Yichun Chi

Optimal Reinsurance under VaR and CVaR Risk Measures: A Simplified Approach

In this paper, we study two classes of optimal reinsurance models by minimizing the total risk exposure of an insurer under the criteria of value at risk (VaR) and conditional value at risk (CVaR). We assume that the reinsurance premium is calculated according to the expected value principle. Explicit solutions for the optimal reinsurance policies are derived over ceded loss functions with increasing degrees of generality. More precisely, we establish formally that under the VaR minimization model, (i) the stop-loss reinsurance is optimal among the class of increasing convex ceded loss functions; (ii) when the constraints on both ceded and retained loss functions are relaxed to increasing functions, the stop-loss reinsurance with an upper limit is shown to be optimal; (iii) and finally under the set of general increasing and left-continuous retained loss functions, the truncated stop-loss reinsurance is shown to be optimal. In contrast, under CVaR risk measure, the stop-loss reinsurance is shown to be always optimal. These results suggest that the VaR-based reinsurance models are sensitive with respect to the constraints imposed on both ceded and retained loss functions while the corresponding CVaR-based reinsurance models are quite robust.

Optimal Reinsurance with General Premium Principles

In this paper, we study two classes of optimal reinsurance models from the perspective of an insurer by minimizing its total risk exposure under the criteria of value at risk (VaR) and conditional value at risk (CVaR), assuming that the reinsurance premium principles satisfy three basic axioms: distribution invariance, risk loading and stop-loss ordering preserving. The proposed class of premium principles is quite general in the sense that it encompasses eight of the eleven commonly used premium principles listed in Young (2004). Under the additional assumption that both the insurer and reinsurer are obligated to pay more for larger loss, we show that layer reinsurance is quite robust in the sense that it is always optimal over our assumed risk measures and the prescribed premium principles. We further use the Wang’s and Dutch premium principles to illustrate the applicability of our results by deriving explicitly the optimal parameters of the layer reinsurance. These two premium principles are chosen since in addition to satisfying the above three axioms, they exhibit increasing relative risk loading, a desirable property that is consistent with the market convention on reinsurance pricing.

Are Flexible Premium Variable Annuities Under-Priced?

A variable annuity (VA) is a deferred annuity that allows an annuitant to invest his/her contributions into a range of mutual funds. A separate account termed as sub-account is set up for the investment. Unlike a mutual fund, a VA offers a guaranteed minimum death benefit or GMDB and often offers a guaranteed minimum living benefit or GMLB during the accumulation phase of the VA contract. Almost all the research to date has focused on single premium variable annuities (SPVA), i.e. it is assumed that an annuitant makes a single lump-sum contribution at the time of issue. In this paper, we study flexible premium variable annuities (FPVA) that allow contributions during the accumulation phase. We derive a valuation formula for guarantees embedded in FPVA and show that the delta hedging strategy for an FPVA is substantially different from that for a SPVA. The numerical examples illustrate that the cost in the form of mortality and expense (M&E) fee for an FPVA in many situations is significantly higher than the cost for a similar SPVA. This finding suggests that the current pricing practice by most VA providers that charges the same M&E fee for both should be re-examined.

Optimal Reinsurance Design: A Mean-Variance Approach

In this article, we study an optimal reinsurance model from the perspective of an insurer who has a general mean-variance preference. In order to reduce ex post moral hazard, we assume that both parties in a reinsurance contract are obligated to pay more for a larger realization of loss. We further assume that the reinsurance premium is calculated only based on the mean and variance of the indemnity. This class of premium principles is quite general in the sense that it includes many widely used premium principles such as expected value, mean value, variance, and standard deviation principles. Moreover, to protect the insurers profit, a lower bound is imposed on its expected return. We show that any admissible reinsurance policy is dominated by a change-loss reinsurance or a dual change-loss reinsurance, depending upon the coefficient of variation of the ceded loss. Further, the change-loss reinsurance is shown to be optimal if the premium loading increases in the actuarial value of the coverage; while it becomes decreasing, the optimal reinsurance policy is in the form of dual change loss. As a result, the quota-share reinsurance is always optimal for any variance-related reinsurance premium principle. Finally, some numerical examples are applied to illustrate the applicability of the theoretical results.

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.