Question

2. Suppose that zero interest rates with quarterly compounding are as follows: Maturity (months) Rate(%) 3 8.0 6 8.4 9 8.8 12 9.0 i. Calculate the forward interest rates for the second, third, and fourth quarters. ii. You should have found that the forward rate over the fourth quarter is 9.6006%. Carefully explain the available arbitrage strategy and calculate your profit) if you found a bank willing to lend to you forward at 9.0000% over the fourth quarter.

Answer 1

i) Interest rate for 1st quarter = 8%/4 = 2% per quarter

Interest rate for 1st two quarters = 8.4%/4 = 2.1% per quarter

Interest rate for 1st three quarters = 8.8%/4 = 2.2% per quarter

Interest rate for 1st four quarters = 9%/4 = 2.25% per quarter

Forward interest rate f2 (from 1st quarter to 2nd quarter) per quarter = (1+0.021)^2/(1+0.02) -1 =0.022001

So, forward interest rate f2 (from 1st quarter to 2nd quarter) per year  =0.022001 * 4 = 0.088004 or 8.8004%

Forward interest rate f3 (from 2nd quarter to 3rd quarter) per quarter = (1+0.022)^3/(1+0.021)^2-1 =0.024003

So, forward interest rate f3 (from 2nd quarter to 3rd quarter) per year = 0.024003 * 4 = 0.096012 or 9.6012%

Forward interest rate f4 (from 3rd quarter to 4th quarter) per quarter = (1+0.0225)^4/(1+0.022)^3-1 =0.024001

So, forward interest rate f4 (from 3rd quarter to 4th quarter) per year = 0.024001 * 4 = 0.096006 or 9.6006%

ii) If a bank is willing to lend at 9.0000% over the 4th quarter, there is an arbitrage opportunity

Arbitrage Strategy has the following steps

1) Today, Use the bank's offer and borrow an amount of $1 forward at 9% for the 4th quarter. Get the amount of $1 at the start of the 4th quarter (end of 3 quarters). At the end of the 4th quarter, the amount to be repaid is

= $1 * (1+0.09/4) = $1.0225

2) Today, Sell 3 quarter zero bonds of Face Value $1 at 1/1.022^3 =$0.9368 and simultaneously buy 4 year zero for bond with the amount $0.9368 (FV =0.9368*1.0225^4 = $1.024001)

3) After three quarters, get the loan amount of $ 1 from bank and repay the Face value of 3 quarter zero bond

4) After 4 quarters, get the Face value of 4 year zero bond purchased = $1.024001 and repay the loan amount of $1.0225 and pocket the remaining $ 0.001501 as arbitrage profit