Question

At your favorite bond store, you see the following prices: (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, no liquidity premium exists, and the bonds are equally risky. What is the imply one-year rate one years from now? (20 points; use exact formula for all questions)

(2) If there is a liquidity premium of 0.5% for the two-year long rate (i2t), what is the imply one-year rate one years from now? (10 points)

(3) If 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 set to sell the bonds at par? (30 points)

Answer 1

Zero coupon Yield:

Price of Bond = F/(1+r)^t

Where,

r is discount rate

F is par value

t is time to maturity

95.2381 = 100 / (1+r)

r1= 5.00%

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

Price of Bond D as per zero bond yields:

Price of Bond = C/(1+r1) + C/(1+r2)²

Where, C is Coupon payment

rt is spot rate/ zero coupon yield

F is par value

t is time to maturity

1000 = 80 / (1+5%) + 1080/(1+r2)²

r2 = 0.08124 = 8.12%

(1)

Forward one-year rate one years from now

Forward rate (2,1) = {(1+r2)² / (1+r1)} -1

Maturity	Spot Rate/zero coupon yield	Forward Rate(t-1,1)
1	5.00%
2	8.12%	11.34%

One-year rate one years from now = 11.33%

(2)

Liquidity premium = 0.5%

One-year rate one years from now i.e. R

(1+R)²= (1+r1)*(1+r2) + 0.5% = (1.05*1.0812)+ 0.005

R = 6.78%

(3)

If your company plans to issue two-year coupon bonds at par but the current one-year rate suddenly increase to 10% and the two-year long rate becomes 9%.

Price of Bond = C/(1+r1) + C/(1+r2)²

1000 = C/(1+10%) + C/(1+9%)² + 1000/(1+9%)²

C = 90.43

So coupon rate that you need to set to sell the bonds at par = 90.43/1000 = 9.04%