Zero Coupon Bond Forward Contract

FI 4000

Zero Coupon Bond Forward Contract

Are Forward Rates Good For Anything?

Suppose you can contract today for delivery of a one period zero coupon bond one period hence. This is like the T-Bill future. For simplicity we express terms as a percent of par value of the zero coupon bonds.

Let denote the one period spot rate and let denote the two period spot rate. We claim that the forward price must satisfy

= ,(*)

where is the one period forward rate one period hence. For suppose . Then buy the bond forward and short a two period zero coupon bond now. Invest the proceeds of the short sale in one period zero coupon bonds. Explicitly, purchase of the one period zero. The cash flows are as follows.

Cash FlowsNowOne Period

Buy Forward0- + bond

Sell Shortcover short using bond bought forward Invest Proceeds -

Net0 -

The net now is 0 because = = = , by the definition of the spot rates and the forward rate. The net in one period of - is by assumption positive and is therefore arbitrage profit.

Similarly, if , then sell forward and purchase a two period zero coupon bond by borrowing (shorting of the one period bonds) . The cash flows are as follows.

Cash FlowsNowOne Period

Sell Forward0 - bond

Buy -deliver bond on forward contract

Short Sell-

Net0 -

Again, the net now is 0 because = = = . There is arbitrage profit of - > 0 (by assumption) in one period.

So, in the absence of arbitrage, (*) must hold. The implications of (*) are that

Buying forward a zero coupon bond is tantamount to contracting now to lend at the rate for one period, one period from now. You pay in one period and you receive a one period zero coupon bond that pays one period from then.

Selling forward a zero coupon bond is tantamount to contracting now to borrow at the rate for one period, one period from now. In one period, you receive and you must borrow a one period zero coupon bond to deliver on the forward sale. In one period, you owe the lender of the borrowed bond .

More generally, consider a forward contract for a one period zero coupon bond to be delivered in t periods. Then by the same argument used above, but with spot positions in a zero coupon bond with t + 1 periods to maturity, to avoid arbitrage, the forward price must be given by the equation

= .(**)

So, in the absence of arbitrage, (**) must hold. The implications of (**) are that

Buying forward a one period zero coupon bond in t periods is tantamount to contracting now to lend at the rate for one period, t periods from now. You pay in t periods and you receive a one period zero coupon bond that pays you one period from then.

Selling forward a one period zero coupon bond in t periods is tantamount to contracting now to borrow at the rate for one period, t periods from now. In t periods, you receive and you must borrow a one period zero coupon bond to deliver on the forward sale. In one period, you owe the lender of the borrowed bond .

You can actually create these forward positions synthetically using put and call options on Treasury bills. See the handout notes “Synthetic Treasury Bill Forward Contracts.”