Question

The current yield curve for default-free zero-coupon bonds is as follows:

Maturity (Years)	YTM
1	10%
2	11%
3	12%

All bonds considered in this question have a face value of $1,000. Assume that the pure expectations hypothesis of the term structure holds.

If market expectations are accurate, what are the expected yields to maturity on 1- and 2-year zero coupon bonds next year?

If you purchase a 3-year zero-coupon bond now, what is the expected total rate of return over the next year assuming that you will sell the bond at the expected price (price that matches the expected yield in part a))? Ignore taxes.

What should be the current price of a 3-year maturity bond with a 12% coupon rate paid annually?

If you purchase the coupon bond at the price you calculated in part c), what would your total expected rate of return over the next year be (coupon plus price change)? Ignore taxes.

Answer 1

a) Pure Expectations theory postulates that long-term interest rates contain a prediction of future short-term interest rates.

In our question, in order to find YTM for a 1-yr zero coupon bond next year,

(1 + 11%)² = (1 + 10%) * (1 + r%)

This basically means, that according to pure expectations theory, the amount of money invested in a 2-year bond today should yield same rate as money invested in 1-year bond today and post maturity (after 1 year), same invested in 1 -year bond again.

r = 12.01%. (Expected YTM for 1 -year bond)

For a 2 year bond starting next year,

(1 + 12%)³ = (1 + 10%) * (1 + r)²

(1 + r)² = 1.2772

(1 + r) = 1.13

r = 13.01%. (Expected YTM for 2 -year bond)

b. Now that this is a zero coupon bond, there would be no coupon payments involved. Only capital appreciation would be there. We need to calculate the price of bond at issue and after 1 year of issue.

Price of the bond (3 year) at issue = 1000/(1 + 12.01%)³ = $711.59

After one year, the YTM for this bond would be equal to YTM of the bond with two years to maturity (calculated in oart a).

Price of bond (2 years to maturity) = 1000/(1 + 13.01%)² = $782.96

Hence total return = (783.15 - 711.78)/711.78 = 10.03%

c. Price of 3-year bond with 12% coupon paid annually

Now, YTM for a 3 year bond = 12%. Since YTM = Coupon Rate, bond would be priced at par.

But let me show that mathematically to you.

Price of a bond is mathematically denoted by:

where P is price of a bond with periodic coupon C, YTM i, n years to maturity and M face value. Substituting values from our question:

P = $288.2198 + $711.7802 = $1000

d) Price of bond next year would be at YTM 13.01%

P = $983.15

Total Return = (983.15 + 120 - 1000)/1000 = 10.315%