Pricing Insurance Risk. Stephen J. Mildenhall

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СКАЧАТЬ S X ΔS S ΔX 0 0 1 0.25 0.75 0 0.75 1 1 7 0.125 0.625 0.125 4.375 2 8 1 0.125 0.5 1 0.5 3 9 1 0.0625 0.4375 0.563 0.4375 4 10 1 0.125 0.3125 1.25 0.3125 5 11 79 0.0625 0.25 0.688 19.75 6 90 8 0.125 0.125 11.25 1 7 98 2 0.0625 0.0625 6.125 0.125 8 100 0.0625 0 6.25 Sum 1 27.25 27.25

      Exercise 30 Apply the Algorithm to X + 100.

Solution. Step (1) now introduces the 0 row, see Table 3.3.

Table 3.3 Solution to Exercise 30.

j	X	ΔX	PX=ΔS	S	X ΔS	S ΔX
0	0	100	0	1	0	100
1	100	1	0.250	0.75	25	0.75
2	101	7	0.125	0.625	12.625	4.375
3	108	1	0.125	0.5	13.5	0.5
4	109	1	0.0625	0.4375	6.8125	0.4375
5	110	1	0.125	0.3125	13.75	0.3125
6	111	79	0.0625	0.25	6.9375	19.75
7	190	8	0.125	0.125	23.75	1
8	198	2	0.0625	0.0625	12.375	0.125
9	200		0.0625	0	12.5
Sum					127.25	127.25

      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 СКАЧАТЬ