An Integrated Approach to Pricing Catastrophe Reinsurance

Carolyn W Chang,Jack S K Chang

doi:10.3390/risks5030051

Carolyn W Chang, Jack S K Chang

Open Access

https://doi.org/10.3390/risks5030051

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Abstract

We propose an integrated approach straddling the actuarial science and the mathematical finance approaches to pricing a default-risky catastrophe reinsurance contract. We first apply an incomplete-market version of the no-arbitrage martingale pricing paradigm to price the reinsurance contract as a martingale by a measure change, then we apply risk loading to price in—as in the traditional actuarial practice—market imperfections, the underwriting cycle, and other idiosyncratic factors identified in the practice and empirical literatures. This integrated approach is theoretically appealing for its merit of factoring risk premiums into the probability measure, and yet practical for being applicable to price a contract not traded on financial markets. We numerically study the catastrophe pricing effects and find that the reinsurance contract is more valuable when the catastrophe is more severe and the reinsurer’s default risk is lower because of a stronger balance sheet. We also find that the price is more sensitive to the severity of catastrophes than to the arrival frequency; implying (re)insurers should focus more on hedging the severity than the arrival frequency in their risk management programs.

Highlights

Catastrophe reinsurance contracts are not traded on financial markets, instead they are one-shot deals made between reinsurers and insurers
Catastrophe reinsurance is conventionally priced according to a two-stage process where in the first stage, catastrophe modelers like RMS (Risk Management Solutions), AIR (AIR Worldwide) and EQE (EQECAT) produce loss distributions through catastrophe modeling and calibration, and in the second stage insurers and reinsurers negotiate prices based on the given loss distributions with additional consideration of non-modeled market factors
We have proposed an integrated approach to price a default-risky cat reinsurance contract by straddling the traditional actuarial science and the mathematical finance approaches

Summary

Introduction

Catastrophe (cat hereafter) reinsurance contracts are not traded on financial markets, instead they are one-shot deals made between reinsurers and insurers. Since the reinsurer’s asset-liability structure and catastrophe loss specification are important factors in modeling the reinsurer’s default, we begin by employing Merton (1974) structural approach to model the balance sheet of the reinsurer in a continuous-time no-arbitrage martingale framework. In the one-factor interest rate model, Equation (1) can be√ risk-neutralized to the following risk-neutralized asset price process with a measure change, dZt∗ = dZt +. As WV,t is orthogonal to Zt by construction and is considered as part of the interest rate risk, we can risk-neutralize Equation (8) by making use of the market price of the interest rate risk alone so that μV + φV κm − φV κrt = rt + φV rt , and a measure change from the physical measure to the λ equivalent martingale measure Q that dZt∗ = dZt +. The loss process, Equation (13), retains its original distributional characteristics after changing from the physical probability measure to the risk-neutralized measure

Pricing a Cat Reinsurance Contract and the Monte Carlo Simulation Results

Concluding Remarks