Term Structure of Interest Rates

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1 Appendix 8B Term Structure of Interest Rates To explain the process of estimating the impact of an unexpected shock in short-term interest rates on the entire term structure of interest rates, FIs use the theory of the term structure of interest rates or the yield curve. The term structure of interest rates compares the market yields or interest rates on, assuming that all characteristics (default risk, coupon rate, etc.) except are the same. The change in required interest rates as the of a security changes is called the premium (MP). The MP, or the difference between the required yield on long- and shortterm of the same characteristics, except, can be positive, negative, or zero. The yield curve for U.S. Treasury is the most commonly reported and analyzed yield curve. The shape of the yield curve on U.S. Treasury has taken many forms over the years, but the three most common shapes are shown in Figure 8B 1. In graph (a), the yield curve on August 12, 2004, yields rise steadily with when the yield curve is upward sloping. This is the most common yield curve, so on average the MP is positive. Graph (b) shows an inverted or downward-sloping yield curve, reported on November 27, 2000, for which yields decline as increases. Inverted yield curves do not generally last very long. Finally, graph (c) shows a flat yield curve, reported on August 31, 2006, in which the yield is virtually unaffected by the term. Note that these yield curves may reflect facrs other than invesrs preferences for the of a security, since in reality there may be liquidity differences among the traded at different points along the yield curve. For example, newly issued 20-year U.S. Treasury bonds offer a rate of return less than (seasoned issues) 10-year Treasury bonds if invesrs prefer new ( on the run ) previously issued ( off the run ). Specifically, since the Treasury (hisrically) issues new 10-year notes and 20- year bonds only at the long end of the spectrum, an existing 10-year Treasury bond would have have been issued 10 years previously (i.e., it was originally a 20-year bond when it was issued 10 years previously). The increased demand for the newly issued liquid 20-year Treasury bonds relative the less liquid 10-year Treasury bonds can be large enough push the equilibrium interest rate on the 20-year Treasury bonds below that on the 10-year Treasury bonds and even below short-term rates. Explanations for the shape of the yield curve fall predominantly in three theories: the unbiased expectations theory, the liquidity premium theory, and the market segmentation theory. UNBIASED EXPECTATIONS THEORY According the unbiased expectations theory for the term structure of interest rates, at a given point in time, the yield curve reflects the market s current expectations of future short-term rates. Thus, an upward-sloping yield curve reflects the market s expectation that short-term rates will rise throughout the relevant time period (e.g., the U.S. Federal Reserve is expected tighten monetary policy in the future). Similarly, a flat yield curve reflects the expectation that shortterm rates will remain constant over the relevant time period. As illustrated in Figure 8B 2, the intuition behind the unbiased expectations theory is that if invesrs have a four-year investment horizon, they either could buy a current four-year bond and earn the current yield on a four-year bond ( R 4, if held ) each year or could invest in four successive one-year bonds (of which they know only the current one-year rate, R 1, but form expectations of the unknown future one-year rates). In equilibrium, the return holding a fouryear bond should equal the expected return investing in four successive one-year bonds. Similarly, the return on a three-year bond should equal the expected return on investing in three successive one-year bonds. If future oneyear rates are expected rise each successive year in the future, then the yield curve will slope upward. Specifically, the current four-year T-bond rate or return will exceed the three-year bond rate, 8B-1

2 8B-2 Appendix 8B Term Structure of Interest Rates FIGURE 8B 1 U.S. Treasury Curves Source: U.S. Treasury, Daily Treasury Rates, (a) Curve, August 12, month 3 month 6 month 1 year 2 year 3 year 5 year 7 year 10 year 20 year Time 7.00 (b) Curve, November 27, month 6 month 1 year 2 year 3 year 5 year 7 year 10 year 20 year 30 year Time 5.50 (c) Curve, August 31, month 6 month 1 year 2 year 3 year 5 year 7 year 10 year 20 year 30 year Time FIGURE 8B 2 Unbiased Expectations Theory of the Term Structure of Interest Rates (1+ 1 R 1 ) [1+E( 2 r 1 )] [1+E( 3 r 1 )] [1+E( 4 r 1 )] Buy 4 one-year bonds (1 + 1 R 4 ) 4 Buy a four-year bond Year

3 Appendix 8B Term Structure of Interest Rates 8B-3 which will exceed the two-year bond rate, and so on. Similarly, if future one-year rates are expected remain constant each successive year in the future, then the four-year bond rate will be equal the three-year bond rate; that is, the term structure of interest rates will remain constant over the relevant time period. Specifically, the unbiased expectations theory posits that long-term rates are a geometric average of current and expected shortterm interest rates. That is, the interest rate that equates the return on a series of short-term security investments with the return on a long-term security with an equivalent reflects the market s forecast of future interest rates. The mathematical equation representing this relationship is R N 2 N R 1 2[1 1 E1 2 r 1 2] p [1 1 E1 N r 1 2] where 1 R N Actual N -period rate N Term 1 R 1 Current one-year rate E (t r 1 ) Expected one-year (forward) yield during period t Notice that upper-case interest rate terms 1 R t are the actual current interest rates on purchased day with a of t years. Lowercase interest rate terms t r 1 are estimates of future one-year interest rates starting t years in the future. For example, suppose the current one-year spot rate and expected one-year U.S. Treasury bill rates over the following three years (i.e., years 2, 3, and 4, respectively) are as follows: 1R % E1 2 r % E1 3 r % E1 4 r % This would be consistent with the market s expecting the Federal Reserve increasingly tighten monetary policy. With the unbiased expectations theory, current long-term rates for one-, two-, three-, and four-year Treasury should be 1R % 1R 2 5 [ ] 1> % 1R 3 5 [ ] 1> % 1R 4 5 [ ] 1> % And the yield curve should look like this: (%) Term (years) Thus, the upward-sloping yield curve reflects the market s expectation of consistently rising oneyear (short-term) interest rates in the future. 1 LIQUIDITY PREMIUM THEORY The unbiased expectations theory has the shortcoming that it neglects recognize that forward rates are not perfect predicrs of future interest rates. If forward rates were perfect predicrs of future interest rates, future prices of Treasury would be known with certainty. The return over any investment period would be certain and independent of the of the instrument initially purchased and of the time at which the invesr needs liquidate the security. However, with uncertainty about future interest rates (and future monetary policy actions) and hence about future security prices, these instruments become risky in the sense that the return over a future investment period is unknown. In other words, because of future uncertainty of return, there is a risk in holding long-term, and that risk increases with the security s. The liquidity premium theory of the term structure of interest rates allows for this future uncertainty. It is based on the idea that invesrs will hold long-term maturities only if they are offered a premium compensate for the future uncertainty in a security s value, which increases with an asset s. Specifically, in a world of uncertainty, short-term provide greater marketability (due their more active secondary market) and have less price risk (due smaller price fluctuations for a given change in interest rates) than long-term. As a result, 1 That is E ( 4 r 1 ) > E ( 3 r 1 ) > E ( 2 r 1 ).

4 8B-4 Appendix 8B Term Structure of Interest Rates invesrs prefer hold shorter-term because they can be converted in cash with little risk of a capital loss, that is, a fall in the price of the security below its original purchase price. Thus, invesrs must be offered a liquidity premium buy longer-term that have a higher risk of capital losses. This difference in price or liquidity risk can be directly related the fact that longer-term are more sensitive interest rate changes in the market than are shorter-term see Appendix 9A for a discussion on bond interest rate sensitivity and the link a bond s. Because the longer the on a security the greater its risk, the liquidity premium increases as increases. The liquidity premium theory states that longterm rates are equal the geometric average of current and expected short-term rates (as with the unbiased expectations theory) plus a liquidity or risk premium that increases with the of the security. Figure 8B 3 illustrates the difference in the shape of the yield curve under the unbiased expectations theory versus the liquidity premium theory. For example, according the liquidity premium theory, an upwardsloping yield curve may reflect the invesr s expectations that future short-term rates will rise, be flat, or fall, but because the liquidity premium increases with, the yield curve will nevertheless increase with the term. The liquidity premium theory may be mathematically represented as 1R N {(1 1 R 1 )[1 E( 2 r 1 ) L 2 ] [1 E( N r 1 ) L N ]} 1/N 1 where L t liquidity premium for a period t and L 2 < L 3 < < L N. For example, suppose that the current one-year rate (one-year spot rate) and expected one-year T-bond rates over the following three years (i.e., years 2, 3, and 4, respectively) are as follows: 1R % E1 2 r % E1 3 r % E1 4 r % In addition, invesrs charge a liquidity premium on longer-term such that L % L % L % Using the liquidity premium theory, current rates for one-, two-, three-, and four-year- Treasury should be FIGURE 8B 3 Curve under the Unbiased Expectations Theory (UET) versus the Liquidity Premium Theory (LPT) (a) Upward-sloping yield (b) Inverted or downward-sloping yield LPT UET Time UET LPT Time (c) Flat yield LPT UET Time

5 Appendix 8B Term Structure of Interest Rates 8B-5 1R % 1R 2 [( )( )] 1/ % 1R 3 [( )( )( )] 1/ % 1R 4 [( )( )( )( )] 1/ % and the current yield curve will be upward sloping as shown: (%) Term (years) Comparing the yield curves in the example above (using the unbiased expectations hypothesis) and here, notice that the liquidity premium in year 2 ( L %) produces a 0.05 premium on the yield on a two-year T-note, the liquidity premium for year 3 ( L %) produces a 0.10 premium on the yield on the three-year T-note, and the liquidity premium for year 4 ( L %) produces a 0.15 premium on the yield on the four-year T-note. MARKET SEGMENTATION THEORY Market segmentation theory argues that individual invesrs have specific preferences. Accordingly, with different maturities are not seen as perfect substitutes under the market segmentation theory. Instead, individual invesrs have preferred investment horizons dictated by the nature of the assets and liabilities they hold. For example, banks might prefer hold relatively short-term U.S. Treasury bills because of the short-term nature of their deposit liabilities, while insurance companies might prefer hold long-term U.S. Treasury bonds because of the long-term nature of their life insurance contractual liabilities. As a result, interest rates are determined by distinct supply and demand conditions within a particular bucket or market segment (e.g., the short end and the long end of the market). The market segmentation theory assumes that neither invesrs nor borrowers are willing shift from one secr another take advantage of opportunities arising from changes in yields. Figure 8B 4 demonstrates how changes in the supply curve for short- versus long-term bonds result in changes in the shape of the yield curve. Such a change may occur if fewer FIGURE 8B 4 Market Segmentation and Determination of the Slope of the Curve r S r L S S S L D S D L curve Short-term Long-term S L Time r S r L S S S L D S D L curve Short-term Long-term S L Time

6 8B-6 Appendix 8B Term Structure of Interest Rates short-term Treasury bills and more long-term government bonds are issued ( lengthen the average of government debt outstanding). Specifically in Figure 8B 4, the higher the yield on, the higher the demand for them. 2 Thus, as the supply of decreases in the short-term market and increases in the longterm market, the slope of the yield curve becomes steeper. If the supply of short-term had increased while the supply of long-term had decreased, the yield curve would have become flatter (and may even have sloped downward). Indeed, the large-scale repurchases of longterm Treasury bonds (i.e., reductions in supply) by the U.S. Treasury in 2000 have been viewed as the major cause of the inverted yield curve that appeared in FORECASTING INTEREST RATES As interest rates change, so do the values of financial. Accordingly, the ability predict or forecast interest rates is critical the profitability of FIs. For example, if interest rates rise, the value of investment portfolios of FIs will fall, resulting in a loss of wealth. Thus, interest rate forecasts are extremely important for the financial wealth of FIs. The discussion of the unbiased expectations theory above indicated that the shape of the yield curve is determined by the market s current expectations of future short-term interest rates. For example, an upward-sloping yield curve suggests that the market expects future short-term interest rates increase. Given that the yield curve represents the market s current expectations of future short-term interest rates, the unbiased expectations theory can be used forecast (short-term) interest rates in the future (i.e., forward one-year interest rates). A forward rate is an expected or implied rate on a short-term security that is be originated at some point in the future. With the equations representing unbiased expectations theory, the market s expectation of forward rates can be derived directly from existing or actual rates on currently traded in the spot market. To find an implied forward rate on a one-year security be issued one year from day, we can rewrite the unbiased expectation theory equation as follows: where 1R 2 5 { R 1 2[ f 1 2]} 1/ f 1 Expected one-year rate for year 2, or the implied forward one-year rate for next year Therefore, 2 f 1 is the market s estimate of the expected one-year rate for year 2. Solving for 2 f 1, we get 2 f 1 5 { R /[ R 1 2]} 2 1 In general, we can find the one-year forward rate for any year, N years in the future, using the following equation: N f 1 5 { R N 2 N /[ R N21 2] N21 } 2 1 For example, on August 31, 2006, the existing or current (spot) one-year, two-year, three-year, and four-year zero-coupon U.S. Treasury security rates were as follows: 1R % 1R % 1R % 1R % With the unbiased expectations theory, one-year forward rates on zero-coupon Treasury bonds for years 2, 3, and 4 as of August 31, 2006, were 2 f 1 5 [ > ] % 3 f 1 5 [ > ] % 4 f 1 5 [ > ] % Thus, the expected one-year rate one year in the future was ; the expected one-year rate two years in the future was ; and the expected one-year rate three years in the future was In general, the price and yield on a bond are inversely related. Thus, as the price of a bond falls (becomes cheaper), the demand for the bond will rise. This is the same as saying that as the yield on a bond rises, the bond becomes cheaper and the demand for it increases. See Appendix 9A.

7 Appendix 8B Term Structure of Interest Rates 8B-7 Questions and Problems 1. The current one-year Treasury bill rate is 5.2, and the expected one-year rate 12 months from now is 5.8. According the unbiased expectations theory, what should be the current rate for a two-year Treasury security? 2. Suppose that the current one-year rate (one-year spot rate) and expected one-year T-bill rates over the following three years (i.e., years 2, 3, and 4, respectively) are as follows: 1R 1 5 6% E1 2 r % E1 3 r % E1 4 r % Using the unbiased expectations theory, calculate the current (long-term) rates for one-, two-, three-, and four-year- Treasury. Plot the resulting yield curve. 3. A recent edition of the Report on Business reported interest rates of 6, 6.35, 6.65, and 6.75 for three-year, four-year, five-year, and six-year Government of Canada notes, respectively. According the unbiased expectations theory, what are the expected one-year rates for years 4, 5, and 6? 4. The Report on Business reports that the rate on threeyear government of Canada notes is 5.60 and the rate on four-year Government of Canada notes is According the unbiased expectations hypothesis, what does the market expect the one-year Government of Canada notes rate be three years from day, E ( 3 r 1 )? 5. How does the liquidity premium theory of the term structure of interest rates differ from the unbiased expectations theory? In a normal economic environment (that is, an upward-sloping yield curve), what is the relationship of liquidity premiums for successive years in the future? Why? 6. Based on economists forecasts and analysis, oneyear Treasury bill rates and liquidity premiums for the next four years are expected be as follows: R % E1 2 r % L % E1 3 r % L % E1 4 r % L % Using the liquidity premium hypothesis, plot the current yield curve. Make sure you label the axes on the graph and identify the four annual rates on the curve both on the axes and on the yield curve itself. 7. The Financial Post reports that the rate on three-year Government of Canada notes is 5.25 and the rate on four-year Government of Canada notes is The one-year interest rate expected in three years is E (3 r 1 ), According the liquidity premium hypothesis, what is the liquidity premium on four-year Government of Canada notes, L 4? 8. You note the following yield curve in the Report on Business. According the unbiased expectations hypothesis, what is the one-year forward rate for the period beginning two years from day, 2 f 1? Maturity One day 2.00% One year 5.50 Two years 6.50 Three years 9.00

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