Pricing Insurance Risk. Stephen J. Mildenhall

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СКАЧАТЬ capital risk measure determines the assets necessary to back an existing or hypothetical portfolio at a given level of confidence. This exercise is also reverse engineered: given existing or hypothetical assets, what are the constraints on business that can be written? Capital risk measures are used by management to determine economic capital, by a regulator to determine a minimum capital requirement, and by a rating agency to opine on the adequacy of held capital. Value at Risk (VaR) or Tail Value at Risk (TVaR) at some high confidence level, such as 99.5% or 1 in 200 years, are both popular, but other possible measures exist. These are both explored in Chapter 4.

      A pricing risk measure determines the expected profit insureds need to pay in total to make it worthwhile for investors to bear the portfolio’s risk. Pricing risk measures are called premium calculation principles (PCP) in actuarial texts (Goovaerts, De Vylder and Haezendonck 1984). It is not expected that the loss be less than premium with a high degree of confidence—capital provides the cushion. But the premium must include a margin sufficient for the insurer to raise the needed capital cushion.

      Risk measures typically have at a free parameter encoding their conservatism. As a result, there are two modes of use:

       Given a level of conservatism, they determine a premium or capital requirement.

       Given price or capital requirement, they evaluate its implied level of conservatism.

      For example, a capital risk measure can be applied to capital model output to determine the amount of capital needed to be sufficient for 99% of outcomes. Or, it can evaluate what percentage of outcomes a given amount of capital covers.

      By determine capital or premium, we mean to derive a technical or benchmark quantity—usually the actual held capital or market premium differ. The benchmark value can be used to evaluate an actual one.

      The capital and premium applications both determine a monetary quantity, so it is helpful if the risk measure is denominated in monetary units. The mean and standard deviation are monetary but variance or probability of default are not. If the risk measure is not monetary, then we need an undesirable extra step get to the answer.

      In practice, the capital and pricing applications usually require different risk measures with different properties. Capital risk measures must be sensitive to tail risk to ensure solvency. Management, concerned with solvency and earnings risk, often looks for pricing risk measures that are sensitive to volatility. They can bemoan the fact that tail risk measures fail to see volatility risk, just as measures of height fail to quantify weight. As we discuss in Section 4.1, the solution is to use two risk measures. The interplay of the pricing and capital risk measures is a central theme of the book.

      Example 34 You have a flight to catch and want to allow sufficient time for your trip to the airport. George Stigler, a winner of the 1982 Nobel Prize in economics, said, “If you never miss the plane, you’re spending too much time in airports.” How much time do you allow?

      If you’re going on vacation you are very concerned about missing your flight. You don’t want to lose any vacation time—there’s only one flight per day—and you’ll pay the cost of flight changes. You use a conservative capital risk measure approach. You build in a big time margin so that even in extreme traffic conditions you still make it to the airport on time.

      If you’re returning from a business trip you are much more concerned about wasting time—your company will pay for flight changes, and there’s a flight every hour. You use a similar approach to a pricing risk measure. You consider the average journey times you have experienced, factoring in the time of day, day of the week, and weather, and add a small margin for minor traffic issues.

Example 35 Stress tests are often used as a risk measure: what are my results in a stressed environment? Again, there are capital and pricing flavors. Lloyd’s Realistic Disaster Scenarios, Example 3.4.4, define a capital stress test. They quantify the impact of extreme but realistic events. By ensuring adequate capital to pay all claims from RDS events, Lloyd’s can communicate its risk paying capacity to a nontechnical audience. Setting capital using a risk measure equal to the worst outcome over the RDS set achieves this goal. Insurers can use alternative climate scenarios in catastrophe models to stress test pricing. These act to increase mean event frequency. An insurer may want to price with a margin reflecting the uncertainty in expected mean frequency. Here, the stress test applies to pricing.

      3.6.4 Risk Measure Functional Forms

      What functional forms should we consider for a risk measure? The representation of ρ(X) should be related to the potential outcomes described by X. The most basic summarization of X is the mean, which is the probability weighted sum of outcomes

       (3.15)

There are many different ways we could generalize the mean; see Figure 3.14.

      Figure 3.14 Potential functional forms for risk measures inspired by the expected value. The left shows the stochastic model, with many different states of the world mapping to the same loss outcome. These extensions overlap: a given risk measure can often be written in multiple ways.

      1 Adjust outcomes by a factor depending on the outcome value and the explicit sample point ω. In finance the sample point is often called the state of the world or simply the state.

      2 Adjust the probabilities to create a new measure. The new measure can make previously impossible events possible and vice versa.

      3 Scale existing probabilities to create a new measure using a function of the explicit sample point (i.e. a random variable) p with p(ω)≥0 and E[p]=1. A specific scenario has this form. In this case events which are impossible under the original probability remain impossible. This approach is developed in Section 8.6.

      4 Adjust with a function of loss and not ω. Standard deviation has this form, h(x)=x and g(x)=(x−μ)2.

      5 Adjust outcomes independently of ω and leave probability unchanged. If the function u is increasing and convex then this form is called an expected utility risk measure. When outcomes can take positive and negative values we can adjust them with an S-shaped value function (Kahneman and Tversky 1979). The value function reflects attitudes to changes in wealth rather than final wealth.

      6 Adjust probabilities by a function of the rank of the loss, but leaves the loss amount unchanged. This form is called dual utility theory and leads to spectral risk measures; see Section 10.7. (The function g:[0,1]→[0,1] must satisfy g(0)=0 and g(1)=1. It is used to adjust probabilities. Integration by parts shows that ∫0∞xg′(S(x))dF(x)=∫0∞g(S(x))dx since d(g(S(x)))/dx=−g′(S(x))dF/dx. S(x) is the rank of x.)

      7 A combination of (e) and (f) leads to rank-dependent utility (Machina, Teugels and Sundt 2004; Quiggin 2012). Other attempts to adjust probabilities independent of ω are hard and often lead to measures that do not uniformly prefer more of a good to less, counter to any intuitive notion of behavior (Quiggin (2012); Section СКАЧАТЬ