Economics
Term Structure of Interest Rates
The term structure of interest rates refers to the relationship between the interest rates and the time to maturity for a given type of debt. It is typically depicted in a yield curve, which shows the interest rates for different maturities. This concept is important for understanding the expectations and perceptions of market participants regarding future interest rates.
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11 Key excerpts on "Term Structure of Interest Rates"
eBook - PDF- Sudipto Bhattacharya, George Michael Constantinides(Authors)
- 2005(Publication Date)
- World Scientific(Publisher)
Econometrica, Vol. 53, No. 2 (March, 1985) A THEORY OF THE Term Structure of Interest Rates' BY JOHN C. COX, JONATHAN E. INGERSOLL, JR., AND STEPHEN A. Ross This paper uses an intertemporal general equilibrium asset pricing model to study the Term Structure of Interest Rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing. 1. INTRODUCTION THE Term Structure of Interest Rates measures the relationship among the yields on default-free securities that differ only in their term to maturity. The determinants of this relationship have long been a topic of concern for economists. By offering a complete schedule of interest rates across time, the term structure embodies the market's anticipations of future events. An explanation of the term structure gives us a way to extract this information and to predict how changes in the underlying variables will affect the yield curve. In a world of certainty, equilibrium forward rates must coincide with future spot rates, but when uncertainty about future rates is introduced the analysis becomes much more complex. By and large, previous theories of the term structure have taken the certainty model as their starting point and have proceeded by examining stochastic generalizations of the certainty equilibrium relationships. The literature in the area is voluminous, and a comprehensive survey would warrant a paper in itself. It is common, however, to identify much of the previous work in the area as belonging to one of four strands of thought. First, there are various versions of the expectations hypothesis.
eBook - ePubThe Creators of Inside Money
A New Monetary Theory
- D. Gareth Thomas(Author)
- 2018(Publication Date)
- Palgrave Macmillan(Publisher)
The Term Structure of Interest Rates, however, recognises a common link between them all in the form of expectations of the future, whether long or short, which determines the holding of various financial assets of maturity as well as influencing the determination of aggregate demand and supply within the real economy. The term structure, therefore, is important for Central Bank ’s policymakers. If these monetary instruments affect short-term rates of interest in the first instance, which leads to the determination of long-term rates of interest, which drives capital and consumption expenditure, then analysing the term structure is crucial for understanding the transmission mechanism of monetary policy (Fender 2012). 1 Nevertheless, it is also relevant for many households in terms of the portfolio choice of assets. Suppose a family requires expenditure on private school fees in ten years’ time and decides to save now. There are a number of options. They could save by investing into a ten-year bond. Alternatively, they could purchase a short-term bill and then take the earnings into another bond each time it matures, until the ten years are up. Clearly, the important components determining the choice will be the expected return (or cost) and the risk involved, embodied in the term structure. Therefore, the analysis must consider the various theoretical models put forward in the literature to explain the relationship between interest rates on bonds (or bills) of differing maturity, although the hypothesis can applied to other assets as diverse as housing and the mortgage rate. The foremost theory of the Term Structure of Interest Rates is the so-called expectations hypothesis, which focuses on the rôle of expectations of future short-term interest rates in the determination of prices and yields on longer-term bills (or bonds). There a number of ways in which the theory in the literature differs in terms of the length of the bills (or bonds) included in the analysis
eBook - PDF- Brian Kettell(Author)
- 2001(Publication Date)
- Butterworth-Heinemann(Publisher)
56 Economics for Financial Markets corresponding to a 4 per cent yield, relative eagerness to sell securities (to borrow) would drive the price of securities down, and would drive the rate of interest up. Now let us drop the assumption about a single rate of interest and consider why it is that rates of interest or yields on different financial instruments vary. Why do some borrowers pay more than others? Why are the yields on long-dated securities different from those on short-dated ones? Why do interest rates on different currency denominated assets vary? It is to these questions that we now turn. The Term Structure of Interest Rates The Term Structure of Interest Rates, or maturity structure as it is sometimes referred to, refers to the set of theories designed to explain why practically homogeneous bonds of different maturities have different interest rates. The starting point for understanding the term structure theories is the present value concept (discussed in detail in Chapter 2). Because of the time value of money, one dollar received at a future date has a present value of less than one dollar. If we denote the present (that is, time 0) value of $1 received n periods from now by D n , then the interest rate is the discount rate (denoted by R n ) that solves the following equation. D n = 1 (1 + R n ) n (3.1) D n represents both the present value of $1 received in n period and the spot price of a zero coupon bond with a par value of $1. The purchaser of this zero coupon bond pays the purchase price D n at time zero and receives the par value of $1 at time n . The rate R n is called the zero coupon discount rate or the spot interest rate. The spot market is the market for immediate delivery. Some observers call the spot market the ‘cash’ market. The spot price D n and the spot interest rate R n are inversely related. When the spot interest rate goes up, the spot price goes down because the spot interest rate is the denominator.
eBook - PDF- Roy E. Bailey(Author)
- 2005(Publication Date)
- Cambridge University Press(Publisher)
See the introduction to volume I of Ross (2000) and the references cited there. 328 The economics of financial markets opportunities, and (b) if the model of the term structure were correct (i.e. an acceptably close approximation to the actual evolution of bond prices). The predictions can then be compared with realized bond prices in order to inform investment strategies (i.e. to provide signals about which bonds to buy or sell) or to provide price quotations in negotiations about OTC contracts. Alternatively, from a more disinterested academic perspective, the predictions can be tested using methods similar to those described in chapter 9. Here the goal is to identify which theories of economic behaviour are more (or less) consistent with the observed patterns of bond prices and their rates of return. 13.7 Summary 1. Studies of the Term Structure of Interest Rates seek to reveal the relationship among the yields on bonds with different times to maturity. Commonly, the term structure is expressed by a yield curve, which plots yields as a function of the number of years remaining before redemption of the bonds. 2. Because the time to maturity is not the only dimension across which bonds differ, it is necessary to control for other characteristics of bonds, in particular their coupon payments. The most straightforward approach is to construct yield curves for zero-coupon bonds (bonds that promise to make a single payment at maturity). 3. Another attribute of some bonds is that the promised payments are adjusted for changes in the price level. From the observed prices (yields) of these index-linked bonds, it is possible to estimate real yield curves that express yields to maturity adjusted for future price level changes as a function of time to maturity. It then becomes possible to extract estimates of expected inflation rates, for various periods in the future, from market prices (yields) for real and nominal bonds.
eBook - ePubFixed Income Securities
Concepts and Applications
- Sunil Kumar Parameswaran(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
Technically speaking, a “yield curve” is a graph depicting the relationship between the yield to maturity, which is plotted along the Y-axis, and the time to maturity, which is plotted along the X-axis. For the purpose of constructing the yield curve, it is imperative that the bonds being compared belong to the same credit risk class. This is the most commonly used version of the yield curve for the simple reason that the YTM is the most commonly used measure of the yield from a bond.The expression “Term Structure of Interest Rates,” on the other hand, refers to a graph depicting the relationship between spot rates of interest, as shown along the Y-axis, and the corresponding times to maturity, which are plotted along the X-axis. Once again, to facilitate meaningful inferences, the data used to construct the graph should be applicable to bonds of the same risk class. The “Term Structure of Interest Rates” is also referred to as the “zero coupon yield curve” because the YTM of a zero coupon bond is nothing but the spot rate. The zero coupon yield curve is considered to be the true Term Structure of Interest Rates because there is no reinvestment risk. This is because such bonds do not give rise to any cash flows prior to maturity, and consequently there is no risk that cash flows may have to be reinvested at a lower than anticipated rate.The yield curve is equivalent to the term structure if the term structure is flat, or in other words, the spot rates are the same for all maturities. This is because, when the term structure is flat, the YTM, which is a complex average of spot rates, is equal to the observed spot rate. However, when the term structure is not flat, as is usually the case, the YTM is somewhere between the lowest spot rate, and the highest spot rate.
- Frank Skinner(Author)
- 2004(Publication Date)
- Butterworth-Heinemann(Publisher)
For example, if you The sovereign term structure and the risk structure of interest rates 21 think interest rates are likely to increase, but you are not sure, you will short a number of hedging instruments that are less than N . That way if interest rates do rise, you will lose more on your cash asset than you will gain on your hedging instrument, but at least you get some reduction in the loss. But, of course, yields might also unexpectedly fall, and you will experience an overall gain since gains on the cash bond are not fully offset by losses on the hedging instru-ment. You would go to the extremes and make large bets by deviating from the perfect hedge ratio by large amounts only when you have a very strong belief in the direction of future yields. In this way, you can see yourself controlling the amount of risk that you accept in your investments. 2.2 Introduction to the term and risk structure of interest rates The Term Structure of Interest Rates is the schedule of sovereign interest rates organized by term to maturity. Note that the interest rates contained in the sovereign term structure are not risk free; they are subject to interest rate risk. That is a sovereign bond investor will experience losses if interest rates unexpectedly increase. If the bond investor actually sells a sovereign bond once sovereign interest rates increase then the loss will be realized. But even if they do not sell the bond once interest rates have increased, an opportunity loss is experienced since they now have a bond that is not worth as much as they had hoped earlier. Hence we can say that sovereign interest rates are risky, and so we wish to model this risk recognizing that sovereign interest rates are stochastic (have a variance).
eBook - PDFFinancial Institutions
Markets and Money
- David S. Kidwell, David W. Blackwell, David A. Whidbee, Richard W. Sias(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
6.1 The Term Structure of Interest Rates 167 maturity and that they buy and sell securities consistent with these maturity preferences. As a result, the yield curve is determined by the supply of and the demand for securities at or near a particular maturity. Investors who desire short‐term securities, such as commer- cial banks, determine the short‐term yield curve; investors with preferences for intermedi- ate maturities determine the intermediate‐term yield curve; and investors who prefer long‐term securities, such as pension funds and life insurance companies, determine the long‐term yield curve. On the supply side, security issuers tailor the maturities of their security offerings to the length of time they need the borrowed funds (see Exhibit 6.3). EXHIBIT 6.2 The Effect of Liquidity Premiums on the Yield Curve Liquidity premiums increase as maturity increases. Thus, liquidity premiums cause an upward slope in market yield curves. Years to maturity Interest rate 5 0 10 15 Observed yield curve Liquidity premium Expectation theory yield curve EXHIBIT 6.3 Market‐Segmentation Yield Curve Market‐segmentation theory suggests that borrowers and lenders have strong preferences for securities of a particular maturity. As a result, the supply and demand for securities at or near a particular maturity determine the yield for that maturity. Years to maturity Yield D l S l D m S m S s D s Note: Supply (S) and demand (D) for s = short‐term maturities, m = intermediate‐term maturities, and l = long‐term maturities. 168 CHAPTER 6 The Structure of Interest Rates Thus, the market‐segmentation theory assumes that both issuers and investors have a pref- erence for securities with a narrow maturity range. Changes in interest rates in one segment of the yield curve, therefore, have little effect on interest rates in other maturities. Under the segmentation theory, discontinuities in the yield curve are possible.
eBook - PDF- (Author)
- 2022(Publication Date)
- Wiley(Publisher)
The shift to overnight secured funding benchmarks extends globally—for example, the secured European Short-Term Rate (ESTR) has been recommended to replace Eonia, and the Canadian Overnight Repo Rate Average (CORRA) is proposed to replace the survey-based unsecured Canadian Dollar Offered Rate (CDOR). 6. TRADITIONAL THEORIES OF THE Term Structure of Interest Rates • explain traditional theories of the Term Structure of Interest Rates and describe the implications of each theory for forward rates and the shape of the yield curve This section presents four traditional theories of the underlying economic factors that affect the shape of the yield curve. 6.1. Expectations Theory One branch of traditional term structure theory focuses on interpreting term structure shape in terms of investors’ expectations. Historically, the first such theory is known as the unbiased expectations theory, also called pure expectations theory. It says that the forward rate is an unbiased predictor of the future spot rate; its broadest interpretation is that bonds of any maturity are perfect substitutes for one another. For example, buying a bond with a maturity of five years and holding it for three years has the same expected return as buying a three-year bond or buying a series of three one-year bonds. The predictions of the unbiased expectations theory are consistent with the assumption of risk neutrality. In a risk-neutral world, investors are unaffected by uncertainty and risk pre- miums do not exist. Every security is risk free and yields the risk-free rate for that particular maturity. Although such an assumption leads to interesting results, it clearly is in conflict with the large body of evidence showing that investors are risk averse. A theory that is similar but more rigorous than the unbiased expectations theory is the local expectations theory.
eBook - ePub- Erik Lindström, Henrik Madsen, Jan Nygaard Nielsen(Authors)
- 2018(Publication Date)
- Chapman and Hall/CRC(Publisher)
Table 11.1 is that the yield is not constant. The yield varies with the maturity of the bonds. This should come as no surprise, since this is exactly the same situation as in the bank where a savings account pays a higher interest than a check account. In other words, the longer your investment horizon the higher the return. This observation, that the interest rate depends on the investment horizon, is expressed by the zero-coupon Term Structure of Interest Rates (or, in short, the term structure). It is clear that only a limited number of points on the term structure is available, and our problem is basically to determine a reasonable interpolation method. We also note that the prices and the maturities confirm (11.6 ), namely that the price and the yield move in adverse directions.Table 11.1: Skatkammerbeviser quoted at the Copenhagen Stock Exchange April 3, 1997.Figure 11.2: Typical yield curves.From empirical market data it is observed that yield curves typically come in three distinct shapes (as illustrated in Figure 11.2 ), each associated with different economic conditions:- Increasing: this is the most common form for the yield curve as it shows that future interest rates are higher than the short interest rate, since it should be more rewarding to tie money up for a long time than for a short time. E.g., the interest rate of a savings account is typically larger than for a check account.
- decreasing: this is typical of periods when the short rate is high but expected to fall.
- humped: again the short rate is expected to fall, although in a more complicated manner.
Definition 11.5 (The spot interest rate). The instantaneous (spot) interest rate at time t is defined byr( t ) = f( t , t ) ( 11.18 )where f (t, t ) is given by (11.12 ).Note that the spot interest rate (for which a number of SDE models were proposed in Chapter 10 ) is connected to the forward rate. The spot interest rate r (t ) is simply the forward rate obtained by investing our money in a bond in the time interval [t, t + dt ]. For obvious reasons the spot rate is also called the instantaneous rate of interest or the short rate. The process of continuously investing our holdings at the spot rate r (t ) is referred to as rolling over the money (see the discussion later on page 220
eBook - PDFFixed Income Securities
Valuation, Risk, and Risk Management
- Pietro Veronesi(Author)
- 2015(Publication Date)
- Wiley(Publisher)
But the model can be useful to interpret the yield curve. For instance, according to the model, if we observe a yield curve that is relatively steep, this can be due to any of the following three reasons: 1. Market participants expected high future inflation, and thus high future spot rates. A MACROECONOMIC MODEL OF THE TERM STRUCTURE 643 Figure 18.5 Yield Curves in the Macro Economic Model 0 5 10 15 20 25 30 2 4 6 8 10 Maturity Yield (%) Panel A: Yield Curve for Three Values of Expected Inflation High Expected Inflation Average Expected Inflation Low Expected Inflation 0 5 10 15 20 25 30 0 0.5 1 1.5 Maturity Term Spread (%) Panel B: Term Spread for Three Values of Risk Aversion Low Risk Aversion Medium Risk Aversion High Risk Aversion 644 THE RISK AND RETURN OF INTEREST RATE SECURITIES 2. Market participants have a high aversion to risk, and therefore require a higher yield for longer term bonds. 3. The amount of risk is high, and therefore again market participants require a higher yield for longer term bonds. Only the first point implies that a positively sloped term structure predicts higher future spot rates, while points 2 and 3 do not. 18.4 CASE ANALYSIS: THE RISK IN THE P&G LEVERAGED SWAP In Section 17.9 of Chapter 17 we investigated the valuation of a complicated interest rate swap, whose cash flows would depend on the setting of a spread six months after the initiation of the contract according to a complicated formula. In this section, we show how we can use Monte Carlo simulations to gauge the riskiness of such an investment. To perform the risk analysis, we first need to estimate the risk natural parameters of the Vasicek model of interest rates. These parameters can be estimated by using the history of interest rates. In Section 17.9, we approximated the short-term interest rate r t with the 3-month continously compounded T-bill rate 4 and considered two samples: A 3-year sample and a 10-year sample.
eBook - PDF- Lance Moir(Author)
- 1997(Publication Date)
- Woodhead Publishing(Publisher)
The structure of interest rates and the yield curve 7 7 easily, p a r t i c u l a r l y w i t h t h e r a n g e of derivatives available to m a n a g e interest r a t e risk. W h e n c o m p a r i n g i n s t r u m e n t s it is critical to u n d e r s t a n d w h e n interest is p a i d . S o m e i n s t r u m e n t s p a y interest e v e r y 6 m o n t h s , o t h e r s o n fixed dates, a n d o t h e r s c l a i m to c a r r y n o interest b u t offer t h e r e t u r n b y a n increase in capital v a l u e . F o r e x a m p l e , 2 i n s t r u m e n t s b o t h w i t h q u o t e d inter-est rates (sometimes called t h e ' c o u p o n ' ) of 1 0 % m a y n o t a c -tually p r o v i d e t h e s a m e total r e t u r n . If o n e p a y s interest a n n u a l l y a n d t h e o t h e r s e m i -a n n u a l l y , t h e n t h e investor in t h e o n e w i t h s e m i -a n n u a l interest p a y m e n t s will get t h e b e t t e r d e a l as h e o r she c a n reinvest t h e first interest p a y m e n t at t h e e n d of 6 m o n t h s . H o w e v e r , in g e n e r a l m o n e y -m a r k e t in-s t r u m e n t s p a y interest o n m a t u r i t y unless t h e y last for l o n g e r t h a n a y e a r , in w h i c h case interest is also p a i d a n n u a l l y (see C h a p t e r 4). C a p i t a l m a r k e t i n s t r u m e n t s s u c h as b o n d s also o p e r a t e o n different bases, so, E u r o b o n d s p a y interest a n n u a l l y o n a 3 6 0 -d a y y e a r basis, b u t d o m e s t i c sterling b o n d s p a y interest semi-a n n u a l l y o n a 3 6 5 -d a y y e a r basis. Active investors d o n o t t e n d to invest in a s t r a i g h t f o r w a r d i n s t r u m e n t a n d leave it at t h a t -t h e y also m a n a g e t h e yield b y u s i n g derivatives (see C h a p t e r 8) a n d these all h a v e different c o n v e n t i o n s o n w h e n cash actually m o v e s .
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