Question

3. Suppose that the short-term risk-free interest rate this year is r1 8% and that the expected value of next year's interest rate is r2 7.5%. Suppose that a two-year zero coupon bond with face value $1000 sells for $820. a. What is the yield to maturity of the 2-year zero? b. Your answer to (a) demonstrates that the yield curve can slope upward even if the market thinks that interest rates are likely to fall. To explain this result, calculate the forward rate for year 2. What is the liquidity premium? Given yields on 1 and 2-year zero-coupon bonds, what would be the price and yield to maturity of a 2-year coupon bond with a coupon rate of 10%? c.

Answer 1

Part (a)

Let's assume annual coupon payment convention for all our calculations.

Price of 2 year zero coupon bond = Face Value / (1 + y)² where y is the yield to maturity.

P = 820 = 1,000 / (1 + y)²

hence, y = $\sqrt{\frac{1000}{820}} - 1$ = 10.43%

Part (b)

The yield of 2 year zero coupon bond is also a reflection of spot rate for year 2

S_0,2 = y = 10.43%

S_0,1 = 8%

Hence forward rate F_1,2 can be given by

(1 + S_0,1) x (1 + F_1,2 ) = (1 + S_0,2 )²

Hence, (1 + 8%) x (1 + F_1,2 ) = (1 + 10.43%)²

Hence, F_1,2 = [(1.1043)² / 1.08] -1 = 12.92%

Expected value of next year's interest rate is E(R₂) = 7.5%

Hence, liquidity premium = E(R₂) - F_1,2 = 12.92% - 7.5% = 5.42%

Part (c)

Face Value, FV = 1000

Coupon rate = 10%

Coupon, C = 10% x 1000 = 100

Price of the bond, P = PV of all the future cash flows = C / (1 + S_0,1) + (C + FV) / (1 + S_0,2)² = 100 / (1 + 8%) + (100 + 1000) / (1 + 10.43%)² = 994.6175

For yield to maturity, we use excel formula = RATE(Period, payment, PV, FV)

Period = 2 years, Payment = C, PV = -Price of the bond = -994.6175, FV = 1000

Hence, YTM = RATE(2,100,-994.6175,1000) = 10.31%