Suppose you are given the following information about the? default-free, coupon-

Question

Suppose you are given the following information about the? default-free, coupon-paying yield? curve:

Maturity? (years)

1

2

3

4

Coupon rate? (annual payment)

0.00?%

9.00?%

5.00?%

15.00?%

YTM

1.991?%

4.346?%

6.229?%

6.759?%

a. Use arbitrage to determine the yield to maturity of a? two-year zero-coupon bond.

b. What is the? zero-coupon yield curve for years 1 through? 4?

Note?:

Assume annual compounding.

Maturity? (years)

1

2

3

4

Coupon rate? (annual payment)

0.00?%

9.00?%

5.00?%

15.00?%

YTM

1.991?%

4.346?%

6.229?%

6.759?%

Explanation / Answer

Maturity(Yrs) 1 2 3 4 Coupon Rate(Annual Payment) 0.00% 9.00% 5.00% 15.00% YTM 1.991% 4.346% 6.229% 6.759% The Two year coupon bond Assuming the face value is $1000 The coupon rate for 2 yr bond is 9% Price of two year coupon bond = 1000*0.09/1.04346+(1000+1000*0.09)/1.04346^2 86.25151 + 1001.094 1087.346 Assuming face value of $90 Price of one year bond = 90/1.01991 88.24308 Using the law of one price Price of two year zero bond = Price of 2 year coupon bond - Price of 1 yr coupon bond 1087.346-88.24308 999.10292 The yield to maturity of the zero coupon bond is (1090/999.10292)^1/2 - 1 1.044499 -1 0.044499 4.45% b) The yield to maturity for 1 yr zero coupon bond is 1.991%, for 2 yr zero coupon is 4.45%, now we would calculate yield for 3 yrs and 4 yrs bond 1 2 3 4 r coupon bond (Face value = $1000) 50 50 1050 e-year zero (Face value = $ 60) -50 2-year zero (Face value = $ 60) -50 Ar zero (Face Value = $1050) - - 1050 Price of 3 yrs coupon bond is 50/1.06229 + 50/1.06229^2 + 1050/1.06229^3 (50/1.06229) + (50/(1.06229^2)) + (1050/(1.06229^3)) 967.2874 Using the law of one price rule Price (3 yr zero bond) = Price of 3 yrs coupon bond - Price of one year zero - price of two year zero 967.2874- 50/1.0991-50/1.04346^2 967.2874-45.49177-45.92175 875.87388 Solving for the YTM = (1050/875.87388)^(1/3) -1 = 1.062305-1 0.062305 6.23% Calculation of yield for 4 year bond 1 2 3 4 Coupon bond(face value = $1000) 150 150 150 1150 1 year zero bond(Face value = $150) -150 2 year zero bond(Face value = $150) -150 3 year zero bond(Face value = $150) -150 r zero bond - - - 1150 Price of 4 years coupon bond = 150/1.06759 + 150/(1.06759^2) + 150/(1.06759^3)+1150/(1.06759^4) 1280.665519 Using the law of one price rule Price of 4 yr zero bond = Price of 4 yrs coupon bond - Price of one year zero - price of two year zero - price of three year zero 874.397652 Solving for YTM for 4 yrs (1120/874.397652)^(1/4)-1 0.063842313 6.38% Year Yield to maturity 1 1.99% 2 4.45% 3 6.23% 4 6.38%

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