Question

Suppose a 10-year zero-coupon bond (zero) is trading spot at 5% and a 20-year zero is trading spot at 7%. In lectures (L4.9) we proved that the 10 year forward rate for a 10 year zero must be 0.0904 (annual compounding). All are risk free.

If the rates are not 0.0904 for the forward you can make a free profit by using arbitrage. Suppose you have $0 dollars today but are allowed to sell and buy $100,000 worth of zero coupon bonds (and commit to the forward 10 year zero coupon bond using ​any​ cash you have ​- or need to reborrow​ - after 10 years from your initial trades).
(a) What trades do you execute if the forward rate is 8% - report your profit. (b) What trades do you execute if the forward rate is 10% - report your profit. (c) Comment on why the forward rate must be 9.04% in light of your results.

Answer 1

a) if the forward rate is 8%, it is cheaper than the theoretical rate of 9.04%

Hence the arbitrage trades are as follows

1. Buy $100000 Face value of 20 year zero bonds yielding 7% today for $100000/1.07^20 = $25841.90

2.sell 10 year zero bonds yielding 5% today for $25841.90 for bonds having maturity value of 25841.90*1.05^10 $42093.73

3. Borrow $42093.73 forward after 10 years for a further period of 10 years at 8%

4. After 10 years , pay the maturity value of 10 years bonds by borrowing the maturity amount

5. After 20 years , get $100000 from 20 year bonds, pay $42093.73*1.08^10 = $90877.21 and realise an arbitrage profit of = $100000- $90877.21 = $9122.79

b) If forward rate is 10%,

it is costlier than the theoretical rate of 9.04%

Hence the arbitrage trades are as follows

1. Sell $100000 Face value of 20 year zero bonds yielding 7% today for $100000/1.07^20 = $25841.90

2.Buy 10 year zero bonds yielding 5% today for $25841.90 for bonds having maturity value of 25841.90*1.05^10 $42093.73

3. Invest $42093.73 forward after 10 years for a further period of 10 years at 10%

4. After 10 years , get the maturity value of 10 years bonds = $42093.73 and invest for next 10 years using forward

5. After 20 years , get $42093.73*1.10^10 = $109180.30, pay maturity value of 20 year bonds =$100000 and realise an arbitrage profit of = $109180.30- $100000 = $9180.30

c) Thus, to prevent arbitrage, the forward rate has to be exactly 9.04% , else there will be arbitrage opportunities If forward rate is less as in case a) then there will be demand for forward loans which will increase forward rates and if forward rates are higher as in case b) then there will be demand of forward investments which will again bring the equilibrium forward rate to 9.04%