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3.1 Introduction

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uppose 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 from a zero coupon bond and the time to maturity is referred to as the term structure of interest rates or sometimes as the Treasury yield curve.   You can see the current US Treasury Yield Curve below:

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. In topic 3.7 you will see that the forward curve embodies expectations about the future spot rates and this drives the shape of the spot curve.  In the above plot of the yield curve the implied forward curve is also plotted.

The shape of the yield curve 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 of interest rates inferred from zero coupon bonds. This is also referred to as the zero curve.  The current zero curve for the US economy is relevant for pricing and this curve is plotted above.

In the next topic, Yield to Maturity, we introduce a measure of the average return from buying a bond.