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12 Risk Measures - An Introduction to Risk Measures for...

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An Introduction to Risk Measures for Actuarial Applications Mary R Hardy CIBC Professor of Financial Risk Management University of Waterloo 1 Introduction In actuarial applications we often work with loss distributions for insurance products. For example, in P&C insurance, we may develop a compound Poisson model for the losses under a single policy or a whole portfolio of policies. Similarly, in life insurance, we may develop a loss distribution for a portfolio of policies, often by stochastic simulation. Profit and loss distributions are also important in banking, and most of the risk measures we discuss in this note are also useful in risk management in banking. The convention in banking is to use profit random variables, that is Y where a loss outcome would be Y < 0. The convention in insurance is to use loss random variables, X = - Y . In this paper we work exclusively with loss distributions. Thus, all the definitions that we present for insurance losses need to be suitably adapted for profit random variables. Additionally, it is usually appropriate to assume in insurance contexts that the loss X is non-negative, and we have assumed this in Section 2.5 of this note. It is not essential however, and the risk measures that we describe can be applied (perhaps after some adaptation) to random variables with a sample space spanning any part of the real line. Having established a loss distribution, either parametrically, non-parametrically, analyti- cally or by Monte Carlo simulation, we need to utilize the characteristics of the distribution for pricing, reserving and risk management. The risk measure is an important tool in this 1
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process. A risk measure is a functional mapping a loss (or profit) distribution to the real numbers. If we represent the distribution by the appropriate random variable X , and let H represent the risk measure functional, then H : X R The risk measure is assumed in some way to encapsulate the risk associated with a loss distribution. The first use of risk measures in actuarial science was the development of premium prin- ciples . These were applied to a loss distribution to determine an appropriate premium to charge for the risk. Some traditional premium principle examples include The expected value premium principle The risk measure is H ( X ) = (1 + α )E[ X ] for some α 0 The standard deviation premium principle Let V[ X ] denote the variance of the loss random variable, then the standard deviation principle risk measure is: H ( X ) = E[ X ] + α q V[ X ] for some α 0 The variance premium principle H ( X ) = E[ X ] + α V[ X ] for some α 0 More premium principles are described in Gerber (1979) and B¨uhlmann (1970). Clearly, these measures have some things in common; each generates a premium which is bigger than the expected loss. The difference acts as a cushion against adverse experience.
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