## CHAPTER 5

**5.1 Introduction**

I |

f you buy a forward contract on a bond, you will receive the bond at a fixed date in the future and at a price agreed upon now. This price is called the delivery price (or the forward price), and is paid in the future, *not* at the time at which you buy the forward contract. The bond to be delivered is called the deliverable, or the underlying bond (or underlying instrument, or simply the underlying!).

Because no cash is exchanged, the principles of arbitrage require that the delivery price be set so that the present value of the forward interest contract is zero. If this was not the case, then either the buyer or seller would receive something for nothing (i.e., a positive net present value for zero cash outlay). In Chapter 3, topic 3.9, *forward contracts*fc_tsr, we discussed the valuation problem for forward contracts and derived the arbitrage free forward price.

Some of the largest forward/futures markets in the world are futures markets. These include U.S. Treasury futures and Eurodollar futures. Therefore, it is important to understand the difference between forwards and futures. The difference arises from an institutional feature called marking to market, as we now explain.

**Forward versus Futures Contracts**

A futures contract is a standardized forward contract that is traded on various exchanges. Here, standardized refers to the terms of the contract, such as the delivery date, the bonds on which the futures are traded, and the settlement procedure. We will discuss some of these in detail in this chapter.

An important difference between a forward contract and a futures contract is that the futures contract is **marked to market** periodically, usually every day. Marking to market means that the futures position is closed out and a new position is taken.

To understand marking to market, suppose we have a futures contract that is marked daily. Suppose that on Day 0, when the contract is first traded, the futures price is f0. This price must be set so that the futures contract has zero value (in case of a forward, f0 equals the value of the future cash flows from the underlying bond at the delivery date).

On Day 1, suppose interest rates have changed, and the futures price is now f1. If f1 > f0, then the buyer of the contract (the "long’) has made money, and the seller (the "short") has lost money. Then, the short has to pay the long $(f1 - f0). Likewise, if the futures falls in value, then the long has to pay the short.

Marking to market means that there may be a difference in price between forward and futures contracts on the same bond. With the forward contract, there is no marking to market, and so no cash changes hands until the delivery date. With a futures contract, however, there typically will be cash inflows and outflows, and the magnitude of these cash flows will depend on the path of interest rates. For example, if interest rates continually fall (raising the value of the underlying bond and thus the futures contract), then only the long will get cash inflows. If they rise, then the short will continually benefit.

In fact, it turns out that the only difference between forward and futures prices is due to interest rate uncertainty. An argument attributable to Cox, Ingersoll and Ross (1981) shows that if the path of interest rates is known, then forward prices must equal futures prices. We should note that in practice, the effects of marking to markets are generally considered to be small.

Forward Interest Rates from Forward or Futures Prices

Depending on the problem at hand, you will at times want to take the forward prices as given, and determine the prices of either the discount bonds or the interest rates themselves. In this way, you can think of the forward or futures price as determining a forward interest rate.

This leads to the following (equivalent) way to determine forward prices. Suppose you can deposit funds for one period at the

and, for two periods at the

Then, the implied forward rate between Period 1 and 2 is

Consider a forward agreement to deposit $X in Period 1 for one period, and let *i* be the forward interest rate, agreed to today, on this deposit. Thus, the agreement is for you to deposit

$*X*

tomorrow and for the counter party to pay you

in Period 2. Then, it must be the case that

Otherwise, there is an arbitrage opportunity.

For example, suppose:

To exploit the arbitrage opportunity you should borrow money for two periods at the two period spot rate of interest, deposit the money for one period at the one period spot rate of interest, and enter into the forward agreement. In Period 1, you simply take your deposit plus interest and execute the forward contract. In Period 2, you will have more money than you need to pay off your loan. This excess money is the arbitrage opportunity.

The Implied Relationship between Futures Rates and Forward Interest Rates

The term structure of interest rates that we have constructed to date is based upon Treasury strips (or other Treasury securities). From this yield curve, we are able to calculate all the forward interest rates. However, we can also work backward from the *futures *prices. The futures prices let us compute future interest rates, and we can compare these to the forward interest rates in the spot yield curve. If there is a difference between these rates, then we can attribute the difference to marking to market.

To do this requires us to become familiar with the futures markets themselves. We will start with the U.S. Treasury bill futures ("T*-*bill futures"), and contrast the yield curve constructed from Treasury instruments to the yield curve implied from Treasury futures’ prices. Let us first become acquainted with the T*-*Bill Futures market.