Question

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

Maturity (Years) YTM
1 5%
2 4%
3 3%
4 2%

a. What are the implied 1-year forward rates for year 2, year 3, and year 4? (3 points)
b. Assume that the pure expectations hypothesis of the tern structure is correct. If market expectations are accurate, what will be the pure yield curve (i.e., the YTM on 1-year, 2-year and 3-year zero-coupon bonds) next year? (3 points)
c. If you purchase a 3-year zero-coupon bond now, what is the expected total rate of return over the next year? What if you purchase a 4-year zero-coupon bond? (4 points)
d. What should be the current price of a 4-year maturity bond with a 3% coupon rate paid annually? If you purchased it at that price, what would your total expected rate of return be over the next year (coupon plus price change)? (5 points)

Answer 1

a)

(1 + 2 year spot rate)² = (1 + 1 year spot rate) * (1 + 1 year spot rate 1 year from now)

(1+ 4%)² = (1 + 5%) * (1 + 1 year spot rate 1 year from now)

(1 + 1 year spot rate 1 year from now) = (1+ 4%)² / (1 + 5%)

(1 + 1 year spot rate 1 year from now) = 1.030095

1 year spot rate 1 year from now = 3.0095%

(1 + 3 year spot rate)³ = (1 + 2 year spot rate)² * (1 + 1 year spot rate 2 years from now)

(1+ 3%)³ = (1 + 4%)² * (1 + 1 year spot rate 2 years from now)

(1 + 1 year spot rate 2 years from now) = (1+ 3%)³ / (1 + 4%)²

(1 + 1 year spot rate 2 years from now) = 1.010288

1 year spot rate 2 years from now = 1.0288%

(1 + 4 year spot rate)⁴ = (1 + 3 year spot rate)³ * (1 + 1 year spot rate 3 years from now)

(1+ 2%)⁴ = (1 + 3%)³ * (1 + 1 year spot rate 3 years from now)

(1 + 1 year spot rate 3 years from now) = (1+ 2%)⁴ / (1 + 3%)³

(1 + 1 year spot rate 3 years from now) = 0.990579

1 year spot rate 3 years from now = -0.9421%

b)

YTM of 1 year zero coupon bond next year = 1 year spot rate 1 year from now

YTM of 1 year zero coupon bond next year = 3.0095%

(1 + YTM of 2 year zero coupon bond next year)² = (1 + 1 year spot rate 1 year from now) * (1 + 1 year spot rate 2 years from now)

(1 + YTM of 2 year zero coupon bond next year)² = (1 + 3.0095%) * (1 + 1.0288%)

(1 + YTM of 2 year zero coupon bond next year)² = 1.040693

(1 + YTM of 2 year zero coupon bond next year) = 1.020143

YTM of 2 year zero coupon bond next year = 2.0143%

(1 + YTM of 3 year zero coupon bond next year)³ = (1 + 1 year spot rate 1 year from now) * (1 + 1 year spot rate 2 years from now)

(1 + YTM of 3 year zero coupon bond next year)³ = (1 + 3.0095%) * (1 + 1.0288%) * (1 - 0.9421%)

(1 + YTM of 3 year zero coupon bond next year)³ = 1.030888

(1 + YTM of 3 year zero coupon bond next year) = 1.010192

YTM of 3 year zero coupon bond next year = 1.0192%

c)

Current Price of 3 year zero coupon bond = Face value / (1 + 3 year spot rate)³

Current Price of 3 year zero coupon bond = $1000 / (1 + 3%)³

Current Price of 3 year zero coupon bond = $915.14

Price of 3 year zero coupon bond next year = Face value / (1 + YTM of 2 year zero coupon bond next year)²

Price of 3 year zero coupon bond next year = $1000 / (1 + 2.0143%)²

Price of 3 year zero coupon bond next year = $960.90

Expected total return = (Price of 3 year zero coupon bond next year / Current Price of 3 year zero coupon bond) - 1

Expected total return = ($960.90 / $915.14) - 1

Expected total return = 5%

Current Price of 4 year zero coupon bond = Face value / (1 + 4 year spot rate)⁴

Current Price of 4 year zero coupon bond = $1000 / (1 + 2%)⁴

Current Price of 4 year zero coupon bond = $923.85

Price of 4 year zero coupon bond next year = Face value / (1 + YTM of 3 year zero coupon bond next year)³

Price of 4 year zero coupon bond next year = $1000 / (1 + 1.0192%)³

Price of 4 year zero coupon bond next year = $970.04

Expected total return = (Price of 4 year zero coupon bond next year / Current Price of 4 year zero coupon bond) - 1

Expected total return = ($970.04 / $923.85) - 1

Expected total return = 5%

d)

Coupon = Coupon rate * Face value

Coupon = 3% * $1000

Coupon = $30

Current Bond price = Coupon / (1 + 1 year spot rate)¹ + Coupon / (1 + 2 year spot rate)² + Coupon / (1 + 3 year spot rate)³ + (Coupon + Face value) / (1 + 1 year spot rate)⁴

Current Bond price = $30 / (1 + 5%)¹ + $30 / (1 + 4%)² + $30 / (1 + 3%)³ + ($30 + $1000) / (1 + 2%)⁴

Current Bond price = $1035.32

Bond price next year = Coupon / (1 + YTM of 1 year zero coupon bond next year)¹ + Coupon / (1 + YTM of 2 year zero coupon bond next year)²  + (Coupon + Face value) / (1 + YTM of 3 year zero coupon bond next year)³

Bond price next year = $30 / (1 + 3.0095%)¹ + $30 / (1 + 2.0143%)² + ($30 + $1000) / (1 + 1.0192%)³

Bond price next year = $1057.09

Expected total return = (Bond price next year + Coupon - Current Bond price) / Current Bond price

Expected total return = ($1057.09 + $30 - $1035.32) / $1035.32

Expected total return = 5%