Please use continuous compounding unless stated otherwise in the question.

2. (a) Give all answers to the nearest cent. Assume a yield curve for all maturities of 4% p.a. You have a liability comprising e100,000 to be paid in two years time and e200,000 to be paid in four years time. Find the present value of the liability and its duration. 1+1% You wish to immunise this debt using two available bonds: a 3-year e100 zero-coupon bond and a 5-year e100 zero-coupon bond. Construct a duration immunising portfolio from these bonds (fractions of bonds are permitted). 4% If after exactly one year, all rates rise by 0.5%, find the value of your bond holding (i) using a linear approximation in the rate change and (ii) exactly. Compare these with the new exact value of the liability. Reference all holdings to time 0

(b) Use six-monthly compounding for all rates in this problem. Give all answers to at least 4 significant figures. The 6-month LIBOR spot rate is 4.4% and 12-month LIBOR spot rate is 4.8%. A bank trades swaps where a fixed rate of interest is exchanged for 6-month LIBOR, with payments being exchanged biannually. The 18-month and 24-month swap rates are 4.8% and 5.2% per annum. Estimate the 18-month and 24-month LIBOR rates. (Hint: construct par value bonds with coupon rates equal to the relevant swap rates). Using the extended LIBOR rates, find the 6-month forward rates starting in 6, 12 and 18 months. Using these, show that the present value of 18-month swap is zero.