Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 28

Recreate Table 2.2 in a spreadsheet (or R or Python). Compute and plot the distribution and survival functions, Pr(X≤x) and Pr(X>x) for X.

Solution. Since the data is discrete, the answers are step functions. The survival function is

(2.1)

Table 2.2 Simple Discrete Example with nine possible outcomes

X1	X2	X	P(X1)	P(X2)	P(X)
0	0	0	1/2	1/2	1/4
0	1	1	1/2	1/4	1/8
0	90	90	1/2	1/4	1/8
8	0	8	1/4	1/2	1/8
8	1	9	1/4	1/4	1/16
8	90	98	1/4	1/4	1/16
10	0	10	1/4	1/2	1/8
10	1	11	1/4	1/4	1/16
10	90	100	1/4	1/4	1/16

Loss statistics are shown in Table 2.3. The table includes the impact of aggregate reinsurance on the more volatile unit X₂ limiting losses to 20 and show a gross and net view for some exhibits. Others are left as Practice questions.

Table 2.3 Discrete Example estimated mean, CV, skewness and kurtosis by line and in total, gross and net. Aggregate reinsurance applied to X2 with an attachment probability 0.25 (¤ 20) and detachment probability 0.0 (¤ 100)

	Gross			Net
Statistic	X1	X2	Total	X1	X2	Total
Mean	4.500	22.750	27.250	4.500	5.250	9.750
CV	1.012	1.707	1.435	1.012	1.624	0.991
Skewness	0.071	1.154	1.131	0.071	1.147	0.794
Kurtosis	−1.905	−0.667	−0.649	−1.905	−0.673	−0.501