Pricing Insurance Risk. Stephen J. Mildenhall
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Exercise 31 The loss outcomes are all distinct in Table 3.2. When there are ties, Step (3) is needed. Recompute the table if X1 can take values 1,9,10.
The algorithm’s computations can be visualized using a Lee diagram, as shown in Figure 3.13. The horizontal axis shows events and their cumulative probabilities (in 1/16ths and decimals). The width of each event corresponds to its objective probability, ΔS in the table. The vertical axis shows potential outcomes from 1 to 100. The horizontal black lines show the outcome X for each event. In this case, there are nine events with nine distinct outcomes. The shaded area shows the chances each unit of assets is needed to fund an event. For example, the 99th and 100th units are only at risk from (or needed to fund) event j = 9, which has outcome 100 (upper right-hand corner). We might say these units are potentially consumed by event j = 9. The eight units 91–98 are at risk from events j = 8 and j = 9. The step heights of the shaded area correspond to ΔX in the table and ΔX7 for row j = 7 is shown. At the bottom of the graph, the first unit of assets is needed to fund events j = 2 through 9. Event j = 1 has a zero loss, which does not require any assets. The plots in Figure 3.11 show the same function (different data values) rotated by 90 degrees to make the interpretation as ∫S(x)dx clearer.
Figure 3.13 Lee diagram showing relationship between differen asset layers and the events they fund.
Given the increasing sequence Xj, 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
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
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
| j | X' | ΔX' | ΔS | S | X'ΔS | SΔX' |
|---|---|---|---|---|---|---|
| 0 |