Lecture 12/13 Bond Pricing and the Term Structure of Interest Rates


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1 1 Lecture 1/13 Bond Pricing and the Term Structure of Interest Rates Alexander K. Koch Department of Economics, Royal Holloway, University of London January 14 and 1, 008 In addition to learning the material covered in the reading and the lecture, students should In addition to learning the material covered in the reading and the lecture, students should understand how spot and forward rates are derived; be familiar with theories of the term structure; be able to price defaultfree bonds. Required reading: Bodie, Kane, and Marcus (008) (Chapter 1) Supplementary reading: Elton and Gruber (199) (Chapter 0)
2 Positive yield curve /1/004 09/11/004 08/10/ M 3M M 1Y Y 3Y Y 7Y 10Y 0Y Figure 1: Positive yield curve (source: U.S. Treasury.) 1 The term structure of interest rates In lecture 11 we discussed basic principles of bond pricing, assuming for simplicity that the same constant interest rate can be used to discount future cash flows. In reality, however, we observe that yields vary with the maturity of bonds. That is, interest rates have a term structure. This is manifested in the shape of the yield curve, a plot of bond yields against maturities. Figure 1 reproduces such a plot of yields of US treasury securities for 9 December 004 and dates in the two preceding months. 1 Two features are apparent. First, the yield curve varies over time. Second, yields are lower for shortterm maturities than for longer maturities. This shape is the normal shape that the yield curve has and is called a positive yield curve. Figures 4 depict other possible shapes observed in the market. We will discuss later how these different shapes can arise. Finally, Figure depicts the UK yield curve in January for the years The term structure under certainty As in the last lecture, we focus here on interest rates on defaultfree bonds. Moreover, we start our analysis of the term structure of interest rates by supposing that investors know for certain the path of future interest rates. Example 1 Consider the sequence of 1year (1Y) interest rates given in table 1. 1 Source: U.S. Treasury. Data is available at debtmanagement/interestrate/yield.html. UK yield curve date is available from the Bank of England at This example is taken from the readings: Bodie, Kane, and Marcus (008), Chapter 1.
3 3 Negative (inverted) yield curve. US 7/11/ UK 04/01/ M 3M M 1Y Y 3Y Y 7Y 10Y 0Y Figure : Negative (inverted) yield curve (source: U.S. Treasury/Bank of England.) Flat yield curve (1 March 1990) M 3M M 1Y Y 3Y Y 7Y 10Y 0Y Figure 3: Flat yield curve (source: U.S. Treasury.) Hump shaped yield curve (9 March 000) M 3M M 1Y Y 3Y Y 7Y 10Y 0Y Figure 4: Humpshaped yield curve (source: U.S. Treasury.)
4 4 Yield curves Jan 0 04 Jan 0 04 Jan Jan M 3M M 1Y Y 3Y Y 7Y 10Y 0Y Figure : UK yield curve in January (source: Bank of England.) year 1Y interest rate r t Table 1: Sequence of 1year interest rates What are the fair values of 1Y, Y, 3Y, and 4Y zerocoupon bonds with face value 1,000? As in lecture 11, we simply compute the present value, now applying the appropriate discount factors for the respective periods. Denote the fair value of an nyear zerocoupon bond by ZB n : ZB 1 = ZB = ZB 3 = ZB 4 = 1, 000 = r 1 1, 000 (1 + r 1 ) (1 + r ) = , 000 (1 + r 1 ) (1 + r ) (1 + r 3 ) = , 000 (1 + r 1 ) (1 + r ) (1 + r 3 ) (1 + r 4 ) = These prices imply a term structure of interest rates that we can extract by looking at the yields to maturities of the zerocoupon bonds: ZB 1 ZB ZB 3 ZB 4 Yield to maturity 8.000% 8.99% 9.0% 9.993%
5 yield curve spot rate 1Y interest rate Figure : Yield curve from example 1. The yield to maturity on an nyear zerocoupon bond is called nyear spot rate, y n. Plotting the spot rates (i.e. the YTM of zerocoupon bonds) from the example we obtain the pure (or zerocoupon) yield curve plotted in figure. What is the relation between the 1Y interest rates and the spot rate? In the case of example 1 we see immediately in figure that the yield curve (i.e., the spot rate curve) is not equal to the sequence of 1Y interest rates, plotted as a dotted line. Let us explore this relation further. The nyear spot rate is the YTM of an nyear zerocoupon bond. Since there is no uncertainty, the return on a 1 investment in a Y zerocoupon bond has to equal that on a 1 investment in a 1Y bond that is then rolled over after one year at the future 1Y rate: 1 (1.08) (1.10) = 1 (1 + y ) 1 + y = [ ] 1/. In general, in our special case of certain future rates, the return on an nyear investment has to be equal to that obtained on a 1year investment that is rolled over at the future 1Y rates for nyears: 1 + y n = [(1 + r 1 ) (1 + r )... (1 + r n )] 1/n. (1) That is, the nyear spot rate is the geometric average of the 1Y interest rates for the next n years. 1. The term structure under uncertainty So far we assumed that future 1Y interest rates were known with certainty. However, in reality this is not the case, of course. Nevertheless, we can compute a zerocoupon yield curve from
6 observed bond prices and extract from it interest rates that we can lock in today for a future period. 3 Such a rate for an nperiod investment starting at date t and ending at date t+n is called a forward rate and we will denote it by f n,t. Example The following prices for zerocoupon bonds with face value 1,000 are currently observed: Compute all possible forward rates. ZB 1 ZB ZB 3 ZB To find the 1Y forward rate f 1,1, i.e., the interest rate that we can lock in today for a 1Y loan starting in one year, we construct a 1,000 synthetic forward contract, using zerocoupon bonds. For these we know the yields to maturity since we know the 1Y and Y spot rates. cash flow t=0 t=1 t= long one 1Y zero bond 9.93 = ZB 1 = 1,000 1+y 1 +1, 000 short x Y zero bond = x ZB 1, 000 x sum 0 +1,000 1, 000 x The above table implies that x = 1 + f 1,1 and x = ZB 1 ZB f 1,1 is percent That is, the forward rate Another way to see this is to exploit the noarbitrage condition stating that an investment in the Y zero bond must have the same return as an investment in a 1Y zero bond rolled over in the second year at an interest rate locked in today at f 1,1 : =ZB 1 { }} { 1, = 1 + y f 1,1 { =ZB }} { 1, 000 (1 + y ) f 1,1 = ZB 1 ZB 1 = (1 + y ) 1 + y 1 1. To compute the Y forward rate f,1, we can employ the same strategy of constructing a synthetic portfolio. Alternatively, we can exploit the noarbitrage condition stating that an investment in a 3Y zero bond must have the same return as an investment in a 1Y zero bond data. 3 See Bodie, Kane, and Marcus (008), Section 1. for details on estimating the pure yield curve from bond
7 7 t ZB t y t 8.00 % 9.00 % 9.0 % 9.80 % n f n,t % 10.1 % % 10. % % Table : Forward rates for example rolled over after the first year for a twoyear investment at the forward rate f,1 : =ZB 1 { }} { 1, y 1 (1 + f,1 ) = f,1 = { =ZB 3 }} { 1, 000 (1 + y 3 ) 3 ( ) 1 ZB1 ZB 3 1 = ( (1 + y3 ) y 1 ) 1 1. To compute the Y forward rate f,, we can again exploit the noarbitrage condition stating that an investment in the 4Y zero bond must have the same return as an investment in a Y zero bond rolled over after the second year for a twoyear investment at the forward rate f, : =ZB { }} { 1, 000 (1 + y ) 1 (1 + f, ) = f, = { =ZB 4 }} { 1, 000 (1 + y 4 ) 4 ( ) 1 ZB ZB 4 1 = ( (1 + y4 ) 4 (1 + y ) ) 1 1. Repeating this exercise, it is easy to convince yourself that the general formula for computing the nperiod forward rate for a starting date t is given by: ( ZBt f n,t = ZB t+n ) 1 n 1 = ( (1 + yt+n ) t+n (1 + y t ) t Find the other 1Y and Y forward rates and compare with table. ) 1 n 1. Forward rates versus expected future spot rates Future interest rates are random variables. Hence, in general the spot rate that will realize in the future will not equal today s forward rate. For example, the realized Yspot rate in two years, y will usually not be equal to today s forward rate f,. The simplest theory on the term structure of interest rates is the expectations hypothesis. It starts from the premise that investors are indifferent between a longterm investment and a shortterm investment that
8 8 is rolled over if the two strategies give the same expected rate of return over the investment horizon. Thus, the expectations hypothesis asserts that forward rates are equal to the market s consensus expectation of future spot rates, i.e. that f n,t = E[y n for date t]. Can forward rates really reflect something like a consensus expectation of market participants on future interest rates? The answer to this question is not necessarily. To see why, consider the following example: Example 3 The current 1Y spot rate is r 1 = 0.08 and the 1Y forward rate for in one year is f 1,1 = 0.08 and for in two years is f 1, = Suppose that the market consensus expectation for the 1Y spot rate in one year s time is 8 percent and for the 1Y spot rate in two year s time is 10 percent. Shortterm investors Suppose an investor wants to earn money on her funds but knows that she will need cash in two years. Buying a Y bond matches the investment horizon of this shortterm investor and guarantees a rate of return of 8 percent, since ZB = F V (1+r 1 ) (1+f 1,1 ). In contrast, buying the 3Y bond exposes her to price risk after two years. Again, the bond is priced as if the forward rate was the interest rate prevailing in the future. However, the price of the bond after years will move depending on the interest rate that will actually realize for the third year. The bond will face value then be a 1Y zero bond priced at. Only if the then realized 1+realized 1Y spot rate in year interest rate corresponds exactly to f 1,, which in this example is also today s expectation of the future 1Y spot rate, will she have the same return as with the shortterm investment. Only if the realized 1Y spot rate turns out to exceed today s expectations does the investor gain relative to the safe strategy. Therefore, a riskaverse shortterm investor will only be willing to hold longterm bonds if the forward rate exceeds the expected rate of return by a risk premium: i.e., if f 1, > E[r 1 for date ]. Longterm investors In contrast, an investor wishing to have cash available only after three years faces a reinvestment rate risk when investing in Y bonds and rolling over at the uncertain thirdperiod rate. Buying a 3Y zerocoupon bond can lock in an interest rate of 8 percent for the first two years and 10 percent for the third year, since ZB 3 = F V (1+r 1 ) (1+f 1,1 )(1+f 1, ). In contrast, buying the Y zerocoupon bond and then rolling over the proceeds leaves the investor with the same interest rate for the first two years but an uncertain interest rate for the third year. If the 1Y rate turns out to be less than the forward rate, the A riskaverse longterm investor who shares the market s belief that the future 1Y spot rate will be equal to today s forward rate for that period will not be willing to buy shortterm bonds. To gain relative to the safe longterm
9 9 strategy the expected 1Y rate must exceed the one that can be locked in today, implying a negative risk premium on the forward rate. Longterm investors hold shortterm bonds only if f 1, < E[r 1 for date ]. The preceding example demonstrated why forward rates need not reflect the market s consensus estimate of future interest rates. Depending on the investment horizon of market participants, the forward rates can either be above or below the market s estimate. Let us return now briefly to the different possible shapes of the yield curve, illustrated by the examples in figures 14. An inverted (or negative) yield curve might indicate that market participants expect interest rates to decline. However, most of the time one can observe the normal shape of an upward sloping (or positive) yield curve. A competing theory to the expectations hypothesis for explaining the yield curves rationalizes this shape by a preference of investors for short investment horizons (the preceding example on shortterm investors illustrates the logic). According to the liquidity preference theory longterm bonds need to offer a higher yield than shortterm ones to convince investors to buy them and face the price risk of these bonds. That is, forward rates exceed the market expectation of future spot rates by a positive risk premium. In the example we have seen that longterm investors actually might require a negative premium on forward rates relative to the market expectation. A third theory explaining the term structure of interest rates allows for such different types of investors. The market segmentation hypothesis asserts that investors have a preferred habitat. For example, some institutional investors such as banks prefer shortterm securities while others such as insurance companies prefer the long end of the market. Yields in each market segment are then determined by the supply and demand in that market. Investors might be convinced to leave their preferred habitat if they receive a positive or negative premium relative to the expectation about future rates, depending on the maturity and the investor s investment horizon, as illustrated in example 3. Nominal vs real yields It is worth pointing out that changes in expected nominal interest rates can either be due to changes in expected inflation or changes in expected real interest rates. As Figure 7 illustrates for UK data from Hanuary 00, the nominal yields and the real yields are quite different. For example, a three year investment of 100 would return in January , but once accounting for expected inflation the investment would be anticipated to return in real terms only To how differences in inflation expectations can give rise to different shapes of the nominla
10 Nominal/real yields and implied inflation (UK January 00) nominal real implied inflation Y Y 7Y 10Y 0Y Figure 7: Real vs Nominal yield curve and implied inflation (source: Bank of England.) yield curve, suppose that investors are demanding a constant real interest rate for a 1year investment at any future date. Then, the nominal rates will depend on the expected inflation rates. Specifiaclly, the relation between the nominal interest rate and the real interest rate over a given period is as follows: real interest rate = 1+ nominal interest rate. 1+inflation rate The real interest rate discounts the nominal rate using the inflation rate. For example, suppose that the 1Y real interest rate is expected to remain at percent for all future dates. Moreover, the market expects the annual inflation rate to rise from 1 percent in the first year to 1. percent in the second and third year and then to percent in the fourth year. Table 3 gives examples for nominal, real and inflation rates over the different investment horizons. As we can see, even with constant real interest rates a positive yield curve can result from expected increases in inflation. Summary of important results Let us briefly pause to review our results. 1. Given prices, ZB t, for zerocoupon bonds with maturities t and face value FV we can compute the tperiod spot rate, y t by computing their respective yields to maturity: ( ) 1 F V t y t = 1. ZB t 4 Often the relation given is approximated using the Fisher equation: real interest rate nominal interest rateinflation rate.
11 11 investment annual rates (in %) over investment period period nominal rate real rate inflation rate Table 3: Nominal, real and inflation rates over different investment periods.. The zerocoupon (or pure) yield curve is a graphical representation of these spot rates. 3. Using the spot rates or zero bond prices, we can compute the nperiod forward rate for a starting date t as: ( ZBt f n,t = ZB t+n ) 1 n 1 = ( (1 + yt+n ) t+n (1 + y t ) t ) 1 n 1. Bond pricing Given the spot rates and forward rates derived from the pure yield curve, we can price bonds easily. The technique for determining the fair value of a bond is to treat each part of the cash flow as a separate zerocoupon bond. Example 4 The current spot rates are as follows: 1Y 8 percent, Y 8.99 percent, and 3Y 9. percent. What is the fair price of a 3Y bond with face value 1,000 and annual coupon payments at coupon rate 8 percent? We strip the bond into its year 1, year, and year 3 cash flows, which each are discounted with the appropriate spot rate: fair value = , y 1 (1 + y ) (1 + y 3 ) 3 = Credit risk So far we ignored default risk. As pointed out in lecture 11, bonds are rated by rating agencies in categories of default risk. These ratings provide benchmarks for valuing bonds based on the prevailing government bond yields and spreads for the respective rating. Figure 8 plots This example is taken from the readings: Bodie, Kane, and Marcus (008), Chapter 1.
12 1 December 13, 004. A Corporate Bonds AA Corporate Bonds AAA Corporate Bonds 4. spread 4 3. US Treasury Bonds Figure 8: US yield curves for different credit ratings 004 (source: Yahoo!Finance). January 10, 008 A Corporate Bonds. AA Corporate Bonds AAA Corporate Bonds AAA spread 3. US Treasury Bonds 10 0 Figure 9: US yield curves for different credit ratings 008 (source: Yahoo!Finance) the yield curves for US treasury bonds and corporate bonds with ratings AAA, AA, and A for December 004. A comparison with the situation in January 008 in Figure 9 shows that the risk premium inherent in corporate bond yields has risen significantly. This is a consequence of the recent turmoil in financial markets (subprime crisis which spread to the UK with the sinking of Northern Rock). Many financial institutions had invested in the structured products exposed to the problems in subprime mortage markets. However, the investment strategies used and the assets involved tend to be very complex, so that there was and still is much uncertainty in the marekt as to the exposure of individual financial institutions. This uncertainty has made banks reluctant to lend to each other: first, because no one could know for certain whether the bank one is lending money to will not find losses Data source: Yahoo!Finance
13 Crisis in the UK Interbank Market 7. 1M LIBOR Base rate Northern Rock crisis. interest rate (%pa) Central banks inject liquidity 3. Jan04 Mar04 May04 Jul04 Sep04 Nov04 Jan0 Mar0 May0 Jul0 Sep0 Nov0 Jan0 Mar0 May0 Jul0 Sep0 Nov0 Jan07 Mar07 May07 Jul07 Sep07 Nov07 Jan08 Figure 10: Crisis in the UK interbank money market (source: Data Stream) related to the subprime crisis; second, because a bank itself may find it needs the extra cash and may then have problems finding funds in the interbank market. This has lead to a crisis in interbank money markets, as illustrated in the large jump in spread between the Bank of England base rate 7 and the 1 month interbank offered rate (Libor) shown in Figure 10. The following newspaper quote from the Telegraph illustrates this: The threemonth London interbank offered rate (Libor) dropped basis points to.89pc  just 39 basis points above the current base rate of.pc. The spread has not been so narrow since before the crisis erupted on August 9. At its height in September, Libor was.9pc  11 points above the then base rate of.7pc. Shortterm borrowing costs have fallen sharply since central banks in the US, UK, Switzerland, Canada and the eurozone last month pledged to pump up to $ 00bn ( 300m) of emergency funding into the markets. The measures were taken to relieve pressure in the system and see the banks through the tricky yearend period, when many were unwilling to lend to protect their capital positions. (source: Telegraph online January 4th 008, http: // References Bodie, Zvi, Alex Kane, and Alan J. Marcus, 008, Investments (Irwin McGrawHill: Chicago). Elton, Edwin J., and Martin J. Gruber, 199, Modern Portfolio Theory and Investment Analysis (Wiley: New York). 7 The rate paid by the Bank of England on commercial bank reserves held with it. 13
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