Suppose you buy a bond at its arbitrage-free value. What return do you get? In Chapter 2, we maintained the assumption that the spot rate of interest is constant across periods. Therefore, for this case the answer is easy. Your return equals the spot rate of interest.

Generally, however, the longer the maturity, the higher the spot rate. The relationship between spot interest rates and the time to maturity is referred to as the term structure of interest rates or sometimes as the Treasury yield curve. In topic 3.2, Yield to Maturity, we develop the theory that lets you compute the return from buying a bond for the case where spot rates vary over time to maturity.

You have already seen that the arbitrage-free value of a fixed-income security is the sum of the present value using the spot interest rate that applies to each cash flow component. That is, different yield curve shapes determine different arbitrage-free values. The return from buying a bond at its arbitrage-free value is a measure of the "average" return from each of these components. This return is referred to as the security's yield to maturity.

The shape of the yield curve is so important that it is reported daily in the financial press. To illustrate what a yield curve looks like, Figure 3.1 gives an example of a yield curve for the U.S. as of October 6, 1994.

The curve shows you the spot interest rate for Treasury securities. You can see that the securities mature in February and then every three months thereafter. The last reported maturity date is August 2023.

Treasury securities are generally considered to be free of default risk, so they offer some of the lowest returns among fixed-income securities. Similar yield curves also exist for other types of bonds, such as corporate bonds. Corporate bonds are debt instruments issued by companies; their yields are typically higher than yields on Treasuries to compensate holders for added risk. Corporate bond yields are directly affected by a company’s credit rating.

A typical observed shape for the yield curve has been upward sloping. This means that the spot interest rates for longer-dated maturities exceed the spot interest rates for shorter-dated maturities. That is, if rt denotes the spot interest rate on a t year Treasury security, then an upward sloping curve implies that this spot rate is bigger than any spot rate with shorter maturity (i.e., rt > rt if t > t).

Several theories attempt to explain the shape of the yield curve. These explanations are discussed in the topic Theories of the Term Structure of Interest Rates. Although the yield curve shape can change from upward sloping to downward sloping, flat, and irregular, it is not unusual to see the upward sloping pattern recur over time. The shape of the yield curve is important because it determines the price of every interest rate-dependent security.

Initially, we will take the shape of the term structure as given. Our goal is to first develop the theory that lets you construct the arbitrage-free bond prices and returns relative to an arbitrarily shaped term structure. In the next topic, Yield to Maturity, we introduce a measure of the average return from buying a bond.

(C) Copyright 1999, OS Financial Trading System