Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 55

Risk measures quantify the following three characteristics of a risk random variable.

1 Volume: a smaller risk is preferred. The mean measures volume.

2 Volatility: a risk exhibiting less volatility (or variability) is preferred. Variance and standard deviation measure volatility. We use the word ‘volatility’ in a sense parallel to stock price volatility. Volatility is two-sided.

3 Tail: a risk with a lower likelihood of extreme outcomes is preferred. The level of loss with a 1% probability of exceedance is a tail risk measure. Tail risk is one-sided.

These characteristics overlap.

A risk measure must reflect volume because we want it to mirror a risk preference satisfying the MONO axiom, smaller risks are preferred to larger ones—even if the small risk is volatile and the large risk is certain. A measure of variability or tail risk that ignores volume is called a measure of deviation. The eponymous standard deviation is the, well, standard example. Adding the mean to a measure of deviation creates a risk measure.

Risk-based capital (RBC) formulas are risk measures. Many of them are volume based. They compute target capital by applying factors to premium, reserve, or asset balances. The factors vary according to the risk of each element. Examples include NAIC RBC, the Solvency II Solvency Capital Requirement standard formula, and most rating agency capital adequacy models. Classification rating plans are also risk measures. They compute a premium as a function of risk characteristics, such as building value and location, construction, occupancy, protection, and use for property insurance.

A risk measure sensitive to volatility quantifies mild to moderately adverse outcomes: outcomes frequent enough that most actuaries see examples during their careers. Management is often concerned with quarterly results volatility and can suppose investors are similarly troubled. Standard deviation quantifies volatility risk very well, and a two deviations from the mean rule of thumb turns out to be a surprisingly accurate estimate of a 20-year event in many situations.

Tail risk represents something so extreme it may or may not be experienced during a career, nevertheless it must always be considered possible. A tail risk catastrophe event often doubles or triples the previous worst historical event.

Variability and tail risk are distinct. An outcome of winning ¤1 million or ¤3 million has variability, but much less tail risk for most people than the possibility of gaining or losing ¤1 million. The variability of the two is the same but the tail risk is different. Risk is always relative to a base.

In Section 5.2 we enumerate several mathematical properties that risk measures should have, providing another way to characterize them.