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
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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

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