Here is an explanation - abridged from the original - which is based on Bruce Tuckman. In fact, even though the question may appear simple and trivial to many a full blown mathematical treatment for futures-forward difference incorporating the term structure of the rates is rather complex and challenging.

Also, we will limit our discussion to bond and note futures and leave out the rates futures (such as Eurodollar futures).

**Explanation in English : **

Let's say that a forward contract - which is an over the counter contract - and a futures contract - which is a market settled contract - is priced the same. Daily changes in the value of a forward contract generate will not generate any cash flows, because the forward contract will be settled at maturity. On the other hand, by construction, a futures contract will generate mark-to-market (MTM) payments at the end of the day and hence there is a daily cash flow associated with them. If an investor holds a futures contract and he makes a mark-to-market gain then he can reinvest that gain; on the other hand if he makes mark-to-market losses on his futures contract, he needs to finance those losses. The key insight to this whole process is that on an average the MTM gains and losses do not cancel out. This makes holding of a futures contract less attractive for an investor, as he can easily switch to a forward contract (assuming there is a seller willing to sell him a forward contract) and avoid daily MTM fluctuations in his cash flows.

Further, there is another problematic issue. If you are holding say, bond or T note futures contract. As bond prices fall, short term rates would rise. And hence a long note futures contract will experience a MTM loss and this loss has to be financed at a higher interest rate. On the other hand if the bond prices rise, thereby causing the note futures to rise, the short term rates will fall and hence any MTM gains can only be reinvested at lower interest rates.

Thus if a forward contract and a futures contract are priced the same then an investor will not have any incentive to hold a long futures contract. Hence, to make them fairly priced, the futures should be priced lower than the forward contracts.

**Explanation using Math : **

The forward-futures difference is adequately explained by the covariance term.

Take two random variables, A and B. Then the covariance between them is:

................(1)

Say, the price of a bond today is and the forward price for delivery on date is . By definition, we have

And hence we have the price of a forward contract as:

.........................(2)

Now assign the value of the random variables in the covariance equation as follows and then rearrange the covariance equation (1).

We get the following relationship:

...............(3)

This shows that covariance is equal to the expected discounted value minus the discounted expected value. Further, the expected value of is just the futures contract, or in other words,

Substituting the value of the futures contract and that of the forward contract, as in equation (2), in the above equation and with a bit of algebra we get the following relationship:

..........(4)

Within the fixed income context the covariance term in the above equation (4) is almost always positive (we leave the reasoning to the reader). Thus the futures price is always less than the forward price and the difference is the covariance term.

Actually, the covariance term will consistently and accurately explain the rationale that we presented in the section above where we explained why the two contracts should be fairly priced in terms of market mechanisms.

*Reference: **For an excellent treatment of the above subject we refer the reader to Bruce Tuckman's excellent book "Fixed Income Securities" (2 nd Edition, Wiley Finance). *

Any comments and queries can
be sent through our
web-based form.

__
More on Quantitative Finance >>__

back to top