Optimal Reinsurance Under the Risk-Adjusted Value of an Insurer’s Liability and an Economic Reinsurance Premium Principle

In this article, an optimal reinsurance problem is formulated from the perspective of an insurer, with the objective of minimizing the risk-adjusted value of its liability where the valuation is carried out by a cost-of-capital approach and the capital at risk is calculated by either the value-at-risk (VaR) or conditional value-at-risk (CVaR). In our reinsurance arrangement, we also assume that both insurer and reinsurer are obligated to pay more for a larger realization of loss as a way of reducing ex post moral hazard. A key contribution of this article is to expand the research on optimal reinsurance by deriving explicit optimal reinsurance solutions under an economic premium principle. It is a rather general class of premium principles that includes many weighted premium principles as special cases. The advantage of adopting such a premium principle is that the resulting reinsurance premium depends not only on the risk ceded but also on a market economic factor that reflects the market environment or the risk the reinsurer is facing. This feature appears to be more consistent with the reinsurance market. We show that the optimal reinsurance policies are piecewise linear under both VaR and CVaR risk measures. While the structures of optimal reinsurance solutions are the same for both risk measures, we also formally show that there are some significant differences, particularly on the managing tail risk. Because of the integration of the market factor (via the reinsurance pricing) into the optimal reinsurance model, some new insights on the optimal reinsurance design could be gleaned, which would otherwise be impossible for many of the existing models. For example, the market factor has a nontrivial effect on the optimal reinsurance, which is greatly influenced by the changes of the joint distribution of the market factor and the loss. Finally, under an additional assumption that the market factor and the loss have a copula with quadratic sections, we demonstrate that the optimal reinsurance policies admit relatively simple forms to foster the applicability of our theoretical results, and a numerical example is presented to further highlight our results.     

“Understanding Relationships Using Copulas,” Edward Frees and Emiliano Valdez,January 1998

Abstract

The problem of how to fund a defined-benefit pension plan is detached from the problem of how the cost of such a plan should be recognized. An approach to funding based on the projection of aggregate cash flows and the explicit modeling of new entrants is presented. It is shown that commonly used funding methods can be derived from the cash-flow approach. A generalized funding method for a plan subject to a stationary distribution of new entrants is derived. It is concluded that plan actuaries might need to modify existing funding methods to incorporate useful information about the expected number and distribution of future entrants.     

“Economic Capital Allocation Derived from Risk Measures”, Eddy Van den Borre,January 2003

Abstract

The sustained reduction in mortality rates and its systematic underestimation has been attracting the significant interest of researchers in recent times because of its potential impact on population size and structure, social security systems, and (from an actuarial perspective) the life insurance and pensions industry worldwide. Despite the number of papers published in recent years, a comprehensive review has not yet been developed.

This paper attempts to be the starting point for that review, highlighting the importance of recently published research—most of the references cited span the last 10 years—and covering the main methodologies that have been applied to the projection of mortality rates in the United Kingdom and the United States. A comparative review of techniques used in official population projections, actuarial applications, and the most influential scientific approaches is provided. In the course of the review an attempt is made to identify common themes and similarities in methods and results.

In both official projections and actuarial applications there is some evidence of systematic overestimation of mortality rates. Models developed by academic researchers seem to reveal a trade-off between the plausibility of the projected age pattern and the ease of measuring the uncertainty involved. The Lee-Carter model is one approach that appears to solve this apparent dilemma.

There is a broad consensus across the resulting projections: (1) an approximately log-linear relationship between mortality rates and time, (2) decreasing improvements according to age, and (3) an increasing trend in the relative rate of mortality change over age. In addition, evidence suggests that excessive reliance on expert opinion—present to some extent in all methods—has led to systematic underestimation of mortality improvements.     

“Understanding Relationships Using Copulas,” by Edward Frees and Emiliano Valdez,January 1998

Presenteeism occurs when employees are present at the workplace but cannot perform at their best because of ill-health or other reasons, while absenteeism occurs when employees are absent from the workplace. Although absenteeism is important, researchers now say presenteeism can be more costly to businesses and may be responsible for as much as three times the health-related lost productivity as compared to absenteeism and may cost the U.S. economy as much as $150 billion per year. Given the cost of absenteeism and presenteeism, one of the objectives of this article is to provide actuaries with the techniques and insights needed to design disability insurance policies that take into account the dynamics of absenteeism and presenteeism. To this end we develop a simple multistate sickness-disability model of the evolution of an employee’s health over time. We assume employees receive sick pay, the size of which depends on their health state, and there is a government-sponsored unemployment insurance program. In our model it is possible for employees in good health to avoid work by staying home, which is called shirking. To reduce shirking, the employer decides to check the health status of a certain percentage of employees who “call in sick.” Given the sick-pay structure, the probability of a health check, and the existence of unemployment insurance, employees develop rational strategies about whether to engage in shirking, absenteeism, or presenteeism. These strategies are captured in a set of Volterra integral equations. We use these Volterra integral equations to show how the employer can design a disability insurance plan that can incentivise employees to eliminate shirking and to act in a manner that will maximize the employer’s expected profits.     

“Bonus-Malus Systems: The European and Asian Approach to Merit-Rating”, Jean Lemaire,January 1998

Abstract

This paper presents an integration-by-parts proof of the Hattendorff theorem in the general fully continuous insurance model. The proof motivates a derivation of the theorem in the general fully discrete insurance model. Increments of a martingale over disjoint time intervals are uncorrelated random variables; the paper explains that the Hattendorff theorem can be viewed as an application of this result. A notable feature of the paper is the extensive use of the indicator function.