DUKE UNIVERSITY The Fuqua School of Business. Financial Management Spring 1989 TERM STRUCTURE OF INTEREST RATES*

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1 DUKE UNIVERSITY The Fuqua School of Business Business 350 Smith/Whaley Financial Management Spring 989 TERM STRUCTURE OF INTEREST RATES* The yield curve refers to the relation between bonds expected yield to maturity and their terms to maturity, where the bonds have the same default risk, no imbedded options such as callability, and no differential tax privileges. The Appendix to these notes contains sample yield curves for U.S. Treasury securities for the past ten years. Note that the use of Treasury securities assures a homogeneous degree of default risk (i.e., none) and that callable issues are clearly distinguished from fixed maturity issues. The expected yield to maturity for each Treasury security is computed using the bond valuation formula n Ct Fn B = +. () n t= t ( + y) ( + y) The bid price of the bond ( B ) is substituted on the left-hand-side; the coupons ( C t ), face value ( F t ), and term to maturity ( n ) are substituted on the right-hand-side; and the expected yield to maturity ( y ) is computed. Note that the yield to maturity is intended to be a simple descriptive statistic for the expected rate of return on the bond or note. The yield to maturity of a coupon-bearing bond can be a misleading reflection of its expected rate of return, however. To see this, consider the following situation. Suppose there exist two discount Treasury bonds, one with a one-year maturity and a price of 90.9, and the other with a two-year maturity and a price of 8.6. Assuming each of these bonds is redeemed at a par value of 00 at their respective maturities, the expected yield on the one-year bond is 00/90.9- = 0.00 percent * Copyright 989 by Robert E. Whaley. All rights reserved.

2 and the expected yield on the two-year bond is ( 00 / 8.6) 2 =.00 percent. Note that if the Treasury decided to issue other one-and two-year discount bonds they would have to have the same prices as the existing issues, otherwise costless arbitrage profits could be earned. Now suppose that the Treasury issues a two- year, twelve percent coupon bond. What will be its price? Well, the new twelve percent issue will pay 2 at the end of the one-year, and 2 at the end of the two years. The discount rate to apply to a oneyear bond is 0 percent and the discount rate to apply to a two-year bond precent, so the price of this coupon- bearing Treasury bond should therefore be 2 2 B = + = ( ) 2 But, what will be the bond s yield to maturity? The yield is computed by solving The solution is 0.94 percent = +. + y + ( y) 2 So, what is happening? The two year coupon bond has an expected yield to maturity of 0.94 percent while the two-year discount bond has an expected yield to maturity of percent. Are arbitrage profits possible? No. The answer to the dilemma is that the two-year coupon bond has an actual term to maturity of less than two years. If you buy the coupon bond you will receive 2 at the end of one year and 2 at the end of two years, so in essence you have a portfolio of two discount bonds. The first discount bond is worth 2 B d = = 0.9,.0 and the second discount bond is worth 2 B d = =

3 The average term to maturity of the bond portfolio (or the coupon-bearing bond) is therefore Average term to maturity = + 2 = years. We should not use the same interest rate to discount all cash flows of the bond, as bond valuation formula () implies. The value of a coupon-bearing bond, when viewed as a portfolio of discount bonds, may be written n Ct Fn B = +, (2) n t ( + r ) ( + r ) t= t n where r t is the spot rate of interest on a t period discount bond. Is it fair to plot the yield to maturity as a function of term to maturity? No. The term to maturity of the bond overstates its actual life. The actual life of the bond should be the average life of the individual constituent bonds or Ct Fn t n n ( rt) ( rn) Average term to maturity = + t + n. t= B + (3) B This average term to maturity is referred to as the bond s duration. The relation between expected return and term to maturity should be expressed by either (a) plotting the yield to maturity as a function of average term to maturity, or (b) plotting the spot rate of interest as a function of term to maturity; but not yield to maturity as a function of term to maturity. 3

4 Spot Rates and Forward Rates Spot rate of interest are rates observables today. The spot rate on a three-month, default-free security is the rate of return promised three-month T-bill. The notation that we use to describe the spot rate on a t period bond is r t. Forward rates of interest are interest rates on loans in the future and are implied from the current term structure of spot rates. A forward rate on a t period loan in period n, denoted n f t, is computed using ( f ) t ( + rn+ t) ( + r ) n+ t + =. n t n n (4) The forward rate on a one-year today, for example, is computed as ( + f ) = ( + r ), 0 and the forward rate on a one-year loan in three years is ( + r3+ ) ( + r ) 3+ ( + f ) = Example: Suppose the one-year spot rate of interest is 0 percent and two-year spot rate is percent. What is the implied one-year forward rate in one year? Substituting the parameters of the example into equation (4), we find that or ( + r2 ) ( + r ) ( ) ( ) f = =.0 f = 2.0 percent 4

5 Example 2 What is the implied forward rate on a four-year loan in six years, if the current sixyear spot rate of interest is 2 percent and the current ten-year spot rate of interest is 5 percent? Substituting the parameters of the example into equation (4), we find that or ( + r0 ) ( + r ) ( ) ( ) f ( + ) = = 6 f 4 = 9.65 percent Theories of the Term Structure of Interest Rates The term structure of interest rates refers to the relation between the spot rate of interest and term to maturity. Theories of the term structure of interest rates attempt to explain why the relation between the spot rates of interest and the terms to maturity of various bond issues takes on particular shapes. In these notes, we discuss two of the theories: (a) the unbiased expectations theory, and (b) the liquidity preference theory. Unbiased Expectations Theory: The unbiased expectations theory of the term structure posits that the expected spot rate on a t period loan in period n is equal to the current forward rate on a t period loan in period n implied by the term structure of spot rates, that is, E( r ) = f, n t n t where E ( ) is the expectations operator. This result is driven by the assumptions that investors, at the margin, are riskneutral and hence make their portfolio decisions by maximizing expected rate of return. For example, suppose that you know that the one-year spot rate of interest is 0 precent and that the two-year spot 5

6 rate of interest is percent. (Recall that these were the values used in Example.) Suppose further that you expect the one-year interest rate in one-year, E( r ), to be.50 percent. If you have a two-year investment horizon and are faced with either (a) buying a two-year bond now, or (b) buying a one-year bond now and buying a second one-year bond in one year, which would you do? Well, if you were risk neutral, you would prefer the alternative (a) to alternative (b) because $00 invested in (a) would provide ( r ) ( ) $00 + = $00. = $23.2 at the end of year 2, while $00 invested in (b) would be expected to provide ( r ) E( r ) ( )( ) $00 + [ + ] = $ = $ In fact, what you have is a costless arbitrage opportunity in which you would buy (a) and sell (b) earning an arbitrage profit of $.66. Under the unbiased expectations theory, a costless arbitrage opportunity arises whenever the expected future spot rate differs from the implied forward rate. Assuming no costless arbitrage opportunities exist in the marketplace, the expected future spot rate equals the implied forward rate, that is, ( ) =. E r f n t n t If the expected on-year spot rate of interest in one year was 2.0 percent in the above example, no costless arbitrage opportunities could be earned because a $00 investment under alternative (b) would be expected to provide at the end of year 2. ( r )( f ) ( ) $ = 00.0 (.20) = $23.2 The unbiased expectations theory is not really a theory in the sense that it is does not explain why the term structure of spot rates takes on particular forms. The theory simply implies that the term structure contains certain predictions about where spot interest rates will move. Simply stated, (a) if the term structure is upward sloping, interest rates are expected to rise; and, (b) if the term structure is downward sloping, interest rates are expected to fall. Liquidity Preference Theory: The liquidity preference theory of the term structure argues that lenders of funds (i.e. buyers of bonds) prefer to hold short maturity instruments, while borrowers of funds (i..e. sellers of bonds) prefer to issue long maturity instruments. If such is the case, the only way to induce lenders of funds to buy longer maturity instruments is to offer them higher and higher rates of return as the maturities get longer. If spot rates of interest increase with term to maturity, implied forward rates also rise the further and

7 further they are in the future. The liquidity preference theory, therefore is best summarized by the fact that E( r ) < f n t n t where nft E( nr t) is the liquidity premium. The liquidity preference theory implies that, in general, the term structure is upward sloping. 6

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