal{M}$ and its dependence with the insurable risk X. The effects of $\mathcal{M}$ on the optimal solutions are investigated in detail in Chi et al. (2017).

Until now, the quantity of the reinsurance premium is deterministic and is known at the inception of reinsurance contract. However, in reinsurance practice, sometimes the reinsurance premium is uncertain at the inception and depends heavily on the indemnity. Therefore, Chen et al. (2016) consider a special retrospective premium principle $\begin{array}{rl}\pi \left(Y\right)& =min\left\{\right(B\left(Y\right)+L\times Y)T,(1+\varphi \left)\mathbb{E}\right[Y\left]\right\},\\ & \mathrm{\forall }Y\in \chi ,\end{array}$ where L is the loss conversion factor that covers the loss adjustment expenses, T>1 is the tax multiplier including premium tax, $(1+\varphi )\mathbb{E}\left[Y\right]$ is the maximum premium with $\varphi >\kappa$, and $B\left(Y\right)$ is the basic premium, which is a functional of Y and determined by $\mathbb{E}\left[\pi \right(Y\left)\right]=(1+\kappa )\mathbb{E}\left[Y\right]$. Under this premium principle, the reinsurance premium $\pi \left(f\right(X\left)\right)$ is random, and the insurer's total risk exposure $T_f\left(X\right)$ is shown to be increasing in parameters L and ϕ in the sense of convex order, according to Theorem 3.1 in Chen et al. (2016). They further find that any optimal reinsurance strategy in ${\mathfrak{C}}_2$ is dominated by a stop-loss reinsurance policy for any optimisation criterion preserving the convex order, which includes ${\rho }_{cv}(\cdot )$ as a special case.

5. Distortion risk measures and distorted premium principles

In the previous sections, the optimal reinsurance problems under VaR and CVaR are discussed separately. In fact, VaR and CVaR are special distortion risk measures, which can be formally defined as (15) ${\rho }_g\left(Y\right)={\int }_0^{\mathrm{\infty }}g\left(S_Y\left(t\right)\right)\mathrm{d}t,\mathrm{\forall }Y\in \chi ,$(15) where $g(\cdot ):[0,1]\to [0,1]$ is an increasing function with $g\left(0\right)=0$ and $g\left(1\right)=1$. In particular, when $g\left(x\right)=\mathbb{I}(x>\alpha )$ and $g\left(x\right)=min\{x/\alpha ,1\}$, ${\rho }_g(\cdot )$ recovers VaR and CVaR risk measures, respectively. For more properties on the distortion risk measure, we refer to Dhaene et al. (2006).

Cui et al. (2013) is the first to unify the analyses of optimal reinsurance problems under VaR and CVaR risk measures. More specifically, they investigate the optimal reinsurance design by using the distortion risk measure to quantify the insurer's risk exposure. To facilitate the analysis, they consider a general reinsurance premium, which is called the distorted premium principle and is an extension of Wang's principle. More precisely, it has the same form with ${\pi }_w(\cdot )$ in (11(11) ${\pi }_w\left(Y\right)={\int }_0^{\mathrm{\infty }}w\left(S_Y\right(y\left)\right)\mathrm{d}y,\mathrm{\forall }Y\in \chi$(11) ), but the $w(\cdot )$ is relaxed to be an increasing, bounded and left-continuous function with $w\left(0\right)=0$ and $w\left(x\right)>0$ for all x>0. This relaxation makes their proposed class of premium principles includes the expected value principle as well as Wang's principle. In the absence of a constraint on the reinsurance premium, the optimal reinsurance problem in Cui et al. (2013) is formulated as $\underset{f\in {\mathfrak{C}}_2}{min}{\rho }_g\left(T_f\right(X\left)\right).$ Thanks to the increasing and Lipschitz-continuous properties of $f\in {\mathfrak{C}}_2$, the objective function can be written by $\begin{array}{rl}{\rho }_g\left(T_f\right(X\left)\right)& ={\int }_0^{\mathrm{\infty }}\left(w\right(S_X\left(t\right))-g(S_X\left(t\right)\left)\right)f^{\prime }\left(t\right)\mathrm{d}t\\ & +{\rho }_g\left(X\right).\end{array}$ The choice of the reinsurance becomes a balance between the cost $w\left(S_X\right(t\left)\right)$ and the benefit $g\left(S_X\right(t\left)\right)$ for the insurer, as observed by Cheung and Lo (2017). Noting that $f^{\prime }\left(t\right)\in [0,1]$ for any $f\in {\mathfrak{C}}_2$, it easily follows from the above equation that $\underset{f\in {\mathfrak{C}}_2}{min}{\rho }_g\left(T_f\right(X\left)\right)={\int }_0^{\mathrm{\infty }}min\left\{w\right(S_X\left(t\right)),g(S_X\left(t\right)\left)\right\}\mathrm{d}t,$ and for any $c\left(t\right)\in [0,1]$, the ceded loss function $f^{\ast }\left(t\right)$ satisfying $\begin{array}{rl}{f^{\ast }}^{\prime }\left(t\right)& =\mathbb{I}\left(g\right(S_X\left(t\right))>w(S_X\left(t\right)\left)\right)\\ & +c\left(t\right)\mathbb{I}\left(g\right(S_X\left(t\right))=w(S_X\left(t\right)\left)\right)\end{array}$ is an optimal solution. See also Theorem 2.1 of Cui et al. (2013).

Cui et al. (2013) further analyse the optimal reinsurance design in the presence of a constraint on the reinsurance premium, i.e., $\begin{array}{rl}\underset{f\in {\mathfrak{C}}_2}{min}& {\rho }_g\left(T_f\right(X\left)\right)\\ \mathrm{s}.\mathrm{t}.& {\pi }_w\left(f\right(X\left)\right)\le {\pi }_0.\end{array}$ Obviously, if ${\pi }_w\left(f^{\ast }\right(X\left)\right)\le {\pi }_0$, it has the same optimal ceded loss function $f^{\ast }$ with the model in the absence of the constraint. Otherwise, if ${\pi }_w\left(f^{\ast }\right(X\left)\right)>{\pi }_0$, they define $\begin{array}{rl}\theta & =inf\left\{\phantom{{\int }_{(1+a)w\left(S_X\right(t\left)\right)-g\left(S_X\right(t\left)\right)<0}}a>0:\right.\\ & \left.{\int }_{(1+a)w\left(S_X\right(t\left)\right)-g\left(S_X\right(t\left)\right)<0}w\left(S_X\right(t\left)\right)\mathrm{d}t\le {\pi }_0\right\}.\end{array}$ By comparing $(1+\theta )w\left(S_X\right(t\left)\right)$ and $g\left(S_X\right(t\left)\right)$, they obtain the optimal ceded loss function with $\begin{array}{rl}{f_c^{\ast }}^{\prime }\left(t\right)& =\mathbb{I}\left(g\right(S_X\left(t\right))>(1+\theta \left)w\right(S_X\left(t\right)\left)\right)\\ & +\gamma \left(t\right)\mathbb{I}\left(g\right(S_X\left(t\right))=(1+\theta \left)w\right(S_X\left(t\right)\left)\right)\end{array}$ for some $\gamma \left(t\right)\in [0,1]$. It is necessary to point out that their proofs are greatly simplified by using a Neyman-Pearson approach in Lo (2017b).

When the choice set is changed from ${\mathfrak{C}}_2$ to ${\mathfrak{C}}_3$, these optimal reinsurance problems are investigated by Zheng and Cui (2014), who also show that a piecewise-linear reinsurance strategy is optimal.

6. Pareto-optimal reinsurance

In the above studies, the optimal reinsurance design is analysed from the perspective of an insurer and the reinsurer is left with the option to set the premium principle. However, two parties in a reinsurance contract have conflicting interests such that a contract which is optimal from the perspective of one party may not be accepted by the other party, as pointed out by Borch (1969). In the economic literature, Pareto optimality is often applied to the decision problems which take into account the interests from both sides.

Cai et al. (2016) consider the Pareto-optimal reinsurance solutions when two parties quantify their risk exposures by VaR and the reinsurance premium is calculated by the expected value principle (6(6) $\pi (\cdot )=(1+\kappa )\mathbb{E}[\cdot ]$(6) ). More specifically, their optimal reinsurance model is formulated as $\begin{array}{rl}\underset{f\in {\mathfrak{C}}_2}{min}& \lambda {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(R_f\left(X\right)+(1+\kappa )\mathbb{E}\left[f\right(X\left)\right]\right)\\ & +(1-\lambda ){\mathrm{V}\mathrm{a}\mathrm{R}}_{\text{β}}\left(f\left(X\right)-(1+\kappa )\mathbb{E}\left[f\right(X\left)\right]\right)\\ \mathrm{s}.\mathrm{t}.& {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(R_f\left(X\right)+(1+\kappa )\mathbb{E}\left[f\right(X\left)\right]\right)\le L_1;\\ & {\mathrm{V}\mathrm{a}\mathrm{R}}_{\text{β}}\left(f\left(X\right)-(1+\kappa )\mathbb{E}\left[f\right(X\left)\right]\right)\le L_2\end{array}$ for some $0<\lambda ,\alpha ,\text{β}<1$, $0<L_1\le {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(X\right)$ and $0<L_2\le {\mathrm{V}\mathrm{a}\mathrm{R}}_{\text{β}}\left(X\right)$. Thanks to the increasing and Lipschitz-continuous properties of $f\in {\mathfrak{C}}_2$, the optimal solution to the above minimisation problem can be obtained by using the constructive approach. They show that Pareto-optimal reinsurance solutions are always in the form of a few layers. See Theorems 3.1 and 3.2 in Cai et al. (2016) for more details.

As pointed out by Lo (2017a) and Lo (2017b), this constrained optimal reinsurance problem can be treated as a special case of the following minimisation problem $\begin{array}{rl}\underset{f\in {\mathfrak{C}}_2}{min}& {\int }_0^{\mathrm{\infty }}{h}_0\left(t\right)f\left(t\right)\mathrm{d}t\\ \mathrm{s}.\mathrm{t}.& {\int }_0^{\mathrm{\infty }}{h}_1\left(t\right)f\left(t\right)\mathrm{d}t\le \pi \end{array}$ for some integrable functions $h_0$ and $h_1$ defined on the non-negative real line and a real constant π, which can be solved by the Neyman-Pearson approach.

Notably, this problem is extended by Cai et al. (2017) and Lo (2017b). Cai et al. (2017) study Pareto optimality of reinsurance arrangements under general model settings, provide the necessary and sufficient conditions for a reinsurance contract to be Pareto-optimal, and characterise all Pareto-optimal reinsurance contracts. Lo and Tang (2019) analyse this problem by assuming that both parties perceive risk via distortion risk measures. Furthermore, the Pareto optimality under risk measures is also analysed by Boonen et al. (2016a). Different to Cai et al. (2017), they don't calculate the reinsurance premium by the premium principle and discard the non-negative property of reinsurance price. In addition, another way to consider the interests of both the insurer and reinsurer in optimal reinsurance problems is introduced in Cai et al. (2013), in which the optimal reinsurance is to maximise the joint survival probability and the joint profitable probability for the insurer and the reinsurer.

Recently, the analysis of Pareto-optimal reinsurance problems above has been extended in several directions. In one direction, it is extended by incorporating multiple reinsurers (see Asimit & Boonen, 2018). Another direction is to study Bowley game in which the reinsurer is monopolistic and designs the reinsurance premium principle while the insurer seeks the optimal reinsurance strategy (See Cheung, Yam, & Zhang, 2019). In Cheung, Yam, and Yuen (2019), adverse selection is also incorporated.

7. Multiple risks

Until now, it is implicitly assumed that the insurer is only facing one risk. In practice, the insurer usually runs many lines of business and is asked to manage the risk from the holistic perspective. Therefore, it is interesting to design a strategy for an insurer to purchase reinsurance on multiple risks. Without loss of generality, let the risks faced by the insurer be $(X_1,\dots ,X_n)$. They may be independent or not. The insurer can cede part of the risk $X_i$ to the ith reinsurer with the reinsurance premium principle ${\pi }_i(\cdot )$.

Obviously, the dependence between the risks plays a critical role in the risk transfer. Assuming that $X_1,\dots ,X_n$ are independent and identically distributed, Denuit and Vermandele (1998) show that the stop-loss reinsurance is optimal for each line of business when all the ${\pi }_i(\cdot )$ follow the expected value premium principle and the optimisation objective on the aggregated risk preserves the convex order. Cai and Wei (2012) instead assume a positive dependence structure, which is described as follows:

Definition 7.1

Random vector $(Y_1,\dots ,Y_m)$ is said to be stochastically increasing in random variable Z, denoted as $(Y_1,\dots ,Y_m){\uparrow }_{SI}Z$, if $\mathbb{E}\left[u\right(Y_1,\dots ,Y_m)|Z=z]$ is increasing in $z\in S\left(Z\right)$ for any increasing function $u:{\mathbb{R}}^n\to \mathbb{R}$ such that the expectation exists, where $S\left(Z\right)$ is the support of random variable Z. Furthermore, random vector $(X_1,\dots ,X_n)$ is said to be positively dependent through the stochastic ordering (PDS) if $(X_1,\dots ,X_{i-1},X_{i+1},\dots ,X_n){\uparrow }_{SI}{X}_i$ for any $i=1,\dots ,n$.

Furthermore, they consider another set of ceded loss functions $\begin{array}{rl}{\mathfrak{C}}_3^{\prime }& =\{0\le f(x)\le x:R_f(x\left)\text{ is an increasing function}\right\}\\ & \supset {\mathfrak{C}}_3.\end{array}$

Theorem 7.1

Proposition 3.7 in Cai & Wei, 2012

Assume that $(X_1,\dots ,X_n)$ is a PDS random vector. For any reinsurance strategy $(f_1,\dots ,f_n)$ with $f_i\in {\mathfrak{C}}_3^{\mathrm{\prime }}$ for $i=1,\dots ,n,$ there exists a non-negative retention vector $(d_1,\dots ,d_n)$ such that $\sum _{i=1}^n{X}_i-(X_i-d_i{)}_{+}{\le }_{cx}\sum _{i=1}^n{X}_i-f_i(X_i).$

The above theorem manifests that it is optimal for the insurer to purchase the stop-loss reinsurance for each line of business when the reinsurance premium is calculated by the expected value principle. However, their analysis is confined to the positive dependence and lacks of generality.

Different to Cai and Wei (2012), Zhu et al. (2014) can analyse the optimal reinsurance design for general dependence structures by introducing an interesting optimisation criterion. More specifically, they assume

1. The reinsurance premium principle ${\pi }_i(\cdot )$ satisfies the axioms of law invariance, risk loading and preserving the convex order (Chi, 2012b);

2. The ceded loss function $f_i\in {\mathfrak{C}}_2$ such that $\mathbb{E}\left[T_{f_i}^i\right(X_i\left)\right]\le 0$, where $T_{f_i}^i\left(X_i\right)=R_{f_i}\left(X_i\right)+{\pi }_i\left(f_i\right(X_i\left)\right)-p_i$ and $p_i>0$ is the insurance premium collected from the ith line of business;

3. The optimisation criterion is to minimise the capital $u_f=\sum _{i=1}^n{u}_i$ which is allocated to each line such that $\mathbb{P}\left\{T_{f_i}^i\right(X_i)\le u_i:1\le i\le n\}\ge 1-\alpha .$

Surprisingly, they obtain the similar results to Chi (2012b) in favour of one-layer or two-layer reinsurance, irrespectively of the dependence structure and the premium principle. For more details, we refer to Theorem 3.2 and Corollary 3.1 in Zhu et al. (2014). It is necessary to emphasise that the optimal parameters of layer reinsurance are indeed greatly affected by the specific assumptions of the dependence structure and the reinsurance premium principle.

Since the dependence structure of $(X_1,\dots ,X_n)$ is often hard to be estimated, Cheung, Sung, and Yam (2014) consider a minimax optimal reinsurance decision formulation in which the worst case scenario is identified under a general law invariant convex risk measure. More specifically, when the probability measure $\mathbb{P}$ is atomless, the risk measure can be represented by ${\rho }_c\left(Y\right)=\underset{\mu \in \mathcal{P}}{sup}\left({\int }_0^1{\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }\left(Y\right)\mu \left(\mathrm{d}\alpha \right)-\text{β}\left(\mu \right)\right),$ where $\mathcal{P}$ is a set of probability measures on $[0,1]$ and $\text{β}:\mathcal{P}\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ is a proper convex function. Further, they assume that the risks are ceded to a reinsurer who prices the contract by the expected value principle, i.e., $P=(1+\kappa )\mathbb{E}\left[\sum _{i=1}^n{f}_i\left(X_i\right)\right].$ Therefore, their optimal reinsurance model is formulated by $\begin{array}{rl}\underset{(f_1,\dots ,f_n)\in {\mathfrak{C}}_2^n}{min}& \left\{\underset{(X_1,\dots ,X_n)\in \mathcal{R}(F_1,\dots ,F_n)}{max}{\rho }_c\left(\sum _{i=1}^n{R}_{f_i}\left(X_i\right)+P\right)\right\}\\ \mathrm{s}.\mathrm{t}.& P\le {\pi }_0,\end{array}$ where $X_i$ has distribution $F_i$ and $\mathcal{R}(F_1,\dots ,F_n)$ denotes the Fréchet space of all n-dimensional random vectors with marginal distributions $F_1,\dots ,F_n$. Proposition 1 in Cheung, Sung, and Yam (2014) show that this optimisation problem is equivalent to $\begin{array}{rl}\underset{(f_1,\dots ,f_n)\in {\mathfrak{C}}_2^n}{min}& {\rho }_c\left(\sum _{i=1}^n{R}_{f_i}\left(X_i^C\right)+P\right)\\ \mathrm{s}.\mathrm{t}.& P\le {\pi }_0,\end{array}$ where $(X_1^C,\dots ,X_n^C)\in \mathcal{R}(F_1,\dots ,F_n)$ is comonotonic. For detailed discussions on the comonotonicity, we refer to Dhaene et al. (2002). Note that $(X_1^C,\dots ,X_n^C)$ is a PDS random vector and the law-invariant convex risk measure preserves the convex order. It follows from Cai and Wei (2012) that the stop-loss reinsurance is optimal for each line of business. See also Theorem 1 in Cheung, Sung, and Yam (2014).

8. Concluding remarks and future studies

In this article, we review several main contributions motivated by Cai and Tan (2007). Due to the limited space, we could not mention all the contributions in this direction. In fact, many other interesting extensions of Cai and Tan (2007) also have appeared in the literature. For instance, the optimal reinsurance design with multiple reinsurers is studied in Asimit, Badescu, and Tsanakas (2013), Chi and Meng (2014), Boonen et al. (2016b) and references therein. The optimal reinsurance contract with default risk can be found in Asimit, Badescu, and Cheung (2013) and Cai et al. (2014). The optimal randomised reinsurance design is investigated in Albrecher and Cani (2019). In addition, there are many other related topics on optimal reinsurance discussed in Balbás et al. (2009), Asimit et al. (2015) and Zhuang et al. (2016).

Although great advances have been made in the past decade, there are many interesting and/or unsolved problems left for future studies. We list a few as follows:

1. Although high-dimensional optimal reinsurance problems have been studied by Cai and Wei (2012) and Zhu et al. (2014), this research is far from complete. In particular, when the risks are negatively dependent such that they have an internal hedge, the optimal reinsurance strategy has not been well investigated.

2. In reinsurance practice, the reinsurer usually returns part of profit to the insurer or sets a retrospective rating. On the other hand, the insurer will consider the demand for reinsurance under the constraints of the available internal capital and the cost of raising external capital. While Chen et al. (2016) make an attempt to incorporate the retrospective premium principle in the optimal reinsurance model, few other papers have been devoted to this topic by taking into account these practical issues.

3. The optimal reinsurance design is often treated as a risk-sharing problem. However, there are significant differences between these two problems. On one hand, the reinsurer can issue many reinsurance contracts and reduce the risk exposure by retrocession. On the other hand, all the risks the reinsurer is facing may not be independent and suffer a systemic shock such as inflation risk. Therefore, the reinsurance pricing should take into account the diversification as well as the systematic risk.

4. In the afore-mentioned studies, they often try to derive the optimal reinsurance solutions explicitly. However, few papers have carried out the comparative static analysis. For example, how does the optimal reinsurance contract vary with the change of the risk the insurer is facing?

5. In most of optimal reinsurance problems, it is assumed that the distributions of the insurer's risks are given or known. However, in practice, the exact distributions of the insurer's risks are difficult to be obtained. Often, only incomplete information, such as mean and variance, on the distributions is available. How to obtain optimal reinsurance contracts with incomplete distribution information is also an interesting topic. An attempt to such a problem has been made in Hu et al. (2015). Solutions under more general model setting than Hu et al. (2015) are worth investigating, and techniques in Asimit et al. (2017) are helpful for solving such problems. Although some papers have been devoted to deriving optimal reinsurance under model uncertainty, the optimal reinsurance with uncertainty still lacks of tractable models and the available analysis tools. This has to be resolved.

6. It is well-known in decision theory that an individual often makes a decision by using a personal view of probability. Due to the information asymmetry, the insurer and the reinsurer often have different beliefs about the loss distribution. Therefore, it is interesting to investigate the optimal reinsurance design with belief heterogeneity. Some formulations and techniques, which are developed to study the optimal insurance design in Chi (2019), Ghossoub (2019) and references therein, are helpful for analysing optimal reinsurance problems when agents have different beliefs about the loss distribution. See also Boonen and Ghossoub (2019).

Acknowledgments

The authors thank the two anonymous referees for the helpful comments and suggestions that improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Jun Cai is grateful to the support from the Natural Sciences and Engineering Research Council of Canada (NSERC) (grant No. RGPIN-2016-03975). Chi's work was supported by grants from the National Natural Science Foundation of China (Grant No. 11971505) and 111 Project of China (No. B17050).

Notes on contributors

Jun Cai

Jun Cai is Professor of Actuarial Science in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. His current main research interest is in the area of quantitative risk management for insurance and finance such as optimal reinsurance, optimal allocation, risk measure with application, and risk management under model uncertainty.

Yichun Chi

Yichun Chi is Professor of Actuarial Science in the China Institute for Actuarial Science at the Central University of Finance and Economics, People's Republic of China. His current research interest is the optimal insurance contract design.

Notes

1 In the literature, the optimal reinsurance design and the optimal insurance design often share the same formulations and use the same techniques. In addition, reinsurance can be treated as a special type of insurance. However, they have significant differences in practice. In particular, an insurance company (as the insured in the optimal reinsurance design) is often subject to the solvency regulation, while the insured in the optimal insurance design usually has no such a solvency requirement. The solvency regulation motivates many researchers to analyse the optimal reinsurance design based on regulatory risk measures. In addition, different to insurance, a reinsurance contract usually contains the profit commission which is paid by the reinsurer to the insurance company if the reinsurer's profit is over a threshold. The existence of the profit commission leads to that the actual reinsurance premium paid by the insurer depends on the reimbursement. It is consistent with the retrospective premium principle discussed in Section 4.

2 For more details on the stochastic orders used in this article, we refer to Shaked and Shanthikumar (2007).

3 Throughout the article, ‘increasing’ and ‘decreasing’ mean ‘non-decreasing’ and ‘non-increasing’, respectively.

4 Random variable $Z_1$ is said to be smaller than random variable $Z_2$ in the convex order, denoted as $Z_1{\le }_{cx}{Z}_2$, if $\mathbb{E}\left[h\right(Z_1\left)\right]\le \mathbb{E}\left[h\right(Z_2\left)\right]$ for any convex function $h(\cdot )$ provided that the expectations exist. See Shaked and Shanthikumar (2007) for more properties on the convex order.