Question

1.Suppose that a 1-year zero-coupon bond with face value $100 currently sells at $91.20, while a 2-year zero sells at $84.28. You are considering the purchase of a 2-year-maturity bond making annual coupon payments. The face value of the bond is $100, and the coupon rate is 6% per year.

a.What is the yield to maturity of the 2-year zero?

b.What is the yield to maturity of the 2-year coupon bond?

c.What is the forward rate for the second year? That is what is f_1,2?

2. Assume you purchase (at par) one 11-year bond with a 6.05 percent coupon and a $1,000 face value. Suppose you are only able to reinvest the coupons at a rate of 4.25 percent. If you sell the bond after 6 years when the yield to maturity is 7.05 percent, what is your realized yield?

(Hint: Think naturally)

Answer 1

Let us calculate the 1 year spot rate

Bond price for 1 year zero coupon bond = Face value / (1 + YTM)

$91.20 = $100 / (1 + YTM)

(1 + YTM) = $100 / $91.20

(1 + YTM) = 1.09649

Yield to Maturity (YTM) = 9.65%

1 year spot rate = 9.65%

a)

Let us calculate the 2 year spot rate

Bond price for 2 year zero coupon bond = Face value / (1 + YTM)²

$84.28 = $100 / (1 + YTM)²

(1 + YTM)² = $100 / $84.28

(1 + YTM)² = 1.1865

(1 + YTM) = 1.1865

(1 + YTM) = 1.08927

Yield to Maturity (YTM) of the 2-year zero = 8.93%

2 year spot rate = 8.93%

b)

Bond Price = Coupon / (1 + 1 year spot rate) + Coupon / (1 + 2 year spot rate)² + Face value / (1 + 2 year spot rate)²

Bond Price = $6 / (1 + 9.65%) + $6 / (1 + 8.93%)² + $100 / (1 + 8.93%)²

Bond Price = $94.80

Bond Price = Coupon / (1 + YTM / 4)^period + Face value / (1 + YTM / 4)^period

$94.80 = $6 / (1 + YTM)¹ + $6 / (1 + YTM)² + $100 / (1 + YTM)²

Using Texas Instruments BA 2 plus calculator

SET N = 2, PV = -94.80, FV = 100, PMT = 6

CPT --> I/Y = 8.95

Yield to Maturity (YTM) of the 2-year coupon bond = 8.95%

c)

(1 + 2 year spot rate)² = (1 + 1 year spot rate) * (1 + forward rate)

(1 + 8.93%)² = (1 + 9.65%) * (1 + forward rate)

(1 + forward rate) = (1 + 8.93%)² / (1 + 9.65%)

(1 + forward rate) = 1.0821

Forward rate for the second year = 8.21%

2)

Coupon = Coupon rate * Face value

Coupon = 6.05% * $1000

Coupon = $60.5

Reinvested coupon amount = Coupon * ((1 + reinvestmnet rate)^{no of periods} - 1) / reinvestmnet rate

Reinvested coupon amount = $60.5 * (1 + 4.25%)⁶ - 1) / 4.25%

Reinvested coupon amount = $403.83

Let us compute the bond price after 6 years, there are 5 years left to maturity

Bond Price = Coupon / (1 + YTM)^period + Face value / (1 + YTM)^period

Bond Price = $60.5 / (1 + 7.05%)¹ + $60.5 / (1 + 7.05%)² + ...+ $60.5 / (1 + 7.05% / 4)⁵ + $1000 / (1 + 7.05%)⁵

Using PVIFA = ((1 - (1 + Interest rate)^{- no of periods}) / interest rate) to value coupons

Bond Price = $60.5 * ((1 - (1 + 7.05%)^-5) / 7.05%) + $1,000 / (1 + 7.05%)⁵

Bond Price = $247.73 + $711.32

Bond Price = $959.05

Holding period return = (Bond price after 6 years + Reinvested coupon amount - Purchase price) / Purchase price

Holding period return = ($959.05 + $403.83 - $1000) / $1000

Holding period return = 0.36288 or 36.288%

Realized yield = (1 + Holding period return) ^{(1 / no of periods)} - 1

Realized yield = (1 + 36.288%)^{(1 / 6)} - 1

Realized yield = 1.0530 -1

Realized yield = 5.30%