As already noted, various factors such as default risk, liquidity and taxes affect the price of bonds and, at the same time, interest rates. However, the most important factor influencing interest rates is that of the bond maturities. Bonds with identical risk, liquidity and taxes may have very different interest rates due to the fact that the time until maturity is not the same.
Thus, the graph presenting the interest rates of different bonds with the same risk, liquidity and tax but different maturities is known as the yield curve or term structure of interest rates (TSIR). More formally, the TSIR can be defined as the function relating interest rates to terms until the securities mature for bonds with similar credit ratings. Obviously, for each level of credit rating there is a curve of different rates, and thus the worse the credit rating is, the higher the interest rates are and vice versa.
The term structure of interest rates can be defined with different types of bond that have very different payment structures, but for consistency the TSIR is normally created with the interest rates of zero-coupon bonds.
Thus, TSIRs that are created in this way are homogeneous and unless otherwise stated, from here onwards the TSIR will refer to its zero-coupon bonds.
Figure 5.2 shows the zero-coupon yield curve of government debt in Spain on three specific dates.
When the yield curve rises, short-term rates are lower than long-term rates; when it isflat, short-term and long-term rates are equal; and when it falls, short-term rates are higher than long-term rates. Moreover, as can be seen in Fig.5.3, other shapes can be achieved (for example, rising at the beginning and falling at the end).
Fig. 5.2 Zero-coupon curve government debt in Spain (Data source:
Bloomberg; Author’s own composition)
Fig. 5.3 Zero-coupon curve (Author’s own composition)
5.2.1 Term Structure Theories of Interest Rates
Detailed study of the TSIR is very complex because each interest rate for each of the many bond price terms has a particular dynamic but is closely related to the dynamics of the other interest rates. Moreover, apart from having an interest rate based on each maturity, as will be discussed later, there is an interest rate for each issuer with its own dynamics but also closely related to the other interest rates at different maturities and from different issuers. Finally, note that interest rates are closely related to exchange rates.
For the reasons detailed above, a coherent characterisation of the general dynamics of interest rates and particularly its TSIR far exceeds the scope of this book and therefore will not be discussed. Regarding the TSIR, some brief points about its dynamics will be outlined. Specifically, the question we will try to answer is: why does the yield curve generally rise but sometimes take other forms?
As well as explaining why the yield curve assumes a certain form or another, at this level a good TSIR theory is considered to be one that explains the following three facts:
a) Interest rates on bonds of different maturities tend to move in the same direction (if one rises, all of them rise; if one falls, all of them fall).
b) When short-term interest rates are low, the yield curve is likely to rise, whereas when short-term interest rates are high, the yield curve is likely to decrease.
c) The yield curve almost always rises.
As only three events must be explained, instead of outlining a coherent and well-grounded financial theory of interest rates, three qualitative theories will be presented in an attempt to shed some light on the subject.
In particular, note that there are three main theories to explain the behaviour of the TSIR, that is, the relationship between such rates at different maturities: the expectations theory, the segmentation theory and the liquidity premium theory (or the preferred habitat theory). The expectations theory explains thefirst two events, while the segmentation
theory explains the third. For this reason, fusing both theories brings us to the third (liquidity premium theory) which manages to explain all three events.
The expectations theory of the TSIR is based on the following (obvi- ous) statement: the interest rates of a long-term bond are equal to the average expected short-term interest rates over the life of the bond. In other words, if short-term interest rates are expected to rise, long-term interest rates today will be higher than short-term interest rates. The key assumption behind this is that bond buyers do not prefer bonds according to their terms, but if a bond has a lower expected return than another at a different maturity, they will purchase the bond with the highest expected return; this means that bonds at different maturities are perfect substitutes.
Under this assumption, the following investment strategies are equiv- alent: buying a one-year bond and after this period buying another one-year bond or buying a two-year bond. Therefore, if the strategies are equivalent, the profitability of both must be the same. The profitabil- ity of thefirst one is: (1 +r0 , 1) (1 +r1 , 1e) 1ẳr0 , 1+r1 , 1e+r0 , 1*r1 , 1e, wherer0,1, is the interest rate today (tẳ0) of the one-year bond whiler1,1e
is the interest rate that one-year bonds are expected to have within a year (tẳ1). The yield of the second is (1 +r0 , 2)21ẳ2r0 , 2+ (r0 , 2)2 where r02is the interest rate of the bond today in two years.
Equating the return of both strategies and assuming for simplicity that bothr0 , 1*r1 , 1eand (r0,2)2are zero, because the interest rates are usually around 2–5 % and therefore (5 %)2ẳ0.25 % which is much lower than 5 %, it is calculated that r0 , 1+r1 , 1eẳ2r0 , 2 or equally r0, 2ẳr0,1ỵr2 1e,1. The interest rate of the zero-coupon bond in two years should be equal to the average interest rate of the zero-coupon bond in one year and the interest rate that the one-year zero-coupon bond is expected to have within a year. The reasoning is identical with longer maturity bonds.
The expectancy theory is an elegant theory that gives us an explanation of why the term structure of interest rates changes over time. When the yield curve increases, the expectancy theory tells us that short-term interest rates are expected to rise in the future, as easily demonstrated from what has been seen so far: if r0 , 2>r0 , 1, thenr1,1e must be greater
thanr0,1 and thus r0, 2ẳr0,1ỵr2 1e,1. This implies that the interest rate of one-year bonds is expected to rise within a year. By the same reasoning, if the yield curve falls, short-term rates are expected to fall in the next year.
This theory explains fact (a), which states that interest rates of bonds with different maturities tend to move in the same direction. The reason is that historically it has been observed that when short-term interest rates rise, they are expected to continue rising in the future. If short-term rates rise, they will be expected to continue rising and for this reason, as long- term rates are the average between short-term rates and their expectations, when short-term rates and their expectations rise, long-term rates also rise.
Therefore, if short-term rates rise, long-term rates also rise and vice versa, and for this reason the rates move together, that is, if short-term rates move in one direction, long-term rates move in the same direction.
In addition, this expectancy theory explains fact (b), which states that when short-term interest rates are low, the yield curve is likely to increase, whereas when short-term interest rates are high, the yield curve is likely to fall. The reason is that historically, when short-term interest rates are low, the market expects them to rise and, therefore, following the same reasoning as before, which is that r1t+ 1e>r1tso r2t>r1t when short- term interest rates are low, the yield curve rises and, by the same reason- ing, when short-term interest rates are high, the yield curve falls.
The expectancy theory is an attractive theory because it gives us a simple explanation for the behaviour of the yield curve but it has one drawback: it fails to account for fact (c), which states that the yield curve usually rises. Following the logic of this theory, if the yield curve usually rises, then generally short-term interest rates are expected to rise in the future; however, it is noted that in practice short-term interest rates are equally likely to fall as they are to rise, and therefore the probability of the curve rising should be equal to the probability of it falling, which contra- dicts fact (c).
As its name suggests, the segmentation theory of the TSIR assumes that the markets of different maturity bonds are completely segmented and separated. The price and the interest rate of a bond with a given maturity is determined by the intersection of the supply curve and the demand curve for that bond, and these curves for a bond with a given maturity
have nothing to do with the supply and demand curves for another bond with a different maturity. For this reason, in this theory the key assump- tion is that bonds at different maturities are not substitutes for each other, which is why the change in interest rates of a bond at a given maturity does not affect the interest rate of bonds with a different maturity. This theory is the extreme opposite of the segmentation theory, in which it was assumed that bonds at different maturities are perfect substitutes.
According to this theory, bonds with different maturities are not sub- stitutes for each other, as investors have very strong preferences for certain types of bonds as opposed to others and therefore each investor looks for bonds with the maturity that interests them. An investor that needs to recover their money in the short term will invest in bonds maturing shortly, while if they want to invest in paying for their children’s educa- tion they will focus on long-term maturities.
On the other hand, considering the fact that investors are risk averse and, therefore, prefer short-term investments, the segmentation theory can explain fact (c), which states that the yield curve rises. As indicated, the reason is that the demand for short-term bonds is usually higher than for long-term bonds and thus, having less demand, the price of the long- term bond will be less than that of the short-term bond. However, the short-term interest rates will be lower than the long-term rates and consequently the yield curve tends to rise.
Although this theory can explain fact (c), it fails when attempting to explain facts (a) and (b). As there is no relationship between bonds with one maturity and bonds with another if some interest rates move in one direction, there is no reason for all the others to move in the same direction and vice versa (fact a). In the same way, there is also no reason to apply fact (b), which states that if short-term rates are low, the yield curve rises and if they are high, it falls.
The liquidity premium theory of the TSIR assumes that the interest rate of a long-term bond is the average of the short-term interest rate and of its expectations over the life of the long-term bond plus a liquidity premium which is determined by supply and demand in the long-term bond. This theory assumes that bonds of different maturities are sub- stitutes, although not perfect substitutes, because changes in interest rates of a bond at one maturity affect those of other bond maturities. However,
it also allows for the fact that some investors prefer bonds of one maturity instead of bonds of another.
Investors generally prefer short-term bonds because they have less interest rate risk and, therefore, long-term bonds offer a positive liquidity premium to increase the demand for them. As a result of this theory, we have something similar to what was seen in the expectations theory but with an added liquidity premium:r0,nẳr0,1ỵr1e,1ỵrn2e,1ỵỵrne,1ỵλ0,nwhere r1tis the interest rate in a year,r1,1e
, is the annual interest rate expected in a year,r2,1e
is the annual interest rate expected in two years. . .and,. . .y λ0,n,nis the liquidity premium today (tẳ0) for the bond which matures in“n”years.
In turn, it is assumed that this liquidity premium is always positive and grows with“n”and because of this premium the resulting yield curve tends to rise and to do so more steeply than that derived from the expectations theory.
The preferred habitat theory is very similar to the liquidity premium theory and is based on a less direct approach to modifying the expectation theory, but it comes to the same conclusions. This theory assumes that investors have a preference for one type of bond as opposed to others—
they prefer short-term bonds to long-term bonds—but it also assumes that if they notice differences between short-term and long-term rates, they may also want to buy long-term bonds for the extra return these provide. This leads to an equation like that expressed previously and, of course, also leads to the same conclusion.
This liquidity premium theory (or preferred habitat theory) is consistent with the three empirical facts discussed previously. In regards to fact (a) it can be said that, as in the expectations theory, when short-term interest rates rise, the expectations of short-term rates do too; then long-term rates rise and, therefore, interest rates tend to move in the same direction.
It also explains why the yield curve tends to rise when short-term rates are low and fall when they are high, that is, fact (b). As in the expectations theory, when short-term rates are low, they are expected to rise and when they are high, they are expected to fall; for this reason, when short-term rates are low, long-term rates are high—a rising curve, whereas when short-term rates are high, long-term rates are low—a decreasing curve.
Unlike the expectations theory, the liquidity premium theory and the preferred habitat theory may also explain fact (c), which states that the yield curve usually rises. The reason for this is the liquidity premium, which is always positive and increases with the maturity of the bond. Even if short-term interest rates are expected to remain constant, due to the liquidity premium, the long-term rates will outweigh the short-term rates and will rise according to their maturity.
At this point, the question might arise regarding how these theories can explain the fact that the yield curve sometimes decreases. The answer is easy: this occurs when expectations that short-term rates will fall are so great that their decreasing effect on long-term rates exceeds the effect of the liquidity premium.
Another attractive conclusion of the liquidity premium theory is that the market prediction of future short-term rates can be deduced from the shape of the yield curve: if the yield curve rises with a large slope, it can be concluded that the market expects a rise in short-term rates in the future, whereas if the yield curve rises with a moderate slope, it shows that the market expects short-term rates to remain more or less stable (with very slight increases or decreases). Conversely, if the yield curve is flat, mod- erate decreases in short-term rates are expected while if it falls, large declines are expected in these short-term rates.
5.2.2 The Implicit Interest Rate, the Forward Interest Rate and IRR
As in most markets, including in the case of interest rates, there are spot rates and forward rates. The spot interest rate,r0,n, is the annual cash goal of a simple operation in which a price P0 is paid and is repaid within n years obtaining a value of Pn, that is, P0 ẳ 1ỵrPn
0,n
ð ịn, while the term interest rate or forward rate,rt,Te
, is the interest rate at a given maturity,
“T”, which it is expected to have within a time“t”.
Accordingly, given the close relationship between interest rates, the implicit interest rate, rt,Ti, is also defined as the rate that is implicit between two spot rates, particularly between the spot rate maturing at
“t”and the spot rate maturing at“T+t”. In other words,rt,Tiis defined such that: (1 +r0 ,T+t)ẳ(1 +r0 ,t)*(1 +rt,Ti
If the dynamics of the TSIR were defined by the expectations theory, it ).
is evident that the forward interest rate and the implicit rate would be the same. However, considering the theory which bestfits the dynamics of the TSIR is the liquidity premium, the implicit interest rate will always be higher than the forward interest rate, as the risk premium is always positive. As this risk premium increases in line with the maturity, the greater the maturity is, the greater the difference between the implicit interest rate and future interest rate will be.
Finally, and tofinish characterising the interest rate, a concept known as the internal rate of return (IRR) must be defined. Once the TSIR for a level of risk is known, the price of a payment structure similar to the TSIR risk should then be calculated by simply discounting each expected payment at the initial time with the interest rate of the corresponding TSIR: Pẳ Payment1
1þr0,T1
ð ịT1ỵ:::ỵ Paymentn
1þr0,Tn
ð ịTn, “payment” being the payment expected to occur in each of theT1,T2,. . .,Tnperiods. Thus, for a given payment structure, the IRR,“r”, is defined as the discount rate such that PẳPaymentð1ỵrịT11ỵ:::ỵPaymentð1ỵrịTnn, that is, the internal rate of an investment is simply the geometric average of expected future returns from that investment.
For a given payment structure, this rate“r”(IRR)financially equals the price paid and the profits made, which is why it is the interest rate that the market assigns to these securities or equivalently the internal point at which this operation occurs.