Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 39

Insurance is based on diversification, where the risk of the sum is less than the sum of the risks. It is important to understand whether a risk is diversifiable, also known as idiosyncratic. Risks diversify when each unit is small relative to the total and their losses exhibit a material degree of independence from one another. A diversification benefit occurs when adding independent units to a portfolio increases its risk by much less than what the standalone risks represent. The central limit theorem ensures that pooling is an effective mechanism to manage diversifiable risk.

The opposite of diversifiable risk is nondiversifiable, also known as systematic risk. The failure to diversify usually means that there is a common underlying cause or other source of dependence risk to multiple unit losses, or there is a single unit heavily influencing the total loss. Catastrophes affecting multiple units simultaneously are an example of the former. A catastrophe line of business with outsize losses compared to the other lines is an example of the latter. The presence of systematic risk means there is less diversification benefit than in its absence.

Dependence risk between units can manifest itself in different ways, some more dangerous than others. It is easy to identify in a simulation context: where are you sharing variables? Variables resimulated in each iteration for each unit diversify, at least to some extent. Any variable whose value is shared between units introduces dependence and systematic risk. Weather and loss trend assumptions are examples of shared variables.

Remark 2 In finance, systematic risk usually refers to the common variation of stock prices over time whereas idiosyncratic risk refers to the deviation of individual stock prices from the common movement. By adding many stocks to a portfolio, idiosyncratic risk—but not systematic risk—can be diversified away. This leads to pricing principles where only systematic risk matters because the well-diversified investor can make idiosyncratic risk “go away.” In Section 12.4 we will see that this simplification does not apply to the problem of pricing insurance risk.

Example 3 You live in a Dystopian Dictatorship and are a member of a group of 100 prisoners. Evil Leader decides to execute 1% of the prisoners. EL considers two approaches. Execute each prisoner with a 1% probability or with 1% probability execute all prisoners. The number of deaths is diversifiable in the former approach and nondiversifiable in the latter. The expected number of executions is the same for each approach.

Remark 4 The discounting impact of timing remains even when amount risk diversifies. Timing risk tends to be quite tame since payout patterns follow a predictable claim settlement process, regulated by the cadences of medicine and law. The historical development of insurance pricing reflects this distinction: in many cases amount risk is largely irrelevant but estimating the appropriate discount rates and investor and insured cash flows remains paramount; see the discounted cash flow and internal rate of return models in Chapter 8.