# (Papers) ACET Paper June 2015 "CT6 Statistical Models"

## (Papers) ACET Paper June 2015 "CT6 Statistical Models"

**Q.1 An insurance company operates a No Claims Discount system for its motor insurance business, with discount levels 0%, 15%, 30% and 50%. The full annual premium is Rs100. The rules for moving between discount levels are:·**

- If no claims are made during a year, the policyholder moves to the next higher level of discount or remains at the maximum discount level.
- If one or more claims are made during a year, a policyholder at the 30% or 50% discount level moves to the 15% discount level and a policyholder at the 15% or 0 discount level moves to, or remains at, the 0% discount level.

When an accident occurs, the distribution of the loss is exponential with mean Rs500. In the event of an accident, a policyholder will claim only if the loss is greater than the total extra premiums that would have to be paid over the next three years, assuming that no further accidents occur.

For each discount level, calculate:

i) the smallest loss for which a policyholder will make a claim.

ii) the probability of a claim being made in the event of an accident occurring.

**Q.2 The total amount claimed for a particular risk in a portfolio is observed
for each of 3
consecutive years. From past knowledge of similar portfolios, an insurer
believes
that the claims are normally distributed with mean q and variance 16, and that
the
prior distribution of the mean is normal with mean 100 and variance 49.**

i) Derive the Bayesian estimate under quadratic loss, and show that it can be written in the form of a credibility estimate combining the mean observed claim size for this risk with the prior mean forq .

ii) State the credibility factor, and calculate the credibility premium if the
mean claim
size over the 3 years is 110.

iii) Comment on how the credibility factor and the credibility estimate change
if the
variance of 16 is decreased.

**Q.3 A portfolio of crop insurance has claims distributed as Pareto with mean 500
and
standard deviation of 1500. In the following Kharif season, 200 claims are
expected,
with claims being assumed to be occurring according to a Poisson process.
To limit its losses, the insurer decides to introduce a policy excess of Rs100.**

**Calculate the percentage reduction in the mean of aggregate claims to the
insurer
following the introduction of the policy excess. **

It may be assumed that payments are made in the middle of a calendar year. Inflation rate on these policies has been estimated to be 10% per annum over the relevant period.

i) What are the assumptions for the inflation adjusted chain ladder method?

ii) Use the inflation adjusted chain ladder method to estimate the total
outstanding payments, up to the end of development year 3, for accident year
2004 in mid-2004 prices, ignoring future inflation.