Question

The price of a 1-year zero coupon bond is 0.97. The price of a 2-year zero coupon bond is 0.93. The price of a 3-year zero coupon bond is 0.85. The price of a 4-year zero coupon bond is 0.78. You are a borrower who anticipates needing to borrow $100,000 for one year at the end of year 3 and would like to guarantee the rate on your upcoming loan (i.e., after 3 years you will need a $100,000 loan which will last for one year).  Which is closest to the interest rate you will be able to guarantee if there are no transactions costs and you get a fair deal in the interest rate forward market?

a.    1%

b.    2.5%

c.     4%

d.    5.5%

e.    7%

Answer 1

Price of 3 year zero coupon bond = $1,000*0.85 = $850

3 years interest rate from today = ($1000/$850)^1/3 – 1 = 1.05567 – 1 = 0.05567

Price of 4 year coupon bond = $1,000 * 0.78 = $780

4 year interest rate from today = ($1000/$780)^1/4 - 1 = 1.06409 – 1 = 0.06409

4 year interest rate from today = 3 year interest rate from today * 1 year interest rate at 3 years from now

Let 1 year interest rate at 3 years from now be r

1.06409⁴ = 1.05567³ * (1+r)

1.282051 = 1.176471*(1+r)

1+r = 1.282051/1.176471

r = 1.08974 – 1 = 0.08974 = 8.97%