Question
You observe that the current three-year discount factor for default-risk free cash time t, i.e. = (1 + ), where is the t-year spot interest
You observe that the current three-year discount factor for default-risk free cash
time t, i.e. = (1 + ), where is the t-year spot interest rate (annual
flows is 0.68. Remember, the t-year discount factor is the present value of $1 paid at
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.
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a) What is the price of a zero-coupon bond maturing in exactly 3 years? (5 marks)
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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)
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c) Inadditiontothebondin(a),youobservethefollowing:a2-yearcouponbond 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)d) Assuming annual compounding, determine the implied one-period forward rates
(i.e. between year 1 and 2) and (i.e. between year 2 and 3) in this economy.23
What inference can you make about the markets estimate of the one-year spot interest rate at = 1 if the liquidity preference theory is correct? (5 marks)
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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. (6 marks)
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f) Consider discount factors such that d1 < d2 < d3. Explain why it would be odd to observe such a situation in a competitive market. (4 marks)
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