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 z1 denote the one period spot rate and let z2 denote the two period spot rate.
We claim that the forward price F0 must satisfy
F0 =
where
100
,
1  1 f1
f is the one period forward rate one period hence. For suppose F0 <
1 1
(*)
100
.
1  1 f1
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
1 1  1 f1  of the one period zero. The cash flows are as follows.
Cash Flows
Now
One Period
Buy Forward
0
- F0 + bond
Sell Short
P2
cover short using bond bought forward
Invest Proceeds
-
P1
1  1 f1
100
1  1 f1
Net
0
100
- F0
1  1 f1
The net now is 0 because P2 =
100
1  z2 
2
=
P1
100
=
, by the definition of
1  z1 1  1 f1  1  1 f1
the spot rates and the forward rate. The net in one period of
100
- F0 is by assumption
1  1 f1
positive and is therefore arbitrage profit.
Similarly, if F0 >
100
, then sell forward and purchase a two period zero coupon
1  1 f1
bond by borrowing (shorting 1 1  1 f1  of the one period bonds)
P1
. The cash flows
1  1 f1
are as follows.
Cash Flows
Now
One Period
Sell Forward
0
F0 - bond
Buy
- P2
deliver bond on forward contract
Short Sell
P1
1  1 f1
-
Net
0
F0 -
Again, the net now is 0 because P2 =
arbitrage profit of F0 -
100
1  1 f1
100
1  z2 
2
100
1  1 f1
=
P1
100
=
. There is
1  z1 1  1 f1  1  1 f1
100
> 0 (by assumption) in one period.
1  1 f1
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 1 f1 for one period, one period from now. You pay F0 in one period and
you receive a one period zero coupon bond that pays F0 1  1 f1  one period from
then.
Selling forward a zero coupon bond is tantamount to contracting now to borrow at
the rate 1 f1 for one period, one period from now. In one period, you receive F0
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 F0 1  1 f1  .
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
F0 =
100
.
1  t f1 
(**)
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 t f1 for one period, t periods from now. You
pay F0 in t periods and you receive a one period zero coupon bond that pays you
F0 1  t f1  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 t f1 for one period, t periods from now. In t
periods, you receive F0 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 F0 1  t f1  .
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.”