Pricing Insurance Risk. Stephen J. Mildenhall

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Figure 3.13 Lee diagram showing relationship between differen asset layers and the events they fund.

      Given the increasing sequence X_j, it is convenient to define j(a)=max{j:Xj<a} and j(0)=0. It is the index of the largest observation strictly less than a. For example, j(90)=6 and j(91)=7. It is used in calculations as follows. To compute the limited expected value of X at a > 0, the survival function form evaluates

       (3.12)

      because ΔXj is the forward difference. It computes the integral as a sum of horizontal slices, e.g. the ΔX7 block in Figure 3.13. For a = 0 obviously E[X∧0]=0. For a=∞, j is set to j + 1, where j is the maximum index with S(Xj)>0, resulting in the unlimited E[X].

      The outcome-probability form is

       (3.13)

      It computes the integral as a sum of vertical slices, e.g. the ΔS5 block in Figure 3.13.

When a = 80, Table 3.4 shows the above calculations through the simple expedient of replacing X values with X∧a and recomputing other columns that depend on X. Notice that columns involving S still use X’s survival function. Numbers changed from Table 3.2 are displayed in bold.

Table 3.4 Computing the limited expected value of X, limited to a = 80. X’ refers to X∧ a but values related to S are unchanged

3.6 Risk Measures

      An ice cream manufacturer wants to introduce a product that customers will prefer over existing ones. It would be very helpful to have a way of predicting customer ice cream preferences. As far as a customer is concerned their preference may be very simple and intuitive: “this brand tastes better!” That preference is not expressed in a way that the manufacturer can use to predict customer responses to a new product. However, through taste tests СКАЧАТЬ