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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

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