Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 15

Part I is about risk. What is risk, and how can it be measured and compared? We discuss the mathematical formalism and practical application of representing an insured risk by a random variable. We define a risk measure as a functional taking a random variable to a real number representing the magnitude of its risk. We give numerous examples of risk measures and the different properties they exhibit.

Some properties are more or less mandatory for a useful risk measure, and they lead us to coherent risk measures. Coherent risk measures have an intuitive representation, providing us with some guidance on forming and comparing them. Spectral risk measures (SRMs)—also known as distortion risk measures—are a subset of coherent measures. They have additional properties and a particularly straightforward representation via a distortion function. Spectral risk measures can be viewed in four equivalent ways:

1 as expected values with varying distorted probabilities,

2 as a weighted sum of TVaRs at different thresholds,

3 as a weighted sum of VaRs at different thresholds, where the weights have specific properties, and

4 as the worst expected value across a set of different probability scenarios.

Spectral risk measures alter or distort the underlying pattern of probabilities and compute expected values based on the new probabilities, analogous to the effect of stochastic discount factors in modern finance. The distorted probability treats large losses as more likely, creating a positive pricing margin. TVaR is the archetypal SRM. It is simple yet powerful and has many desirable properties. We gain analytical insights into the nature of SRMs because they are all weighted averages of TVaRs. For example, we can allocate any SRM-derived quantity by bootstrapping a TVaR allocation.