1. (i) State the main assumptions of mean-variance portfolio theory. [4]

Consider a mean-variance portfolio model with two securities, with respective returns SA and SB, where the expected return E[SB] = 0.25E[SA] and the variance of return

V[SB] = 0.25V[SA].

Let the correlation between the returns on the two securities be .

(ii) Determine, in terms of E[SA], the expected return on the minimum variance

portfolio if:

(a) = 0

(b) = 1

[4]

(iii) (a) Calculate the variance of the return on the minimum variance portfolio

for part (ii)(b).

(b) Comment on the risk in this portfolio.

2. An insurance company's business consists only of policies covering a specified event and which pay a sum assured of Z immediately on occurrence of this event. Claims on the portfolio of policies are considered to occur in accordance with a Poissonprocess with annual rate .

The insurance company currently has assets of S (Z > S). It charges a premiumwhich is to be 50% more than the expected outgo. Premiums can be assumed to bereceived continuously. The insurance company's expenses are small and can beignored.

(i) Derive the total annual premium charged by the insurance company on the portfolio. [1]

(ii) Show that the probability that the insurance company has insufficient assets topay the next claim made is given by:1 1 exp 1 .5

3. Assume that the numbers of accidents for three different risks in five years are as

follows:

Year 1 Year 2 Year 3 Year 4 Year 5 Total

Risk A 1 4 5 0 2 12

Risk B 1 6 4 6 5 22

Risk C 5 6 4 9 4 28

An actuary is modelling each risk according to a Poisson distribution.

(i) Determine the Poisson parameter for each risk using the method of maximumlikelihood estimation. [5]

(ii) Test the hypothesis that the three risks have the same claim rate, using thescaled deviances. [5]