Question 11.3: Suppose the policy for the car pool of 100 cars that was dis......
Suppose the policy for the car pool of 100 cars that was discussed has a €500 excess and a maximum payout of €12,500. Calculate the expected claims payment and the insurer’s risk.
The claim payment distribution is determined from Table 11.4.
C(y)=\begin{cases} loss=0\ or\ 500, 0.9\ y=0\\ loss=5000, 0.08\ y=4500\\ loss=15,000, 0.02\ y=12,500\end{cases}The expected claims payment and standard deviation for a single policy are:
E[Y] = 0.9 * 0 + 0.08 * 4500 + 0.02 * 1250
= 360 + 250
= €610
{{\sigma_{Y}^{2}}}\ {{=}}\ {{0.9*(0-610)^{2}+0.08*(4500-610)^{2}+0.02*(12500-610)^{2}}}= 4, 372, 900
\sigma_{Y} = √4372900 = 2091
The expected claim payment for the 100 policies is then €61,000 with the variance 437,290,000 and the standard deviation 20,911. That is, by employing a policy excess and maximum limit the insurer’s expected claim payments has fallen from €75,000 to €61,000 and the standard deviation has fallen from 24,418 to 20,917.
The insurance company would need to take inflation into account as the cost of repairs will increase over a period, so there will be a need to adjust the excess and benefit limit to reflect inflation.
| TABLE 11.4 | |||||
| Claim Payment Function (Excess/Limit) | |||||
| F(x) | S(x) | L(x) | C(y) | ||
| 0.8 | – | 0.8, 0 | 0.9, 0 | ||
| 0.2 | 0.5 | 0.1, 500 | 0.08, 4500 | ||
| 0.4 | 0.08, 5000 | 0.02, 12500 | |||
| 0.1 | 0.02, 15000 | ||||