Ka Chun Cheung

Optimal Reinsurance Revisited – A Geometric Approach

In this paper, we reexamine the two optimal reinsurance problems studied in Cai et al. (2008), in which the objectives are to find the optimal reinsurance contracts that minimize the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risk exposure under the expectation premium principle. We provide a simpler and more transparent approach to solve these problems by using intuitive geometric arguments. The usefulness of this approach is further demonstrated by solving the VaR-minimization problem when the expectation premium principle is replaced by Wangs premium principle.

Optimal reinsurance under general law-invariant risk measures

In recent years, general risk measures play an important role in risk management in both finance and insurance industry. As a consequence, there is an increasing number of research on optimal reinsurance decision problems using risk measures beyond the classical expected utility framework. In this paper, we first show that the stop-loss reinsurance is an optimal contract under law-invariant convex risk measures via a new simple geometric argument. A similar approach is then used to tackle the same optimal reinsurance problem under Value at Risk and Conditional Tail Expectation; it is interesting to note that, instead of stop-loss reinsurances, insurance layers serve as the optimal solution. These two results highlight that law-invariant convex risk measure is better and more robust, in the sense that the corresponding optimal reinsurance still provides the protection coverage against extreme loss irrespective to the potential increment of its probability of occurrence, to expected larger claim than Value at Risk and Conditional Tail Expectation which are more commonly used. Several illustrative examples will be provided.

Bounds for sums of random variables when the marginal distributions and the variance of the sum are given

In this paper, we establish several relations between convex order, variance order, and comonotonicity. In the first part, we extend Cheung (2008b) to show that when the marginal distributions are fixed, a sum with maximal variance is in fact a comonotonic sum. Thus the convex upper bound is achieved if and only if the marginal variables are comonotonic. Next, we study the situation where besides the marginal distributions; the variance of the sum is also fixed. Intuitively one expects that adding this information may lead to a bound that is sharper than the comonotonic upper bound. However, we show that such upper bound does not even exist. Nevertheless, we can still identify a special dependence structure known as upper comonotonicity, in which case the sum behaves like a convex largest sum in the upper tail. Finally, we investigate when the convex order is equivalent to the weaker variance order. Throughout this paper, interpretations and significance of the results in terms of portfolio risks will be emphasized.

Characterizations of optimal reinsurance treaties: a cost-benefit approach

This article investigates optimal reinsurance treaties minimizing an insurer’s risk-adjusted liability, which encompasses a risk margin quantified by distortion risk measures. Via the introduction of a transparent cost-benefit argument, we extend the results in Cui et al. [Cui, W., Yang, J. & Wu, L. (2013). Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles. Insurance: Mathematics and Economics 53, 74–85] and provide full characterizations on the set of optimal reinsurance treaties within the class of non-decreasing, 1-Lipschitz functions. Unlike conventional studies, our results address the issue of (non-)uniqueness of optimal solutions and indicate that ceded loss functions beyond the traditional insurance layers can be optimal in some cases. The usefulness of our novel cost-benefit approach is further demonstrated by readily solving the dual problem of minimizing the reinsurance premium while maintaining the risk-adjusted liability below a fixed tolerance level.

Ordered Random Vectors and Equality in Distribution

Abstract In this paper we show that under appropriate moment conditions, the supermodular ordered random vectors and with equal expected utilities (or distorted expectations) of the sums and for an appropriate utility (or distortion) function, must necessarily be equal in distribution, that is . The results in this paper can be considered as generalizations of some recent results on comonotonicity, where necessary conditions related to the distribution of are presented for the random vector to be comonotonic.

Robust and Pareto optimality of insurance contracts

The optimal insurance problem represents a fast growing topic that explains the most efficient contract that an insurance player may get. The classical problem investigates the ideal contract under the assumption that the underlying risk distribution is known, i.e. by ignoring the parameter and model risks. Taking these sources of risk into account, the decision-maker aims to identify a robust optimal contract that is not sensitive to the chosen risk distribution. We focus on Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)-based decisions, but further extensions for other risk measures are easily possible. The Worst-case scenario and Worst-case regret robust models are discussed in this paper, which have been already used in robust optimisation literature related to the investment portfolio problem. Closed-form solutions are obtained for the VaR Worst-case scenario case, while Linear Programming (LP) formulations are provided for all other cases. A caveat of robust optimisation is that the optimal solution may not be unique, and therefore, it may not be economically acceptable, i.e. Pareto optimal. This issue is numerically addressed and simple numerical methods are found for constructing insurance contracts that are Pareto and robust optimal. Our numerical illustrations show weak evidence in favour of our robust solutions for VaR-decisions, while our robust methods are clearly preferred for CVaR-based decisions.

Explicit Solutions for a General Class of Optimal Allocation Problems

We revisit the problem of minimizing a separable convex function with a linear constraint and box constraints. This optimization problem arises naturally in many applications in economics, insurance, and finance. Existing literature exclusively tackles this problem by using the traditional Kuhn-Tucker theory, which leads to either iterative schemes or yields explicit solutions only under some special classes of convex functions. Instead, we present a new approach of solving this constrained minimization problem explicitly by using the theory of comonotonicity. The key step is to apply an integral representation result to express each convex function as the expected stop-loss of some suitable random variable.

Multivariate countermonotonicity and the minimal copulas

FrchetHoeffding upper and lower bounds play an important role in various bivariate optimization problems because they are the maximum and minimum of bivariate copulas in concordance order, respectively. However, while the FrchetHoeffding upper bound is the maximum of any multivariate copulas, there is no minimum copula available for dimensions d3. Therefore, multivariate minimization problems with respect to a copula are not straightforward as the corresponding maximization problems. When the minimum copula is absent, minimal copulas are useful for multivariate minimization problems. We illustrate the motivation of generalizing the joint mixability to d-countermonotonicity defined in Lee and Ahn (2014) through variance minimization problems and show that d-countermonotonic copulas are minimal copulas.

On Partial Hedging and Counter-Monotonic Sums

In this article, we show, in the context of partial hedging, that some important relationships about comonotonicity and convex order cannot be translated to counter-monotonicity in general because of the possibility of over-hedging. We propose a new notion called proper hedge that can e effectively avoid over-hedging. Diff erent characterizations of a proper hedge are given, and we show that this notion is useful in translating relationship between comonotonicity and convex order to the case of counter-comonotonicity.As an application, we apply our results to identify desirable structural properties of insurance indemnities that make an insurance contract appealing to both the policyholder and the insurer.

Probabilistic solutions for a class of deterministic optimal allocation problems

Abstract We revisit the general problem of minimizing a separable convex function with both a budget constraint and a set of box constraints. This optimization problem arises naturally in many resource allocation problems in engineering, economics, finance and insurance. Existing literature tackles this problem by using the traditional Kuhn–Tucker theory, which leads to either iterative schemes or yields explicit solutions only under some special classes of convex functions owe to the presence of box constraints. This paper presents a novel approach of solving this constrained minimization problem by using the theory of comonotonicity. The key step is to apply an integral representation result to express each convex function as the stop-loss transform of some suitable random variable. By using this approach, we can derive and characterize not only the explicit solution, but also obtain its geometric meaning and some other qualitative properties.