You observe that the current three-year discount factor for default-risk free cash flows is 0.68. Remember, the t-year discount factor is the present value of $1 paid at time t, i.e. = (1 + ), where is the t-year spot interest rate (annual compounding). Assume all bonds have a face value of $100 and that all securities are default-risk free. All cash flows occur at the end of the year to which they relate.

a) What is the price of a zero-coupon bond maturing in exactly 3 years? (5 marks)

b) Your friend makes the following observation about the above bond: Since there is no risk of default and there are no coupons to re-invest, buying the 3-year zero coupon bond today is a risk-free investment; that is, you are guaranteed to earn an annual return of 13.72% (i.e. 3-year spot rate). Explain why your friend is not entirely correct and how you would modify the statement to make it correct. (5 marks)

c) In addition to the bond in (a), you observe the following: a 2-year coupon bond paying 10% annual coupons with a market price of $97, and two annuities that are trading at the same market price as each other. The first annuity matures in 3 years and pays annual cash flows of $20, while the second annuity pays annual cash flows of $28 and matures in 2 years. Using this information:

i. Complete the term structure of interest rates, i.e. determine the one- and two-year discount factors, d₁ and d₂, respectively. (6 marks)

ii. Determine the price of the annuities. (2 marks)