Ch07 Understanding Interest Rates The Term Structure of Interest Rates.docx

CHAPTER 7: Understanding Interest Rates: The Term Structure of Interest Rates FOCUS OF THE CHAPTER Various theories of the term structure of interest rates, such as the expectations hypothesis, market segmentation hypothesis, and liquidity premium theory, are the focus of this chapter. Possible shapes of the yield curve and some stylized facts about the yield curve are also discussed. Learning Objectives: Explain why bond yields differ according to the length of time before they mature List some of the key properties of a term structure of interest rates Explain how a yield curve is defined Describe how the expectations hypothesis is formulated Identify some of the anomalies in yield curve behaviour Explain how the liquidity premium hypothesis is formulated Determine how the segmented markets and preferred habitat hypotheses are formulated SECTION SUMMARIES The Term Structure of Interest Rates The term structure of interest rates is the structure of interest rates on instruments that differ only in their term to maturity. On a diagram, the term structure of interest rates is given by a yield curve which depicts the relationship between the yield (yield to maturity) and the term to maturity. On such a diagram, normally, the yield is measured along the vertical axis, and the term to maturity is given by the horizontal axis. In general, the yield on a long-term bond is higher than the yield on a short-term bond, and therefore, the yield curve is upward-sloping. However, the yield curve can take other shapes, such as downward-sloping or flat. Several hypotheses have been developed to explain the possible shapes of the yield curve (or the term structure of interest rates). The Expectations Hypothesis: The decision of investors to hold short-term or long-term bonds depends on their expectations about future interest rates. According to the expectations hypothesis, the yield on a long-term bond is the average of expected short-term interest rates. Investors are indifferent between long-term and short-term bonds as long as the annual yield on the long-term bond is equal to the arithmetic average of the expected short-term yields, a condition given by the following equation: Rn = (R1 + E11 + E12 +... + E1n-1)/n where n is the number of years to maturity, Rn is the annual yield on an n-year bond (say, a long-term bond), R1 is the yield on a one-year bond (say, a short-term bond) maturing in one year, and E11, E12, and E1n-1 are the expected yields on a short-term bond maturing in one, two, and n-1 years from today, respectively. Note that the above equation is an approximation of the technically more correct equation 7.3a in the text in which long term yields are a geometric average of the expected short term yields. Over a two year period this simplifies to: R2 = (R1 + E11 ) / 2 If the expectations hypothesis is correct, the spread (or the difference between the long-term and short-term interest rates) provides investors with a forecast of future short-term interest rates and helps them to predict the future course of short-term rates. Expectations of higher (lower) future short-term interest rates result in an upward-sloping (downward-sloping) yield curve, while expectations of constant future short-term rates produce a flat yield curve. We can calculate the expected future one year rates by solving equations like the one above for the expected future rates. For years two through five that would be: E11 = 2 R2 - R1 E12 = 3 R3 - 2R2 E13 = 4 R4 – 3R3 E14 = 5 R5 – 4R4 The Role of the Real Interest Rate: The yield curve conveys information about changes in expected future inflation and the real interest rate. With the help of the Fisher equation, one can show how changes in inflation expectations and real interest rates influence the position and shape of the yield curve. Consider the Fisher equation (R = ? + ?e), which states that the nominal rate of interest (R) is the sum of the real interest rate (?) and the expected rate of inflation (?e)). If the real interest rate (?) is constant, the change in nominal interest rate (?R) must equal the change in the expected rate of inflation (??e). Using the Fisher equation, it can also be shown that the spread between the nominal interest rates of a short-term bond and those of a long-term bond equals the difference between the expected rates of inflation. Suppose the real interest rate (?) is constant. Using the Fisher equation, the nominal rate of interest on a short-term (one-year) bond (R1) can be written as R1 = ? + ?e1 , and the nominal rate of interest on a long-term (two-year) bond (R2) can be written as R2 = ? + ?e2. The difference between R2 and R1 (?R) equals the difference between the expected rates of inflation ?e2 and ?e1 (??e ): R2 - R1= (?+?e2 ) - (?+?e1) = ?e2 - ?e1 = ??e ?R = ??e A sudden and permanent increase (decrease) in the expected rate of inflation increases (decreases) the nominal interest rates on both long-term and short-term bonds by the increase (decrease) in the expected rate of inflation, and, therefore, the yield curve shifts upward (downward). If the expected rate of inflation (?e) is unchanged, changes in nominal interest rates can be explained only by changes in real interest rates. Yield Curve Puzzles: Many believe that the expectations hypothesis provides an accurate explanation for various possible shapes of the yield curve. However, the expectations hypothesis may not be easily or satisfactorily applied to explain certain stylized facts about the yield curves. Three such stylized facts are: 1) The yield curve is generally upward-sloping. According to the expectations hypothesis, an upward-sloping yield curve implies that investors almost always expect higher future interest rates. 2) The yield curve tends to shift over time. This may be due to the government or to individuals’ decisions to shift their bond holdings from long-term to short-term, or vice-versa. Such decisions are often prompted by expectations of future inflation. 3) The slope of the yield curve tends to predict future economic activity. It has been found that the yield curve has the ability to predict recessions and, thus, is linked to the state of the business cycle. It has also been found that the size of the spread is linked to the future economic growth rate. In Canada, recessions have been preceded by a falling or negative spread and expansions (recovery phase of business cycle) have been preceded by a rising or positive spread. Competing Views of the Term Structure: The liquidity premium theory, market segmentation theory, and preferred habitat theory provide alternative views to the expectations theory in explaining the term structure of interest rates. The Liquidity Premium Theory: Long-term bonds are relatively less liquid than short-term bonds. This is a fact that arises because the market values of longer term bonds fluctuates relatively more than do the market values of shorter term bonds for any given change in the market rate of interest. (See Problem 4 below.) The liquidity premium theory assumes that investors prefer more liquid investments over less liquid investments. They prefer greater certainty of market value even though that means having less certainty over interest income. All else equal therefore, the holders of long-term bonds expect a liquidity premium (an additional yield for accepting lower liquidity, also called term premium), as indicated in the following equation: Rn = (R1 + E11 + E12 +... + E1n-1)/n + LP where LP is the liquidity premium. The rest of the terms are defined earlier in the section on the expectations hypothesis. The liquidity premium theory does not deny that yields on long term bonds do not reflect expected future interest rates as hypothesized by the expectations hypothesis, but presumes a bias of investors in favour of short term investments. The term premium compensates the investor for this liquidity risk. Market Segmentation and Preferred Habitat Views: In both of the views which follow, it is recognized that liquidity, a feature of shorter term bonds is not universally desirable. Many investors in bonds seek a stable income from their bonds and have very little expectation of a need to sell those bonds. These individuals prefer long to short term bonds and would have to receive a premium to make them willing to hold short term bonds. It should also be observed that issuers or suppliers of bonds have preferences as well. Companies financing long term projects will typically prefer to issue long term bonds so as to have greater certainty over their financing costs. Market Segmentation View: According to the market segmentation view, long-term and short-term bonds are not substitutes. As such, their markets are separated (segregated) from each other. Changes in market forces (supply and demand) in one market have no effect on the other. The short-term interest rate is determined in the short-term bond market, while the long-term interest rate is determined by supply and demand in the long-term bond market. Any shape of yield curve is compatible with this view of bond markets. Preferred Habitat View: According to this view, a certain degree of substitutability exists between long-term and short-term bonds. But investors and borrowers have a distinct preference for certain maturities. An investor’s preferred habitat is the term to maturity (short-term or long-term) for which that investor has a distinct preference. An investor's willingness to accept a term to maturity different from the preferred habitat (a less preferred habitat) will depend on the size of the term premium the investor can earn. A term premium is an additional yield for accepting a less-preferred maturity. If the term premium is large enough, an investor who has a particular preference for a short-term bond will purchase a long-term bond. MULTIPLE-CHOICE QUESTIONS 1. The term structure of interest rates is the structure of interest rates on bonds that differ only in terms of a) purchase price. b) income risk. c) term to maturity. d) liquidity. 2. The graphic display of the relationship between the rate of return and the term to maturity is called a) the yield curve. b) the supply curve for bonds. c) the Phillips curve. d) the demand curve for bonds. 3. The yield on a government bond maturing in one year is 6%, and the expected yield on a government bond maturing in one year from today is 8%. According to the expectations hypothesis, the yield on a government bond maturing in two years is: a) 14%. b) 2%. c) 7%. d) 8%. 4. Which of the following is not a theory of the term structure of interest rates? a) Expectations hypothesis b) Permanent income hypothesis c) Liquidity premium theory d) Market segmentation hypothesis 5. According to the expectations hypothesis the yield on a) a short-term bond is an average of the expected yields on long-term bonds. b) a long-term bond is the average of expected yields on short-term bonds. c) a short-term bond is the difference between the long-term yield and the inflation rate. d) a long-term bond is the sum of the short-term yield and the risk premium. 6. According to the expectations hypothesis, the expectation of falling short-term interest rates results in a) a downward-sloping yield curve. b) an upward-sloping yield curve. c) a flat (horizontal) yield curve. d) a hump-shaped yield curve. 7. If the real interest rate remains constant, the spread between short-term and long-term yields is equal to a) changes in expected future inflation rates. b) the liquidity premium. c) the difference between long-term and short-term bond prices. d) changes in the foreign rate of inflation. 8. The liquidity premium theory of the term structure of interest rates assumes that a) short-term and long-term bonds are perfect substitutes. b) long-term bonds are relatively less liquid (less marketable) than short-term bonds. c) short-term bonds are relatively less liquid (less marketable) than long-term bonds. d) short-term and long-term bonds are perfect complements. 9. Through the operation of the Fisher Effect, a permanent increase in the expected rate of inflation a) shifts the yield curve downward. b) shifts the yield curve upward. c) leaves the yield curve unaffected. d) changes the slope of the yield curve from positive to negative. 10. According to the market segmentation hypothesis, short-term and long-term bonds a) are perfect substitutes. b) are imperfect substitutes. c) are not substitutes for each other. d) are complementary to each other. 11. According to the preferred habitat view, short-term and long-term bonds a) are perfect substitutes. b) are imperfect substitutes. c) are not substitutes for each other. d) are complementary to each other. PROBLEMS 1. Suppose the following interest rate structure was observed in 2000 and 2001: Yield (%) Bond 2004 2005 3-month treasury bill 4.00 5.00 1-year government bond 4.50 5.50 3-year government bond 4.75 5.75 5-year government bond 5.00 6.00 10-year government bond 5.30 6.30 a) In light of the expectations hypothesis, explain what the yield curves say about future interest rates. b) If real interest rates were the same in both years, what might have caused the shift in the yield curve? 2. Consider the following forecasts of future interest rates: Yield R1 E11 E12 E13 E14 % 6.6 6.4 5.6 5.2 5 Using the expectations hypothesis, determine the yields on two through five year bonds and plot the yield curve. 3. Consider the following forecasts of future interest rates: Yield R1 R2 R3 R4 R5 % 3.8 4.1 4.3 4.4 4.5 Using the expectations hypothesis, determine E11 E12 E13 E14 % ___ ___ ___ ___ 4. a) Calculate the market value $10,000 one year, zero-coupon bond when the market rate of interest is 5% and then recalculate that value if the market rate is 6%. Calculate the percentage loss on that bond as a result of the increase in the market rate of interest. b) Calculate the market value $10,000 10 year, zero-coupon bond when the market rate of interest is 5% and then recalculate that value if the market rate is 6%. Calculate the percentage loss on that bond as a result of the increase in the market rate of interest. c) Calculate the market value $10,000 20 year, zero-coupon bond when the market rate of interest is 5% and then recalculate that value if the market rate is 6%. Calculate the percentage loss on that bond as a result of the increase in the market rate of interest. d) Compare the effect of an increase in market rates of interest on the percentage change in market values on one year, 10 year and 20 year bonds. What do you observe? 5. What would be the possible explanation for a flat yield curve given by the following theories? a) expectations hypothesis b) market segmentation hypothesis c) liquidity premium theory 6. Consider the following question: “An economic expansion is widely regarded as being imminent. Assuming that the expectations hypothesis is true, would you be better off holding short-term or long-term bonds? Explain.” Now consider the following answer: “The expectation of an economic expansion generates an increase in the supply of bonds, in particular long-term bonds. This implies a decrease in the price of long-term bonds. The price of a bond and the yield are inversely related. Therefore, the decrease in the price of long-term bonds implies an increase in the yield on long-term bonds. Therefore, I would be better off holding long-term bonds rather than short-term bonds.” Explain why the conclusions reached in this answer are incorrect. ANSWER SECTION Answers to multiple-choice questions: c (see page 117) a (see pages 117, and 122-123) c (see pages 120-121) b (see pages 127-130) b (see pages 119-120) a (see pages 121-122) a (see pages 124-125) b (see pages 127) b (see pages 126-127) c (see page 128-129) b (see pages 130-131) Answers to problems: 1. a) In each of the two years, the yield increases as the term to maturity increases. This shows that the yield curves in the two years are normal (upward-sloping). According to the expectations hypothesis, a normal yield curve is the result of expectations of rising future short-term interest rates. Therefore, this normal yield curve indicates increases in future interest rates. b) On a diagram, the yield curve for 2001 lies above the yield curve for 2000, indicating an upward shift in the yield curve. Using the Fisher equation, the yield (rate of interest) for a given term to maturity can be stated as the sum of the real interest rate and the expected rate of inflation. The data show that, between the two years, the yield on each type of bond has increased by 2%, causing a parallel shift in the yield curve. If the real interest rate remained the same, the increase in yield must be equal to the change in the expected rate of inflation. Therefore, the shift in the yield curve may have been caused by an increase in the expected rate of inflation. 2. According to the given data, the expected future rates of return (E11, E12, E13, and E14 ) are decreasing. The expectations hypothesis predicts a downward-sloping yield curve in this situation. This can be verified by the long-term rates of return predicted by the expectations hypothesis. The following rates of return can be calculated from the data using the equation: Rn = (R1 + E11 + E12 +... + E1n-1)/n Yield R2 R3 R4 R5 % 6.4 5.6 5.2 5 These data show clearly that the yield curve is downward-sloping (has a negative slope). 3. E11= 2(4.1) - 3.8 = 4.4 E12 = 3(4.3) - 2(4.1) = 4.7 E13= 4(4.4) - 3(4.3) = 4.7 E14= 5(4.5) - 4(4.4) = 4.9 4. Value at 5% Value at 6% Percentage decrease. 1 year bond $9,524 $9,434 0.94% 10 year bond $6,139 $5,584 9.04% 20 year bond $3,769 $3,118 17.27% Obviously longer term bonds have far more volatile market values. 5. a) According to the expectations hypothesis, the yield curve is flat because future short-term interest rates are expected to remain the same. b) According to the market segmentation hypothesis, the yield curve is flat because market forces in short-term and long-term bond markets determine the same interest rate (yield), even though the markets are segregated from each other. c) According to the liquidity premium theory, a flat yield curve will still reflect a liquidity premium. This implies that expected future interest rates get progressively lower as they get farther into the future. 6. “The expectation of an economic expansion generates an increase in the supply of bonds, long-term bonds in particular.” This part of the answer is correct. “This implies a decrease in the price of long-term bonds. The price of a bond and the yield are inversely related. Therefore, the decrease in the price of long-term bonds implies an increase in the yield on long-term bonds.” These statements are also essentially correct. “Therefore, I would be better off holding long-term bonds rather than short-term bonds.” This conclusion is wrong on two counts. First, if you buy long term bonds and the interest rate rises, you will suffer a capital loss. Second, if you understand and believe the expectations hypothesis, you will know that long term rates are based on expected future short term interest rates, and therefore, your expected return will be the same regardless of whether you invest in short or long term bonds assuming that you reinvest the proceeds from your short term bonds in more short term bonds.