Question:

(a) In any year, the rate of interest on funds invested with a given insurance company is independent of the rates on interest in all previous years.

Each year the value of (1+i_t) where i_t is the rate of interest earned in the t^th year is log-normally distributed. The mean and standard deviation of i are 0.07 and 0.20 respectively.

(i) Determine the parameters μ and σ² of the lognormal distribution of 1+ i_t

(ii) Determine the distribution of S15, where S15 denotes the accumulation of one unit of money over 15 years.

(iii) Determine the probability that S15 >2.5.

(b) A pension fund has a liability to pay 100,000 at the end of one year, 105,000 at the end of two years, and so on, the amount increasing by 5,000 each year to 195,000 at the end of 20 years, this being the last payment. The fund values these payments using an effective rate of 7% per annum. This is also the interest rate at which the current prices of all bonds are calculated.

The fund invests an amount equal to the present value of these liabilities in the following two assets:

(A) a zero coupon bond redeemable in 25 years, and

(B) a fixed interest bond redeemable at par in 12 years' time which pays a coupon of 8% per annum annually in arrears.

(i) Calculate the present value and the duration of the liabilities.

(ii) Calculate the amount of cash that should be invested in each asset if the duration of the assets is to equal that of the liabilities.