Chengguo Weng

Ruin probabilities in a discrete time risk model with dependent risks of heavy tail

This paper establishes some asymptotic results for both finite and ultimate ruin probabilities in a discrete time risk model with constant interest rates, and individual net losses in , the class of regular variation with index α>0. The individual net losses are allowed to be generally dependent while they have zero index of upper tail dependence, so that our results partially generalize the counterparts in Tang (2004). The procedure of deriving our results also demonstrates a new approach of achieving asymptotic formulation for ruin probabilities when the individual risks are dependent.

Optimal reinsurance with expectile

In this paper, we study optimal reinsurance treaties that minimize the liability of an insurer. The liability is defined as the actuarial reserve on an insurer’s risk exposure plus the risk margin required for the risk exposure. The risk margin is determined by the risk measure of expectile. Among a general class of reinsurance premium principles, we prove that a two-layer reinsurance treaty is optimal. Furthermore, if a reinsurance premium principle in the class is translation invariant or is the expected value principle, we show that a one-layer reinsurance treaty is optimal. Moreover, we use the expected value premium principle and Wang’s premium principle to demonstrate how the parameters in an optimal reinsurance treaty can be determined explicitly under a given premium principle.

Empirical Approach for Optimal Reinsurance Design

This article proposes a novel and practical approach of addressing optimal reinsurance via an empirical approach. This method formulates reinsurance models using the observed data directly and has advantages including (1) transformation of an infinite dimensional optimization problem to a finite dimension, (2) no required explicit distributional assumption on the underlying risk, and (3) many empirical-based reinsurance models can be solved efficiently using the second-order conic programming. This allows insurers to incorporate many desirable objective functions and constraints while still retaining the ease of obtaining optimal reinsurance strategies. Numerical examples, including applications to actual Danish fire loss data, are provided to highlight the efficiency and the practicality of the proposed empirical models. The stability and consistency of the empirical-based solutions are also analyzed numerically.

Constant proportion portfolio insurance under a regime switching exponential Lévy process

The constant proportion portfolio insurance is analyzed by assuming that the risky asset price follows a regime switching exponential Levy process. Analytical forms of the shortfall probability, expected shortfall and expected gain are derived. The characteristic function of the gap risk is also obtained for further exploration on its distribution. The specific implementation is discussed under some popular Levy models including the Merton’s jump–diffusion, Kou’s jump–diffusion, variance gamma and normal inverse Gaussian models. Finally, a numerical example is presented to demonstrate the implication of the established results.

CDF Formulation for Solving an Optimal Reinsurance Problem

An innovative cumulative distribution function (CDF)-based method is proposed for deriving optimal reinsurance contracts to maximize an insurer’s survival probability. The optimal reinsurance model is a non-concave constrained stochastic maximization problem, and the CDF-based method transforms it into a functional concave programming problem of determining an optimal CDF over a corresponding feasible set. Compared to the existing literature, our proposed CDF formulation provides a more transparent derivation of the optimal solutions, and more interestingly, it enables us to study a further complex model with an extra background risk and more sophisticated premium principle.

Conditional Value-at-Risk-Based Optimal Partial Hedging

In this paper, we consider the problem of optimal partial hedging for a contingent claim subject to a preset hedging budget constraint. Under some technical assumptions on the hedged loss function and the market pricing functional, the optimal partial hedging strategy, which minimizes the conditional value-at-risk (CVaR) of the hedger’s total risk exposure, is derived explicitly. Some in-depth analysis is conducted for a utility-based indifference pricing functional. Ample numerical examples are presented to highlight the comparative advantages of the proposed CVaR-based hedging strategy relative to other hedging strategies including expected shortfall hedging, VaR-based hedging strategies and the CVaR hedging strategy of Melnikov and Smirnov. Among these hedging strategies, the numerical examples demonstrate that our proposed CVaR-based hedging is more robust and more effective in terms of managing the tail risk of the hedger’s risk exposure.

Response to Hans U. Gerber on His Comments on Our Paper Entitled ”Empirical Approach for Optimal Reinsurance Design”

We are grateful to Professor Gerber’s interest in our work. We wish to point out that Professor Gerber may have misinterpreted our statement, and as a result underscored the advantage of our model. In the passage quoted we were not referring to the model in this article, but rather, we were pointing out the shortcomongs in prior work that motivated us. As we pointed out, “simplifying assumptions (as well as the objective function and constraints) are often imposed in the model to ensure its tractability. As such, the insurers are tremendously restricted in exploring the optimal reinsurance strategy in managing and mitigating their risk exposures” (Tan and Weng 2014). We are pleased that Professor Gerber recognizes, with us, the need for models to be developed with practical applications in mind. In our article, we developed an empirical approach with the key objective that it can be used to solve a wide variety of optimal reinsurance models, including the more practical ones. As we illustrated, our proposed empirical approach directly formulates the optimal reinsurance model based on empirically observed data (or simulated data, if the underlying distribution is fully known), and we discovered that a wide range of optimal reinsurance models can be cast as the Second-Order-Conic programming (SOCP), which can be solved by various available solvers efficiently. As we noted: