Chapter 3
THE YIELD CURVE
Understanding and appreciating the yield curve is important to all capital market participants. It is especially important to debt capital market participants, and even more especially important to bank ALM practitioners. So for anyone reading this book it is safe to assume that the yield curve is a very important subject! This is a long chapter but well worth getting to grips with. In it, we discuss the basic concepts of the yield curve, as well as its uses and interpretation. We show how to calculate the zero-coupon (or spot) and forward yield curve, and present the main theories that seek to explain its shape and behaviour. We will see that the spread of one different curve to another, such as the swap curve compared with the government curve, is itself important. We begin with an introduction to the curve and interest rates.
Importance of the yield curve
Banks deal in interest rates and credit risk. These are the two fundamental tenets of banking – just as fundamental today as they were when banking first began. The first of these – interest rates – is an explicit measure of the cost of borrowing money and is encapsulated in the yield curve. For bankers, understanding the behaviour and properties of the yield curve is an essential part of the ALM process. The following are some, but not all, of the reasons that this is so:
- changes in interest rates have a direct impact on bank revenue; the yield curve captures the current state of term interest rates and also presents the current market expectation of future interest rates;
- the interest rate gap reflects the state of bank borrowing and lending; gaps along the term structure are sensitive to changes in the shape and slope of the yield curve;
- current and future trading strategy, including the asset allocation and credit policy decision, will impact interest rate risk exposure and therefore will take into account the shape and behaviour of the yield curve.
We can see then that understanding and appreciating the yield curve is a vital part of ALM operations. This chapter is a detailed look at the curve from the banker’s viewpoint.
The yield curve is an important indicator and knowledge source of the state of a debt capital market. It is sometimes referred to as the term structure of interest rates, but strictly speaking this is not correct, as this term should be reserved for the zero-coupon yield curve only. But we don’t need to worry about this.
The analysis and pricing activity that takes place in financial markets revolves around the yield curve. The yield curve describes the relationship between a particular yield and its term to maturity. So, plotting yields of a set of bonds along the maturity structure will give us our yield curve. The primary yield curve in any domestic capital market is the government bond yield curve – for example, in the US market it is the US Treasury yield curve. Outside government bond markets, yield curves are plotted for Eurobonds, money market instruments, off-balance-sheet instruments – in fact, virtually all debt market instruments. So, it is always important to remember to compare like for like when analysing yield curves across markets.
Using the yield curve
The yield curve tells us where the bond market is trading now. It also implies the level of trading for the future, or at least what the market thinks will be happening in the future. In other words, it is a good indicator of the future level of the market. It is also a much more reliable indicator than any other used by private investors, and we can prove this empirically. But, for the moment take my word for it!
As an introduction to yield curve analysis, let us first consider its main uses. All participants in debt capital markets will be interested in the current shape and level of the yield curve, as well as what this information implies for the future. The main uses are summarized below.
Setting the yield for all debt market instruments. The yield curve essentially fixes the price of money over the maturity structure. The yields of government bonds from the shortest maturity instrument to the longest set the benchmark for yields for all other debt instruments in the market, around which all debt instruments are priced. What does this mean? Essentially, it means that if a government 5-year bond is trading at a yield of 5.00%, all other 5-year bonds, whoever they are issued by, will be issued at a yield over 5.00%. The amount over 5.00% that the other bond trades is known as the spread. Therefore, issuers of debt use the yield curve to price bonds and all other debt instruments. Generally, the zero-coupon yield curve is used to price new-issue securities, rather than the redemption yield curve.
Acting as an indicator of future yield levels. As we discuss later in this chapter, the yield curve assumes certain shapes in response to market expectations of future interest rates. Bond market participants analyse the present shape of the yield curve in an effort to determine implications regarding the direction of market interest rates. This is perhaps one of the most important functions of the yield curve. Interpreting it is a mixture of art and science. The yield curve is scrutinized for its information content not just by bond traders and fund managers but also by corporate financiers as part of their project appraisals. Central banks and government Treasury departments also analyse the yield curve for its information content, not just regarding forward interest rates but also inflation levels. They then use this information when setting interest rates.
Measuring and comparing returns across the maturity spectrum. Portfolio managers use the yield curve to assess the relative value of investments across the maturity spectrum. The yield curve indicates returns that are available at different maturity points and is therefore very important to fixed interest fund managers, who can use it to assist them to assess which point of the curve offers the best return relative to other points.
Indicating the relative value between different bonds of similar maturity. The yield curve can be analysed to indicate which bonds are ‘cheap’ or ‘dear’ (expensive) to the curve. Placing bonds relative to the zero-coupon yield curve helps to highlight which bonds should be bought or sold, either outright or as part of a bond spread trade.
Pricing interest rate derivative instruments. The price of derivatives such as futures and swaps revolves around the yield curve. At the shorter end, products such as forward rate agreements are priced off the futures curve, but futures rates reflect the market’s view on forward 3-month cash deposit rates. At the longer end, interest rate swaps are priced off the yield curve, while hybrid instruments that incorporate an option feature such as convertibles and callable bonds also reflect current yield curve levels. The ‘risk-free’ interest rate – one of the parameters used in option pricing – is the T-bill rate or short-term government repo rate, both constituents of the money market yield curve.
Yield-to-maturity yield curve
Yield curve shapes
The most commonly occurring yield curve is the yield-to-maturity yield curve. The process of calculating a debt instrument’s yield to maturity is described in countless finance textbooks. The curve itself is constructed by plotting yield to maturity against term to maturity for a group of bonds of the same class.
Curves assume many different shapes; Figure 3.1 shows three hypothetical types. Bonds used in constructing the curve will only rarely have an exact number of whole years to redemption; however, it is often common to see yields plotted against whole years on the -axis. This is because once a bond is designated the benchmark for that term, its yield is taken to be the representative yield. A bond loses benchmark status once a new benchmark for that maturity is issued.
Figure 3.1 Yield-to-maturity yield curves.
The yield-to-maturity yield curve is the most commonly observed curve simply because yield to maturity is the most frequent measure of return used. The business sections of daily newspapers – if they quote bond yield at all – usually quote bond yields to maturity.
The yield-to-maturity yield curve contains some inaccuracies. This is because the yield-to-maturity measure has one large weakness: the assumption of a constant discount rate for coupons during the bond’s life at the redemption yield level. In other words, we discount all the cashflows of the bond at one discount rate. This is not a realistic assumption to make because we know, just as night follows day, that interest rates in 6 month’s time (used to discount the coupon due in 6 months) will not be the same as the interest rate prevailing in 2 years’ time (used to discount the 2-year coupon). But we make this assumption, nevertheless – for the sake of convenience. However, the upshot of all this is that redemption yield is not the true interest rate for its particular maturity.
By the way, this gives rise to a feature known as reinvestment risk: the risk that – when we reinvest each bond coupon as it is paid – the interest rate at which we invest it will not be the same as the redemption yield prevailing on the day we bought the bond. We must accept this risk, unless we buy a strip or zero-coupon bond. Only zero-coupon bondholders avoid reinvestment risk as no coupon is paid during the life of their bond.
For the reasons we have discussed, the professional wholesale market often uses other types of yield curve for analysis when the yield-to-maturity yield curve is deemed unsuitable – usually, the zero-coupon yield curve. This is the yield curve constructed from zero-coupon yields; it is also known as the term structure of interest rates. We construct a zero-coupon curve from bond prices and redemption yields.
Analysing and interpreting the yield curve
From observing yield curves in different markets at any time, we notice that a yield curve can adopt one of four basic shapes:
- normal or conventional – in which yields are at ‘average’ levels and the curve slopes gently upwards as maturity increases;
- upward-sloping or positive or rising – in which yields are at historically low levels, with long rates substantially greater than short rates;
- downward-sloping or inverted or negative – in which yield levels are very high by historical standards, but long-term yields are significantly lower than short rates;
- humped – where yields are high with the curve rising to a peak in the medium-term maturity area, and then sloping downwards at longer maturities.
Sometimes yield curves incorporate a mixture of the above features. A great deal of effort is spent by bond analysts and economists analysing and interpreting yield curves. There is considerable information content associated with any curve at any time.
The very existence of a yield curve indicates that there is a cost associated with funds of different maturities, otherwise we would observe a flat yield curve. The fact that we very rarely observe anything approaching a flat yield curve suggests that investors require different rates of return depending on the maturity of the instrument they are holding. In the next section we will consider the various explanations that have been put forward to explain the shape of the yield curve at any one time. Why do we need to do this? Because an understanding of why the yield curve assumes certain shapes will help us understand the information that a certain shape implies.
None of the theories can adequately explain everything about yield curves and the shapes they assume at any time, so generally observers seek to explain specific curves using a combination of accepted theories.
Theories of the yield curve
No one mathematical explanation of the yield curve explains its shape at all times. At the same time, some explanations are mutually exclusive. That said, practitioners often seek to explain the shape of a curve by recourse to a mixture of theories.
The expectations hypothesis
The expectations hypothesis suggests that bondholder expectations determine the course of future interest rates. There are two main competing versions of this hypothesis, the local expectations hypothesis and the unbiased expectations hypothesis. The return-to-maturity expectations hypothesis and yield-to-maturity expectations hypothesis are also quoted (see Ingersoll, 1987). The local expectations hypothesis states that all bonds of the same class – but differing in term to maturity – will have the same expected holding period rate of return. This suggests that a 6-month bond and a 20-year bond will produce the same rate of return, on average, over the stated holding period. So, if we intend to hold a bond for 6 months, we will get the same return no matter what specific bond we buy. The author feels that this theory is not always the case, despite being mathematically neat; however, it is worth spending a few moments discussing it and related points. Generally, holding period returns from longer dated bonds are on average higher than those from short-dated bonds. Intuitively, we would expect this, with longer dated bonds offering higher returns to compensate for their higher price volatility (risk). The local expectations hypothesis would not agree with the conventional belief that investors, being risk-averse, require higher returns as a reward for taking on higher risk; in addition, it does not provide any insight into the shape of the yield curve. Essentially though, in theory one should expect that the return from holding any bond for a 6-month period will be the same irrespective of the term to maturity and yield that the bond has at time of purchase.
In his excellent book Modelling Fixed Income Securities Professor Robert Jarrow (1996, p. 50) states
‘…in an economic equilibrium, the returns on similar maturity zero-coupon bonds cannot be too different. If they were too different, no investor would hold the bond with the smaller return. This difference could not persist in an economic equilibrium.’
This is true, but in practice other factors can impact holding period returns between bonds that do not have similar maturities. For instance, investors have restrictions as to which bonds they can hold – for example, banks and building societies are required to hold short-dated bonds for liquidity purposes. In an environment of economic disequilibrium, these investors would still have to hold shorter dated bonds, even if the holding period return was lower.
This is noted by Mark Rubinstein (1999, pp. 84–85) who states in his book Rubinstein on Derivatives,
‘In the real world … it is usually the case that annualised shorter-term riskless returns are lower than longer-term riskless returns … Real assets with shorter-term payouts will tend to have a “liquidity” advantage. In aggregate this advantage will be passed on to shorter-term financial claims on real assets [which results in them having a lower return].’
A related theory is the pure or unbiased expectations hypothesis, which states that current implied forward rates are unbiased estimators of future spot interest rates.1 It assumes that investors act in a way that eliminates any advantage of holding instruments of a particular maturity. Therefore, if we have a positive-sloping yield curve, the unbiased expectations hypothesis states that the market expects spot interest rates to rise. Equally, an inverted yield curve is an indication that spot rates are expected to fall. If short-term interest rates are expected to rise, then longer yields should be higher than shorter ones to reflect this. If this were not the case, investors would only buy the shorter dated bonds and roll over the investment when they matured. Likewise if rates are expected to fall then longer yields should be lower than short yields. The unbiased expectations hypothesis states that the long-term interest rate is a geometric average of expected future short-term rates.
Using elementary mathematics we can prove this theory. Indeed, its premise must be so, to ensure no arbitrage opportunities exist in the market. The hypothesis can be used to explain any shape in the yield curve.
Therefore, a rising yield curve is explained by investors expecting short-term interest rates to rise. A falling yield curve is explained by investors expecting short-term rates to be lower in the future. A humped yield curve is explained by investors expecting short-term interest rates to rise and long-term rates to fall. Expectations, or views on the future direction of the market, are a function mainly of the expected rate of inflation. If the market expects inflationary pressures in the future, the yield curve will be positively shaped, while if inflation expectations are inclined towards disinflation, then the yield curve will be negative. Several empirical studies including one by Fama (1976) have shown that forward rates are essentially biased predictors of future spot interest rates, and often overestimate future levels of spot rates. The unbiased hypothesis has also been criticized for suggesting that investors can forecast (or have a view on) very long-dated spot interest rates, which might be considered slightly unrealistic. As yield curves in most developed country markets exist to a maturity of up to 30 years or longer, such criticisms may have some substance. Are investors able to forecast interest rates 10, 20 or 30 years into the future? Perhaps not, nevertheless this is indeed the information content of, say, a 30-year bond; since the yield on the bond is set by the market, it is valid to suggest that the market has a view on inflation and future interest rates for up to 30 years forward.
The expectations hypothesis is stated in more than one way; we have already encountered the local expectations hypothesis. Other versions include the return-to-maturity expectations hypothesis, which states that total return from holding a zero-coupon bond to maturity will be equal to total return that is generated by holding a short-term instrument and continuously rolling it over the same maturity period. A related version – the yield-to-maturity hypothesis – states that the periodic return from holding a zero-coupon bond will be equal to the return from rolling over a series of coupon bonds, but refers to annualized return earned each year rather than total return earned over the life of the bond. This assumption enables a zero-coupon yield curve to be derived from the redemption yields of coupon bonds. The unbiased expectations hypothesis of course states that forward rates are equal to the spot rates expected by the market in the future. Cox, Ingersoll and Ross (1981) suggest that only the local expectations hypothesis describes a model that is purely arbitrage-free, as under the other scenarios it would be possible to employ certain investment strategies that would produce returns in excess of what was implied by today’s yields. Although it has been suggested2 that differences between the local and unbiased hypotheses are not material a model that describes such a scenario would not reflect investors’ beliefs, which is why further research is required in this area.
The unbiased expectations hypothesis does not in itself explain all the shapes of the yield curve or the information content contained within it, which is why it is often combined with other explanations when seeking to explain the shape of the yield curve, including the liquidity preference theory.
Liquidity preference theory
Intuitively, we might feel that longer maturity investments are more risky than shorter ones. An investor lending money for a 5-year term will usually demand a higher rate of interest than if she were to lend the same customer money for a 5-week term. This is because the borrower may not be able to repay the loan over the longer time period as he may, for instance, have gone bankrupt in that period. For this reason longer dated yields should be higher than short-dated yields, to recompense the lender for higher risk exposure during the term of the loan.3
We can consider this theory in terms of inflation expectations as well. Where inflation is expected to remain roughly stable over time, the market would anticipate a positive yield curve. However, the expectations hypothesis cannot in itself explain this phenomenon, as under stable inflationary conditions one would expect a flat yield curve. The risk inherent in longer dated investments, or the liquidity preference theory, seeks to explain a positive-shaped curve. Generally, borrowers prefer to borrow over as long a term as possible, while lenders will wish to lend over as short a term as possible. Therefore, as we first stated, lenders have to be compensated for lending over the longer term; this compensation is considered a premium for a loss in liquidity for the lender. The premium is increased the further the investor lends across the term structure, so that longest dated investments will, all else being equal, have the highest yield. So, the liquidity preference theory states that the yield curve should almost always be upward-sloping, reflecting bondholders’ preference for the liquidity and lower risk of shorter dated bonds. An inverted yield curve could still be explained by the liquidity preference theory when it is combined with the unbiased expectations hypothesis. A humped yield curve might be viewed as a combination of an inverted yield curve together with a positive-sloping liquidity preference curve.
The difference between a yield curve explained by unbiased expectations and an actual observed yield curve is sometimes referred to as the liquidity premium. This refers to the fact that in some cases short-dated bonds are easier to transact in the market than long-term bonds. It is difficult to quantify the effect of the liquidity premium, because it is not static and fluctuates over time. The liquidity premium is so called because, in order to induce investors to hold longer dated securities, the yields on such securities must be higher than those available on short-dated securities, which are more liquid and may be converted into cash more easily. The liquidity premium is the compensation required for holding less liquid instruments. If longer dated securities then provide higher yields, as is suggested by the existence of the liquidity premium, they should generate on average higher total returns over an investment period. This is not consistent with the local expectations hypothesis.
Segmentation hypothesis
Capital markets are made up of a wide variety of users, each with different requirements. Certain classes of investors will prefer dealing at the shorter end of the yield curve, while others will concentrate on the longer end of the market. The segmented markets theory suggests that activity is concentrated in certain specific areas of the market and that there are no interrelationships between these parts of the market; the relative amounts of funds invested in each of the maturity spectra causes differentials in supply and demand, which results in humps in the yield curve. That is, the shape of the yield curve is determined by supply and demand for certain specific maturity investments, each of which has no reference to any other part of the curve.
For example, banks and building societies concentrate a large part of their activity at the short end of the curve, as part of daily cash management (known as asset and liability management) and for regulatory purposes (known as liquidity requirements). However, fund managers such as pension funds and insurance companies are active at the long end of the market. But, few institutional investors have any preference for medium-dated bonds. This behaviour on the part of investors will lead to high prices (low yields) at both the short and long ends of the yield curve and lower prices (higher yields) in the middle of the term structure.
According to the segmented markets hypothesis a separate market exists for specific maturities along the term structure, hence interest rates for these maturities are set by supply and demand.4 Where there is no demand for a particular maturity, the yield will lie above other segments. Market participants do not hold bonds in any other area of the curve outside their area of interest5 so that short-dated and long-dated bond yields exist independently of each other. The segmented markets theory is usually illustrated by reference to banks and life assurance companies. Banks and building societies usually hold their funds in short-dated instruments for no longer than 5 years in maturity. This is because of the nature of retail banking operations, with a large volume of instant access funds being deposited at banks, and also for regulatory purposes. Holding short-term, liquid bonds enables banks to meet any sudden or unexpected demand for funds from customers. The classic theory suggests that – as banks invest their funds in short-dated bonds – the yields on these bonds are driven down. When they then liquidate part of their holding, perhaps to meet higher demand for loans, the yields are driven up and the prices of the bonds fall. This affects the short end of the yield curve but not the long end.
The segmented markets theory can be used to explain any particular shape of the yield curve, although it perhaps fits best with positive-sloping curves. However, it cannot be used to interpret the yield curve whatever shape it may be, and therefore offers no information content during analysis. By definition, the theory suggests that – for investors – bonds with different maturities are not perfect substitutes for each other. This is because different bonds would have different holding period returns, making them imperfect substitutes for one another.6 As a result of bonds being imperfect substitutes, markets are segmented according to maturity.
The segmentations hypothesis is a reasonable explanation of certain features of a conventional positive-sloping yield curve, but by itself is not sufficient. There is no doubt that banks and building societies have a requirement to hold securities at the short end of the yield curve, as much for regulatory purposes as for yield considerations; however, other investors are probably more flexible and will place funds where value is deemed to exist. Nevertheless, the higher demand for benchmark securities does drive down yields along certain segments of the curve.
A slightly modified version of the market segmentation hypothesis is known as the preferred habitat theory. This suggests that different market participants have an interest in specified areas of the yield curve, but can be induced to hold bonds from other parts of the maturity spectrum if there is sufficient incentive. Hence, banks may at certain times hold longer dated bonds once the price of these bonds falls to a certain level, making the return on the bonds worth the risk involved in holding. Similar considerations may persuade long-term investors to hold short-dated debt. So, higher yields will be required to make bondholders shift out of their usual area of interest. This theory essentially recognizes the flexibility that investors have – outside regulatory or legal constraints (such as the terms of an institutional fund’s objectives) – to invest in whatever area of the yield curve they identify value.
The flat yield curve
Conventional theories do not seek to explain a flat yield curve. Although it is rare – certainly for any length of time – to observe flat curves in a market, at times they do emerge in response to peculiar economic circumstances. In conventional thinking, a flat curve is not tenable because investors should in theory have no incentive to hold long-dated bonds over shorter dated bonds when there is no yield premium, so that the yield at the long end should rise as they sell off long-dated paper, producing an upward-sloping curve. In previous occurrences of a flat curve, analysts have produced different explanations for their existence. In November 1988 the US Treasury yield curve was flat relative to the recent past; researchers contended that this was the result of the market’s view that long-dated yields would fall as bond prices rallied upwards.7 One recommendation is to buy longer maturities when the yield curve is flat, in anticipation of lower long-term interest rates, which is diametrically opposite to the view that a flat curve is a signal to sell long bonds. In the case of the US market in 1988, long bond yields did in fact fall by approximately 2% in the following 12 months. This would seem to indicate that one’s view of future long-term rates should be behind the decision to buy or sell long bonds, rather than the shape of the yield curve itself. A flat curve may well be more heavily influenced by supply and demand factors than anything else, with the majority opinion eventually winning out and forcing the curve to change into a more conventional shape.
Further views on the yield curve
In this discussion we have assumed the economist’s world of a perfect market (also sometimes called a frictionless financial market). Such a perfect capital market is characterized by
- perfect information;
- no taxes;
- bullet maturity bonds;
- no transaction costs.
Of course, markets are not perfect in practice. However, assuming perfect markets makes the discussion of the term structure easier to handle. When we analyse yield curves for their information content, we have to remember that the markets that they represent are not perfect, and that frequently we observe anomalies that cannot be explained by conventional theories.
At any one time it is probably more realistic to suggest that a range of factors contribute to the yield curve being a particular shape. For instance, short-term interest rates are greatly influenced by the availability of funds in the money market. The slope of the yield curve (usually defined as 10-year yield minus 3-month interest rate) is also a measure of the degree of tightness of government monetary policy. A low, upward-sloping curve is often thought to be a sign that an environment of cheap money, due to looser monetary policy, is to be followed by a period of higher inflation and higher bond yields. Equally, a high downward-sloping curve is taken to mean that a situation of tight credit, due to stricter monetary policy, will result in falling inflation and lower bond yields. Inverted yield curves have often preceded recessions; for instance, an article in The Economist in April 1998 remarked that in the United States every recession since 1955 bar one has been preceded by a negative yield curve. The analysis is the same: if investors expect a recession they also expect inflation to fall, so the yields on long-term bonds will fall relative to short-term bonds. So, the conventional explanation of an inverted yield curve is that the markets and the investment community expect either a slowdown of the economy – if not an outright recession.8 In this case one would expect monetary policy to ease the money supply by reducing the base interest rate in the near future: hence, an inverted curve. At the same time, a reduction in short-term interest rates will affect short-dated bonds, which are then sold off by investors, further raising their yield.
While the conventional explanation for negative yield curves is expectation of economic slowdown, on occasion other factors are involved. In the UK between July 1997 and June 1999 the gilt yield curve was inverted. However, there was no general view that the economy was heading for recession; in fact, the new Labour government (or should that be New Labour?) inherited an economy believed to be in good health. Instead, the explanation behind the inverted shape of the gilt yield curve focused on two other factors: first, the handing of responsibility for setting interest rates to the Monetary Policy Committee (MPC) of the Bank of England and, second, the expectation that the UK would abandon sterling over the medium term and adopt the euro. The yield curve at this time suggested that the market expected the MPC to be successful and keep inflation at a level around 2.5% over the long term (its target is actually the 1% range either side of 2.5%); it also suggested that sterling interest rates would need to come down over the medium term as part of convergence with conditions in Europe’s euro currency area. However, these were both medium-term expectations and in the author’s view not tenable at the short end of the yield curve. In fact, the term structure moved to a positive-sloped shape up to the 6-to-7-year area, before inverting out to the long end of the curve, in June 1999. By the beginning of 2002 it had assumed a conventional positive-sloping shape. This is a more logical shape for the curve to assume.
There is therefore significant information content in the yield curve, and economists and bond analysts will consider the shape of the curve as part of their policy-making and investment advice. The shape of parts of the curve, whether the short end or long end, as well as that of the entire curve, can serve as useful predictors of future market conditions. As part of an analysis it is also worthwhile considering yield curves across several different markets and currencies. For instance, the interest rate swap curve, and its position relative to that of the government bond yield curve, is also regularly analysed for its information content. In developed country economies the interest rate swap market is invariably as liquid as the government bond market – if not more so – hence, it is common to see the swap curve analysed when making predictions about, say, the future level of short-term interest rates.9
Government policy will influence the shape and level of the yield curve, including its policy on public sector borrowing, debt management and open-market operations. The market’s perception of the size of public sector debt will influence bond yields – for instance, an increase in the level of debt can lead to an increase in bond yields across the maturity range. Open-market operations – that is, the Bank of England’s daily operations to control the money supply (to which end the Bank purchases short-term bills and also engages in repo dealing) – can have a number of effects. In the short term they can tilt the yield curve both upwards and downwards; in the longer term, changes in the level of the base rate will affect yield levels. An anticipated rise in base rates can lead to a drop in prices for short-term bonds, whose yields will be expected to rise; this can lead to a temporary inverted curve. Finally, debt management policy will influence the yield curve. Much government debt is rolled over as it matures, but the maturity of the replacement debt can have a significant influence on the yield curve in the form of humps in the market segment in which the debt is placed, as long as the debt is priced by the market at a relatively low price and hence high yield.
The zero-coupon yield curve
The zero-coupon (or spot) yield curve plots zero-coupon yields (or spot yields) against the term to maturity. A zero-coupon yield is the yield prevailing on a bond that has no coupons. In the first instance – as long as there is a liquid zero-coupon bond market – we can plot the yields from these bonds if we wish to construct this curve. However, it is not necessary to have a set of zero-coupon bonds in order to construct this curve, as we can derive it from a coupon or par yield curve; in fact, in many markets where zero-coupon bonds are not traded, a spot yield curve is derived from the conventional-yield-to-maturity-yield curve. This is of course a theoretical zero-coupon (spot) yield curve, as opposed to a market or observed spot curve that can be constructed using the yields of actual zero-coupon bonds trading in the market.
Spot yields must comply with equation (3.1)#. This equation assumes annual coupon payments and that the calculation is carried out on a coupon date such that accrued interest is zero:
where
In equation (3.1), is the current 1-year spot yield, the current 2-year spot yield and so on. Theoretically, the spot yield for a particular term to maturity is the same as the yield on a zero-coupon bond of the same maturity, which is why spot yields are also known as zero-coupon yields.
This last result is important. It means spot yields can be derived from redemption yields that have been observed in the market.
As with the yield-to-redemption yield curve the spot yield curve is commonly used in the market. It is viewed as the true term structure of interest rates because there is no reinvestment risk involved; the stated yield is equal to actual annual return. That is, the yield on a zero-coupon bond of years maturity is regarded as the true -year interest rate. Because the observed government bond redemption yield curve is not considered to be the true interest rate, analysts often construct a theoretical spot yield curve. Essentially, this is done by breaking down each coupon bond being observed into its constituent cashflows, which become a series of individual zero-coupon bonds. For example, 100 nominal of a 5% 2-year bond (paying annual coupons) is considered equivalent to 5 nominal of a 1-year zero-coupon bond and 105 nominal of a 2-year zero-coupon bond.
Let us assume that there are 30 bonds in the market all paying annual coupons. The first bond has a maturity of 1 year, the second bond of 2 years, and so on out to 30 years. We know the price of each of these bonds, but we wish to determine what the prices imply about the market’s estimate of future interest rates. We naturally expect interest rates to vary over time and that all payments being made on the same date are valued using the same rate. For the 1-year bond we know its current price and the amount of the payment (comprising one coupon payment and the redemption proceeds) we will receive at the end of the year; therefore, we can calculate the interest rate for the first year. Assume the 1-year bond has a coupon of 5%. If the bond is priced at par and we invest 100 today we will receive 105 in one year’s time, hence the rate of interest is apparent and is 5%. For the 2-year bond we use this interest rate to calculate the future value of its current price in 1 year’s time: this is how much we would receive if we had invested the same amount in the 1-year bond. However, the 2-year bond pays a coupon at the end of the first year; if we subtract this amount from the future value of the current price, the net amount is what we should be giving up in 1 year in return for the one remaining payment. From these numbers we can calculate the interest rate in Year 2.
Assume that the 2-year bond pays a coupon of 6% and is priced at 99.00. If 99.00 was invested at the rate we calculated for the 1-year bond (5%), it would accumulate 103.95 in 1 year, made up of the 99 investment and interest of 4.95. On the payment date in 1 year’s time, the 1-year bond matures and the 2-year bond pays a coupon of 6%. If everyone expected the 2-year bond at this time to be priced at more than 97.95 (which is 103.95 minus 6.00), then no investor would buy the 1-year bond, since it would be more advantageous to buy the 2-year bond and sell it after 1 year for a greater return. Similarly, if the price was less than 97.95 no investor would buy the 2-year bond, as it would be cheaper to buy the shorter bond and then buy the longer dated bond with the proceeds received when the 1-year bond matures. Therefore, the 2-year bond must be priced at exactly 97.95 in 12 months’ time. For this 97.95 to grow to 106.00 (the maturity proceeds from the 2-year bond, comprising the redemption payment and coupon interest), the interest rate in Year 2 must be 8.20%. We can check this using the present value formula covered earlier. At these two interest rates, the two bonds are said to be in equilibrium.
This is an important result and shows that there can be no arbitrage opportunity along the yield curve; using interest rates available today the return from buying the 2-year bond must equal the return from buying the one-year bond and rolling over the proceeds (or reinvesting) for another year. This is the known as the breakeven principle.
Using the price and coupon of the 3-year bond we can calculate the interest rate in Year 3 in precisely the same way. Using each of the bonds in turn, we can link together the implied 1-year rates for each year up to the maturity of the longest dated bond. The process is known as boot-strapping. The ‘average’ rate over a given period is the spot yield for that term: in the example given above, the rate in Year 1 is 5%, and in Year 2 is 8.20%. An investment of 100 at these rates would grow to 113.61. This gives a total percentage increase of 13.61% over 2 years, or 6.588% per annum. The average rate is not obtained by simply dividing 13.61 by 2, but – using our present value relationship again – by calculating the square root of ‘1 plus the interest rate’ and then subtracting 1 from this number. Thus, the 1-year yield is 5% and the 2-year yield is 8.20%.
In real-world markets it is not necessarily as straightforward as this; for instance, on some dates there may be several bonds maturing, with different coupons, and on some dates there may be no bonds maturing. It is most unlikely that there will be a regular spacing of bond redemptions exactly 1 year apart. For this reason it is common for analysts to use a software model to calculate the set of implied spot rates which best fits the market prices of the bonds that do exist in the market. For instance, if there are several 1-year bonds, each of their prices may imply a slightly different rate of interest. We choose the rate which gives the smallest average price error. In practice, all bonds are used to find the rate in Year 1, all bonds with a term longer than 1 year are used to calculate the rate in Year 2, and so on. The zero-coupon curve can also be calculated directly from the coupon yield curve using a method similar to that described above; in this case, the bonds would be priced at par and their coupons set to par yield values.
The zero-coupon yield curve is ideal to use when deriving implied forward rates, which we consider next, and when defining the term structure of interest rates. It is also the best curve to use when determining the relative value, whether cheap or dear, of bonds trading in the market, and when pricing new issues, irrespective of their coupons.
Arithmetic
Having introduced the concept of the zero-coupon curve in the previous section, we can illustrate the mathematics involved more formally. When deriving spot yields from redemption yields, we view conventional bonds as being made up of an annuity (the stream of fixed coupon payments) and a zero-coupon bond (the redemption payment on maturity). To derive the rates we can use equation (3.1), setting 00 and, as shown in equation (3.2). This has coupon bonds trading at par, so that the coupon is equal to the yield:
where is par yield for a term to maturity of years, the discount factor is the fair price of a zero-coupon bond with a par value of 1 and a term to maturity of years, and
is the fair price of an annuity of 1 per year for years (with by convention). Substituting equation (3.3) into equation (3.2) and re-arranging will give us the expression for the -year discount factor shown in equation (3.4):
If we assume 1-year, 2-year and 3-year redemption yields for bonds priced at par to be 5%, 5.25% and 5.75%, respectively, we will obtain the following solutions for the discount factors:
We can confirm that these are the correct discount factors by substituting them back into equation (3.2); this gives us the following results for the 1-year, 2-year and 3-year par value bonds (with coupons of 5%, 5.25% and 5.75%, respectively):
Now that we have found the correct discount factors it is relatively straightforward to calculate the spot yields using equation (3.1):
Equation (3.1) discounts the -year cashflow (comprising the coupon payment and/or principal repayment) by the corresponding -year spot yield. In other words is the time-weighted rate of return on a -year bond. Thus, as we said in the previous section the spot yield curve is the correct method for pricing or valuing any cashflow, including an irregular cashflow, because it uses the appropriate discount factors. That is, it matches each cashflow to the discount rate that applies to the time period in which the cashflow is paid. Compare this with the approach for calculating the yield-to-maturity, which discounts all cashflows by the same yield to maturity. This neatly illustrates why the -period zero-coupon interest rate is the true interest rate for an -year bond.
The expressions above are solved algebraically in the conventional manner, although those wishing to use a spreadsheet application such as Microsoft Excel can input the constituents of each equation into individual cells and solve using the ‘Tools’ and ‘Goal Seek’ functions.
There is a very large literature on the zero-coupon yield curve. A small fraction of it – as referred to in this chapter – is given in the Bibliography at the end of the chapter.
Example calculation illustrations
In this section we illustrate some elementary uses of the yield curve by providing some example calculations.
Forward rates: Breakeven principle
Consider the following spot yields:
| 1-year | 10% |
| 2-year | 12% |
Assume that a bank’s client wishes to lock in today the cost of borrowing 1-year funds in 1 year’s time. The solution for the bank (and the mechanism to enable the bank to quote a price to the client) involves raising 1-year funds at 10% and investing the proceeds for 2 years at 12%. The no-arbitrage principle means that the same return must be generated from both fixed rate and reinvestment strategies.
In effect, we can look at the issue in terms of two alternative investment strategies, both of which must provide the same return:
| Strategy 1 | Invest funds for 2 years at 12%. |
| Strategy 2 | Invest funds for 1 year at 10%, and reinvest the proceeds for a further year at the forward rate calculated today. |
The forward rate for Strategy 2 is the rate that will be quoted to the client. Using the present value relationship we know that the proceeds from Strategy 1 are:
while the proceeds from Strategy 2 would be:
We know from the no-arbitrage principle that the proceeds from both strategies will be the same, therefore this enables us to set:
This enables us to calculate the forward rate that can be quoted to the client (together with any spread that the bank might add) as follows:
This rate is the 1-year forward–forward rate, or the implied forward rate.
Further examples
If a 1-year AAA Eurobond trading at par yields 10% and a 2-year Eurobond of similar credit quality, also trading at par, yields 8.75%, what should the price of a 2-year AAA zero-coupon bond be? Note that Eurobonds pay coupon annually:
| (a) Cost of 2-year bond (per cent nominal) | 100 |
| (b) less amount receivable from sale of first coupon on this bond (i.e., its present value) | = 8.75 / = 7.95 |
| (c) equals amount that must be received on sale of second coupon plus principal in order to break even | 92.05 |
| (d) calculate the yield implied in the cashflows below (i.e., the 2-year zero-coupon yield) receive 92.05 pay out on maturity 108.75 Therefore Gives R equal to 8.69% | 92:05 = 108:75/(1 + R) |
| (e) What is the price of a 2-year zero-coupon bond with nominal value 100, to yield 8.69%? | = (92.05/108.75) × 100 = 84.64 |
A highly-rated customer asks you to fix a yield at which he could issue a 2-year zero-coupon USD Eurobond in 3 years’ time. At this time the US Treasury zero-coupon rates were:
| 1 year | 6.25% |
| 2 year | 6.75% |
| 3 year | 7.00% |
| 4 year | 7.125% |
| 5 year | 7.25% |
(a) Ignoring borrowing spreads over these benchmark yields, as a market-maker you could cover the exposure created by borrowing funds for 5 years on a zero-coupon basis and placing these funds in the market for 3 years before lending them on to your client. Assume annual interest compounding (even if none is actually paid out during the life of the loans):
(b) The key arbitrage relationship is:
Therefore, breakeven forward yield is:
FORWARD RATE CALCULATION FOR MONEY MARKET TERM
Consider two positions:
| Borrowing | 100 million today for 30 days at 5.875% |
| Lending | 100 million today for 60 days at 6.125% |
The two positions can be viewed as a 30-day forward 30-day interest rate exposure (a 30-day versus 60-day forward rate). It is usually referred to as an interest rate gap position. What forward rate must be used if the trader wished to hedge this exposure?
The 30-day by 60-day forward rate can be calculated using the following formula:
(3.5) 
where
Using this formula we obtain a 30-day versus 60-day forward rate of 6.3443%.
This interest rate exposure can be hedged using interest rate futures or forward rate agreements (FRAs). Either method is an effective hedging mechanism, although the trader must be aware of
- the basis risk that exists between cash market rates and the forward rates implied by futures and FRAs;
- date mismatches between expiry of futures contracts and the maturity dates of cash market transactions.
Understanding forward rates
Spot and forward rates calculated from current market rates follow mathematical principles to establish what the arbitrage-free rates for dealing today will be at some point in the future. In other words, forward rates and spot rates are actually saying the same thing. However, as we have already noted, forward rates are not a prediction of future rates. It is important to be aware of this distinction. If we were to plot the forward rate curve for the term structure in 3 months’ time, and then compare it in 3 months with the actual term structure prevailing at the time, the curves would certainly not match. However, this has no bearing on our earlier statement: that forward rates are the mathematical expectation of future rates. The main point to bear in mind is that we are not comparing like for like when plotting forward rates against actual current rates at a future date. When we calculate forward rates, we use the current term structure. The current term structure incorporates all known information, both economic and political, and reflects the market’s views. This is exactly the same as when we say that a company’s share price reflects all that is known about the company and all that is expected to happen with regard to the company in the near future, including expected future earnings. The term structure of interest rates reflects everything the market knows about relevant domestic and international factors. It is this information, then, that goes into forward rate calculation. An instant later, though, there will be new developments that will alter the market’s view and therefore alter the current term structure; these developments and events were (by definition, as we cannot know what lies in the future!) not known at the time we calculated and used 3-month forward rates. This is why rates actually turn out to be different from what the term structure mathematically constructed at an earlier date. However, for dealing today we use today’s forward rates, which reflect everything we know about the market today.
BIBLIOGRAPHY
Campbell, J. (1986) ‘A Defence of Traditional Hypotheses about the Term Structure of Interest Rates,’ Journal of Finance, March, 183–193.
Choudhry, M. (1998) ‘The information content of the United Kingdom gilt yield curve,’ unpublished MBA assignment, Henley Management College.
Choudhry, M. (2001) The Bond and Money Markets, Butterworth Heinemann, Chapters 51–53.
Choudhry, M. (2009) ‘The value of introducing structural reform to improve bond market liquidity: Experience from the U.K. gilt market,’ European Journal of Finance and Banking Research, 2(2).
Cox, J., Ingersoll, J.E., and Ross, S.A. (1981) ‘A re-examination of traditional hypotheses about the term structure of interest rates,’ Journal of Finance, 36, September, 769–799.
Culbertson, J.M. (1957) ‘The term structure of interest rates,’ Quarterly Journal of Economics, 71, November, 485–517.
The Economist (1998) ‘Admiring those shapely curves,’ 4 April, p.117.
Fama, E.F. (1976) ‘Forward rates as predictors of future spot interest rates,’ Journal of Financial Economics, 3 (4), October, 361–377.
Fama, E.F. (1984) ‘The information in the term structure,’ Journal of Financial Economics, 13, December, 509–528.
Fisher, I. (1986) ‘Appreciation of Interest,’ Publications of the American Economic Association, August, 23–39.
Hicks, J. (1946) Value and Capital, Oxford University Press, 1946.
Ingersoll, J. (1987) Theory of Financial Decision Making, Rowman & Littlefield, Chapter 18.
Jarrow, R. (1996) Modelling Fixed Income Securities and Interest Rate Options, McGraw-Hill.
Levy, H. (1999) Introduction to Investments, Second Edition, South-Western.
Livingstone, M. (1990) Money and Capital Markets, Prentice Hall.
Lutz, F. (1940) ‘T#he Structure of Interest Rates,’ Quarterly Journal of Economics, November, 36–63.
McCulloch, J.H. (1975) ‘An estimate of the liquidity premium,’ Journal of Political Economy, 83, January/February, 95–119.
Meiselman, D. (1962) The Term Structure of Interest Rates, Prentice Hall.
Rubinstein, M. (1999) Rubinstein on Derivatives, RISK Publishing.
1 For the original discussion, see Lutz (1940) and Fisher (1986), although the latter formulated his ideas earlier.
2 For example, Campbell (1986) and Livingstone (1990).
3 For original discussion, see Hicks (1946).
4 See Culbertson (1957).
5 For example, retail and commercial banks hold bonds for short dates, while life assurance companies hold long-dated bonds.
6 Ibid.
7 See Levy (1999).
8 A recession is formally defined as two successive quarters of falling output in the domestic economy.
9 Interest rate swaps are derivative instruments used in professional wholesale markets to change the basis of an interest rate liability; they are also used for speculative trading purposes. We don’t need to worry about them.