Gradients of Risk Measures: Theory and Application to Catastrophe Risk Management and Reinsurance Pricing

As part of pricing, many reinsurers would like to know the incremental impact that adding a new contract or canceling an existing contract might have on the capital needed to support the entire portfolio of business. Typically, catastrophe models take hours or days to run, ruling out a straightforward approach. A method of assessing incremental impact which did not require repeatedly simulating losses to the entire portfolio would therefore be quite useful. Efficient procedures for calculating the first derivatives of widely-used risk measures (such as Value at Risk and Tail Value at Risk) with respect to portfolio parameters would support the development of such a method. This paper presents general formulas for gradients of risk measures including VaR and TVaR. While the derivative of VaR in the case of linear risk models is widely known, this paper presents the general solution applicable not only to linear portfolio weights, but also to nonlinear parameters, such as retentions and limits. Implementation of the theoretical formulas within existing catastrophe simulation models is elaborated. A normal mixture approximation leads to a closed form solution for the incremental impact on VaR or TVaR of adding or removing a contract from a portfolio of excess-of-loss contracts.