The government of a country has just issued a series of zero-coupon bonds maturing at the end of years 1, 2, 3 and 4. Suppose the spot rates (or continuously compounded yields per annum) on each of the zero-coupon bonds are as follows:

Let B(0; T) denote the current price of the zero-coupon bonds and R(0; T) denote the corresponding spot rates maturing at the end of year T.

(a) State the relationship between B(0; T) and R(0; T), and determine the prices of the four zero-coupon bonds issued by the government.

(b) The government wants to issue a 4-year bond with xed annual coupons payable in arrears. The bond will be issued at par, i.e. at a price of $1 for $1 nominal. Determine the annual coupon rates to ensure that there are no arbitrage opportunities.

(c) An investor is planning to invest a sum of money at the end of the third year for a period of one year. She wants to enter into a  contract now and x the rate of interest for that period.

(i) Show that the arbitrage-free (continuously compounded) rate of interest for the contract is 3.9%.

(ii) A bank has quoted a (continuously compounded) rate of 4% for such a contract. Devise a strategy to take advantage of any arbitrage opportunity.