# The Term Structure of Interest Rates

The term structure of interest rates is one of the most important and central topics in the study of economics and finance.

First, let us define the term structure. The term structure is the relationship between the interest rates and the maturities of bonds/loans. This is the standard definition but one that requires some qualification. Strictly speaking the concept of the term structure should apply only to zero coupon bonds. This is not often the case in practice as we often see plots of the yields on coupon bonds against their maturities. Most of the principles applying to the term structure do not apply in a straight forward manner to coupon bonds. Personally, I use the expression “”term structure” only to apply to zero coupon bonds and when I use the expression “yield curve” I make no restrictions on the type of bond.

Perhaps the reason why the term structure and yield curve are often confused is that until about ten years ago, the U.S. government did not offer zero coupon bonds of maturities greater than one year. Thus, it was not possible to construct more than a one-year term structure except by using analytical or statistical methods. Consequently, a yield curve was often used as a proxy. It is, however, a biased proxy because the effect of coupons introduces a new factor that can affect the rates.

In addition most of the theories of the term structure should apply only to default-free bonds. There is, of course, a term structure of default risky bonds, but any attempt to explain the relationship among the interest rates on such bonds of different maturities must take into account default risk. It is quite difficult to incorporate default risk and term structure effects in one model so typically we handle them separately. The relationship between an option pricing approach that handles default risk.

Let P(0,1) be the price of a one-year zero coupon bond. Let P(0,2) be the price of a two-year zero coupon bond, etc. We can define the term structure in terms of prices or rates. The price of the one-year bond P(0,1) = 1/(1+r(0,1). The price of the two-year bond P(0,2) = 1/(1+r(0,2))^2.

Here, r(0,1), r(0,2),…., r(0,T) make up the term structure going out T periods. These are called the spot rates because they represent transactions such as loans or bonds undertaken on the spot.

Closely allied to the concept of a spot rate is that of a forward rate. Consider a market in which one can contract today (time 0) to borrow or lend at a future date at a rate agreed-upon today. This rate is called the forward rate. Let f(0,1,2) be the rate that can be locked in today for a bond that would be issued in one year and matures in 2 years. This bond would have a one-year maturity and its price is specified as f(0,1,2) Now consider the following transactions: Purchase a one year bond at the price p(0,1) and enter into a forward contract to buy a one-year bond in one year at the price F(0,1,2). At the end of two years you will have [1/P(0,1)][1/F(0,1,2)] for every dollar invested. This must equal the return per dollar from buying a two-year bond today and holding it two years, which is 1/P(0,2); otherwise one transaction would be more profitable than the other and, therefore, the less profitable transaction could be reversed and used to finance the more profitable transaction.

In addition to the forward price F(0,1,2), there are other forward prices such as F(0,1,3), which is the forward price implied by the term structure today for a bond issued at time 1 and paid back at time 3; thus, it is a two-year bond. Likewise, we could have other forward prices like F(0,1,4),…, F(0,1,T). In addition, we have forward prices like F(0,2,3), which is the forward price implied by the term structure today for a bond issued at time 2 and paid back in time 3. We also have F(0,2,4), etc as well as F(0,3,5), etc. A special case of the set of forward prices is F(0,0,1), f(0,0,2), etc which is nothing more than the set of spot prices. In other words, the forward prices for transactions set to begin today are simply the spot prices.

Using the same arbitrage arguments as we did above, any forward price F(0,i,j) is given as P(0,j)/P(0,i). It is important to recognize that this statement, which defines the entire term structure in terms of forward prices, holds by definition. It does not require any assumption about investor utility. It is not conditional on investors being risk neutral and by no means does it make any statement about what investors expect in the future. Two investors with wildly divergent expectations would agree on the relationship between the spot rates and the forward rates. It is also important to note that specifying the spot rates or prices completely defines the structure of forward rates or prices but not vice verse. In other words, with F(0,1,2) = P(0,2)/P(0,1) there are many combinations of P(0,2) and p(0,1) that could lead to a given forward rate.

It is important to note that the product of successive forward prices will give us a spot price. For example, consider the product F(0,1,2)F(0,2,3). We know from our arbitrage examples above that F(0,1,2)=P(0,2)/P(0,1) and also that F(0,2,3)=F(0,3)/P(0,2). Using these two relationships,w e have F(0,2,3)=P(0,3)/[P(0,1)F(0,1,2)], which can be rewritten as P(0,1)F(0,1,2)F(0,2,3) = P(0,3). Thus, the price today of a three three-period bond is the product of the price today of a one-period bond and the forward prices, as implied by the term structure today, of a one-period bond starting at time 1 and another one-period bond starting at time 2.

Again, I emphasize that these relationships have nothing to do with expectations about the future rates or prices.

In the next article we will attempt to look at the shape of the term structure.

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