Chapter 13: The Term Structure of Interest Rates PDF

Chapter 13 The term structure of interest rates 13.1 Introduction With the exceptions of Chapters 11 and 12, throughout this guide we have only ever considered ‘the’ interest rate, that is the one set by the monetary authorities, or that is allowed to change to clear the money markets if the money...

Chapter 13 The term structure of interest rates 13.1 Introduction With the exceptions of Chapters 11 and 12, throughout this guide we have only ever considered ‘the’ interest rate, that is the one set by the monetary authorities, or that is allowed to change to clear the money markets if the money supply is the instrument of choice. Even in Chapters 11 and 12, there was only ever one market interest rate. In reality, there are a large number of interest rates, from those on debt that mature overnight to interest rates on debt that mature up to 30 years in the future. This chapter will examine in more detail the links between short-term and long-term interest rates, explaining why such links are important and providing theories that explain the short-termlong-term interest rate relationship. 13.2 Aims This chapter will introduce the relevance of a rich variety of interest rates in financial and macroeconomic decisions. We will also discuss different term structure theories. 13.3 Learning outcomes By the end of this chapter, and having completed the Essential reading and activities, you should be able to: explain what a yield curve is and why it is important for policy makers list and discuss the different empirical regularities of the yield curve list and describe the three different theories of the term structure outlined in this chapter, noting any differences and similarities and stating which theories can explain which empirical facts describe the relationship between rates of return and bond prices and why this is so important for the absence of arbitrage conditions on which the expectations hypothesis focuses. 175 13. The term structure of interest rates 13.4 Reading advice The main readings for this section are the chapters in the textbooks by Goodhart (1989), Harris (1985) and especially Mishkin (2003). Chapter 7 of Mishkin should be read before starting this part of the subject as it starts by giving a general introduction to the ideas behind the term structure and then develops the relevant theory. The two entries in the New Palgrave Dictionary by Malkiel and Mishkin are very useful but the chapter by Shiller in the Handbook of Monetary Economics is difficult and should only be read if you feel comfortable with the material. 13.5 Essential reading Goodhart, C.A.E. Money, Information and Uncertainty. (London: Macmillan, 1989) Chapter 11. Harris, L. Monetary Theory. (New York; London: McGraw-Hill, 1985) Chapter 17. Malkiel, B.G. ‘The term structure of interest rates’, in Newman, P., M. Milgate and J. Eatwell (eds) The New Palgrave Dictionary of Money and Finance. (London: Macmillan, 1994). Mishkin, F.S. The Economics of Money, Banking and Financial Markets. (Boston, Mass.; London: Addison Wesley, 2003) Chapter 7. Mishkin, F.S. ‘The yield curve’, in Newman, P., M. Milgate and J. Eatwell (eds) The New Palgrave Dictionary of Money and Finance. (London: Macmillan, 1994). 13.6 Further reading Books Shiller, R.J. ‘The term structure of interest rates’, in Friedman, B. and F. Hahn (eds) Handbook of Monetary Economics. (Amsterdam: North-Holland, 1990). Walsh, C.E. Monetary Theory and Policy. (Cambridge, Mass.: MIT Press, 2003) Chapter 10. Journal articles Aksoy, Y. and H. Basso, ‘Liquidity, term spreads and monetary policy’, Economic Journal, 124, 2014, 1234–1278. Mankiw, N.G. and L.H. Summers ‘Do long-term interest rates overreact to short-term interest rates?’, Brookings Papers on Economic Activity (1984) 1, pp.223–247. McCallum, B.T. ‘Monetary policy and the term structure of interest rates’, National Bureau of Economic Research working paper, w4938, (1994). Shiller, R.J. ‘The volatility of long-term interest rates and expectations models of the term structure’, Journal of Political Economy 87(5, Part 2) 1979, 1190–1219. 176 13.7. The yield curve 13.7 The yield curve Governments issue a number of differently dated bonds, from short-term paper through to very long-dated bonds that mature in a number of years in the future. Each bond will pay a different rate of return, also known as yield to maturity. If we plot on a graph the time to maturity of government debt on the horizontal axis and the yield to maturity on the vertical axis, this will give us what is known as the yield curve. An example of such a curve is shown in Figure 13.1. Figure 13.1 shows that, for this example, government debt that matures in one month’s time pays an (annualised) interest rate of 4%. One-year paper pays 6% and debt that matures in 10 years from now pays an annual rate of return of 9%. Activity 13.1 What does the yield curve look like in your country? Figure 13.1: 13.8 Why is the yield curve of importance to policy-makers? Central Banks, when setting interest rates, only set one interest rate, normally at the very short end of the spectrum. In the US, the Federal Reserve sets overnight rates. However, investment decisions and aggregate demand in the economy will tend to depend on long-term interest rates since firms will compare the rate of return on investment projects that accrue over the entire life of the project (15 years plus), to the 177 13. The term structure of interest rates alternative of buying equivalent dated bonds maturing in 15 years’ time or so. If aggregate demand depends on long-term interest rates and the monetary authorities set short-term rates but wish to affect aggregate demand, they will need to know the relationship between short-term and long-term rates of return (i.e. they need to know the shape of the yield curve). But why then do Central Banks not just set long-term rates of return and allow the market to determine the rates of return along the rest of the yield curve? If the authorities set long-term rates, they may be faced with large capital losses due to fluctuations of bond prices in the market. To avoid the possibility of facing large price variations, the authorities instead set the rate of return on short-dated paper, on which the possibility of capital gains/ losses is very small or non-existent. However, not only is the yield curve important for analysing the effects of monetary policy on the economy, it is also important because it gives information on expected future inflation, as will be explained below. 13.9 Bond prices and the rate of return There are, in general, two types of bonds a government can issue. These are: Coupon bonds. Typically, coupon bonds promise to pay a fixed sum of money, the coupon, every period, whether it is every month, quarter or year, and also promise to pay the holder of the bond its face value on maturity. For example, a ten-year bond may be bought from the government for £100 that promises to pay £5 (the coupon payment) every year for the next ten years. In the final year, both the last £5 coupon and the maturity value of the bond, £100, are paid. Discount bonds. Discount bonds do not offer any interest or coupon payments. Instead, they are sold at a ‘discount’ and pay a larger amount on maturity. For example, a ten-year discount bond may be sold today for £60 that pays no coupons at all but pays £100 on maturity, ten years from now. Consider a coupon bond that pays C every year from now to infinity. Such bonds, without a maturity date, are called perpetuities or Consols (the UK term). The price of this bond, P , will be the future cash flow, discounted back to today by the interest rate, R, assumed constant here for convenience.   C C C 1 C C + + ··· = + = . (13.1) P = 2 3 1 + R (1 + R) (1 + R) 1 + R 1 − 1/(1 + R) R There is then a negative relationship between the price of a bond and the interest rate. This was discussed in Chapter 2 when examining Keynes individual money demand function. (For bonds that do have a maturity date, the relationship between bond prices and the interest rate, although still negative, is not as simple as (13.1). Intuitively, if the interest rate increased, people would be willing to pay less for a bond that pays a fixed coupon payment each period. If a perpetuity bond that paid a coupon of £5 had a price of £100, this implies the market interest rate is 5%. If the market interest rate doubled to 10%, no investor would be willing to pay more than £50 for the same bond since the £5 coupon on a £50 bond is 10%. Hence the bond price and interest rate are negatively related. 178 13.10. Empirical regularities of the term structure 13.10 Empirical regularities of the term structure There are three features of the term structure that any theory of the yield curve should be able to explain: 1. Rates of return on short- and long-dated bonds move together over time. 2. When short rates are low, the yield curve is likely to be upward sloping and vice versa. 3. Yield curves generally have a persistent upward slope. We now examine a number of theories that try to explain these features. 13.11 The expectations hypothesis The expectations hypothesis of the yield curve links the rate of return on short-term bonds to that on long-term bonds, essentially by assuming that short and long-dated bonds are perfect substitutes. Consider someone who wishes to save a fixed amount of money for n periods. She could either buy long-dated bonds now, at date t, that pay t Rt+n per cent per year. The left-hand subscript denotes the time when the bond is bought and the right-hand subscript denotes when the bond will mature. Alternatively, she could buy a bond that matures at date t + 1 (a one-period bond) paying a rate of return t rt+1 . When this matures she will buy another one-period bond (at date t + 1 that matures at date t + 2) paying a rate of return t+1 rt+2 and will continue doing this until date t + n. If the return from a portfolio made up of long-dated bonds is not identically equal to the expected return from continually rolling over one-period debt, then arbitrage opportunities will be exploited to bring the two returns together. Suppose for example that a two-year bond paid a rate of return of 4% per year. If the rate of return on a one-year bond was 3% today and expected to remain at 3% next year, investors will rush in to buy the longer-dated bond as it pays a higher rate of return. The increased demand for the two period bonds will push the price up and, as explained above, will push the rate of return down from 4%. Also, the reduction in demand for one-period bonds will push the price down, causing the one-period rate of return to increase from 3%. In equilibrium, under the expectations hypothesis, the total return from each portfolio must be equal to avoid arbitrage opportunities. Using the same notation as above, but noting that at date t when the saving decision is made, all future one-period rates are not known and have to be estimated, then the discount bond reads: e e e )(1 + t+2 rt+3 ) · · · (1 + t+n−1 rt+n ). (1 + t Rt+n )n = (1 + t rt+1 )(1 + t+1 rt+2 (13.2) The left-hand side is the total return from holding n period bonds until they mature. The right-hand side is the total expected return from holding and continually rolling over oneperiod bonds, n times. The rates of return on all bonds bought at date t + 1 onwards are not known at date t, hence the e (expectation) superscript. Taking logs of 179 13. The term structure of interest rates (13.2), noting that ln(1 + X) ≈ X for small X, gives: n · t Rt+n = ⇒ t Rt+n t rt+1 e e e + · · · + t+n−1 rt+n + t+2 rt+3 + t+1 rt+2 t+n−1 1 X e = . sr n s=t s+1 (13.3) This is one of the main results of the expectations hypothesis; the long-term interest rate is an average of the current and all future expected short-term interest rates. The expectations hypothesis does have a number of criticisms, however: It cannot explain the empirical fact that the yield curve has a persistent upward slope. If the long rate is an average of current and expected future short rates, this can only be explained if the short rate increases through time. This clearly is not the case. If the long rate is an average of current and future short rates, the long rate must be a smoother series (when plotted through time) than the short rate. This is not true; the long rate is just as volatile. In order to avoid arbitrage opportunities and keep total returns equal, if the rate of return on long-term bonds is greater than the return on short bonds, then holding long-dated bonds must be accompanied by a capital loss (i.e. the price of long-term bonds must fall). If the price of long-term bonds falls, the rate of return must increase still further in the next period. Putting this another way: if the long rate is higher than the short rate, the long rate must increase. This does not happen in reality. The expectations hypothesis and expected inflation If the long rate is an average of the current and expected future short rates, then an upward-sloping yield curve suggests that the short rate is expected to increase in the future (see (13.3)). If the real interest rate is constant then the increase in expected future nominal interest rates must be associated with an increase in expected future inflation, as per the Fisher equation. The greater the slope of the yield curve, the more short-term rates are expected to rise and so the faster is inflation expected to increase. 13.12 The segmentation hypothesis Whereas the expectations hypothesis assumes short- and long-term bonds are perfect substitutes so the decision as to whether to hold long- or short-dated debt depends entirely on expected returns, the segmentation hypothesis assumes short- and long-term bonds are not substitutes in any way. The rate of return on m period bonds will not depend on the market for, or the return on, m − j period bonds at all. Instead, its rate of return will depend entirely on the demand for and supply of credit that matures in m periods’ time. If the demand for m period bonds increased, caused by more people wanting to save for m periods, then the price of m period bonds will increase and the 180 13.13. Preferred habitat theory rate of return will fall. The markets for m, m + 1, m + 2, etc. period debt are said to be segmented. Whereas the expectations hypothesis could not explain the persistent upward sloping nature of the yield curve, this feature can be easily explained if we use the segmentation hypothesis. If people generally prefer to hold short-dated debt (i.e. save by buying shortterm bonds), then this will cause the price of short-term bonds to be high, relative to longterm bonds, and so the rate of return on short-term debt will then be lower than the longterm rate, implying an upward-sloping yield curve. Activity 13.2 Why do you think people would prefer to hold short-term, rather than long-term, debt? (Hint: think of the desire to lock up money in long-term bonds and the risk of capital gains or losses when such holdings have to be liquidated.) Despite being able to explain persistent upward-sloping yield curves, the segmentation hypothesis cannot explain the fact that interest rates move together since the markets are completely segmented. Also, the theory cannot explain why yield curves are upward (downward)-sloping when short rates are low (high). 13.13 Preferred habitat theory The preferred habitat theory lies in between the expectations and segmentations hypotheses. It assumes bonds are neither fully substitutable nor non-substitutable. Instead, people have a preferred bond maturity they wish to hold (as in the segmentation hypothesis) but will be willing to move to another bond maturity if the gains from doing so are significant (so exploiting excess arbitrage opportunities as in the expectations hypothesis). The preferred habitat theory can be represented as (13.4): t Rt+n = t+n−1 1 X e + t kn . sr n s=t s+1 (13.4) This is exactly the same as the expectations hypothesis, (13.3), except for the inclusion of a term premium, t kn . If n period bonds were not the bond of choice, then people would need an extra rate of return in order to encourage them to hold such debt. In this case t kn would be positive. If people generally tend to prefer short-dated debt then the term premium will be a monotonic function of maturity; in order to hold longer and longer dated debt, an increasing term premium must be offered. When combined with the ‘expectations hypothesis’ component, (13.4) can explain the persistent upward slope of the yield curve, along with the other empirical regularities: the co-movement of short and long rates, and the fact the yield curve tends to be upward (downward)-sloping when short rates are low (high). However, using the term premium to fix the problems of the expectations and segmentations hypotheses, is arguably a not a proper solution. A theory of the term premium should be provided, rather than simply assumed in order to make the data consistent with the ‘fixed’ theory. For a recent paper on a theory of term premium that builds upon maturity transformation risk faced by the banks, see Aksoy and Basso (2014). 181