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This part of the book is about two important and related topics in modeling insurance business: measuring risk and computing the likelihood of ruin. In Chapter 4 we introduce various measures of risk, which are constructed with the purpose of setting premium or capital. We discuss the axiomatic approach of identifying risk measures that are coherent. Specific measures such as Value-at-Risk, conditional tail expectation, and the distortion-function approach are discussed. Chapter 5 analyzes the probability of ruin of an insurance business in both discrete-time and continuous-time frameworks. Probabilities of ultimate ruin and ruin before a finite time are discussed. We show the interaction of the initial surplus, premium loading, and loss distribution on the probability of ruin.
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