Question

Question 1 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. ?t = (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?

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.

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.

ii. Determine the price of the annuities. d) Assuming annual compounding, determine the implied one-period forward rates ??2 (i.e. between year 1 and 2) and ??3 (i.e. between year 2 and 3) in this economy. What inference can you make about the market’s estimate of the one-year spot interest rate at ?? = 1 if the liquidity preference theory is correct?

e) Suppose you decide to purchase a 1-year zero-coupon bond today and also contract today to re-invest the proceeds from the bond for the following two years at 16.5% per year. Show that this arrangement presents an arbitrage opportunity. Demonstrate how you would take advantage of this opportunity.

f) Consider discount factors such that d1 < d2 < d3. Explain why it would be odd to observe such a situation in a competitive market.

Answer 1

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