1
Mathematical Modelling
Worksheet 13
Insurance with Claims of Constant Size
- Consider an insurance portfolio in which each person is insured for the same sum S. An example would be a friendly society in which the benefits are constant amounts payable on death for funeral expenses. The number of claims in a year may be taken to have a Poisson distribution with mean n (n = expected number of claims per year).
(i) We can use the information that the premium income in one year is where is the safety loading, to find how large a reserve capital, U, is needed so that the probability that the claims in one year exhaust U is equal to . We can approximate a Poisson random variable with mean n by a Normal random variable with mean n and variance n.
P[(Reserve Capital (U) + Total Income-Total Claims)<0] =
Total amount lost through claims = Number of claims S =
Now considering
And substituting this into the equation above:
(ii)We can show that if , and then U is proportional to .
put
(iii)We can show that if and , then no reserve U is required since the safety loading is more than enough to cover fluctuations of claims with probability .
,,
Since S is constant U0, therefore safety loading is enough.
iv)When plot U/S against n for
U/S against n has been plotted for n up to 1000.
2a)A friendly society has 1000 members (N). In the event of death a fixed sum of £500 is paid. The mean value of the rate of mortality is 0.01 and the safety loading is .
i)What is the annual premium charged per member?
n = expected number of claims/year
= N rate of mortality
n = 1000 0.01
= 10
Therefore the premium charged per member is
ii)The actuarial status of the society is examined every five years. How large a security reserve U should the society have to be sure, at the 99% probability level, that after a five year period the balance does not show any deficit?
n = N rate of mortality 5
=5 0.01 1000
=50
b)How many members (N) should the society have for no security reserve to be necessary under the conditions mentioned in (a)?
c)The same as (a(ii)) except that the status is examined every year instead of every five years. Apply both the Poisson distribution and the Normal approximation and compare the results.
Normal distribution:
Poisson distribution:
Using tables with n=10, X=17.5
Comparing the two distributions it is clear that it is better to go with the Normal distribution because it works out the cheapest.
d)A friendly society grants funeral expense benefits on the death of a member, the benefit being fixed at £100. The expected number of claims is n>1. The society has a stop loss reinsurance in accordance with which, if the number of deaths exceeds two, the reinsurer pays the third and subsequent benefits. What is the net premium for the reinsurance?
n=1 Pay out when X=1,2
[1-(P(0)+P(1)+P(2))] 100 = £10.36