Pricing Insurance Risk. Stephen J. Mildenhall

Figure 2.4 Cat/Non-Cat Case Study, gross (top) and net (bottom) densities on a nominal (left) and log (right) scale.

      Figure 2.5 Cat/Non-Cat Case Study, bivariate densities: gross (left), net (center), and a sample from gross (right). Impact of reinsurance is clear in net plot.

      Figure 2.6 Hu/SCS Case Study, gross (top) and net (bottom) densities on a nominal (left) and log (right) scale.

      Figure 2.7 Hu/SCS Case Study, bivariate densities: gross (left), net (center), and a sample from gross (right). Impact of reinsurance is clear in net plot.

      We strongly recommend that the reader reproduce the Examples and Cases. We suggest a general-purpose programming language such as R or Python, although SQL or even a spreadsheet suffices, with a bit of ingenuity. See Section 2.4.5 for a discussion of the implementation we used.

      2.4.1 The Simple Discrete Example

Ins Co. writes two units taking on loss values X1=0, 8, or 10, and X2=0, 1, or 90. The units are independent and sum to the portfolio loss X=X1+X2. The outcome probabilities are 1/2,1/4, and 1/4, respectively, for each marginal. The nine possible outcomes, with associated probabilities, are presented in Table 2.2. The output is typical of that produced by a catastrophe, capital, or pricing simulation model—albeit much simpler.

      Exercise 1

      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