Question

The current yield curve for default-free zero-coupon bonds is as follows: Maturity (years) YTM 10.1% 11.1 12.1 a. What are the implied one-year forward rates? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Maturity (yers) YTM 10.1% Forward Rate 12.1% b. Assume that the pure expectations hypothesis of the term structure is correct. If market expectations are accurate, what will the pure yield curve (that is, the yields to maturity on one- and two-year zero-coupon bonds) be next year? There will be a shift upwards in next year's curve. O There will be a shift downwards in next year's curve. There will be no change in next year's curve. c-1. If you purchase a two-year zero-coupon bond now, what is the expected total rate of return over the next year? (Hint: Compute the current and expected future prices.) Ignore taxes. (Do not round intermediate calculations. Round your answer to 2 decimal places.) Expected total rate of return c-2. If you purchase a three-year zero-coupon bond now, what is the expected total rate of return over the next year? (Hint: Compute the current and expected future prices.) Ignore taxes. (Do not round intermediate calculations. Round your answer to 2 decimal places.) Expected total rate of return

Answer 1

Answer (a)

• A zero coupon bond (ZCB) pays nothing during the lifetime.
• Current yield is the yield expected when you hold the bond for a year.
• Yield to maturity (YTM) is the yield that will be expected if you hold the bond till maturity.
• Now, since a zero coupon bond pays nothing during the lifetime and pays only at the expiration, the current yield and YTM are the same.
• Hence, the YTM of a ZCB are the spot rates

Using the spot rates, we can find out forward rates as below:-

Answer (b)

The idea behind pure expectations theory is that - the forward rates = expected spot rates of the future, i.e. the market expectations are accurate.

Now, as per the table above we see, the term structure is rising, the forward rates are also rising.

The way the system functions is as follows:-

• A 2-year spot rate, for example, is an approximate average of 1-year spot rates and 1-year forward rate (1 year from today).
• Hence, if 1-year spot rate is 10.1% and 1 year forward rate is 12.2% - it implies that their average will naturally be higher than 10.1%.
• Thus, we see the 2-year spot rate is 11.1% which is more than 10.1%

Hence, if pure expectations theory is in place, the forward rates are accurate measurement of future expectations and the spot rates or YTM of ZCB will be rising.

Hence, answer (b) = there will be shift upwards in next year's curve.

answer (c)

The 2 year spot rate is 11.1%. This is the YTM of ZCB

Using the YTM we can find the price of the bond using the formula as below:-

The price of the bond at any time is present value of future cash flows discounted at YTM

Hence, at T-0 (today) the only cash flow to be received is $100 at the expiration after 2 years (T=2)

Hence, current price = 100/(1+11.11%)^2

= $81.016202

Price at future = $100

Hence, the return % = (100-81.016/100)= 18.98%

Similarly, for a 3 year ZCB =

Current price = 100/(1+12.1%)^3 = $70.98

Future price = $100

Hence, the return % = (100-70.98)/100 = 29.01%

Remember, these are holding period returns and not annual returns.