Question 11.2: 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. Calculate the expected claims payment and the insurer’s risk.
The claim payment distribution is determined from Table 11.3.
C(y)=\begin{cases} loss=0\ or\ 500, 0.9\ y=0\\ loss=5000, 0.08\ y=4500\\ loss=15,000, 0.02\ y=14,500\end{cases}The expected claims payment and standard deviation for a single policy are:
E[Y] = 0.9 * 0 + 0.08 * 4500 + 0.02 * 14500
= 360 + 290
= €650
{{\sigma_{Y}^{2}}}\ {{=}}\ {{0.9*(0-650)^{2}+0.08*(4500-650)^{2}+0.02*(14500-650)^{2}}}= 5,402,500
\sigma_{Y} = √5402500 = 2324
That is, the expected claim payments for the 100 policies would be €65,000 and the variance would be 540,250,000 and the standard deviation would be 23,243.
This example shows that the policy excess eliminated the need to process small claims below €500, and the probability of a claim has fallen from 20% to 10% of policies. Further, the excess reduces the amount of the expected claim payments from €75,000 to €65,000, and the standard deviation has fallen from 24,418 to 23,243.
We mentioned that it is quite common for a maximum benefit limit to be set in the insurance policy, and this reduces the risk taken on by the insurance company as well as ensures that the insurer has the reserves to meet all claims. There may be a choice of benefit limits that the policyholder may choose with a corresponding reduction in premium for choosing a lower limit. There are various ways in which benefit limits may be set in a policy.
| TABLE 11.3 | |||||
| Claim Payment Function (Excess) | |||||
| 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, 14500 | |||
| 0.1 | 0.02, 15000 | ||||