Question

there are two bonds on the market:

(a)One-year $100 zero selling for $95.2381

(b)Two-year 8% coupon $1000 par bond selling for $1000

(1) Assume that the pure expectations theory for the term structure of interest rates holds and no liquidity premium exists. Find implied 1 year rate 1 year from now? show formulas

(2) for the same bonds, assume a liquidity premium of 0.5% for the 2-year long rate (i2t), what is the implied 1-year rate 1 years from now?

(3) your company plans to issue two-year coupon bonds but the current one-year rate suddenly increase to 10% and the two-year long rate becomes 9%, what coupon rate that you need to sell at par?

Answer 1

there are two bonds on the market: (a)​One-year $100 zero selling for $95.2381 (b)​Two-year 8% coupon $1000 par bond selling for $1000

(1) Assume that the pure expectations theory for the term structure of interest rates holds and no liquidity premium exists. Find implied 1 year rate 1 year from now? show formulas

a. 1 year yield = Sale price = Face Value / (1+ Interest)

95.2381 = 100 / (1 + Interest)

1 + Interest = 1.05

Interest = 1 year yield = 5%

b. 2 year yield = 8% because when the bond is selling at par, the coupon rate equals the yield to maturity

Thus implied 1 year rate 1 year from now: (1 + 1 year yield) * (1 + 1 year rate 1 year from now) = (1 + 2 year yield)^2

(1 + 0.05) * (1 + 1 year rate 1 year from now) = (1 + 0.08)^2

(1 + 1 year rate 1 year from now) = 1.1664 / 1.05

(1 + 1 year rate 1 year from now) = 1.1109

1 year rate 1 year from now = 11.09%

(2) for the same bonds, assume a liquidity premium of 0.5% for the 2-year long rate (i2t), what is the implied 1-year rate 1 years from now?

implied 1 year rate 1 year from now: (1 + 1 year yield) * (1 + 1 year rate 1 year from now) = (1 + 2 year yield)^2

(1 + 0.05) * (1 + 1 year rate 1 year from now + liquidity premium) = (1 + 0.08)^2

(1 + 1 year rate 1 year from now + 0.50%) = 1.1664 / 1.05

(1 + 1 year rate 1 year from now) = 1.1109 - 1.005

1 year rate 1 year from now = 10.59%

3. Assume Face Value = 1000%

Computation of Coupon Rate:

Face Value = Year 1 coupon / (1 + 1 year yield)] + [(Year 2 coupon + Face Value) / (1 + 2 year yield)^2]

1000 = 1000 * Coupon Rate / 1.10 + [(1000 * Coupon rate + 1000) / (1 + 9)^2]

1000 = 1000 * Coupon Rate / 1.10 + [(1000 * Coupon rate + 1000) / 1.881

1000 * 1.1 * 1.881 = 1881 * Coupon rate + 1100 * Coupon Rate + 1100

206.91 = 2981 * Coupon rate

Coupon Rate to issue at par = 6.94%