VaR or TVaR, which measure of risk should insurance and risk managers be using?

By Asad Khalil FIA

Originally published on LinkedIn on 18 April 2017.

Probability-based models provide a description of risk exposure. The level of exposure to risk is often described by one number, or at least a small set of numbers. These numbers are functions of the model and are often called “key risk indicators” or “measures of risk.” Such measures of risk inform actuaries and other risk managers about the degree to which the company is subject to particular aspects of risk. One such measure of risk is Value-at-Risk (VaR) and it is often used by risk managers to look at “the chance of an adverse outcome.”

A more informative and more useful measure of risk is Tail-Value-at-Risk (TVaR). It has arisen independently in a variety of areas and has been given different names, including Conditional-Value-at-Risk (CVaR), Conditional Tail Expectation (CTE), and Expected Shortfall (ES).

Comparing different measures of risk

In order to compare the VaR and TVaR we need to agree on the properties that a good measure of risk should demonstrate. The most commonly agreed upon are:

• Monotonicity: if a portfolio has lower returns than another portfolio for every state of the world, its risk measure should be greater.
• Translation invariance: if we add an amount of cash K to a portfolio, its risk measure should go down by K.
• Homogeneity: changing the size of a portfolio by a factor λ while keeping the relative amounts of different items in the portfolio the same should result in the risk measure being multiplied by λ.
• Sub-additivity: the risk measure for two portfolios after they have been merged should be no greater than the sum of their risk measures before they were merged.

The VaR satisfies the first three conditions but does not always satisfy the forth. It is often the case that VaR for the portfolio of a particular percentile is not less than the sum on the individual lines of businesses. For example, the portfolio VaR at the 50th percentile can be greater than the sum of the 50th percentiles of the individual lines of business.

Risk measures that satisfy all four conditions stated previously are referred to as coherent. As a result, the VaR cannot always be considered as a coherent measure of risk. On the other hand the TVaR can be considered as a coherent measure of risk as it satisfies all four properties.

Risk and Insurance managers should be aware of some of the shortcomings of VaR, however given the extensive use by insurers and regulators its importance is unlikely to diminish anytime soon.

How Perrenial can help

Perrenial uses stochastic modelling to measure the cost of risk at different retentions and to help clients manage tail risk through (re)insurance programme design optimisation. We also support captive insurers in setting retentions that match their risk appetite.

Want to understand the tail risk in your programme? Contact Perrenial