193 CHAPTER 10 The Short-Rate Process and the Shape of the Term Structure G iven the initial term structure and assumptions about the true interest rate process for the short-term rate, Chapter 9 showed how to derive the risk-neutral process used to determine arbitrage prices for all fixed in- come securities. Models that follow this approach and take the initial term structure as given are called arbitrage-free models of the term structure. Another approach, to be described in this and subsequent chapters, is to derive the risk-neutral process from assumptions about the true interest rate process and about the risk premium demanded by the market for bear- ing interest rate risk. Models that follow this approach do not necessarily match the initial term structure and are called equilibrium models. The benefits and weaknesses of each class of models are discussed throughout Chapters 11 to 13. This chapter describes how assumptions about the true interest rate process and about the risk premium determine the level and shape of the term structure. For equilibrium models an understanding of the relation- ships between the model assumptions and the shape of the term structure is important in order to make reasonable assumptions in the first place. For arbitrage-free models an understanding of these relationships reveals the assumptions implied by the market through the observed term structure. Many economists might find this chapter remarkably narrow. An economist asked about the shape of the term structure would undoubtedly make reference to macroeconomic factors such as the marginal productiv- ity of capital, the propensity to save, and expected inflation. The more modest goal of this chapter is to connect the dynamics of the short-term rate of interest and the risk premium with the shape of the term structure. While this goal does fall short of answers that an economist might provide, it is more ambitious than the derivation of arbitrage restrictions on bond and derivative prices given the prices of a set of underlying bonds. The first sections of this chapter present simple examples to illustrate the roles of interest rate expectations, volatility and convexity, and risk premium in the determination of the term structure. A more general, math- ematical description of these effects follows. Finally, an application illus- trates the concepts and describes the magnitudes of the various effects in the context of the U.S. Treasury market. EXPECTATIONS The word expectations implies uncertainty. Investors might expect the one- year rate to be 10%, but know there is a good chance it will turn out to be 8% or 12%. For the purposes of this section alone the text assumes away uncertainty so that the statement that investors expect or forecast a rate of 10% means that investors assume that the rate will be 10%. The sections following this one reintroduce uncertainty. To highlight the role of interest rate forecasts in determining the shape of the term structure, consider the following simple example. The one-year interest rate is currently 10%, and all investors forecast that the one-year interest rate next year and the year after will also be 10%. In that case, in- vestors will discount cash flows using forward rates of 10%. In particular, the price of one-, two-, and three-year zero coupon bonds per dollar face value (using annual compounding) will be (10.1) (10.2) (10.3) From inspection of equations (10.1) through (10.3), the term structure of spot rates in this example is flat at 10%. Very simply, investors are willing to lock in 10% for two or three years because they assume that the one- year rate will always be 10%. P() . 3 1 110 110 110 1 110 3 = ()()() = P() . 2 1 110 110 1 110 2 = ()() = P() . 1 1 110 = 194 THE SHORT-RATE PROCESS AND THE SHAPE OF THE TERM STRUCTURE Now assume that the one-year rate is still 10%, but that all investors forecast the one-year rate next year to be 12% and the one-year rate in two years to be 14%. In that case, the one-year spot rate is still 10%. The two- year spot rate, r ˆ (2), is such that (10.4) Solving, r ˆ (2)=10.995%. Similarly, the three-year spot rate, r ˆ (3), is such that (10.5) Solving, r ˆ (3)=11.998%. Hence, the evolution of the one-year rate from 10% to 12% to 14% generates an upward-sloping term structure of spot rates: 10%, 10.995%, and 11.988%. In this case investors require rates above 10% when locking up their money for two or three years because they assume one- year rates will be higher than 10%. No investor, for example, would buy a two-year zero at a yield of 10% when it is possible to buy a one-year zero at 10% and, when it matures, buy another one-year zero at 12%. Finally, assume that the one-year rate is 10%, but that investors fore- cast it to fall to 8% in one year and to 6% in two years. In that case, it is easy to show that the term structure of spot rates will be downward-slop- ing. In particular, r ˆ (1)=10%, r ˆ (2)=8.995%, and r ˆ (3)=7.988%. These simple examples reveal that expectations can cause the term structure to take on any of a myriad of shapes. Over short horizons, one can imagine that the financial community would have specific views about the future of the short-term rate. The term structure in the U.S. Treasury market on February 15, 2001, analyzed later in this chapter, implies that the short-term rate would fall for about two years and then rise again. 1 At the time this was known as the “V-shaped” recovery. At first, the economy would continue to weaken and the Federal Reserve would continue to re- duce the federal funds target rate in an attempt to spur growth. Then the P r () (. )(. )(. ) ( ˆ ()) 3 1 110 112 114 1 13 3 == + P r () ( . )( . ) ( ˆ ()) 2 1 110 112 1 12 2 == + Expectations 195 1 Strangely enough, Eurodollar futures at the same time implied that rates would fall for less than one year before rising again. (Chapter 17 will describe Eurodol- lar futures.) economy would rebound sharply and the Federal Reserve would be forced to increase the target rate to keep inflation in check. 2 Over long horizons the path of expectations cannot be as specific as those mentioned in the previous paragraph. For example, it would be diffi- cult to defend the position that the one-year rate 29 years from now will be substantially different from the one-year rate 30 years from now. On the other hand, one might make an argument that the long-run expectation of the short-term rate is, for example, 5% (2.50% due to the long-run real rate of interest and 2.50% due to long-run inflation). Hence, forecasts can be very useful in describing the level and shape of the term structure over short time horizons and the level of rates over very long horizons. This conclusion has important implications for extracting expectations from observed inter- est rates (see the application at the end of this chapter), for curve fitting tech- niques not based on term structure models (see Chapter 4), and for the use of arbitrage-free models of the term structure (see Chapters 11 to 13). VOLATILITY AND CONVEXITY This section drops the assumption that investors believe their forecasts are realized and assumes instead that investors understand the volatility around their expectations. To isolate the implications of volatility on the shape of the term structure, this section assumes that investors are risk neutral so that they price securities by expected discounted value. The next section drops this assumption. Assume that the following tree gives the true process for the one-year rate: 14 12 10 10 8 6 1 2 1 2 1 2 % % %% % % ← ← ← ← ← ← 1 2 1 2 1 2 196 THE SHORT-RATE PROCESS AND THE SHAPE OF THE TERM STRUCTURE 2 Those who thought the economy would take some time to recover predicted a “U- shaped” recovery. Those even more pessimistic expected an “L-shaped” recovery. Note that the expected interest rate on date 1 is .5×8%+.5×12% or 10% and that the expected rate on date 2 is .25×14%+.5×10%+.25×6% or 10%. In the previous section, with no volatility around expectations, flat expectations of 10% imply a flat term structure of spot rates. That is not the case in the presence of volatility. The price of a one-year zero is, by definition, 1 / 1.10 or .909091, imply- ing a one-year spot rate of 10%. Under the assumption of risk neutrality, the price of a two-year zero may be calculated by discounting the terminal cash flow using the preceding interest rate tree: Hence, the two-year spot rate is such that .82672=1/(1+r ˆ (2)) 2 , implying that r ˆ (2)=9.982%. Even though the one-year rate is 10% and the expected one-year rate in one year is 10%, the two-year spot rate is 9.982%. The 1.8-basis point difference between the spot rate that would obtain in the absence of uncer- tainty, 10%, and the spot rate in the presence of volatility, 9.982%, is the effect of convexity on that spot rate. This convexity effect arises from the mathematical fact, a special case of Jensen’s Inequality, that (10.6) Figure 10.1 graphically illustrates this equation. The figure assumes that there are two possible values for r, r Low and r High . The curve gives values of 1/(1+r) for the various values of r. The midpoint of the straight line connect- ing 1/(1+r Low ) to 1/(1+r High ) equals the average of those two values. Under the assumption that the two rates occur with equal probability, this average equals the point labeled E[1/(1+r)] in the figure. Under the same assumption, the point on the abscissa labeled E[1+r] equals the expected value of 1+r and the corresponding point on the curve equals 1/E[1+r]. Clearly, E[1/(1+r)] is E r Er Er 1 1 1 1 1 1 +       > + [] = + [] 1 892857 826720 1 925926 1 1 2 1 2 1 2 1 2 1 2 1 2 . . . ← ← ← ← ← ← Volatility and Convexity 197 greater than 1/E(1+r). To summarize, equation (10.6) is true because the pricing function of a zero, 1/(1+r), is convex rather than concave. Returning to the example of this section, equation (10.6) may be used to show why the one-year spot rate is less than 10%. The spot rate one year from now may be 12% or 8%. According to (10.6), (10.7) Dividing both sides by 1.10, (10.8) The left-hand side of (10.8) is the price of the two-year zero coupon bond today. In words, then, equation (10.8) says that the price of the two-year zero is greater than the result of discounting the terminal cash flow by 10% over the first period and by the expected rate of 10% over the second pe- riod. It follows immediately that the yield of the two-year zero, or the two- year spot rate, is less than 10%. 1 110 5 1 112 5 1 108 1 110 2 . . . . ×+×       > . . . . 5 1 112 5 1 108 1 5112 5108 1 110 ×+×> ×+× = + 198 THE SHORT-RATE PROCESS AND THE SHAPE OF THE TERM STRUCTURE FIGURE 10.1 An Illustration of Convexity 1+r High E[1+r] 1+r Low 1/(1+r High ) 1/E[1+r] E[1/(1+r)] 1/(1+r Low ) The tree presented at the start of this section may also be used to price a three-year zero. The resulting price tree is The three-year spot rate, such that .752309=1/(1+r ˆ (3)) 3 , is 9.952%. There- fore, the value of convexity in this spot rate is 10%–9.952% or 4.8 basis points, whereas the value of convexity in the two-year spot rate was only 1.8 basis points. It is generally true that, all else equal, the value of convexity increases with maturity. This will be proved shortly. For now, suffice it to say that the convexity of the price of a zero maturing in N years, 1/(1+r) N , in- creases with N. In other words, if Figure 10.1 were redrawn for the func- tion 1/(1+r) 3 , for example, instead of 1/(1+r), the resulting curve would be more convex. Chapters 5 and 6 show that bonds with greater convexity perform bet- ter when yields change a lot but mentioned that this greater convexity is paid for at times that yields do not change very much. The discussion in this section shows that convexity does, in fact, lower bond yields. The mathematical development in a later section ties these observations to- gether by showing exactly how the advantages of convexity are offset by lower yields. The previous section assumes no interest rate volatility and, conse- quently, yields are completely determined by forecasts. In this section, with the introduction of volatility, yield is reduced by the value of convexity. So it may be said that the value of convexity arises from volatility. Further- more, the value of convexity increases with volatility. In the tree intro- duced at the start of the section, the standard deviation of rates is 200 basis 1 877193 797448 1 752309 909091 857633 1 943396 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 . . . . ← ← ← ← ← ← ← ← ← ← ← ← Volatility and Convexity 199 points a year. 3 Now consider a tree with a standard deviation of 400 basis points a year: The expected one-year rate in one year and in two years is still 10%. Spot rates and convexity values for this case may be derived along the same lines as before. Figure 10.2 graphs three term structures of spot rates: one with no volatility around the expectation of 10%, one with a volatility of 200 basis points a year (the tree of the first example), and one with a volatility of 400 basis points per year (the tree preceding this paragraph). Note that 18 14 10 10 6 2 1 2 1 2 1 2 1 2 1 2 1 2 % % %% % % ← ← ← ← ← ← 200 THE SHORT-RATE PROCESS AND THE SHAPE OF THE TERM STRUCTURE FIGURE 10.2 Volatility and the Shape of the Term Structure 9.75% 9.80% 9.85% 9.90% 9.95% 10.00% 10.05% 10.10% 10.15% 10.20% 10.25% 123 Term Rate Volatility = 0 bps Volatility = 200 bps Volatility = 400 bps 3 Chapter 11 describes the computation of the standard deviation of rates implied by an interest rate tree. the value of convexity, measured by the distance between the rates assum- ing no volatility and the rates assuming volatility, increases with volatility. Figure 10.2 also shows that the value of convexity increases with maturity. For very short terms and realistic volatility, the value of convexity is quite small. Simple examples, however, must use short terms, so convexity effects would hardly be discernible without raising volatility to unrealistic levels. Therefore, this section is forced to choose unrealistically large volatility values. The application at the end of this chapter uses realistic volatility to present typical convexity values. RISK PREMIUM To illustrate the effect of risk premium on the term structure, consider again the second interest rate tree presented in the preceding section, with a volatility of 400 basis point per year. Risk-neutral investors would price a two-year zero by the following calculation: (10.9) By discounting the expected future price by 10%, equation (10.9) implies that the expected return from owning the two-year zero over the next year is 10%. To verify this statement, calculate this expected return directly: (10.10) Would investors really invest in this two-year zero offering an ex- pected return of 10% over the next year? The return will, in fact, be ei- ther 6% or 14%. While these two returns do average to 10%, an investor could, instead, buy a one-year zero with a certain return of 10%. Pre- sented with this choice, any risk-averse investor will prefer an investment with a certain return of 10% to an investment with a risky return that av- erages 10%. In other words, investors require compensation for bearing interest rate risk. 4 . . . . .%. % % 5 877193 827541 827541 5 943396 827541 827541 56 514 10 × − +× − =× +× = . . 827541 5 1 1 14 1 1 06 1 1 5 877193 943396 1 1 =+ [] =+ [] Risk Premium 201 4 This is a bit of an oversimplification. See the discussion at the end of the section. Risk-averse investors demand a return higher than 10% for the two- year zero over the next year. This return can be effected by pricing the zero coupon bond one year from now at less than the prices of 1 / 1.14 or .877193 and 1 / 1.06 or .943396. Equivalently, future cash flows could be discounted at rates higher than the possible rates of 14% and 6%. The next section shows that adding, for example, 20 basis points to each of these rates is equivalent to assuming that investors demand an extra 20 basis points for each year of modified duration risk. Assuming this is in- deed the fair market risk premium, the price of the two-year zero would be computed as follows: (10.11) First, this is below the price of .827541 obtained in equation (10.9) by as- suming that investors are risk-neutral. Second, the increase in the discount- ing rates has increased the expected return of the two-year zero. In one year, if the interest rate is 14% then the price of a one-year zero will be 1 / 1.14 or .877193. If the interest is 6%, then the price of a one-year zero will be 1 / 1.06 or .943396. Therefore, the expected return of the two-year zero priced at .826035 is (10.12) Hence, recalling that the one-year zero has a certain return of 10%, the risk-averse investors in this example demand 20 basis points in expected return to compensate them for the one year of modified duration risk in- herent in the two-year zero. 5 Continuing with the assumption that investors require 20 basis points for each year of modified duration risk, the three-year zero, with its ap- proximately two years of modified duration risk, 6 needs to offer an ex- pected return of 40 basis points. The next section shows that this return . . . .% 5 877193 943396 826035 826035 10 20 + [] − = 826035 5 1 1 142 1 1 062 1 1=+ [] 202 THE SHORT-RATE PROCESS AND THE SHAPE OF THE TERM STRUCTURE 5 The reader should keep in mind that a two-year zero has one year of interest rate risk only in this stylized example: It has been assumed that rates can move only once a year. In reality rates can move at any time, so a two-year zero has two years of interest rate risk. 6 See the previous footnote. [...]... Chapter 10, it is difficult to make a case for rising expected rates These interpretive difficulties arise because Model 2 is still not flexible enough to explain the shape of the term structure in an economically 228 THE ART OF TERM STRUCTURE MODELS: DRIFT 8.00% 7 .50 % Forward 7.00% Spot Rate 6 .50 % Par 6.00% 5. 50% 5. 00% 4 .50 % 4.00% 0 5 10 15 20 25 30 Term FIGURE 11 .5 Rate Curves from Model 2 and Selected... section, to be called Model 2, adds a drift to Model 1, interpreted 1.20% Volatility 1. 15% 1.10% 1. 05% 1.00% 0. 95% 0 5 10 15 20 25 30 Term FIGURE 11.3 Par Rate Volatility from Model 1 and Selected Implied Volatilities, February 16, 2001 226 THE ART OF TERM STRUCTURE MODELS: DRIFT 15 Shift (bps) 10 5 0 0 5 10 15 20 25 30 Term FIGURE 11.4 Sensitivity of Spot Rates to a 10 Basis Point Change in the Factor,... compounded par, spot, and forward rate curves for the numerical example along with data from U.S dollar swap par rates as of February 16, 2001.3 The initial value 7.00% 6.00% Par Spot 5. 00% Forward Rate 4.00% 3.00% 2.00% 1.00% 0.00% 0 5 10 15 20 25 30 Term FIGURE 11.2 Rate Curves from Model 1 and Selected Market Swap Rates, February 16, 2001 3 Swaps will be discussed in Chapter 18 For now, the reader may... twoyear volatility was used So, for example, the convexity effect on a 20-year security is computed as follows The convexity of a 20-year par bond at a yield of 5. 67% is about 194 .5 Interpolating 91 .5 basis points per year for a 10-year yield and 68.6 basis points per year for a 30-year yield gives an approximation for 20-year volatility of about 80 basis points per year Therefore, the magnitude of the... dr = λdt + σdw (11 .5) The process (11 .5) differs from that of Model 1 by adding a drift to the short-term rate equal to λdt For this section, consider the values r0 =5. 138%, λ=.229%, and σ=1.10% If the realization of the random variable, dw, is again 15 over a month, then the change in rate is ( ) dr = 229% × 1 12 + 1.10% × 15 = 1841% (11.6) Starting from 5. 138%, the new rate is 5. 322% The drift of... horizon 40 35 30 Density 25 20 15 10 5 0 2.790% 3.920% 5. 050 % 6.180% Rate 7.310% 8.440% 9 .57 0% FIGURE 11.1 Distribution of Short Rates after One Year, Model 1 2 Actually, the interest rate could be slightly negative if a security or bank account were safer than holding cash and provided some transactional advantages relative to cash 223 Normally Distributed Rates, Zero Drift: Model 1 Over 10 years, for example,... the other hand, that the expectation curve rises above 5% , to a maximum of about 5. 23%, before falling back to 5% violates, at least to some extent, the second objective of the previous paragraph Lastly, using the volatilities given in Table 10.1, the magnitude of the risk premium gives Sharpe ratios ranging from 9.4% for 5- year bonds to 13.1% for 30-year bonds These values are in the range of historical... risk premium tends to dominate in 7 This is an artifact of this example in which rates change only once a year 2 05 Risk Premium 10. 25% 10.20% Volatility = 400 bps 10. 15% Risk Premium = 40 bps 10.10% Rate 10. 05% 10.00% Risk Premium = 20 bps 9. 95% 9.90% 9. 85% 9.80% Risk Premium = 0 bps 9. 75% 1 2 Term FIGURE 10.3 Volatility, Risk Premium, and the Shape of the Term Structure the short end while convexity... three-year zero in one year are ( ) (10.13) ( ) (10.14) 769067 = 5 847 458 + 909091 1.142 and 88 958 7 = 5 909091 + 980392 1.062 Finally, then, the expected return of the three-year zero over the next year is ( ) 5 769067 + 88 958 7 − 751 184 751 184 = 10.40% (10. 15) To summarize, in order to compensate investors for about two years of modified duration risk, the return on the three-year zero is about 40 basis... by –(1/2 )Cσ 2 The convexity of a particular par bond may be computed using the formulas given in Chapter 6 6.00% Par Curve 5. 00% Expectations 4.00% Rate 3.00% 2.00% 1.00% Risk Premium 0.00% 5 10 15 20 25 Convexity –1.00% Term FIGURE 10.4 Expectations, Convexity, and Risk Premium Estimates in the Treasury Market, February 15, 2001 APPLICATION: Expectations, Convexity, and Risk Premium 213 Choosing the . require compensation for bearing interest rate risk. 4 . . . . .%. % % 5 877193 82 754 1 82 754 1 5 943396 82 754 1 82 754 1 56 51 4 10 × − +× − =× +× = . . 82 754 1 5 1 1 14 1 1 06 1 1 5 877193 943396. why risk premium tends to dominate in . . . .% 5 769067 88 958 7 751 184 751 184 10 40 + () − = .88 958 7 5 909091 980392 1 062=+ () .769067 5 847 458 909091 1 142=+ () 204 THE SHORT-RATE PROCESS. negatively Risk Premium 2 05 FIGURE 10.3 Volatility, Risk Premium, and the Shape of the Term Structure 9. 75% 9.80% 9. 85% 9.90% 9. 95% 10.00% 10. 05% 10.10% 10. 15% 10.20% 10. 25% 123 Term Rate Risk Premium