Question

9. (10 points) Suppose that the spot in spot interest rate on a two-year zero-com year Zero-coupon bond is 40 ar zero-coupon bond is 3.0%, the Assume annual compound a. (6 points) Based on the pure expectat is the expected one-year inte approximation from class.) se that the spot interest rate one-year zero-coupon Zero-coupon bond is 4.0 and the spot interest rate on a three al compounding throughout the problem. points) Based on the nume r ations theory of the term structure of interes od one-year interest rate during year three, i.e. EG ). (You should use the cture of interest rates, what b. (4 points) Using the same set of interest rates, what is the price of a three-year, 6%-CO bond with face value of 1000? (Use annual compounding.)

Answer 1

a)

Spot Interest Rate on 1 year ZCB = 3%

Spot Interest Rate on 2 year ZCB = 4%

Using the above information, we can compute 1 year forward rate starting from Year 1 (i.e Expected One Year interest rate during year two) using the following equation :-

(1+Spot Interest Rate on Year 1) * (1 + One Year Forward Rate starting from Year 1) = (1 + Spot Interest Rate on Year 2) ^ 2

(1 + 0.03) * (1 + One Year Forward Rate from Year 1) = (1 + 0.04) ^ 2

1.03 * (1 + One Year Forward Rate from Year 1) = (1.0816)

(1 + One Year Forward Rate from Year 1) = (1.0816) / 1.03 = 1.050097

One Year Forward Rate from Year 1 = (1.050097 - 1) = 0.050097  =5.009%

Similarly, expected One Year Interest Rate during Year 3 will be "One Year Forward Rate starting from Year 2". The relationship can be expressed in the following manner :-

(1+Spot Interest Rate on Year 1) * (1 + One Year Forward Rate starting from Year 1) * (1 + One Year Forward Rate starting from Year 2) = (1 + Spot Interest Rate on Year 3) ^ 3

(1.03) * (1.05009) * (1 + One Year Forward Rate starting from Year 2) = (1 + 0.04) ^ 3

(One Year Forward Rate starting from Year 2) = (1.12486) / (1.0816) -1 = 0.0400

One Year Forward Rate starting from Year 2 = 4%

Hence, expected One Year interest rate during Year 3 = 4%.

b)

We can find the price of 3 year bond, by discounting the cash flows using the Spot rate for Year 1, Year 2 & Year 3 respectively.

Coupon amount for Year 1 = 60

Coupon amount for Year 2 = 60

Cash Flow Year 3 = ( 60 + 1,000) = 1,060

Price = 60 / ( 1.03) + 60 / (1.04) ^ 2 + 1,060 / (1.04) ^ 3

= 1,056.06