# 3.5 Measures of Risk

## 3.5.1 Introduction

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.” Such key risk indicators inform actuaries and other risk managers about the degree to which the company is subject to particular aspects of risk. In particular, Value-at-Risk (VaR) is a quantile of the distribution of aggregate losses. Risk managers often look at “the chance of an adverse outcome.” This can be expressed through the VaR at a particular probability level. VaR can also be used in the determination of the amount of capital required to withstand such adverse outcomes.

Investors, regulators, and rating agencies are particularly interested in the company’s ability to withstand such events.

VaR suffers from some undesirable properties. 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).

While these measures have been developed in a risk management context, they are useful in assessing any random variable.

## 3.5.2 Risk measures and coherence

A risk measure is a mapping from the random variable representing the loss associated with the risks to the real line (the set of ...

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