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Figure 8.11 The Weighting Function of Chartists. Source: De Grauwe et a1 (1993).
by permission of Blackwell Publishers
If for simplicity we assume E,D,+1 = constant and substitute for E,P,+1 (i.e. eq
E,S,+1) from (8.33) we have a non-linear difference equation in P,. However
to our exchange rate example, we see in Chapter 13 that the Euler equation li
E,(S,+1 j is of the form:
s, = Xf[Jw,+llh
where X , = fundamentals, for example relative money supplies, that inf
exchange rate (and De Grauwe et a1 set b = 0.95 and X , = 1, initially). De G
simulate the model for particular parameter values and find that when the ex
parameter y is sufficiently high, chaotic behaviour ensues. This can be clea
Figure 8.12(a) where the equilibrium exchange S rate is normalised to unity.
T
initial condition is changed by 1 percent the time series is similar for about th
periods but then the two patterns deviate quite substantially (e.g. compare the
Figures 8.12(a) and 8.12(b) for periods 200-400).




Figure 8.12 Sensitivity to Initial Conditions Generated by a Difference in the Initia
Rate. (a) Base Run; (bj A 1 Percent Change in the Initial Exchange Rate Compared
Run. Source: De Grauwe et a1 (1993). Reproduced by permission of Blackwell Publis
closely enough) the form of the chaotic system, it may be possible to under
term forecasting. Note also that there has been no random events or ˜news™ tha
required to produce the graphs in Figures 8.12(a) and 8.12(b), which are the
pure deterministic non-linear system. Hence the RE hypothesis is not require
to yield apparent random behaviour in asset prices and asset returns. Finally
recalling that ˜fundamentals™ X , have been held constant in the above simulat
is the inherent dynamics of the system that yield the random time path. Hence
world does contain chaotic dynamics, then agents may discard fundamentals (i
of X,) when trying to predict future asset prices and instead concentrate on m
methods, as used, for example, by chartists and other NTs.
Clearly, the above analysis is a long way from providing a coherent theory of
movements but it does alert one to alternative possibilities to the RE paradigm
agents have homogeneous expectations, know the true model of the economy
and use all available useful information, when forecasting. The above poin
developed further in Chapter 13 when we look in more detail at models of th
rate and discuss tests which attempt to distinguish between systems that are pre
deterministic yet yield chaotic solutions and those that are genuinely stochasti


SUMMARY
8.4
It is now time to summarise this rather diverse set of results on market inef
would appear that some of the anomalies in the stock market are merely manife
a small firm effect. Thus the January effect appears to be concentrated prima
small firms as are the profits to be made from closed end mutual funds, where th
available is highly correlated with the presence of small firms in the portfolio
Thus it may be the case that there is some market segmentation taking place w
smart money only deals in large tranches of frequently traded stocks of large c
The market for small firms™ stocks may be rather ˜thin™ allowing these an
persist. The latter is, strictly speaking, a violation of the efficient markets
However, once we recognise the real world problems of transactions costs, i
costs, and costs of acquiring information in particular markets, it may be tha
money, at the margin, does not find it profitable to deal in the shares of smal
eliminate these arbitrage opportunities. Nevertheless, where the market consis
frequent trades undertaken primarily by institutions, then violation of varian
tests, and the existence of very sharp changes in stock prices in the absence
still need to be explained.
The idea of noise traders coexisting with smart money is a recent and impo
retical innovation. Here price can diverge from fundamentals simply because o
uncertainty introduced by the noise traders. Also the noise traders coexist alo
smart money and they do not necessarily go bankrupt. Though this theory can i
explain sharp movements in stock prices (i.e. bull and bear markets) and indee
volatility experienced in the stock market, nevertheless it has to embrace so
assumptions about behaviour. We have seen that one way to obtain the abo
changes and indeed whether such changes go through tranquil or turbulent pe
noise-trader model also allows stock prices to undergo persistent swings if w
assumption that there is persistence in agents™ perceptions of volatility. Such
in volatility is often found in empirical research (e.g. ARCH models, Chapter
however, this is merely an empirical regularity and there is no rational optimi
of why this should be the case. It all boils down to the mass psychology a
behaviour of participants in the market.
Chaos theory demonstrates how a non-linear deterministic system can gene
ently random behaviour. This applies a fortiori to noisy chaotic systems. What
most important about this strand of the literature is that it alerts us to the po
non-linearities in economic behavioural equations (e.g. see Pesaran and Pott
However, at present it is very much a ˜technique in search of a good econom
although it has been outlined how it can be used to develop a plausible econo
of asset price movements. Another major difficulty in trying to analyse econom
in terms of chaotic models is the very large amount of data required to detect a
opposed to a stochastic) process. A reasonable conjecture might be that chaoti
(and its allied companion, neural networks) could become important statistic
short-term forecasting of asset prices, but unless they are allied to economic the
their usefulness in general policy analysis will be very limited.
Of course none of the models discussed in this chapter are able to expla
a crucial fact, as far as public policy implications are concerned. That is to
do not tell us how far away from the fundamental price a portfolio of partic
might be. For example, if the deviation from fundamental value is only 5 pe
portfolio of stocks, then even though this persists for some time it may not
substantial misallocation of investment funds, given other uncertainties that
the economy. Noise-trader behaviour may provide an apriori case for publi
the form of trading halts, during specific periods of turbulence or of insisting
margin requirements. The presence of noise traders also suggests that hostile
may not always be beneficial for the predators since the actual price they p
stock of the target firm may be substantially above its fundamental value.
establishing a prima facie argument for intervention is a long way short of
specific government action in the market is beneficial.


APPENDIX 8.1
The De Long et a Model of Noise lkaders
1
The basic model of De Long et a1 (1990) is a two-period overlapping generations m
are no first-period consumption or labour supply decisions: the resources agents have
therefore exogenous. The only decision is to choose a portfolio in the first-period (i.e. w
to maximise the expected utility of end of period wealth. The ˜old™ then sell their ris
the ˜new young™ cohort and use the receipts from the safe asset to purchase the consum
The safe asset s is in perfectly elastic supply. The supply of the uncertainhisky ass
and normalised at unity. Both assets pay a known real dividend r (riskless rate) so
of the expected price is denoted p* and at any point in time the actual misperception
according to:
N(P*,02)
pr
Each agent maximises a constant absolute risk aversion utility function in end of perio
U = -exp(-2yw)
If returns on the risky asset are normally distributed then maximising (2) is equivalent to
-
W-P:

where F = expected final wealth, y = coefficient of absolute risk aversion. The SM there
the amount of the risky asset to hold, A; by maximising

E ( U ) = co + ˜ f [ + r q + 1 - pt(1+ r ) ] - Yoif)2rai,+,
r
where co is a constant (i.e. period zero income) and is the one-period ahead
expected variance of price:
ot,+, = ˜ r [ p , +- ˜ r ˜ r + t ] ˜
I
The NT has the same objective function as the SM except his expected return has a
+
term AFpf (and of course A“ replaces A; in (4)). These objective functions are of the
as those found in the simple two-asset, mean-variance model (where one asset is a sa
discussed in Chapter 3.
Setting aE(U)/aAr = 0 in (4) then the objective function gives the familar mean-va
demand functions for the risky asset for the SM and the NTs




+ +
where K+l = rr rPr+1 - (1 r)Pr. The demand by NTs depends in part on their abn
of expected returns as reflected in pt. Since the ˜old™ sell their risky assets to the you
fixed supply of risky assets is 1, we have:
+ pA; = 1
(1 - /%)A;
Hence using (6) and (7), the equilibrium pricing equation is:



The equilibrium in the model is a steady state where the unconditional distribution of
that for P,.Hence solving (9) recursively:




Only Pt is a variable in (10) hence:
p202
2
= o2 = -
(1 + r)2
p,+l
APPENDIX 8.2
The Shleifer-Vishny Model of Short-Termism
This appendix formally sets out the Shleifer-Vishny (1990) model whereby long-term
subject to greater mispricing than short-term assets even though arbitrageurshmart mon
rationally. As explained in the text this may then lead to managers of firms pursuing
projects with short-horizon cash flows to avoid severe mispricing and the risk of a ta
model may be developed as follows.
There are three periods, 0, 1, 2, and firms can invest either in a ˜short-term™ invest
with a $ payout of V , in period 2 or a ˜long-term™ project also with payout only in per
The key distinction between the projects is that the true value of the short-term proj
known in period 1 , but the true value of the long-term project doesn™t become known un
Thus arbitrageurs are concerned not with the timing of the cash flows from the proj
the timing of the mispricing and in particular the point at which such mispricing is r
hence disappears. The market riskless interest rate = 0. All investors are risk neutral.
There are two types of trader, noise traders (NT) and smart money (SM) (arbitrag
traders can either be pessimistic (Si > 0) or optimistic at time t = 0 about the payoffs V
types of project (i = s or g ) . Hence both projects suffer from systematic optimism o
We deal only with the pessimistic case (i.e. ˜bearish™ or pessimistic views by NTs).
for the equity of fr engaged in project i(= s or g) by noise traders is:
im


+
For the bullishness case 4 would equal ( V ; S ; ) / P i .Smart money (arbitrageurs) face
constraint of $b at an interest rate R > 1 (i.e. greater than one plus the riskless ra
traders are risk neutral so they are indifferent between investing all $ b in either of
Their demand curve is:
q ( S M , i) = n i b / P ,
where ni = number of SM traders who invest in asset i (= s or g). There is a unit su
asset i so equilibrium is given by:



and hence using (1) and (2) the equilibrium price for each asset is given by:



It is assumed that nib < Si so that both assets are mispriced at time t = 0. If SM in
t = 0, he can obtain b / y shares of the short-term asset. At t # 1 the payoff per
short-term asset V , is revealed. There is a total $ payoff in period 1 of V , ( b / P : ) . Th
NR, in period 1 over the borrowing cost of bR is:




(where we have used equation 4. Investing at t = 0 in the long-term asset the SM pur
)
shares. In period t = 1 , he does nothing. In period t = 2, the true value V, per shar
+
which discounted to t 1 at the rate R implies a $ payoff of bV,/P,R. The amount o
The only difference between ( 5 ) and (6) is that in (6) the return to holding the (misprice
share is discounted back to t = 1, since its true value is not revealed until t = 2.
In equilibrium the returns to arbitrage over one period, on the long and short as
equal (NR, = NR,) and hence from (5) and (6):




Since R > 1, then in equilibrium the long-term asset is more underpriced (in perce
than the short-term asset (when the noise traders are pessimistic, S , > 0). The diffe
mispricing occurs because payoff uncertainty is resolved for the short-term asset in per
the long-term asset this does not occur until period 2. Price moves to fundamental valu
short asset in period 1 but for the long asset not until period 2. Hence the long-term
value V , has to be discounted back to period 1 and this ˜cost of borrowing™ reduces
holding the long asset.


ENDNOTE
1. To see this, note that E,Yt+* = ut, and ErYt+2 = but after n periods ut
from EtYt+n+l which then equals zero. So if Y , starts at zero, a single po
uf results in a higher expected value for EYr+k for a further It periods on


FURTHER READING
There is a vast and ever expanding journal literature on the topics covered in
will concentrate on accessible overviews of the literature excluding those fou
finance texts.
On predictability and efficiency, in order of increasing difficulty, we ha
(1991), Scott (1991) and LeRoy (1989). A useful practitioner™s viewpoint
lent references to the applied literature is Lofthouse (1994). Shiller (1989),
Section 11: ˜The Stock Market™, is excellent on volatility tests. On ˜bubbles™
section in the Journal of Economic Perspectives (1990 Vol. 4, No. 2) is a use
point with the collection of papers by Flood and Garber (1994) providing a
nical overview.
At a general level Thaler™s (1994) book provides a good overview of
while Thaler (1987) and De Bondt and Thaler (1989) provide examples of
in the finance area. Finally, Shiller (1989) in parts I: ˜Basic Issues™ and V
Models and Investor Behaviour™ are informative and entertaining. Gleick (198
a general non-mathematical introduction to chaos, Baumol and Benhabib (1989
basics of non-linear models while Barnett et a1 (1989) provide a more technica
with economic examples. Peters (1991) provides a clear introduction to chaos
financial markets while De Grauwe et a1 (1993) is a very accessible accoun
exchange rate as an example. The use of neural networks in finance is clearly
in Azoff (1994).
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PART 3
1
I
I 2
The Bond Market
Most governments at some time or another attempt to influence short-term in
as a lever on the real economy or in an attempt to influence the rate of inflat
usually accomplished by the monetary authority either engaging in open market
(i.e. buying or selling bills) or threatening to do so. Changes in short rates (with
inflationary expectations) may influence real inventory holdings and consumer
ture, particularly on durable goods. Short-term interest rates may have an effect
of long-term interest rates on government (and corporate) bonds: this is the
relationship. Corporate bond rates may affect real investment in plant and
Hence the government™s monetary policy can influence real economic activi
allied to this is the idea that governments may be able to ˜twist™ the yield
is, to raise short-term interest rates (to encourage personal saving) while sim
lowering long-term rates to encourage fixed investment. However, if the so-ca
tations hypothesis of the yield curve holds then the authorities cannot alter the r
between short-term and long-term rates and must accept the ˜free market™ co
for long rates of any change in short rates they might engineer. The authorities
influence future short rates through various mechanisms, such as a declared an
policy. Also its open market operations at short maturities may have a direct
short rates and hence, via the expectations hypothesis, on long rates. Changes i
short rates may influence capital flows, the exchange rate and hence price comp
the volume of net trade (exports minus imports) and the level of output and em
Part 4 deals with the link between short rates and the exchange rate.
Another reason for governments being interested in the determinants of mo
bond prices is that bonds constitute a substantial proportion of the portfolio o
of financial institutions (e.g. life insurance and pension funds). Variations in b
influence the balance sheets of financial institutions, while Central Banks, when
a statutory role as the regulatory authority, will wish to know whether this
put the financial viability of such institutions at risk. If the bond market can
to be excessively volatile, that is the degree of volatility exceeds that which w
from the behaviour of agents who use RE, then there is an added reason for g
intervention in such markets over and above any supervisory role (e.g. introduc
breakers™ or ˜cooling-off procedures whereby the market is temporarily clos
changes exceed a certain specified limit in a downward direction).
earlier chapters).
Before we plunge into the details of models of the behaviour of bond prices
it is worth giving a brief overview of what lies ahead. Bonds and stocks ha
features in common and, as we shall see, many of the tests used to assess
in stock prices and returns can be applied to bonds. For stocks we investigat
(one-period holding period) returns are predictable. Bonds have a flexible m
just like stocks. Unlike stocks they pay a knownjixed ˜dividend™ in nominal te
is the coupon on the bond. The ˜dividend™ on a stock is not known with certain
the stream of nominal coupon payments is. The one-period holding period y
on a bond is defined analogously to the ˜return™ on stocks, namely as the pr
(capital gain) plus the coupon payment, denoted Hj:), for a bond with term
of n periods. Hence we can apply the same type of regression tests as we did
to see if the excess HPY on bonds is predictable from information at time t o
The nominal stock price under the EMH is equal to the DPV of expecte
payments. Similarly, the nominal price of a bond may be viewed as the DPV
nominal coupon payments. However, since the nominal coupon payments are k
certainty the only source of variability in bond prices under rational expectatio
about future one-period interest rates (i.e. the discount factors). We can constru
foresight bond price P t using the DPV formula for bonds with actual (ex-p
rates (rather than expected interest rates) as the discount factors, in the same
did for stocks. By comparing P , and PT for bonds, we can then perform Shill
bounds tests, and also examine whether P, - P: is independent of informat
t , using regression tests. A similar analysis can be applied to the yield on
give either the perfect foresight yield R or the perfect foresight yield spre
T
time series behaviour of the ex-post variables R and S; can then be compared
T
actual values R, and S,, respectively.
Chapter 9 discusses various theories of the determination of bond prices,
returns and ˜yields™, while Chapter 10 discusses empirical tests of these theorie
a plethora of technical terms used in analysing the bond market and therefore w
defining such concepts as pure discount bonds, coupon paying bonds, the hol
yield (HPY), spot yields, the yield to maturity and the term premium. We the
the various theories of the determination of one-period HPYs on bonds of diff
rities (e.g. expectations hypothesis (EH), liquidity preference, market segme
preferred habitat hypotheses).
Models of the term structure are usually applied to spot yields and it will
strated how the various hypotheses about the determination of the spot yie
period bond Rj”™ can be derived from a model of the one-period HPYs on z
bonds. Under the pure expectations hypothesis (PEH), it will be shown how
between the long rate and a short rate is the optimal predictor of both the expec
in long rates and the expected change in (a weighted average) of future short r
relationships provide testable predictions of the expectations hypothesis, und
expectations. Strictly speaking the expectations hypothesis only applies to spo
term premium) are deferred to Chapter 14 and tests involving time varying t
to Chapter 19.
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9
Bond Prices and the Term Struct
of Interest Rates
The main aim in this chapter is to present a set of tests which may be applied t
validity of the EMH in the bond market. However, several preliminaries need
before we are in a position to formulate these hypotheses. The procedure is a
We analyse zero coupon and coupon paying bonds, spot yields, c
compounded spot yields, the holding period and the yield to maturity
We see how the rational valuation formula may be applied to the deter
bond prices
It will be demonstrated how a model of the one-period HPY on pure disc
can lead to the term structure relationship, namely that the yield on an n-p
is equal to a weighted average of expected future short rates plus a term
Various hypotheses of the term structure applied to holding period yields,
and the yield to maturity are examined. Theories include the expectations
the liquidity preference hypothesis, the market segmentation hypothesis and
(applied to HPYs)


9.1 PRICES, YIELDS AND THE RVF
The investment opportunities on bonds can be summarised not only by the hol
yield but also by spot yields and the yield to maturity. Hence, the ˜return™ on a b
defined in a number of different ways and this section clarifies the relationshi
these alternative measures. We then look at various hypotheses about the b
participants in the bond market based on the EMH,under alternative assump
expected ˜returns™.
Bonds and stocks have a number of basic features in common. Holders
expect to receive a stream of future dividends and may make a capital gai
given holding period. Coupon paying bonds provide a stream of income cal
payments Cr+i, which are known (in nominal terms) for all future periods, at
bond is purchased. In most cases Cr+i is constant for all time periods but it is
useful to retain the subscript for expositional purposes. Most bonds, unlike
+
redeemable at a fixed date in the future (= t n) for a known price, nam
and known at the time of issue. The return on the bill is therefore the differenc
its issue price (or market price when purchased) and its redemption price (ex
a percentage). A bill is always issued at a discount (i.e. the issue price is le
redemption price) so that a positive return is earned over the life of the bil
therefore often referred to as pure discount bonds or zero coupon bonds. Mos
are traded in the market are for short maturities (i.e. they have a maturity at iss
months, six months or a year). Coupon paying bonds, on the other hand, are
maturities in excess of one year with very active markets in the 5-15 year ba
This book is concerned only with (non-callable) government bonds and bil
assumed that these carry no risk of default. Corporate bonds are more risky th
ment bonds since firms that issue them may go into liquidation, hence consid
the risk of default then enter the analysis.
Because coupon paying bonds and stocks are similar in a number of respect
the analytical ideas, theories and formulae derived for the stock market can be
the bond market. As we shall see below, because the ˜return™ or ˜yield™ on a b
measured in several different ways, the terminology (although often not the
ideas) in the bond market differs somewhat from that in the stock market.

Spot YieldslRates
The spot yield (or spot rate) is that rate of return which applies to funds
borrowed or lent at a known (usually risk-free) interest rate over a given ho
example, suppose you can lend funds (to a bank, say) at a rate of interest
applies to a one-year loan. For an investment of $A the bank will pay out M1
r s ( l ) )after one year. Suppose the bank™s rate of interest on ˜two-year mon
expressed at a proportionate annual compound rate. Then $A invested will
A42 = A(l + rs(2))2 after two years. The spot rate (or spot yield) therefore as
the initial investment is ˜locked in™ for a fixed term of either one or two year
An equivalent way of viewing spot yields is to note that they can be used
a discount rate applicable to money accruing at specific future dates. If you
$M2 payable in two years then the DPY of this sum is M2/(1 + rs(2))2 where
two-year spot rate.
In principle, a sequence of spot rates can be calculated from the observ
price of pure discount bonds (i.e. bills) of different maturities. Since these a
a fixed one-off payment (i.e. the maturity value M ) in ˜n* years™ time, t
(expressed at an annual compound rate, rather than as a quoted simple interest
sequence of spot rates rsJ™), rs!2™. . ., etc. For example, suppose the redemption
discount bonds is $M and the observed market price of bonds of maturity n
are P:™)*Pj2™. . ., etc. Then each spot yield can be derived from P!™) = M/
+
pi2™= M/(I rs!2™)2, etc.
+
For a n-period pure discount bond P , = M/(1 rs,)” where rs (here w
superscript) is the n-period spot rate, hence:

InP, = InM - n ln(1 + rs,)
or
-
Pt = Mexp(-rc, n)
In practice (discount) bills or pure discount bonds often do not exist at the long
maturity spectrum (e.g. over one year). However, spot yields at longer maturi
approximated using data on coupon paying bonds (although the details need n
us here, see McCulloch (1971) and (1990)).
If we have an n-period coupon paying bond and market determined spot
for all maturities, then the market price of the bond is determined as:




+
where M = maturity (redemption) value and Vi = Cf+i/(l for i = 1,
+ +
and Vn = (Cr+n M n ) / ( l r ˜ ; ! $ ) ˜ . market price is the DPV of future co
The
maturity value) where the discount rates are spot yields. If the above formula do
then riskless arbitrage profits can be made by ˜coupon stripping™. To illustrate
consider a two-period bond and assume its market price Pj2™ is less than V1
current market price of two, zero coupon bonds with payouts of Ct+l and (C1
will be V1 and V2, respectively. The coupon paying bond can be viewed as a
+
coupon bonds. If Pj2™ < V1 V2 then one could purchase the two-year coupo
sell a claim on the ˜coupon payments™ in years 1 and 2, that is Ct+l and (Ct
to other market participants. If zero coupon bonds are correctly priced then th
could be sold today for V1 and V2, respectively. Hence an instantaneous ris
+
of ( V l V2 - Pj”) can be made. In an efficient market, the increased dema
two-year coupon paying bond would raise Pj2),while sales of the coupons wou
prices of one and two-year zero coupon bonds. Hence this riskless arbitrage
to the restoration of the equality in (9.3).

Holding Period Yield (HPY) and the Rational Valuation Formula (RVF)
As with a stock, if one holds a coupon paying bond between t and t + 1 th
made up of the capital gain plus any coupon payment. For bonds this measure
is known as the (one-period) holding period yield (HPY).




Note that in the above formula the n-period bond becomes an (n - 1) period
one period. (In empirical work on long-term bonds (e.g. with n > 10 years) r
often use P: in place of PI:;™™ since for data collected weekly, monthly o
!
:
they are approximately the same.)
At any point in time there are bonds being traded which have different time
maturity or ˜term to maturity™. For example, a bond which when issued had
to form expectations of P,+l and hence of the expected HPY. (Note that fut
payments are known and fixed at time t for all future time periods and are usua
so that C , = C).

Rational Valuation Formula (RVF) for Bonds
In our most general model for determining stock prices in Chapter 4 it was a
expected one-period return required by investors to willingly hold the stock
over time. The required one-period return (or HPY) on an n-period bond w
and hence for bonds we have:




This equation can be solved forward (as demonstrated in Chapter 4) to give
bond price as the DPV of future known coupon payments discounted at th
one-period spot returns k,+i. For an n-period bond:




where A = the redemption price and it is usually assumed that C,+j = C (a c
4
Note that the only variable the investor has to form expectations about is h
one-period return k,+i in (all) future periods. Also no transversality conditio
be imposed since after iterating forward for n periods the expected price is e
known redemption value, M .
As with our analysis of stocks we can ˜split™ the per period required retu
one-period risk-free (interest) rate r, and a risk premium. However, when ta
the government bond market the risk premium on an n-period bond is frequen
to as the termpremium, 7˜;”) Hence(2):



+ ++
where r,+i is the one-period rate applicable between t i and t i 1. Becau
ment bonds carry little or no risk of default, the only ˜risk™ attached to such b
eyes of the investor arises because they have different terms to maturity; henc
˜term premium™. One aspect of risk arises if the investor wishes to liquidate h
before the maturity date of the bond; the investor faces ˜price uncertainty™. Al
if the investor holds the bond to maturity there is no (nominal) price uncertai
rate at which future coupon payments can be reinvested is uncertain; this is
as ˜reinvestment risk™. Note that (9.7) is an identity which is used to defin
premium and there is as yet no behavioural content in (9.7).
J
L i=O i=O
j=1

The variance bounds test is then var(P,) < var(PT). We can also test the E
this explicit assumption for k,+i) by running the regression


where Qr is any information available at time t . Under the null of the EM
a = b = c = 0.
Strictly speaking one cannot have both (9.3) and (9.6) determining the price
since there is only one quoted market price. Rather, equation (9.3) determine
of the bond via riskless arbitrage. Equation (9.6) may then be viewed as det
set of expected one-period returns that yield the same bond price given by (9.
preferences working via the supply and demand for bonds of different mat
establish a set of equilibrium one-period rates k,+i. Of course, when there doe
a set of market determined spot rates for all maturities n then equation (9.6) m
tually be viewed as determining bond prices. Since for n > 2 years, zero cou
are usually not available, equation (9.6) can be used legitimately to examine
of long-term bonds. The reason equations (9.3) and (9.6) give different expr
the price of the bond is that they are based on different behavioural assumpti
derivation of (9.6) agents are only concerned with the sequence of one-perio
returns k,+i whereas the bond pricing equation (9.3) involving spot rates rsf
under the assumption of (instantaneous) riskless arbitrage(”1.

Yield to MaturitylRedemption Yield
For coupon paying bonds the rate which is quoted in the market is the yield
(YTM). Investors know the current market price of the bond P,, the stream
coupon payments C, the redemption value of the bond (= M )and its maturity d
ussume that the coupon payments at different horizons are discounted using
discount rate 1/(1 + R : ) . Note that RT has a subscript t because it may vary
(but it does not vary in each period in the DPV formula). If we now equate t
the coupon payments with the current market price we have



The bond may be viewed as an investment for which a capital sum P, is paid
and the investment pays the known stream of dollar receipts (C and M )in the
constant value of R: which equates the LHS and RHS of (9.10) is the ˜inter
return™ on this investment and when the investment is a bond Rr it is referred to
to maturity or redemption yield on the (n-period) bond. Clearly one has to ca
each time the market price changes and this is done in the financial press which
report bond prices, coupons and yields to maturity. It is worth noting that th
for each bond (whereas (9.3) involves a sequence of spot rates). The yield to m
an n-period bond and another bond with q periods to maturity will generally b
at any point in time, since each bond may have different coupon payments
course the latter will be discounted over different time periods (i.e. n and q). I
see from (9.10) that bond prices and redemption yields move in opposite dir
that for any given change in the redemption yield R; the percentage change in
long bond is greater than that for a short bond. Also, the yield to maturity form
reduces to P, = C/R;"for aperpetuity (i.e. as n -+ 09).
Although redemption yields are widely quoted in the financial press they a
what ambiguous measure of the 'return' on a bond. For example, two bon
identical except for their maturity dates will generally have different yields t
Next, note that in the calculation of the yield to maturity, it is implicitly as
agents are able to reinvest the coupon payments at the constant rate R: in all fut
over the life of the bond. To see this consider the yield to maturity for a two-p
given by (9.10) rearranged to give:



The LHS is the terminal value (in two years' time) of $P,invested at the con
+ +
alised rate R;".The RHS consists of the amount (C M ) paid at t 2 and
+ +
C( l R r ) which accrues at t 2 after the first year's coupon payments have
vested at the rate R;. Since (9.10) and (9.11) are equivalent, the DPV formu
that the first coupon payment is reinvested in year 2 at a rate R:. However, th
reason to argue that investors always believe that they will be able to reinves
coupon payments at the constant rate Ry. Note that the issue here is not tha
have to form a view of future reinvestment rates for their coupon paymen
they choose to assume, for example, that the reinvestment rate applicable on
bond, between years 9 and 10 say, will equal the current yield to maturity
20-year bond.
There is another inconsistency in using the yield to maturity as a measure o
on a coupon paying bond. Consider two bonds with different coupon payme
CiL:, C::: but the same price, maturity date and maturity value. Using (9.1
imply two different yields to maturity R:, and R;,. If an investor holds both of t
in his portfolio and he believes equation (9.10) then he must be implicitly ass
+ ++
he can reinvest coupon payments for bond 1 between time t j and t j 1
R:, and at the different rate Ri, for bond 2. But in reality the reinvestment ra
++
t + j and t j 1 will be the same for both bonds and will equal the one-
rate applicable between these years.
In general, because of the above defects in the concept of the yield to mat
curves based on this measure are usually difficult to interpret in an unambigu
(see Schaefer (1977)). However, later in this chapter we see how the yield
may be legitimately used in tests of the term structure relationship.
in value of its liabilities. It is easiest to examine the concept of duration using
uously compounded YTM, which we denote as y. Duration is a measure of t
time one has to wait to receive coupon payments. For a zero coupon bond th
in n years the duration is also n years. However, a coupon paying bond ma
years has a duration less than n since some of the cash payments are receiv
n. Let us determine the price response of a coupon paying bond to a small ch
YTM. The price of the bond with coupons Ci (where Cn also includes the
price) is:
n n


i= 1 i= 1

where PV; = C;exp(-yt;) is the present value of cash flows Cj. Differenti
respect to y and dividing both sides by P:




If we now define duration D as:
n
D= ˜˜[Pv˜/PI
i= 1

then d PIP = -Ddy. From the definition of D we see that it is a time weight
of the present value of payments (as a proportion of the price). If we know th
of a bond then we can calculate the capital gain or loss consequent on a small
the yield. The simplest case of immunisation is when one has a single liabili˜y
DL. Here immunisation is most easily achieved by purchasing a single bond
has a duration equal to DL. However, one can also immunise against a sing
with duration DL by purchasing two bonds with maturity D1 and 0 2 such that
2


i= 1

and wi are the proportions of total assets held in the two bonds




Clearly this principle may be generalised to bond portfolios of any size. Ther
tations when using duration to immunise a portfolio of liabilities, the key o
that the calculations only hold for small changes in yields and for parallel sh
(spot) yield curve. Also since D alters as the remaining ˜life™ of the bond(s)
immunised portfolio must be continually rebalanced, so that the duration of
liabilities remains equal (for further details see Fabozzi (1993) and Schaefer (
The RVF can also be written in terms of real variables. In this case, the no
price P, and nominal coupon payments are deflated by a (nominal) goods pric
they are then measured in real terms. The required real rate of return which w
is determined by the real rate of interest (i.e. the nominal one-period rate less th
one-period rate of inflation Efnr+l) the term premium:
and

+ +
k: = (rt - Efnr+1) T , = Ef(rrf) Tt
where rr, = real rate of interest (and is assumed to be independent of the rate o
It should be fairly obvious that the nominal price of the bond is influenced
degree by forecasts of future inflation. Coupon payments are (usually) fixed
terms, so if higher expected inflation is reflected in higher nominal discount
the nominal price will fall (see equation (9.6)). More formally, this can be
+
easily by taking a typical term in (9.6), for example the term at t 2 where w
+
+ +
(1 h+i)= (1 k;+j)(l rTi
T t)+




where k,? is the real rate of return (discount factor). Equation (9.13) can be re




Hence for constant real discount factors k;+i the nominal bond price depends
on the expected future one-period rates of inflation over the remaining life of
Let us now demonstrate that the real bond price is independent of the rate o
+
as one might expect. If we define the real coupon payment at t 2 as CT+2 =
+ +
where Ir+2 is the goods price index at t 2, and note that Ir+2 = (1 nr+l)(
then from (9.14) the real bond price has a typical term:



Hence as one would expect, the real bond price depends only on real variables
if the variables are all measured in real terms or all in nominal terms, the term
is invariant to this transformation of the data.


9.2 THEORIES OF THE TERM STRUCTURE
We now examine various theories of the term structure based on different a
made about the required rate of return k,. These are summarised in Table 9.1
with theories of the term structure based on the one-period HPY as a mea
return on the bond. In subsequent sections we then consider how these same t
be implemented using spot yields and the yield to maturity(4). Our aim is to
RI"' - E,(r,+,(s)= 0
E,H!:\ - r, = 0
2. Expectations Hypothesis or Constant Term Premium
(i) Expected excess return equals a constant which is the same for all maturities
or (ii) The term premium 'T' is a constant and the same for all maturities
RI"' - E,(r,+,rs)= T
E,Hi:), - r, = T
3. Liquidity Preference Hypothesis
(i) Expected excess return on a bond of maturity n is a constant but the value o
constant is larger the longer the period to maturity
or (ii) The term premium increases with n, the time period to maturity
ErHj:), - rr = T'"' RI"' - Er(r,+,rS)= T'")
where T(")> T("-'). . ., etc.
4. Time Varying Risk
(i) Expected excess return on a bond of maturity n varies both with n and over
(ii) The term premium depends on the maturity n and varies over time
RI")
- E,(r,+,/s) ˜ ( nz,)
E,Hj:), - r, = ˜ ( nz,) = ,
,
where T ( ) is some function of n and a set of variables z,.
5. Market Segmentation Hypothesis
(i) Excess returns are influenced at least in part by the outstanding stock of asse
different maturities
(ii) The term premium depends in part on the outstanding stock of assets of diffe
maturities
RI"' - E,(r,+,!s) T(zj"')
E,H!:{ - r, = T(zj"') =
where z:") is some measure of the relative holdings of assets of maturity 'n'
proportion of total assets held.
6. Preferred Habitat Theory
(i) Bonds which mature at dates which are close together should be reasonably c
substitutes and hence have similar term premia


clearly the tests implied by the various hypotheses: actual empirical results are
in the next chapter and more advanced tests in Chapter 14.

Using the HPY
The theories based on the HPY include the expectations hypothesis, the liquid
ence hypothesis, market segmentation and the preferred habitat hypothesis. T
only in their treatment of the term premium. Each will be dealt with in turn.

The Expectations Hypothesis
If all agents are risk neutral and concerned only with expected return then the
one-period HPY (over say one month, or one quarter) on all bonds, no matter
This is the pure expectations hypothesis (PEH). The term premium T is z
maturities and the discount factor in the RVF (9.6) is simply kt+j = rf+,,the s
one-period risk-free rates. All agents at the margin are 'plungers'. For example
bond with three years to maturity has a HPY in excess of that on a bond with
maturity. Agents would sell the two-year bond and purchase the three-year
pushing up the current price of the three-year bond and reducing its one-pe
The opposite would occur for the two-period bond and hence all holding per
would be equalised, To (9.16) we now add the assumption of rational expectatio
+
E&::: qf;;, where E f ( q ˜ ˜ ) l l = ,0 and : is the (one-period) rational e
Q) q
forecast error:
(n ) (n )
H I + , - r, = v,+˜ (for all n)
+
Hence a test of the PEH RE is that the ex-post excess holding period y
have a zero mean, be independent of all information at time t(C2,) and should
uncorrelated.
It seems reasonable to assert that because the return on holding a long bo
period) is uncertain (because its price at the end of the period is uncertai
excess holding period yield ought to depend on some form of 'reward for ri
premium T!") .
+
E,H;I_; = r, ˜ j " )
Without a model of the term premium, equation (9.18) is a tautology. The sim
trivial) assumption to make about the term premium is that it is (i) constan
and (ii) that it is also independent of the term to maturity of the bond (i.e. Ty
constitutes the expectations hypothesis (EH) (Table 9.1). Obviously this yie
predictions as the PEH, namely no serial correlation in excess yields and th
should be independent of 52,. Note that the excess yield is now equal to the co
+
premium T and the discount factor in the RVF is k, = rf T .
Under RE and a time invariant term premium we obtain (see (9.18) with T
the following variance inequality.
2 var(rt)
var[H,(:;l
Thus the variance of the HPY on an n period bond should be greater than or
variance of the one-period safe rate such as the interest rate on Treasury bills

Liquidity Preference Hypothesis (LPH)
Here, the assumption is that the term premium does not vary over time but it d
on the term to maturity of the bond (i.e. TI"' = T'")). For example, bonds
periods to maturity may be viewed as being more 'risky' than those with a s
to maturity, even though we are considering a fixed holding period for both
might arise because the price change is larger for any given change in the yield
with longer maturities. Consider the case where the one-month HPY on 20-
The liquidity preference hypothesis asserts that the excess yield is a const
given maturity but for those bonds which have a longer period to maturity
premium will increase. That is to say the expected excess HPY on a 10-year b
exceed the expected excess HPY on a 5-year bond but this gap would rema
over time. Thus, for example, 10-year bonds might have expected excess return
above those on 5-year bonds, for all time periods. Of course, in the data, actu
excess returns will vary randomly around their expected HPY because of (
forecast errors in each time period. Under the liquidity preference hypothes
+
k, = r, T ( ” )as the discount factor in the RVF and expected excess HPYs ar



Under rational expectations, the liquidity preference hypothesis predicts that ex
are serially uncorrelated and independent of information at time t . Thus, apart f
innocuous constant term, the main testable implications using regressions ana
PEH, EH and the LPH are identical. For the PEH we have a zero constant t
EH, we have T = constant and for the LPH we have a different constant for
of maturity n , as in (9.20).

Time Varying Risk
If the risk or term premium varies over time and varies differently for bonds
maturities then
+
k, = r, T ( n , z,)
where z, is a set of variables that influences investors™ perceptions of risk.
most general model so far, but unless one specifies an explicit form of the func
model of expected excess HPYs is non-operational. Below is an illustration
CAPM provides a model of the term premium, and Chapter 19 examines the be
HPYs on bonds when the term premium is assumed to depend on time varying
and covariances.

CAPM and HPY
An obvious theoretical approach to explain the excess HPY on bonds and to
explicit form for the risk premium are the CAPM-type models which we ha
discussed when examining stock returns. Thus, for those who are familiar wit
theory, it should not come as a surprise that the term premium may vary both f
maturities and over time. The CAPM predicts that the expected holding period
any asset, which of course includes an n period bond, is given by
constant over time and in general one might therefore expect time variation
premium.
According to Merton™s (1973) model, the excess return on the market
proportional to the conditional variance of the forecast errors on the market p


Combining equations (9.22) and (9.23):

E , H ! ˜ \- r, = AE*COV(H˜:\, R;˜)
Hence the excess HPY depends upon the covariance between the holding perio
the bond and the return on the market portfolio. It is possible that this covaria
varying and that it may be serially correlated, implying serial correlation in
HPY. Notice that the CAPM version of the determination of term premia doe
that expected holding period returns on 10-year bonds should necessarily be
those on 5-year bonds. To take an extreme (non-general equilibrium) examp
10-year bonds have a negative covariance with the market portfolio, while 5-
have a zero or positive covariance, then the CAPM implies that expected ex
on 10-year bonds should be below those on 5-year bonds.
The standard one-period CAPM model assumes the existence of a risk-free
(1972) developed the zero-beta CAPM to cover the case where there is no ris
and this results in the following equation:


Here the expected HPY on the n period bond equals a weighted average o t f
return on the market portfolio and the so-called ˜zero-beta portfolio™. A zero-be
is one which has zero covariance with the market portfolio. The next chapte
empirical results from the CAPM applied to bonds under the assumption of
beta, but empirical results from CAPM models which incorporate time vari
betas of bonds of different maturities are not discussed until Chapter 17.
The above taxonomy of models might at first sight seem rather bewilderin
main all we are doing is repeating the analysis of valuation models that we
stocks in Chapters 4 and 6. We have gradually relaxed the restrictive assump
return required by investors to hold bonds (i.e. k,) and naturally as we do so
become more general (see Table 9.1) but also more difficult to implement
Indeed, Table 9.1 contains two further hypotheses which will be dealt with b

Market Segmentation Hypothesis
The market segmentation hypothesis may be viewed as a reduced form or mar
rium solution of a set of standard asset demand equations. To simplify, suppo
only two risky assets (i.e. bonds B1 and B2) and the proportion of wealth hel
B2 is given by their respective demand functions

- r, H; - r )
( B ˜ / w =˜ I ( H ?

+ B2)/W
(TBIW) = 1 - (Bi
and need not concern us. If we now assume that the supply of B1 and B2 is
then market equilibrium rates of return, given by solving (9.26), result in e
the form
- r = G i [ B I / W ,B 2 / W ]
H;
He2 - r = G 2 [ B l / W ,B 2 / W ]
Hence the expected excess HPYs on the two bonds of different maturities
the proportion of wealth held in each of these assets. This is the basis of
segmentation hypothesis of the determination of excess HPYs.
Tests of the market segmentation hypothesis based on (9.28) are often a gro
plification of the rather complex asset demand functions usually found in th
literature in this area. In general, the demand functions (9.26) contain many
pendent variables than holding period yields; for example, the variance of r
wealth, price inflation and their lagged values and lags of the dependent va
appear as independent variables (see Cuthbertson (1991)). Hence the reduced
librium equations (9.28) should also include these variables. However, tests
segmentation hypothesis usually only include the proportion of debt held i
maturity ˜n™ in the equation to explain excess holding period yields.

Preferred Habitat Hypothesis
The preferred habitat theory is, in effect, agnostic about the determinants o
premium. It suggests that we should only compare ˜returns™ on governmen
similar maturities and one might then expect excess holding period yields to m
together.

9.2.1 Theories Using Spot Yields
The term structure of interest rates deals with the relationship between the
bonds of different maturities. The yields in question are spot yields and therefo
tually, the analysis applies to pure discount bonds or zero coupon bonds. Pu
government bonds for long maturities do not exist, nevertheless as we noted e
yields can be derived from a set of coupon paying bonds (McCulloch, 199
measure of the return on a bond is the yield to maturity and our analysis o
structure in terms of spot yields can be applied with minor modification (and
of approximation) to yields to maturity.
For the moment we proceed as follows. First, we examine how the term
relationship can be derived from a model of expected one-period HPYs and th
our derivation of the RVF for stock prices in Chapter 4, since it involves
difference equation in one-period rates. This derivation requires the use of co
compounded rates. We then examine the economic behaviour behind the term
before demonstrating how it may be rewritten in a number of equivalent way
+ T!"'
Er[ln P:?;') - In P:"'] = r
r
Erh,'?:
Equation (9.29), although based on the HPY, leads to a term structure rela
terms of spot yields. For continuously compounded rates we have In PI"' = In
and substituting in (9.29) this gives the forward difference equation:

+ rr + ˜j"'
n ˜ ! " ) (n - I)E˜R:;;')
=
Leading (9.30) one period:
+ +T,+˜
( n - I)R!:;') = (n - 2)˜,+1˜j:;˜) r,+l ("-1)


Taking expectations of (9.31) using E,E,+1 = Et and substituting in (9.30)
+ E,(rr+l + rt) + E,("q;" + 7';"')
nRI") = (n - 2)E,R,'q;2'
Continually substituting for the first term on the RHS of (9.32) and noting
j)EtRf;;'' = 0 for j = n we obtain:


where
n-1


i=O

n-1


i=l

Hence the n-period long rate equals a weighted average of expected future sho
plus the average risk premium on the n-period bond until it matures, @In'. T
R:'"' is referred to as the perfect foresight rate since it is a weighted aver
outturn values for the one-period short rates, rr+i. Subtracting r from both side
r
we obtain an equivalent expression:


where




Equation (9.35) states that the actual spread SjnV1)
between the n-period and
rate, equals a weighted average of expected changes in short rates plus a term
Equations (9.33) and (9.35) are general expressions for the term structure relat
they are non-operational unless we assume a specific form for the term prem
The PEH applied to spot yields assumes investors are risk neutral, that is, the
ferent to risk and base their investment decision only on expected returns. T
variability or uncertainty concerning returns is of no consequence to their
decisions. In terms of equations (9.33) and (9.35) the PEH implies @(") =
We can impart some economic intuition into the derivation of (9.33) when w
zero term premium. To demonstrate this point we revert to using per-period
than continuously compounded rates, but we retain the same notation so that
are now per-period rates.
Consider investing $A in a (zero coupon) bond with n years to maturity. T
value (TV) of the investment is:
+ R,'"')"
TV, = $A(1

where R,'"' is the (compound) rate on the n-period long bond (expressed a
rate). Next consider the alternative strategy of reinvesting $A and any interes
a series of 'rolled-over' one-period investments, for n years. Ignoring transac
the expected terminal value E,(TV) of this series of one-period investments i


++
+
where rr+j is the rate applicable between periods t i and t i 1. The invest
long bond gives a known terminal value since this bond is held to maturity. In
series of one-year investments gives a terminal value which is subject to uncert
the investor must guess the future values of the one-period spot yields, rr+j
under the PEH risk is ignored and hence the terminal values of the above two
investment strategies will be equalised:

+ rr)(l + Errr+1)(1+ Etrt+2) - - + Errr+n-i)
(1 +I$"))" = (1 (1
The equality holds because if the terminal value corresponding to investment
bond exceeds the expected terminal value of that on the sequence of one-year in
then investors would at time t buy long bonds and sell the short bond. This w
in a rise in the current market price of the long bond and given a fixed mat
a fall in the long (spot) yield R,. Simultaneously, sales of the short bond wo
fall in its current price and a rise in r,. Hence the equality in (9.39) would
(instantaneously) restored.
We could define the expected 'excess' or 'abnormal' profit on a $1 investm
long bond over the sequence of rolled-over short investments as:


where Er(r;+j) represents the RHS of (9.39). The PEH applied to spotyields
implies that the expected excess or abnormal profit is zero. We can go through
+ Etrt+i + Errr+2+ -
RI"' = (l/n)[rt Errr+2]
*



In general when testing the PEH one should use continuously compounded
since then there is no linearisation approximation involved in (9.41)(@.
The PEH forms the basis for an analysis of the (spot) yield curve. For exam
from time t, if short rates are expected to rise (i.e. Errr+, > Errr+j-1) for all j
(9.41) the long rate R!")will be above the current short rate rr. The yield curve
of RI"' against time to maturity - will be upward sloping since R!"' > R!"
Since expected future short rates are influenced by expectations of inflation (Fi
the yield curve is likely to be upward sloping when inflation is expected to
future years. If there is a liquidity premium that depends only on the term to
and T(")> T("-l) > . . ., then the basic qualitative shape of the yield curve wil
described above. However, if the term premium varies other than with 'n', f
varies over time, then the direct link between R(") and the sequence of future
in broken.
ff

PEH: Tests Using Different Maturities
Early tests were based on equation (9.41) and include variance bounds and
based tests. So far, we have analysed the expectations hypothesis only in te
n-period spot rate RI"' and a sequence of one-period short rates. However, the
can also be expressed in terms of the relationship between R!"' and any rn-
R!*' (for which s = n/m is an integer):
s- 1

R!") = (l/s) E,Rj$m
i=O

Two rearrangements of (9.42)can be shown to imply that the spread Sr(n*m) (I?
=
is an optimal predictor of future changes in long rates and the spread is
predictor of (a weighted average of) future changes in short rates. We consi
these in turn using convenient values of (n, rn) for expositional purposes.

Spread Predicts Changes in Long Rates
Consider equation (9.42) for n = 6, m = 2:



this may be shown to be equivalent to(')

ErRj:\ - RI6' = (1/2)S!612'

Hence if Rj6' > Ri2' then the PEH predicts that, on average, this should be fol
+
rise in the long rate between t and t 2. The intuition behind this result can
+ /!3[S!6t2)/2]+ q1+2
- =
RI:\

where we expect = 1. Under RE, S,'6'2' independent of qt+l and therefo
is
(9.45) yields consistent estimates (although a GMM correction to the standa
required if there is heteroscedasticity in the error term, or if qt+2 is serially cor
to the use of overlapping observations). For any (n,m), equation (9.44) can
to be:
E t R("-m) - RI"' = [ m / ( n -
r+m


The Spread Predicts Future Changes in Short Rates
Equation (9.43) may be rearranged to give(')
E ˜ ( 6 * 2 )= ˜ 1 6 ˜ 2 )
*
tt




and AmZr= 2, - 2 r - m . The term Sj6'2'* the perfect foresight spread. Equa
is
implies that the actual spread S16*2' an optimal (best) predictor of a weight
is
of future two-period changes in the short rate Hence if Si6'2' > 0 then ag
on average that future (two-period) short rates should rise. Note that althou
an optimal predictor this does not necessarily imply that it forecasts the RHS
+
accurately. If there is substantial 'news', between periods t and t 4, then Si
a rather poor predictor. However, it is optimal in the sense that, under the ex
hypothesis, no variable other than S, can improve one's forecast of future
short rates. The fact that S , is an optimal predictor will be used in Chapter 1
examine the VAR methodology. The generalisation of (9.47) is
=
E,Sjn*m)* S p m )
where
s- 1
= (-
Sj"*")* c 1 i/s)A( m )R,+im
(m)

i= 1

and equation (9.48) suggests a straightforward regression-based test of the ex
hypothesis (under RE) based on:

S p ) *= a + /!3Sl"*m) + wt

Under the EH we expect /?= 1 and if the term premium is zero a = 0. Sinc
(by RE) independent of information at time t , then OLS yields unbiased estim
parameters (but G M M estimates of the covariance matrix may be required - s
Note, however, that if the term premium is time varying and correlated with
then the parameter estimates in (9.49) will be biased.
are equivalent. Also if (9.46) holds for all ( n , m) then so will (9.48). Howev
is rejected for some subset of values of (n,m) then equation (9.48) doesn't
hold and hence it provides independent information on the validity of the PE
In early empirical work the above two formulations were mainly undertak
2m, in fact usually for three- and six-month pure discount bonds (e.g. Treasu
which data are readily available. Hence the two regressions are statistically
Results from equations (9.46) and (9.48) are, however, available for other ma
n # 2m) and some of these tests are reported in the next chapter.

Variance Bounds Tests on Spot Yields
Shiller (1988) uses an equation similar to (9.33) to perform a very simple vo
for the expectations hypothesis. In the extreme case where agents have perfe
then future expected short rates would equal their ex-post outturn values. If w
perfect foresight long rate as:



+ q,+i) we hav
then under the PEH equation (9.33) and RE (i.e. rf+i = E,r,+i
+
Rf = R,
where
n-1


i= 1

and E(q,+l IQ,) = 0. Using past data we can construct the perfect foresight lo
each year of any sample of data. Under RE, agents' forecasts are unbiased an
sum of the forecast errors 0, should be close to zero, for large n . Hence fo
bonds we expect the perfect foresight rate R to track the broad swings in the
:
rate R, and as we shall see in the next chapter Shiller's evidence on US data i
with this hypothesis.
Clearly, as we found for stock prices, a more sophisticated method of testing
volatility is available. Under the RE the forecast error w, is independent of 52
it is independent of R,. Hence from (9.51) we have:
var[R:] 3 var[Rr]
since by RE, cov(R,, w , ) = 0 and var(w,) 2 0. These inequalities also apply if
a constant term premium T,'"' = T ( " ) .Thus, if (i) the expectations or liquidity
hypothesis (as applied to spot yields) is correct, (ii) agents have RE, (iii) the ter
depends only on n (i.e. is time invariant), then the variance of the actual long
be less than the variability in the perfect foresight long rate.
It is also the case that the expected (abnormal) profit should be independent
tion at time t . In this linear framework the expected abnormal profit is given by
market.) If interest rates are non-stationary I(1) variables then the unconditio
tion variances in (9.52) are undefined but the variance bound inequality can b
in terms of the variables in (9.48) which are likely to be stationary. Hence w
var(S:) b var(St)
and in a regression context we have


where we expect b = 1, c = 0 if the expectations hypothesis is true.

9.2.2 Using the Yield to Maturity
The tests described above are based on spot yields for pure discount (zero cou
either expressed as continuously compounded rates or as per-period rates. A
already noted the calculation of (approximate) spot yields from published data
maturity is often not undertaken. There is generally much more published da
to maturity so it would be useful if our tests of the term structure could be rec
of the yield to maturity. In a pioneering article this has been done by Shiller
the results are summarised below.
The basis of Shiller's approximations can be presented heuristically as fo
price of an n-period coupon paying bond is a non-linear function of the yield
(equation (9.10)) and therefore so is the holding period yield Hi:\.



where Rj") is the yield to maturity (i.e. the superscript 'y' is dropped) and
non-linear function. Shiller linearises (9.56) around the point



where E is the mean value of the yield to maturity. For bonds selling at o
(redemption) value this gives an approximate expression for the holding p
denoted fi!") where



where

and
yn = y(1 - y"-')/(l - y")

+R)
y = 1/(1

Equation (9.58) is an (approximate) identity which defines the HPY in terms o
to maturity. We now add the economic hypothesis that the excess holding p
From (9.58) and (9.59) we obtain a forward recursive equation for RI"' w
= r f + n - l , where r, is the one-period
solved with terminal condition
gives Shiller's approximation formula for the term structure in terms of th
maturity:




+ @"
= E,(R:)
where @ n = f ( y , y", 4") and Rf is the perfect foresight long rate. Hence t
maturity on a coupon bond is equal to a weighted average of expected f
rates rf+k where the weights decline geometrically (i.e. fl < yk-' < . . .). Th
is constant for any bond of maturity n and may be interpreted as the aver
one-period term premia, over the life of the long bond. For the expectations
@ n = @ for all n and for the pure expectations hypothesis = 0. However
alternative theories of the term structure the precise form for any constant term
is of minor significance and the term premium only plays a key role when we
is time varying (see Chapter 17).
The RHS of (9.60) with Errt+k replaced by actual ex-post values rt+k is
foresight long rate (yield to maturity) RI")* and the usual variance bound
regression tests of R!"' on R!")*may be examined. Shiller (1979) also shows
HI"),
using the approximation for the holding period yield the variance bound

var H!") 6 a2var(r,)
where a = (1, - y2)-1/2. Note that for E = 0.064 then y = 0.94 and a2 = 8.6,
variance of H:, can exceed that of the short rate. Equation (9.62a) puts an up
i)
(whereas (9.19) imposes a lower bound). If r, is non-stationary
on varfi!:;
var(r,) is undefined but in this case Shiller (1989) provides an equivalent ex
terms of the stationary series At-,, namely:

- r,) 6 c2var(Ar,)
var<&!!$

where c = 1/R.


9.3 SUMMARY
Often, for the novice, the analysis of the bond market results in a rather bewild
of terminology and test procedures. An attempt has been made to present the
a clear and concise fashion and the conclusions are as follows:
The one-period HPY on coupon paying bonds consists of a capital gain plu
0

payment. This is very similar to stocks. Hence the rational valuation fo
may be viewed as consisting of a risk-free rate plus a term premium.
The ˜return™ on zero coupon bonds of different maturities is measured by the
0

Theories of the term structure imply that continuously compounded long
specific linear weighted average of (continuously compounded) expected f
rates. This linear relationship applies as an approximation if one uses (co
spot yields and a similar approximate relationship applies if one uses yields
The yield to maturity, although widely quoted in the financial press, is a
0

misleading measure of the ˜return™ on a bond particularly when one wishes
the shape and movements in the yield curve.
Alternative theories of the term structure are, in the main, concerned wi
0

term premia are (i) zero, (ii) constant over time and for all maturities, (i
over time but differ for different maturities, (iv) depend on the proportion
held in ˜long debt™. These assumptions give rise to the pure expectatio
esis, the expectations hypothesis, the liquidity preference hypothesis and
segmentation hypothesis, respectively. In addition, term premia may be ti
(this is discussed in Chapter 14).
The key element in the expectations hypothesis is that the long rate is
0

(geometric) weighted average of expected future short rates which can b
mated by a linear relationship. Agents are risk neutral and equalise expec
over all investment horizons, using all available relevant information to pre
short rates. Hence, no abnormal profits can be made by switching between
shorts and the EMH under risk neutrality holds.
The expectations hypothesis (plus RE) applied to HPYs implies that the ex-
0

holding period yield (Hj;), - rt) is independent of information at time t, Q
The expectations hypothesis applied to (continuously compounded) spot yie
0

that the variance of the perfect foresight long rate
n


i

should exceed the variance of the actual long rate R,.
The expectations hypothesis implies that the spread S!n9m) between the n-pe
0

yield R,˜“™and the m-period spot yield R:”) is an optimal predictor of both
change in the long rate and future changes in short rates and gives rise to the
regression tests:




where the perfect foresight spread
s- 1


i=O
are now in a position to examine illustrative empirical results, in the next cha


ENDNOTES
1. Strictly the sequence of future one-period required returns should be deno
+
k(n - 1, t l), etc. but for ease of exposition k,, k , + l , etc. are used.
2. Any complications that arise from Jensen's inequality are ignored.
3. When market determined spot rates exist then riskless arbitrage ensures t
the bond is given by (9.3). Note that rs!') is the spot rate applicable betw
+
+
t and t i , whereas kf+; is the one-period rate of return between t i an
The two formulae give the same price for the bond when, for example, f

+ r s ( l ) )= (1 + ko)
(1
+
(1 + rs(2))2= (1 + ko) E,(1 kl)
*




which implies


where kl is the one-period rate applicable between periods 1 and 2. So
will recognise (iii) as the pure expectations hypothesis. So-called general
models of interest rate determination also deal with these issues but su
are beyond the scope of this book (e.g. Cox et a1 (1981) and Brennan an
(1982)).
4. In discussing the various theories of the term structure, problems that
Jensen's inequality are ignored. In fact, the EH expressed in terms of the
HPY and in terms of the long rate as a weighted average of expected sho
not generally equivalent. However, after linearisation the equivalence hold
et a1 (1981), McCulloch (1990) and Shiller et a1 (1983)).
5 . For notational simplicity the subscript 'c' to denote continuously compo
is not used. Later it is noted that, subject to an element of approximatio
formulae hold for both continuously compounded rates and for discrete c
rates. This analysis also goes through for HPYs for rn > 1 periods, tha
n , m (with n / m an integer). See Shiller et a1 (1983).
6. In terms of continuously compounded rates the same analysis yields



where Rt(n),r,+; are continuously compounded rates. Taking logarithms
relationships (ignoring Jensen's inequality) holds exactly for these c
compounded (spot) rates:
Moving (1) two periods forward, the six-period bond becomes a four-per



Applying the law of iterated expectations to (2), that is Ef(E,+2R,+,)=
have:
+
E,Rj$ = (1/2)(E,RI?2 ErR::):)
Substituting for the RHS of (3) in (1):




8. Use will be made of continuously compounded rates with a maturity v
since this makes the algebra more transparent. We have:




where we have used ln(1) = 0. The (logarithmic) HPY on holding the
+
bond from t to t 2 is


The safe return on the two-period bond is



Equating (4) and (5) and rearranging we obtain equation (9.44) in the tex

+ E,RI:)\ + EfRi:\]
Rj6) = (1/3)[Ri2)
9.

Substracting Rj2™ from both sides and rearranging:




The RHS of (3) is equal to that is equation (9.47b) in the text.
Empirical Evidence on the
Term Structure
The previous chapter dealt with the wide variety of possible tests of the EMH
market which can be broadly classified as regression-based tests and variance bo
These two types of test procedure can be applied to spot yields, yields to mat
prices and holding period yields (HPY). This chapter provides illustrative (
exhaustive) examples of these tests. At the end of this chapter some broad c
are reached on whether such tests support the EMH but the reader should
these conclusions as definitive, given the wide variety of tests not reported.
the EMH requires an economic model of expected (excess) returns on bon
failure of the EMH may be due to having the wrong economic model of (e
returns. In general the theories outlined in the previous chapter and the tests d
this chapter assume a time invariant term premium - models which relax this
are presented in Chapter 17. The stationarity of the variables used in particu
of importance, as noted when applying variance bounds and regression test
prices. The issue of stationarity is noted in this chapter but is discussed in mo
Chapter 14 on the VAR methodology. After presenting a brief account of som
facts for bond returns and the quality of data used, the following will be disc

We investigate the term structure at the short end of the market, name
three-month and six-month bills. These are ˜pure discount™ or ˜zero coup
and this enables us to use quoted spot rates of interest.
We examine bonds at the long end of the maturity spectrum. In particular
the relationship between actual long rates and the perfect foresight lon
undertake the appropriate variance bounds and regression based tests. We
these tests using bond prices.
Again for long maturity bonds, we examine the variance bounds inequaliti
period HPYs and examine whether the zero-beta CAPM can explain the
of HPYs.
A word about notation in this chapter. When using the symbol R, it is no
explicitly to distinguish between spot rates and yields to maturity as this
should be clear from the context/data being discussed. Where no ambiguity wi
superscript ˜n™ on R, and H, will be dropped in order to simplify the notation
reverse is the case.
Table 10.1 presents results for Germany across the whole term structure
to maturity rises monotonically as the term to maturity increases (column 1)
yield spread (R!"' - r f ) (column 7). The one-month holding period return H: I:
excess holding period return (H!:), - r f )both increase with maturity (colum
and the volatility of the price of 'long-maturity bonds' is greatly in excess
short-maturity bonds (column 4). Thus for Germany the results are quite strai
It appears that in order to willingly hold long-term bonds both a higher y
(Rj"' - r l ) or a higher excess holding period yield (Hf:\ - r f ) is required
over the 1967-1986 period and this is broadly consistent with the liquidity
hypothesis. However, agents who hold long bonds in order to obtain a high
expected excess HPY also experience increased 'risk', as measured in terms of
standard deviation (of bond prices). For example, when holding a 10-year b
than a 5-year bond the investor receives an average additional HPY of about
(= 1.62/1.34) but the standard deviation increases by about 43 percent (= 24
(Table 10.2, columns 5 and 6). As we shall see in Chapter 17, the above evid

15

14

13

12

11




h9
E
88
b
e 7

6

5

4

3
69112 74412 79/12 8

Figure 10.1 Government Securities Yields: Germany (Secondary Market - Percent p
Reproduced by permission of Joseph Bisignano
236

Table 10.1 Summary Market Yield and One-Period Holdin
Holding Period
Market Yield Return

Maturity Mean Variance Mean Varia
˜˜



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