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tegrated (with a cointegration parameter equal to 14). The ‘fundamentals onl
Pf = aD, also implies that In P, and In Df are cointegrated with a cointegrat
eter of unity. However, estimates reveal this parameter to be in the range 1
that (In P, - In 0,) not be stationary. Hence taken at face value, these te
may
reject the (no-bubble) fundamentals model. However, Froot and Obstfeld no
OLS cointegrating parameter could be heavily biased (Banerjee et al, 1993), a
power and size of these tests are problematic.
Froot and Obstfeld then consider a direct test for the presence of intrins
based on estimation of (7.25). A representative result is:
+
( P / D ) , = 14.6 0.04D˜â€™6â€śâ€˜â€ť
(2.28) (0.12)
Annual: 1900-1988, R2 = 0.57, ( - ) = Newy-West standard errors.
Although there are some subtle econometric issues involved in testing for
A - 1 = 0 in (7.25) in a finite sample, the evidence above is in part suppor
intrinsic bubble. The joint null, that c and h - 1 equal zero is strongly rejected
the empirical evidence is not decisive since we do not reject the null that c =
5.4
5.2
5.0
4.0
4.6
4.4
4.2
4.0
3.8
3.6
3.4
1900 1910 1920 1930 1940 1950 1970 19
1960



Figure 7.3 Actual and Predicted Stock Prices. Source: Froot and Obstfeld (1991).
by permission of the American Economic Review


simulate values for the fundamentals price P f = 14.60t, and the price with
bubble P , given by (7.23) and compare these two series with the actual pri
path of the intrinsic bubble (Figure 7.3) is much closer to the actual path of s
than is Pf.The size of the bubble can also be very large as in the post-Sec
War period. Indeed, at the end of the period, the bubble element of the S&P
index appears to be large.
Finally, Froot and Obstfeld assess the sensitivity of their results to differen
models (using Monte Carlo methods) and to the addition of various additiona
of D or other deterministic time trends in the regression (7.22). The estim
t
basic intrinsic bubble formulation in (7.23) are quite robust.
Driffill and Sola (1994) repeat the Froot-Obstfeld model assuming divide
undergoes regime shifts, in particular that the (conditional) variance of divide
varies over the sample. A graph of (real) dividend growth for the US show
low variance between 1900 and 1920, followed by periods of fairly rapid ‘s
variance over 1920- 1950 and then relatively low and constant variance post-19
and Sola use the two-state Markov switching model of Hamilton (1989) (see
to model dividend growth and this confirms the results given by ‘eyeballing’
They then have two equations of the form (7.23) corresponding to each of the
of ‘high’ and ‘low’ variance. However, their graph of the price with an intrin
is very similar to that of Froot and Obstfeld (see l i t , Figure 7.3) so this particu
does not appear to make a major difference.
There are a number of statistical assumptions required for valid infere
approach of Froot and Obstfeld (some of which we have mentioned). For
Econometric Issues
There are severe econometric problems in testing for rational bubbles and th
tation of the results is also problematic. Econometric problems that arise
analysis of potentially non-stationary series using finite data sets, the behav
statistics in the presence of explosive regressors as well as the standard p
obtaining precise estimates of non-linear parameters (as in the case of intrins
and of corrections for heteroscedasticity and moving average errors. Some of
are examined further in Chapter 20. Tests for rational bubbles are often co
having the correct equilibrium model of expected returns: we are therefore tes
hypothesis. Rejection of the no-bubbles hypothesis may simply be a manifest
incorrect model based on fundamentals.
Another difficulty in interpreting results from tests of rational bubbles aris
Peso problem which is really a form of omitted variables problem. Suppos
in the market had information, within the sample period studied by the rese
dividends would increase in the future, but the researcher did not (or could n
this ‘variable’ in his model of fundamentals used to forecast dividends. In the
data, the stock price would rise but there would be no increase in dividends for
researcher. Stock prices would look as if they have overreacted to (current) div
more importantly such a rise in price might be erroneously interpreted as a b
problem of interpretation is probably most acute when there are severe price ch
one cannot rule out that the econometrician has omitted some factor (in say
model for dividends or discount rates) which might have a large impact o
dividends (or discount rates) but is expected to occur with a small probabili
to the issue of periodically collapsing bubbles, it appears unlikely that standar
detect such phenomena.


SUMMARY
7.4
Mathematically, rational bubbles arise because, in the absence of an (arbitrar
sality condition, the Euler equation yields a solution for stock prices that eq
mental value plus a ‘bubble term’, where the latter follows as a martingale p
key results in this chapter are as follows:
In the presence of a bubble, stock returns are unpredictable and therefore or
0

tests cannot be used to detect rational bubbles.
In the early literature bubbles were exogenous to fundamentals (e.g. divid
0

‘origin’ of the bubble cannot be explained and only the time path of th
given by these models.
Standard unit root and cointegration tests may be able to detect continuously
0

bubbles but are unlikely to detect periodically collapsing bubbles.
Intrinsic bubbles depend, in a non-linear deterministic way, on economic fu
0

(e.g. dividends) yet still satisfy the Euler equation. The evidence for intrin
bubble which is subject to ‘eruptions’ and subsequent collapse. However,
involves information being passed between a heterogeneous set of agents wi
beliefs, so it does not fit neatly into the type of intrinsic bubble in this cha
assumed a homogeneous set of rational agents.


ENDNOTES
1. Strictly speaking, for the Hausman test to be valid we do not require di
discount rates) and the bubble to be correlated. This is because bias may
constant term of the regression of P, on D,(and lagged values of D)or b
excluded bubble has an exploding variance.
2. This solution arises because D,+l= D,exp(s,+l) and hence E(D
D,E[exp(E,+l )I. The log-normal distribution has the property that E[ex
+
exp(p a 2 / 2 )where &,+I is distributed as independent normal with m
variance 02.
h Anomalies, Noise Traders
I
and Chaos
8.1 THE EMH AND ANOMALIES
In an ‘efficient market’ all ‘players’ have access to the same information, th
the information in the same ‘rational way’ and all have equal opportunities for
and lending. In the real world these conditions are unlikely to be met. Fo
different investors may form different probability assessments about future o
use different economic models in determining expected returns. They may also
ences in transactions costs (e.g. insurance companies versus individuals when
shares), or face different tax rates, and of course they will each devote a differ
of resources (i.e. time and money) in collecting and processing information. O
these heterogeneous elements play a rather minor role then asset prices and rate
will be determined mainly by economic fundamentals and rational behaviour.
prices may deviate substantially and persistently from their fundamental values.
below it is often the assumption of heterogeneity in behaviour which allows us
why markets may not be efficient. In this chapter we examine the EMH from
different angle to the technical statistical research outlined in Chapter 6, in pa

By observing actual behaviour in the stock market one can seek to isolate
trading opportunities which persist for some time. If these ‘profitable trad
reflect a payment for risk and are persistent, then this refutes the EMH. Th
is referred to as stock market anomalies.
Theoretical models are examined in which irrational ‘noise traders’ in
interact with rational ‘smart money’ traders. This helps to explain why s
might deviate from fundamental value for substantial periods and why st
might be excessively volatile.
Continuing the above theme of the interaction of noise traders and smart m
shown how a non-linear deterministic system can yield seemly random be
the time domain: this provides an applied example of chaos theory.

8.1.1 Weekend and January Effects
The weekend effect refers to the fact that there appears to be a systematic
daily rate of return on (some) stocks between the Friday closing and Monda
Monday (price is ‘low’) assuming that the expected profit more than covers t
costs and a payment for risk. This should then lead to a ‘removal’ of the ano
this should result in prices falling on Friday and rising on Monday.
The so-called January effect is a similar phenomenon to the weekend effec
rate of return on common stocks appears to be unusually high during the ea
the month of January. For the USA one explanation is due to year-end selli
in order to generate some capital losses which can be set against capital gai
to reduce tax liability. (This is known as ‘bed and breakfasting’ in the UK.)
investors wish to return to their equilibrium portfolios and therefore move into
to purchase stock. Again if the EMH holds, this predictable pattern of pri
should lead to purchases by non-tax payers (e.g. pension funds) in Decembe
price is low and selling in January when the price is high, thus eliminating th
arbitrage opportunity. The January effect seems to take place in the first five t
of January (Keane, 1983) and also appears to be concentrated in the stocks of
(Reinganum, 1983).

8.1.2 The Small Firm Effect
Between 1960 and the middle of the 1980s all small-capitalized companies
average a higher rate of return than the overall stock market index. Of course
to the CAPM this could be due to the higher risks attached to these small f
should be reflected in their higher beta values. However, Reinganum (1983) su
the rate of return, even after adjustment for risk, is higher on stocks of small
firms. Hence Reinganum has found that stocks of small firms do not lie on t
market line.

8.1.3 Closed End Funds
Closed end funds issue a fixed number of shares at the outset and trading in t
then takes place between investors. Shares which comprise the ‘basket’ in the
mutual fund are generally also traded openly on the stock market. The value
ought therefore to equal the market value of the individual shares in the fun
often the case that closed end mutual funds trade at a discount on their market
violates the EMH, for investors could buy the closed end fund’s shares at t
price and at the same time sell short a portfolio of stocks which are ident
held by the fund. The investor would thereby ensure he earned a riskless pro
the discount. Figure 8.1 shows that the discount on such funds can often be
(see Fortune (1991)) and appears to vary inversely with the return on the st
itself. For example, the bull markets of 1968-1970 and 1982-1986 are asso
declining discount where the bear markets of the 1970s and 1987 are asso
large discounts.
Several reasons have been offered for such closed end fund discounts. First,
fund members face a tax liability (in the form of capital gains tax), if the fund
securities after they have appreciated. This potential tax liability justifies pay
0


-
-10


-
-20



1 --30 1
1982 1986
1970 1974 1978 1990

Figure 8.1 Average Premium (+) or Discount (-) on Seven Closed-End Funds. Sou
(1991), Fig. 3, p. 23. Reproduced by permission of The Federal Reserve Bank of Bos

price than the market value of the underlying securities. Second, some of th
the closed end funds are less marketable (i.e. have ‘thin’ markets). Third, agen
the form of management fees might also explain the discounts. However, Mal
found that the discounts were substantially in excess of what could be expla
above reasons, while Lee et a1 (1990) find that the discounts on closed end
primarily determined by the behaviour of stocks of small firms.
There is a further anomaly. This occurs because at the initial public offe
closed end fund shares, they incur underwriting costs and the shares in th
therefore priced at a premium over their true market value. The value of the
fund then generally moves to a discount within six months. The anomaly is
any investors purchase the initial public offering and thereby pay the underw
via the future capital loss. Why don’t investors just wait six months before
the mutual fund at the lower price?

8.1.4 The Value Line Enigma
The Value Line Investment Survey (VLIS) produces reports on public traded
ranks these stocks in terms of their ‘timeliness’, by which it means the des
purchasing them. In Figure 8.2 is shown the excess return of ‘rank 1’ stocks ov
stocks and of ‘rank 5’ stocks over ‘rank 3’ stocks. ‘Rank 3’ stocks are designa
that are expected to increase in line with the market, while rank 1 stocks are a
and rank 5 stocks a ‘bad buy’, in terms of their expected future returns. Cle
stocks earn a higher return than rank 3 stocks. This could be due to the fact
stocks have higher risk (reflected in higher betas) than do rank 3 stocks. Also
a trading strategy to be profitable account must be taken of transactions costs
(1981) found that even after adjustments for risk and transactions costs, a pass
of purchasing rank 1 stocks at the beginning of the year and selling them at
the year outperformed the passive strategy using rank 3 stocks. The Value Li
System therefore does provide profitable information for a buy and hold strate
is inconsistent with the EMH.
\-
-20-- Less \I
c
Rank 3
1 1 I
1
I




Figure 8.2 Annual Excess Return on Stocks, Classified by Value Link Rank, 19
Source: Fortune (1991)’ Fig. 4, p. 24. Reproduced by permission of The Federal Rese
Boston

8.1.5 Winner’s Curse
There exists a strong negative serial correlation for stock returns for those
have experienced extreme price movements (particularly those which experie
fall followed by ‘a price rise). Thus for some stocks (or portfolios of stocks) the
reversion in stock price behaviour. Put another way there is some predictabili
returns. The issue for the EMH is then whether such predictability can lead to su
profits net of transactions costs and risk.
De Bondt and Thaler (1985) take 35 of the most extreme ‘winners’ and
extreme ‘losers’ over the five years from January 1928 to December 1932
monthly return data from the NYSE) and form two distinct portfolios of these c
shares. They follow these companies for the next five years (= ‘test period’). T
the exercise 46 times by advancing the start date by one year each time. Fi
calculate the average ‘test period’ performance (in excess of the return on the wh
index) giving equal weight (rather than value weights) to each of the 35 compa
find (Figure 8.3):
(i) The five-year price reversals for the ‘loser portfolio’ (at about plus 30 p
more pronounced than for the ‘winner portfolio’ (at minus 10 percent).
(ii) The excess returns on the ‘loser portfolio’ occur in January (i.e. ‘Janua
(iii) The returns on the portfolios are mean reverting (i.e. a price fall is fol
price rise and vice versa).
It is worth emphasising that the so-called ‘loser portfolio’ (i.e. one where p
fallen dramatically in the past) is in fact the one that makes high returns in
a somewhat paradoxical definition of ‘loser’. An arbitrage strategy of selling t
portfolio’ short and buying the ‘loser portfolio’ earns profits at an annual rate
5-8 percent (see De Bondt and Thaler (1989)).
Bremer and Sweeney (1988) find that the above results also hold for very
periods. For example, for a ‘loser portfolio’ comprising stocks where the on
fall has been greater than 10 percent, the subsequent returns are 3.95 percen
days. They use stocks of largefirms only. Therefore they have no problem that t
0.2

a 0.1
a
0
0.0

-0.1

-0.2




Figure 8.3 Cumulative Excess Returns for ‘Winner’ and ‘Loser’ Portfolios. Sourc
and Thaler (1989). Reproduced by permission of the American Economic Association

spread is a large percentage of the price (which could distort the results). Also
problems with ‘the small firm effect’ (i.e. smaller firms are more ‘risky’ and he
a greater than average equilibrium excess return). Hence Bremer and Sweeney
to find evidence of supernormal profits and a violation of the EMH.
One explanation of the above results is that ‘perceived risk’ and actual risk m
That is the perceived risk of the ‘loser portfolio’ is judged to be ‘high’ hence
high excess return in the future, if one is to hold them. Evidence from psycholog
suggests that misperceptions of risk do occur. For example, people rank the
of dying from homicide greater than the risk of death from diabetes but proba
they are wrong.
The above evidence certainly casts doubt on the EMH in that it may be
make supernormal profits because of some predictability in stock prices. Howe
be noted that many of the above anomalies are most prominent among small
January effect and winner’s curse of De Bondt and Thaler (1985) and discounts
End Funds (Lee et al, 1990). If the ‘big players’ (e.g. pension funds) do not trad
firm stocks then it is possible that the markets are too thin and information g
costly, so the EMH doesn’t apply. The EMH may therefore be a better pa
the stocks of large firms. These will be actively traded by the ‘big players
be expected to have the resources to process quickly all relevant information
access to cash and credit to execute trades so that prices (of such stocks) al
fundamental value. Of course, without independent confirmation this is mere


NOISE TRADERS
8.2
The EMH does not require that all participants in the market are ‘efficien
informed. There can be a set of irrational or ‘noise’ traders in the market w
quote prices equal to fundamental value. All the EMH requires is that there i
‘smart money’ around who recognise that P will eventually equal fundam
r
Vt. So, if some irrational traders quote Pt < Vt, the smart money will qui
irrational investors are able to survive in the market (De Long et al, 1990).
If investors have finite horizons then they will be concerned about the pri
future time N . However, if they base their expectations of the value of E,P,+N o
+
future dividends from t N onwards, then we are back to the infinite horizo
tion of the rational investor (see Chapter 4). However, if we allow heterogene
in our model then if agents believe the world is not dominated by rationa
the price at t + N will depend in part on what the rational investor feels th
investors view of P t + will be (i.e. Keynes’ beauty contest). This general arg
˜
applies if rational investors know that other rational investors use different
equilibrium asset returns. Here we are rejecting the EMH assumption that a
instantaneously know the true model, or equivalently that learning by market p
about the changing structure of the economy (e.g. ‘shipbuilding in decline, ch
grow’) is instantaneous. In these cases, rational investors may take the view tha
price is a weighted average of the rational valuation (or alternative rational
and effect on price of the irrational traders (e.g. chartists). Hence price does
+
equal fundamental value at t N . The rational traders might be prevented by
from buying or selling until the market price equals what perhaps only they be
fundamental value. Bonus payments to market traders based on profits over a
period (e.g. monthly) might reinforce such behaviour. The challenge is to dev
models that mimic such noise-trader behaviour.
A great deal of the analysis of financial markets relies on the principle of arb
see Shleifer and Summers (1990)). Arbitrageurs or smart money or rational
continually watch the market and quickly eliminate any divergence between a
and fundamental value and hence immediately eliminate any profitable oppo
a security has a perfect substitute then arbitrage is riskless. For example, a (v
+
mutual fund where one unit of the fund consists of ‘1 alpha 2 beta’ shares sh
the same price that one can purchase this bundle of individual shares in the op
If the mutual fund is ‘underpriced’ then a rational trader should purchase th
simultaneously sell (or short sell) the securities which constitute the fund, o
market, thus ensuring a riskless profit (i.e. buy ‘low’, sell ‘high’). If the sm
has unlimited funds and recognises and acts on this profit opportunity then
quickly lead to a rise in price of the mutual fund (as demand increases) an
price of the securities on the stock exchange (due to increased sales). Riskle
ensures that relative prices are equalised. However, if there are no close sub
hence if arbitrage is risky then arbitrage may not pin down the absolute pric
stocks (or bonds) as a whole.
The smart money may consider short selling a share that appears to be
relative to fundamentals. They do so in the expectation that they can purch
when the actual price falls to the price dictated by fundamentals. If enough o
money acts on this premise then their actions will ensure that the price does
all start short selling. The risks faced by the smart money are twofold. First
may turn out to be ‘better than expected’ and hence the actual price of the shar
further: this we can call fundamentals risk. Second, if arbitrageurs know th
horizon. The smart money may believe that prices will ultimately fall to their fu
value and hence in the long term, profits will be made. However, if arbitra
either to borrow cash or securities (for short sales) to implement their trades
pay per period fees or report their profit position on their ‘book’ to their s
frequent intervals (e.g. monthly, quarterly) then an infinite horizon certainly ca
to all or even most trades undertaken by the smart money.
It may be that there are enough arbitrageurs, with sufficient funds in the ag
that even over a finite horizon, risky profitable opportunities are arbitraged away
of the latter argument is weakened, however, if we recognise that any single ar
unlikely to know either the fundamental value of a security, or to realise whe
price changes are due to deviations from the fundamental price. Arbitrageurs
are also likely to disagree among themselves about fundamental value (i.e.
heterogeneous expectations) hence increasing the general uncertainty they perc
profitable opportunities, even in the long term. Hence the smart money ha
in identijjing any mispricing in the market and if funds are limited (i.e. a
perfectly elastic demand for the underpriced securities by arbitrageurs) or h
finite, it is possible that profitable risky arbitrage opportunities can persist in
for some time.
If one recognises that ‘information costs’ (e.g. man-hours, machines, build
be substantial and that marginal costs rise with the breadth and quantity of tr
this also provides some limit on arbitrage activity in some areas of the m
example, to take an extreme case, if information costs are so high that dea
concentrate solely on bonds or solely on stocks (i.e. complete market segment
differences in expected returns between bonds and stocks (corrected for risk)
arbitraged away.

Noise Traders and Herding
It was explained above why risky arbitrage may be limited and insufficient to k
prices of stocks in line with their fundamental value. We can now discuss wh
might contain a substantial number of noise traders who follow simple ‘rules
or ‘trends’ or waves of investor sentiment (herding behaviour) rather than act o
of fundamentals. In order that noise traders as a group are capable of influenc
prices their demand shifts must broadly move in unison (i.e. be correlated ac
traders).
General information on these issues can be had from psychological experi
Shleifer and Summers (1990) and Shiller (1989) for a summary) which ten
that individuals make systematic (i.e. non-random) mistakes. Subjects are foun
react to new information (news) and they tend to extrapolate past price trends
overconfident, which makes them take on excessive risk.
As the stock market involves groups of traders it is useful to consider some e
on group behaviour (Shiller, 1989). In Sherif‘s (1937) ‘autokinetic experiment’
in total darkness were asked to predict the movement of a pencil of light. In the e
with individuals there was no consensus about the degree of movement (wh
with a group where all other members of the group are primed to give the s
answers. The individual when alone usually gave correct answers but when
group pressure the ‘individual’ frequently gave wrong answers. After the ex
was ascertained that the individual usually knew the correct answer but wa
contradict the group. If there is no generally accepted view of what is the
fundamental price of a given stock then investors may face uncertainty rathe
This is likely to make them more susceptible to investor sentiment.
Models of the diffusion of opinions are often rather imprecise. There is ev
ideas can remain dormant for long periods and then be triggered by some seemi
event. The news media obviously play a role here, but research on persuasion
that informal face-to-face communication among family, friends and co-wo
greater importance in the diffusion of views than is the media.
There are mathematical theories of the diffusion of information based on
epidemics. In such models there are ‘carriers’ who meet ‘susceptibles’ and c
carriers’. Carriers die off at a ‘removal rate’. The epidemic can give rise to
shape pattern if the infection ‘takes off’. If the infection doesn’t take off (i.e.
either a low infection rate or a low number of susceptibles or a high removal ra
number of new carriers declines monotonically. The difficulty in applying su
to investor sentiment is that one cannot accurately quantify the behavioural de
of the various variables (e.g. the infection rate) in the model, which are like
from case to case.
Shiller (1989) uses the above ideas to suggest that the bull market of the
1960s may have something to do with the speed with which general informa
how to invest in stocks and shares (e.g. investment clubs) spreads among indiv
also notes the growth in institutional demand (e.g. pension funds) for stock
period, which could not be offset by individuals selling their own holdings to
total savings constant. This was because individuals’ holdings of stocks were n
evenly distributed (most being held by wealthy individuals): some people in oc
pension funds simply had no shares to sell.
Herding behaviour or ‘following the trend’ has frequently been observed in t
market, in the stock market crash of 1987 (see Shiller (1990)) and in the foreign
market (Frankel and Froot, 1986 and Allen and Taylor, 1989b). Summers (198
and Summers, 1990) also shows that a time series for share prices that is
generated from a model in which price deviates from fundamentals in a pers
does produce a time series that mimics actual price behaviour (i.e. close to
walk) so that some kind of persistent noise-trader behaviour is broadly cons
the observed data.

Survival of Noise Traders
If we envisage a market in which there are smart speculators who tend to set p
to fundamental value and noise traders who operate on rules of thumb, then
arises as to how the noise traders can survive in this market. If noise traders
when the price is above the fundamental value, then the smart money should
money should purchase such assets from the noise traders and they will th
profit as the price rises towards the fundamental value. Hence the net effect
noise traders lose money and therefore should disappear from the market leavi
smart money. When this happens prices should then reflect fundamentals.
Of course, if there were an army of noise traders who continually entered
(and continually went bankrupt) it would be possible for prices to diverge f
mental value for some significant time. One might argue that it is hardly likely
traders would enter a market where previous noise traders have gone bankru
numbers. However, entrepreneurs often believe they can succeed where others h
To put the reverse argument, some noise traders will be successful over a fin
and this may encourage others to attempt to imitate them and enter the marke
the fact that the successful noise traders had in fact taken on more risk and jus
to get lucky.
Can it be explained why an existing cohort of noise traders can still make
market which contains smart money? The answer really has to do with the p
herding behaviour. No individual smart money trader can know that all other sm
traders will force the market price towards its fundamental value in the peri
for which he is contemplating holding the stock. Thus any strategy that the so
traders adopt given the presence of noise traders in the market is certainly n
There is always the possibility that the noise traders will push the price even fu
from fundamental value and this may result in a loss for the smart money.
averse smart money may not fully arbitrage away the influence of the noise
there are enough noise traders who follow common fads then noise-trader r
pervasive (systematic). It cannot be diversified away and must therefore earn a
risk premium, in equilibrium. Noise trading is therefore consistent with an ave
which is greater than that given by the pure CAPM. If noise traders hold a lar
assets subject to noise-trader risk they may earn above average returns and sur
market. If there are some variables at time t which influence the ‘mechanical’
of noise traders and noise-trader behaviour is persistent then such variables ma
expected returns in the market. This may explain why additional variables, w
to the CAPM, prove to be statistically significant.
Shiller (1989) presents a simple yet compelling argument to suggest tha
non-institutional investors are concerned, the smart money may not dominate
He notes that if the smart money investor accumulates wealth at a rate ‘
per annum) greater than the ordinary individual investor (e.g. noise trader) th
bequest at age 50, he can expect to accumulate additional terminal wealth of
If n = 5 percent then the smart investor ends up with 2.1 times as much we
ordinary (noise-trader) investor. Thus if the percentage of smart investors in
is ‘moderate’ then they are unlikely to take over the market completely. Also i
money investor wishes only to preserve the real value of the ‘family wealth’, t
not accumulate any additional wealth, he will spend it. However, given that i
investors play an important role in the market it must be explained why no
influence institutional decisions on portfolio allocation.
volatile than indicated by the volatility in future dividends and discount ra
Shiller’s variance bounds inequalities might not hold in a world which inc
traders. To see this, note that the stock price is now determined by its fundam
V, = E,(CS‘D,+i) (i.e. the DPV of future dividends) formed by rational trad
influence of the noise traders denoted N,:


For simplicity assume cov(V,, N,) = 0 and hence from (8.1) cov(P,, N,) = v
before the perfect foresight price PT differs from the fundamental value by
forecast error (due entirely to rational traders):
+ qr
P; = v,
From (8.1) and (8.2)
+ qr - Nt
P: = P,
+ var(q,) + var(N,) - 2 cov(P,, N,)
var(P;) = var(P,)
= var(P,) + var(qt) - var(N,)
where it has been assumed cov(P,, qr) = 0 by RE and for simplicity we
q,) = 0. From (8.3) we see that if var(N,) is large enough then we expect the
P: to be less than the variance of P,. Hence, the variance bound, var(P:) - v
may not hold in the presence of noise traders. The intuition behind this resul
Noise traders directly influence the variance of actual prices (via (8.1)) but do n
the variance of future dividends and hence the perfect foresight price. Also
noise trader behaviour Nt is mean reverting at long horizons then price chang
be mean reverting and this is consistent with the empirical evidence on mea
in stock returns.
If noise traders are more active in dealing in shares of small firms than for
this may explain why small firms earn an above average return corrected for
CAPM risk. Again, this is because the noise-trader risk is greater for small-
than for large-firm stocks, and as this risk is systematic it is reflected in the hi
on small-firm stocks.
The impact of noise traders on prices may well be greater when most inves
the advice given in finance text books and passively hold the market portfol
traders move into a particular group of shares based on ‘hunch’, the holders of
portfolio will do nothing (unless the movement is so great as to require a ch
‘market value’ proportions held in each asset). The actions of the noise traders
countered by a set of genuine arbitrageurs who are active in the market. In th
if all investors hold the market portfolio but one noise trader enters the mark
to purchase shares of a particular firm then its price will be driven to infinity
Arbitrageurs may not only predict fundamentals but may also divert their
anticipating changes in demand by the noise traders. If noise traders are optim
particular securities it will pay arbitrageurs to create more of them (e.g. j
the business. The arbitrageurs (e.g. an investment bank) can then earn a shar
from the ‘abnormally high priced’ issues of new oil shares which are current
with noise traders.
Arbitrageurs will also behave like noise traders in that they attempt to pick
noise-trader sentiment is likely to favour: the arbitrageurs do not necessarily co
in demand by noise traders. Just as entrepreneurs invest in casinos to exploi
it pays the smart money to spend considerable resources in gathering info
possible future noise-trader demand shifts (e.g. by studying chartists’ foreca
arbitrageurs have an incentive to behave like noise traders. For example, if n
are perceived by arbitrageurs to be positive feedback traders then as prices
above fundamental value, arbitrageurs get in on the bandwagon themselves in th
they can sell out ‘near the top’. They therefore ‘amplify the fad’. Arbitrageurs
prices in the longer term to return to fundamentals (perhaps aided by arbitrage
in the short term, arbitrageurs will ‘follow the trend’. This evidence is cons
findings of positive autocorrelation in returns at short horizons (e.g. weeks or
arbitrageurs follow the short-term trend, and negative correlation at longer ho
over two or more years) as some arbitrageurs take a long horizon view and sell
shares. Also if ‘news’ triggers off noise-trader demand, then this is consistent
overreacting to ‘news’.
So far we have been discussing the implications of the presence of noise trad
general terms. It is now time to examine more formal models of noise-trader
As one might imagine it is by no means easy to introduce noise-trader behav
fully optimising framework since almost by definition noise traders are irratio
misperceive the true state of the world. Noise-trader models therefore contain
arbitrary (non-maximising) assumptions about behaviour. Nevertheless, the o
the interaction between smart money traders (who do maximise a well-define
function) and the (ad-hoc) noise traders is of interest since we can then ascerta
such models confirm the general conjectures made above. Generally speak
shall see, these more formal models do not contradict our ‘armchair specu
outlined above.

8.2.1 Noise Waders and the Rational Valuation Formula
Shiller (1989) provides a simple piece of analysis in which noise-trader dema
as smart money influence the price of stocks. It follows that the smart mone
to predict the noise-trader demand for stocks, if it is to predict price correctly
attempt to eliminate profitable opportunities. The proportionate demand for sh
smart money is Q1.The demand function for the smart money is based (loos
mean-variance model and is given by:


If ErR,+1 = p then demand by the smart money equals zero. If Q, = 1 then
money holds all the outstanding stock and this requires an expected return E t R t
Hence 0 is a kind of risk premium payment to induce the smart money to hold al
substituting (8.4) in (8.5):


Hence the expected return as perceived by the smart money depends on how th
current and future demand by noise traders: the higher is noise-trader demand
are current prices and the lower is the expected return perceived by the sm
Using (8.6) and the definition




+ + 0). Hence by repeated forward substitution:
where 6 = 1/(1 p




Thus if the smart money is rational and recognises the existence of a deman
traders then the smart money will calculate that the market clearing price is
average of fundamentals (i.e. E,D,+i) and of future noise-trader demand, E
weakness of this 'illustrative model' is that noise-trader demand is completely
However, as we see below, we can still draw some useful insights.
If E,Y,+i and hence aggregate noise-trader demand is random around zer
moving average of E,Y,+i (for all future i ) in (8.8) will have little influence on
will be governed primarily by fundamentals. Price will deviate from fundam
only randomly. On the other hand, if demand by noise traders is expected to b
(i.e. 'large' values of Y , are expected to be followed by further large values)
changes in current noise-trader demand can have a powerful effect on curren
price can deviate substantially from fundamentals over a considerable period
Shiller (1989) uses the above model to illustrate how tests of market effici
on regressions of returns on information variables known at time t (Q,), have
to reject the EMH when it is false. Suppose dividends (and the discount rate) a
for all time periods and hence the EMH (without noise traders) predicts tha
price is constant. Now suppose that the market is actually driven entirely by n
and fads. Let noise trader demand be characterised by


and hence:
S3, etc. Because 0 < 6 < 1, price changes are heavily dominated by uf (rath
past U r - j ) . However as uf is random, price changes in this model, which by c
are dominated by noise traders, are nevertheless largely unforecastable.
Shiller generates a AP,+1 series using (8.8) for various values of the per
Y t (given by the lag length n) and for alternative values of p and 8. He the
the generated data for AP,+1 on the information set consisting only of Pt.
EMH we expect the R-squared of this regression to be zero. For p = 0, 0
n = 20 he finds R2 = 0.015. The low R-squared supports the EMH, but it res
model where price changes are wholly determined by noise traders. In additio
level can deviate substantially from fundamentals even though price changes
forecastable. He also calculates that if the generated data includes a constan
price ratio of 4 percent then the ‘theoretical R-squared’ of a regression of the
on the dividend price ratio ( D , / P , ) is only 0.079 even though the noise-tra
is the ‘true model’. Hence empirical evidence that returns are only very wea
to information at time t (e.g. Dt/P,) are not necessarily inconsistent with p
determined by noise traders and not by fundamentals.
Overall, Shiller makes an important point about empirical evidence. Th
using real world data is not that stock returns are unpredictable (as sugges
EMH) but that stock returns are not very predictable. However, the latter evide
not inconsistent with possible models in which noise traders play a part.
If the behaviour of Y , is exogenous (i.e. independent of dividends) but is sta
mean reverting then we might expect returns to be predictable. An above av
of Y will eventually be followed by a fall in Y (to its mean long-run level). H
are mean reverting and current returns are predictable from previous periods’
In addition our noise-trader model can explain the positive association b
dividend price ratio and next periods’ return on stocks. If dividends vary very
time, a price rise caused by an increase in E Y , + j will produce a fall in the divi
ratio. If Yt is mean reverting then prices will fall in the future, so returns Rt+
Hence one might expect a fall in the dividend price ratio at time t to be foll
fall in returns. Hence ( D / P ) , is positively related to returns Rr+l, as found in
studies.
Shiller also notes that if noise trader demand Y f + j is influenced either by p
(i.e. bandwagon effect) or past dividends then the share price might overreact
dividends compared to that given by the first term in (8.8), that is the fundam
of the price response.

8.2.2 An Optimising Model of Noise-’Ikader Behaviour
in the model of De Long et a1 (1990), both smart money and noise traders
expected lifetime utility. Both noise traders and smart money are risk averse
a finite horizon in the model so that arbitrage is risky. The (basic) model is c
so that there is no fundamental risk (i.e. dividends are known with certainty
noise-trader risk. The noise traders create risk for themselves and the smart
generating fads in demand for the risky asset. The smart money forms optima
- N ( P * ,a2)
Pr

If p* = 0, noise traders agree on their forecasts with the smart money (on a
noise traders are on average pessimistic (e.g. in a bear market) then p* < 0, an
price will be below fundamental value. If noise traders are optimistic p* > 0, th
applies. As well as having this long-run view (= p * ) of the divergence of the
from the optimal forecasts, ‘news’ also arises so there can be abnormal but
variations in optimism and pessimism (given by a term, p - p * ) . The specific
is ad hoc but does have an intuitive appeal based on introspection and evid
behavioural/group experiments.
In the DeLong et a1 model the fundamental value of the stock is a cons
arbitrarily set at unity. The market consists of two types of asset: a risky a
safe asset. Both noise traders and smart money are risk averse so their dem
risky asset depends positively on expected return and inversely on the noise-
The noise trader demand also depends on an additional element of return de
whether they feel bullish or bearish about stock prices (i.e. the variable p,)
asset is in fixed supply (set equal to unity) and the market clears to give an e
price P,. The equation which determines P, looks rather complicated but we
it down into its component parts and give some intuitive feel for what is goi
D e h n g et a1 equation for P, is given by



where p = the proportion of investors who are noise traders, r = the riskle
of interest, y = the degree of (absolute) risk aversion and a2 = variance of n
misperceptions.
If there are no noise traders p = 0 and (8.12) predicts that the market p
its fundamental value (of unity) as set by the smart money. Now let us supp
a particular point in time, noise traders have the same long-run view of the
as does the smart money (i.e. p* = 0) and that there are no ‘surprises’ (i.e. n
bullishness or bearishness), so that (pi - p * ) = 0. We now have a position
noise traders have the same view about future prices as do the smart money
the equilibrium market price still does not solely reflect fundamentals and
market price will be less than the fundamental price by the amount given
term on the RHS of equation (8.12). This is because the mere presence of n
introduces an additional element of uncertainty since their potential actions ma
future prices. The price is below fundamental value so that the smart money
traders) may obtain a positive expected return (i.e. capital gain) because of thi
noise-trader risk. Both types of investor therefore obtain a reward for risk
generated entirely by fads and not by uncertainty about fundamentals. This m
probably the key result of the model and involves a permanent deviation of
fundamentals. The effect of the third term is referred to in (8.12) as the amou
mispricing ’.
will perform even better than average. These terms imply that at particular ti
price may be above or below fundamentals.
The duration of the deviation of the actual stock price from its fundamental v
depends upon how persistent the effects on the RHS of equation (8.12) are.
varies so that pr - p* is random around zero then the actual price would deviat
around its ‘basic mispricing’ level. In this case the model would give a moveme
prices which was ‘excessively volatile’ (relative to fundamentals). From (8.
that the variance of prices is:




Hence excess volatility is more severe, the greater is the variability in the mis
of noise traders 0 2 ,the more noise traders there are in the market p and tile l
cost of borrowing funds r .
The above mechanism does not cause stock prices to move away from
mispricing level for long periods of time and to exhibit volatility which is
over time (i.e. periods of tranquillity and turbulence). To enable the mode
duce persistence in price movements and hence the broad bull and bear mo
stock prices, we need to introduce ‘fads’ and ‘fashions’. Broadly speaking th
for example, that periods of bullishness are followed by further periods of b
Statistically this can be represented by a random walk in p,*:



-
where N ( 0 , o ; ) . (Note that o is different from 02, above.)
:
0,
At any point in time the investors’ best guess (optimal forecast) of p* is
value. However, as ‘news’ (a,) arrives noise traders alter their views about p
‘change in perceptions’ persists over future periods. It should be clear from
term in (8.12) that a random walk in p* can generate a sequence of values
also follow a random walk and therefore mimic a stochastic cyclical path (i.e. m
over time which are not smooth and not of equal amplitude and wavelength). T
in ‘bull and bear’ patterns in P,.
Fortune (1991) assumes for illustrative purposes independent normal distri
at and (pt - p;) and uses representative values for r, p, y in (8.12). He then
values for Of and ( p - p;) using a random number generator and obtains a
for P, shown in Figure 8.4. The graph indicates that on this one simulation,
to 85 percent of fundamental value (which itself may be rising) with some dra
and falls in the short run.
An additional source of persistence in prices could be introduced into the
assuming that a2is also autoregressive. The latter, of course, embodies the hyp
variances can be time varying and this may, for example, be modelled using A
GARCH models (see Part 7). It is also not unreasonable to assume that the ‘
rate’ from being a smart money trader to being a noise trader may well tak
54 107 160 213 266 319 372 425 478 531 584
1
Month

Figure 8.4 Simulated 50-Year Stock Price History. Source: Fortune (1991), Fig. 6, p
duced by permission of The Federal Reserve Bank of Boston

move in cycles. This will make p (i.e. the proportion of noise traders) exhibit
and hence so might P I . It follows that price may differ from fundamentals for
periods of time in this type of model. Mispricing can therefore be severe and
because arbitrage is incomplete.

Mean 'Reversion and Predictability of Returns
Waves of optimism and pessimism in noise-trader behaviour could also imply s
tence in pI - p*. The behaviour of (pf - p * ) could be mean reverting so
(pl - p*) is positive it will fall back towards its mean, some time in the f
would imply that prices are mean reverting and that returns on the stock
partly predictable from past returns or from variables such as the dividend
Note that to introduce mean reversion in prices an ad-hoc assumption has
that fads are mean reverting: this latter assumption has not been derived from
formal optimising model.

Can Noise Traders Survive?
D e h n g et al show that where the proportion of noise traders is fixed in each
p is constant) it is possible (although not guaranteed) that noise traders do su
though they tend to buy high and sell low (and vice versa). This is because the
optimistic and underestimate the true riskiness of their portfolio. As a conseq
tend to 'hold more' of the assets subject to bullish sentiment. In addition, if n
risk o2 is large, the smart money will not step in with great vigour, to buy u
assets because of the risk involved.
The idea of imitation can be included in the model by assuming that the
rate from smart money to noise trader depends on the excess returns earne
traders over the smart money (R" - R') in the previous period:
+ q ( R " - R")r
cr+l=

where p is bounded between 0 and 1. De Long et a1 also introduce fundam
into the model. The per period return on risky assets becomes a random vari
since otherwise the newly converted noise traders may influence price and thi
forecast by the ‘old’ noise traders who retire.)

Closed End Fund Anomalies
We have noted that closed end funds often tend to sell at a discount and th
varies over time, usually across all funds. Sometimes such funds sell at a prem
our noise-trader model we can get a handle on reasons for these empirical ano
the risky asset be the closed end fund itself and the safe asset the actual underly
The smart money will try and arbitrage between the fund and the underlying
buy the fund and sell the stocks short, if the fund is at a discount). Howev
p, = p* = 0, the fund (risky asset) will sell at a discount (see equation (8.12))
inherent noise-trader risk. Changes in noise-trader sentiment (i.e. in p* and p,
cause the discounts to vary over time and as noise-trader risk is systematic, d
most funds are expected to move together.
In the noise-trader model a number of closed end funds should also tend to
at the same time, namely when noise-trader sentiment for closed end funds i
p* > 0, p, - p* > 0). When existing closed end funds are at a premium it pay
money to purchase shares (at a relatively low price), bundle them together in
end fund and sell them at a premium to optimistic noise traders.
Again the key feature of the De Long et a1 model is to demonstrate the po
underpricing in equilibrium. The other effects mentioned above depend on one’s
to the possibility of changes in noise-trader sentiment, which are persistent.
‘persistence’ is not the outcome of an optimising process in the model althou
intuitively appealing one.

Changes in Bond Prices
In empirical work on bonds we shall see (Chapter 14) that when the long-sh
(R-r) on bonds is positive, then long rates tend to fall, and hence the prices of
tend to rise. This is the opposite to what one would expect from the pure ex
hypothesis of the term structure, which incorporates the behaviour of rational r
agents only. The stylised facts of this anomaly are consistent with our noise-tra
with the long bond being the risky asset (and the short bond the safe asset). Wh
then the price of long bonds as viewed by noise traders could be said to be a
low. If noise-trader fads are mean reverting they will expect bond prices to
future and hence long rates R to fall. This is observed in the empirical work o
structure. Of course, even though the noise-trader model explains the stylised
still leaves us a long way from a formal test of the noise-trader model in the bo

Short-Termism
In a world of only smart money, the fact that some of these investors take a ‘
view of returns should not lead to a deviation of price from fundamentals. The
chain of short-term ‘rational fundamental’ investors performs the same calcul
investor with an infinite horizon.
With a finite investment horizon and the presence of noise traders the abov
doesn’t hold. True, the longer the horizon of the smart money the more willing
to undertake risky arbitrage based on divergences between price and fundame
The reason being that in the meantime he receives the insurance of dividend
each period and he has a number of periods over which he can wait for the p
to fundamental value. However, even with a ‘long’ but finite horizon there is
resale risk. The share in the total return from dividend payments over a ‘lon
period is large but there is still some risk present from uncertainty about p
‘final period’.
We note from the noise-trader model that if a firm can make its equity
subject to noise-trader sentiment (i.e. to reduce 02) then its underpricing w
severe and its price will rise. This reduction in uncertainty might be accompl

(i) raising current dividends (rather than investing profits in an uncertain
investment project, for example R&D expenditures)
(ii) substitutes debt for equity,
(iii) share buybacks.

Empirical work by Jensen (1986) has shown that items (i)-(iii) do tend to
increase in the firm’s share price and this is consistent with our interpreta
influence of noise traders described above. It follows that in the presence of n
one might expect changes in capital structure to affect the value of the fr (
im
the Modigliani-Miller hypothesis).

Mispricing and Short-Termisrn: Shleifer- Vishny Model
The underpricing of an individual firm’s stock is not a direct result of the fo
trader model of De Long et a1 since the formal model requires noise-trader beha
systematic across all stocks. However, the impact of high borrowing costs on
of mispricing in individual shares has been examined in a formal model
and Vishny (1990). They find that current mispricing is most severe for th
where mispricing is revealed at a date in the distant future (rather than next p
Suppose physical investment projects with uncertain long horizon payoffs a
with shares whose true value is only revealed to the market at long horiz
Shleifer-Vishny model these shares will be severely underpriced. It follows th
might be less willing to undertake such long horizon yet profitable projects. Sho
on the part of the firm’s managers might ensue, that is they choose less profi
term physical investment projects rather than long-term projects since this in
current undervaluation of the share price and less risk of them losing their
a hostile takeover or management reorganisation by the board of directors
misallocation of real resources. We begin our description of this model cons
If the smart money (arbitrageur) has access to a perfect capital market wh
borrow and lend unlimited amounts then he does not care how long it takes a
security to move to fundamental value. Table 8.1 considers a simple case of u
where the cost of borrowing r, and the fundamentals return on the security (i.
return q), are identical at 10 percent. If the mispriced security moves from
fundamental value of $6 after only one period, the price including the divid
+
is $6 (1 q) = $6.6 in period 1. At the end of period 1 the arbitrageur has t
+
the loan plus interest, that is $5(1 r) = $5.5. If the price only achieves its fu
+ +
value in period 2 the arbitrageur receives $6(1 q)2 = $7.26 at t 2 but has
+ +
additional interest charges between t 1 and t 2. However, in present valu
arbitrageur has an equal gain of $1 regardless of when the mispricing is irradic
with a perfect capital market he can take advantage of any further arbitrage p
that arise since he can always borrow more money at any time.

Finite Horizon
In the case of a finite horizon, fundamentals and noise-trader risk can lead
from arbitrage. If suppliers of funds (e.g. banks) find it difficult to assess the
arbitrageurs to pick genuinely underpriced stocks, they may limit the amount
the arbitrageur. Also, they may charge a higher interest rate to the arbitrageur be
have less information on his true performance than he himself has (i.e. the inte
under asymmetric information is higher than that which would occur under
information).
If r = 12 percent in the above example, while the fundamentals return on
remains at 10 percent then the arbitrageur gains more if the mispricing is
sooner rather than later. If a strict credit limit is imposed then there is an add
to the arbitrageur, namely that if his money is tied up in a long-horizon arbitra
then he cannot take advantage of other potentially profitable arbitrage opportu
An arbitrageur earns more potential $ profits the more he borrows and takes
in undervalued stocks. He is therefore likely to try and convince (signal to) the s

lhble 8.1 Arbitrage Returns: Perfect Capital Market
Assumptions:
Fundamental Value = $6
= $5
Current Price
Interest Rate, r = 10% per period
Return on Risky Asset, q = 10% per period (on fundamental value)
Smart Money Borrows $5 at 10% and Purchases Stock at t = 0.

D
Selling Price Repayment of Net Gain
(including Loan o
dividends) (a
+ + r ) = $5.5
Period 1 6(1 4)= $6.60 5(1 $1.10 $
+ + r)* = $6.05
Period 2 6(1 q)* = $7.26 5(1 $1.21 $
The formal model of Shleifer and Vishny (1990) has both noise traders
money (see Appendix 8.2). Both the short and long assets have a payout at the
in the future but the true vahe of the short asset is revealed earlier than that fo
asset. They show that in equilibrium, arbitrageurs’ rational behaviour results
current mispricing of ‘long assets’ when the mispricing is revealed at long ho
terms ‘long’ and ‘short’ therefore refer to the date at which the mispricing
(and not to the actual cash payout of the two assets). Both types of asset are
but the long-term asset suffers from greater mispricing than the short asset.
In essence the model relies on the cost of funds to the arbitrageur being g
the fundamentals return on the mispriced securities. Hence the longer the arbi
to wait before he can liquidate his position (i.e. sell the underpriced security
it costs him. The sooner he can realise his capital gain and pay off his ‘expen
the better. Hence it is the ‘carrying cost’ or per period costs of borrowed fu
important in the model. The demand for the long-term mispriced asset is low
for the short-term mispriced asset and hence the current price of the long-term
asset is lower than that for the short-term asset.
To the extent that investment projects by a firm have uncertain payoffs (pro
accrue in the distant future then such projects may be funded with assets
fundamental value will not be revealed until the distant future (e.g. the Chan
between England and France, where passenger revenues begin to accrue many
the finance for the project has been raised). In this model these assets will be
strongly undervalued.
The second element of the Shleifer and Vishny (1990) argument which yie
outcomes from short-termism concerns the behaviour of the managers of the
conjecture that managers of a firm have an asymmetric weighting of misprici
pricing is perceived as being relatively worse than an equal amount of overpr
is because underpricing either encourages the board of directors to change it
or that managers could be removed after a hostile takeover based on the un
Overpricing on the other hand gives little benefit to managers who usually
large amounts of stock or whose earnings are not strongly linked to the stock p
incumbent managers might underinvest in long-term physical investment proj
A hostile acquirer can abandon the long-term investment project and hen
short-term cash flow and current dividends, all of which reduce uncertainty an
duration of mispricing. He can then sell the acquired firm at a higher price
degree of underpricing is reduced because in essence the acquirer reduces t
of the firm’s assets. The above scenario implies that some profitable (in D
long-term investment projects are sacrificed because of (the rational) short-
arbitrageurs who face ‘high’ borrowing costs or outright borrowing constrain
contrary to the view that hostile takeovers involve the replacement of inefficien
value maximising) managers by more efficient acquirers. Thus, if smart mo
wait for long-term arbitrage possibilities to unfold they will support hostile
which reduce the mispricing and allow them to close out their arbitrage pos
quickly.
mental value and they use aggregative stock price indices. Following Miles (19
examine an explicit model of short-termism by considering variants on the RV
five-year horizon the RVF for the price of equity of the jth company at time




+ +
where djr+i = 1/(1 r,,t+j rpj)’ and rt,r+i is the risk-free rate at horizon t
is the risk premium for company j which is assumed to be constant for ea
Short-termism could involve a discount factor that is ‘too high’ or cash flows
low (relative to a rational forecast). Hence we would have either:

+ rpj)bi
+ with b > 1
djr+i = 1/(1 rt,r+i
or xiE,Dl+i replacing ErDf+i in (8.16). In the above examples, ‘short-termism
+
each time period t i. Another form of short-termism is when the correct (rat
+ +
calculation is undertaken for periods t 1 to t 5, but either all future cash
+ +
discount factors for t 6, t 7, etc. are not weighted correctly, hence the l
(8.16) becomes either
+
rt,r+5 r p j Y 5
˜r˜jr+5/(1+




Short-termism implies either a > 1 or h -c 1, respectively. In order to make
relationships operational we need a model of the risk premium for each comp
assumes rpj depends on that firm’s beta, p j , and the firm’s level of gearing,



Miles uses a cross-section of 477 UK non-financial companies, with Pjf set fo
He then invokes RE and ‘replaces’ the expectations terms in dividends and t
price by their known outturn values in 1985-1989 (and uses instrumental v
estimation). The rr,f+jare measured by the yield to maturity on UK governm
for maturities 1-5 years. Substituting for r p j from (8.20) in any of the vari
(8.18) or (8.19) which are then substituted in (8.16), we have a cross-section
which is non-linear in the unknown parameters a1 and a2 (which appear in all t
and in the unknown short-termism parameters, i.e. either b, x , a or h.
Miles also adjusts the RVF formulae for the taxation of dividends. Some of
can be found in Table 8.2 (for the ‘central’ tax case). All the measures (see
Table 8.2) used indicate that short-termism leads to substantial undervaluation
prices relative to that given by a rational forecast of either dividends or discoun
example, the estimate of b = 1.65 implies that cash flows five years hence are
as if they did not accrue for more than eight years. The value of x = 0.93 impli
b = 1.67
1. All discount -0.4 8.7
factors are high (2.9)
(5.6) (3.5)
x = 0.93 -0.07 14.9
2. All cash flows
pessimistic (30.6) (2.1) (5.3)
CY = 2.0 -0.05
3. ‘Year t + 5 ’ 7.4
discount rates high (4.7) (3.8)
(6.3)
+ 13.2
h = 0.52 -0.10
4. ‘Year t 5’ cash
flows pessimistic (3.7)
(6.1) (5.2)
Source: Miles (1993).
(.) = t statistic.


flows five years hence are ‘undervalued’ by 30 percent (i.e. 1 - x 5 ) and hen
with more than five years to maturity need to be 30 percent more profitable than
If we take the value of the gearing coefficient as a2 x 7-10 then this im
company with average gearing = 57 percent (in the sample) will have a ris
about 5.7 percent higher than a company with zero debt. There is one peculia
namely the negative beta coefficient a1 (which also varies over different spec
Under the basic CAPM,a1 should equal the mean of Rjt - which should b
However, Miles demonstrates that in the presence of inflation the CAPM has t
fied as in section 3.16 and it is possible (but not certain) that the coefficient on
negative. However, when u1 is set to zero the results still indicate short-term
although the robustness of these results requires further investigation there is
evidence of short-termism.

8.2.4 Noise Waders and Contagion
We now discuss a noise trader model based on Kirman (1993). Kirman’s mo
different to that of DeLong et a1 in that it explicitly deals with the interactio
individuals, the rate at which individuals’ opinions are altered by recruitment an
phenomena of ‘herding’ and ‘epidemics’. The basic phenomenon of ‘herding’
by entomologists. It was noted that ants, when ‘placed’ equidistant from tw
food sources which were constantly replenished, were observed to distribute
between each source in an asymmetric fashion. After a time, 80 percent of t
from one source and 20 percent from the other. Sometimes a ‘flip’ occurred wh
in the opposite concentrations at the two food sources. The experiment was rep
one food source and two symmetric bridges leading to the food. Again, initially
of the ants used one bridge and only 20 percent used the other, whereas intu
might have expected that the ants would be split 50-50 between the bridges
of recruitment process in an ant colony is ‘tandem recruiting’ whereby the an
the food returns to the nest and recruits by contact or chemical secretion. Ki
that Becker (1991) documents similar herding behaviour when people are face
similar restaurants in terms of price, food, service, etc. on either side of the ro
majority choose one restaurant rather than the other even though they have
line’ (queue). Note that there may be externalities in being ‘part of the crow
loosely tied to ‘fundamentals’. The parallel with the behaviour of the ants
A model that explains ‘recruitment’, and results in a concentration at one so
considerable time period and then a possibility of a ‘flip’, clearly has releva
observed behaviour of speculative asset prices. Kirman makes the point tha
economists (unlike entomologists) tend to prefer models based on optimising
optimisation is not necessary for survival (e.g. plants survive because they ha
a system whereby their leaves follow the sun, but they might have done muc
develop feet which would have enabled them to walk into the sunlight).
Kirman’s stochastic model of recruitment has the following assumptions:
There are two views of the world, ‘black’ and ‘white’, and each agent
(and only one) of them at any one time.
There are a total of N agents and the system is defined by the numbe
agents holding the ‘black’ view of the world.
The evolution of the system is determined by individuals who meet
and there is a probability (1 - 6) that a person is converted (8 = prob
converted) from black to white or vice versa. There is also a small pr
that an agent changes his ‘colour’ independently before meeting anyon
to exogenous ‘news’ or the replacement of an existing trader by a new
a different view).
The above probabilities evolve according to a statistical process known a
+
chain and the probabilities of a conversion from k to k 1, k - 1 or


-
given by:
+ +
k 1 with probability p1 = p ( k , k 1)
Y
no change with probability = 1 - p1 - p2
k
L - 1 with probability p2 = p ( k , k - 1)
k
where
(1 - 6 ) k


(1 - S)(N - k )
1
k
p2 = N [E+
N-1
In the special case E = 6 = 0 the first person always gets recruited to the secon
viewpoint and the dynamic process is a martingale with a final position at k = 0
Also when the probability of being converted (1 - 8) is relatively low and the
of self-conversion E is high then a 50-50 split between the two ensues (see F
Kirman works out what proportion of time the system will spend in each
the equilibrium distribution). The result is that the smaller the probability of sp
conversion E relative to the probability of not being converted 6, the more time
spends at the extremes, that is 100 percent of people believing the system is in o
of the two states. (The required condition is that E < (1 - 6 ) / ( N - l), see F
0 0
1200 1600 2000
800
400

Figure 8.5 100000 Meetings Every Fiftieth Plotted: E = 0.15,6= 0.8. Source: A. Kirman (199
terly Journal of Economics. 0 1993 by the President and Fellows of Harvard College and the M
Institute of Technology. Reproduced by permission

100


80


60
K
40


20



1200
0 800 1600
400 2000

Figure 8.6 1OOOOO Meetings Every Fiftieth Plotted: E = 0.002, 6 = 0.8. Source: A. Kirman
Quarterly Journal ofEconomics. 0 1993 by the President and Fellows of Harvard College and the
Institute of Technology. Reproduced by permission


The absolute level of 6 , that is how ‘persuasive’ individuals are, is not impo
only that E is small relative to 1 - 6. Although persuasiveness is independ
number in each group, a majority once established will tend to persist. Hence
are more likely to be converted to the majority opinion of their colleagues in
and the latter is the major force in the evolution of the system (i.e. the prob
any single meeting will result in an increase in the majority view is higher th
the minority view).
Kirman (1991) uses this type of model to examine the possible behaviour
price such as the exchange rate which is determined by a weighted average of
talists and noise traders’ views. The proportion of each type of trader wi depe
above evolutionary process of conversion via the Markov chain process. He si
115
t
8 111
$3
{ 107
w
103

99

1 I l
I I l l l I I I I
I I I I I I I
50 70
20 80
11 30 901
40 60
Period

Figure 8.7 Simulated Exchange Rate for 100 Periods with S = 100. Source: Kirm
Taylor, M.P. (ed) Money and Financial Markets Figure 17.3, p. 364. Reproduced by p
Blackwell Publishers

model and finds that the asset price (exchange rate) may exhibit periods of
followed by bubbles and crashes as in Figure 8.7.
In a later paper Kirman (1993) assumes the fundamental’s price, Pf, is det
some fundamentals, P,, while the chartists’ forecast by simple extrapolation,
change in the market view, is


where A Pf, + ˜ v(p, - p r ) ,Ap;+l = p , - pt-l and the weights of depend on
=
eters governing the rate of conversion of market participants. The weights are e
and incorporate Keynes’ beauty queen idea. Individuals meet each other and
converted or not. They then try and assess which opinion is in the majority
their forecasts on who they think is in the majority, fundamentalists or char
the agent does not base his forecast on his own beliefs but on what he perce
majority view. This is rational since it is the latter that determines the ma
not the individual’s minority view. The model is then simulated and exhibit
that resembles a periodically collapsing bubble. When the chartists totally dom
constant and when the fundamentalists totally dominate P, follows a random w
dard tests for unit roots are then applied (e.g. Dickey and Fuller (1979) and P
Perron (1988)) and cointegration tests between P , and pt tend (erroneously)
there are no bubbles present. A modification of the test by Hamilton (1989
designed to detect points at which the system switches from one process to a
only moderately successful. Thus as in the cases studied by Evans (1991, see
when a periodically collapsing bubble is present, it is very difficult to detect.
There is very little hard evidence on the behaviour of noise traders and the d
opinions in financial markets. Allen and Taylor (1989) use survey techniques to
the behaviour of chartists in the FOREX market. These ‘players’ base their v
important. Also there is a tendency for chartists to underpredict the spot rate
market and vice versa. Hence the elasticity of expectations is less than one (i
the actual rate does not lead to expectations of a bigger rise next period). They
the heterogeneity in chartists’ forecasts (i.e. some forecast ‘up’ when others are
‘down’) means that they probably do not as a group influence the market ov
and hence are not destabilising. The evidence from this study is discussed in
in Chapter 12.


8.3 CHAOS
Before commencing our analysis of chaotic systems it is usefal briefly to review
of the solutions we have obtained so far, to explain returns on stocks and the s
In earlier chapters we noted that stochastic linear systems, even as simple as
walk with drift, can generate quite complex time series patterns. In contrast,
dynamic linear deterministic system such as


Equation (8.22) is a second-order difference equation. Given starting values yo,
parameters (a,b, c) we can determine all future values of yt to any degree of a
repeated substitution in (8.22). The time path of yt can converge on a stable e
value 7 = a/(l - b - c ) or for certain parameter values may either have an
path or a monotonic path. For some parameter values the path may either be
and explosive (i.e. cycles of ever increasing amplitude) or monotonic and exp
problem in basing models on deterministic differential equations like (8.22
the ‘real world’ we do not appear to observe deterministic paths for economi
Hence equations like (8.22) require an additional additive (linear) stochastic e

+ +˜ +˜
Yt = a y - ˜
Et

or
- cL2)-’(a+ e t ) = f(e,, et-l, ˜
yt = (1 - bL ˜-2)

If we assume E˜ is white noise then we can see that yt is generated by a infin
average of the random disturbances The latter can produce a time series tha
have cyclical elements (see Part 6) on which are superimposed random shocks
these cycles are not of fixed periodicity (unlike the deterministic case). The ra
with drift is a special case of (8.24) with a # 0, b = 1, c = 0.
In our ‘first look’ at empirical results on stock prices and stock returns we
the apparent randomness in the behaviour of these series. Indeed the EMH su
stock prices and (excess) returns should only change on the arrival of new info
news (about future fundamentals such as dividends). Hence the randomness
in the data is given an explicit theoretical basis and is represented by linear
models such as the random walk.
The course of the bubble is unpredictable, so unpredictability of stock returns
Intrinsic bubbles again yield a solution for stock prices which consists of two



where f ( D , ) is a non-linear function of dividends. An arbitrary linear stochas
for dividends (e.g. random walk with drift) then completes the model and w
(8.25) yields a stochastic process for P,. Because the function f(D,) non-line
is
linear stochastic process for D, is ‘transformed’ by (8.25) to yield a non-linear
process for P,.
So far, therefore, our models to explain the random nature of stock price (re
have involved introducing explicit stochastic processes somewhere into the
example, the equilibrium model in which expected returns are assumed to be co
the assumption of RE yields the random walk model. The latter equilibrium ret
via the RVF implies that stock prices only move in response to news about
that is the random forecast errors in the stochastic dividend process. In con
above, in chaotic models, apparent random patterns observed in real world d
generated by a non-linear system that is purely deterministic.
There is no commonly agreed definition of chaos but loosely speaking chao
are deterministic yet they exhibit seemingly random and irregular time serie
The time series produced by chaotic systems are highly sensitive to the initial
(i.e. the starting point yo of the system) and to slight changes in the parame
However, this sensitivity to initial conditions and parameter values does not r
possibility of producing reasonably accurate forecasts over short horizons. This
the time series from a chaotic system will be broadly repetitive in the early
time series, even if the initial conditions differ slightly.
The ‘sensitivity’ of chaotic systems is such that if the same chaotic system i
on two ‘identical’ computers (which estimate each data point to a precision of
then after a certain time, the path of the two series will differ substantially bec
minute rounding errors reacting with the highly non-linear system. This kind
the source of the observation that if the weather can be represented as a chao
then a butterfly flapping its wings in China can substantially influence weath
and hence may result in a hurricane in Florida.
Although chaotic systems produce apparent random patterns in the time do
nevertheless have a discernible structure (e.g. a specific frequency distribution)
be used to provide statistical tests for the presence of chaos. Space constraint
shall not analyse these tests (but see De Grauwe et a1 1993). As one might ima
be very difficult to ascertain whether a particular ‘random looking’ time serie
generated from a deterministic chaotic system or from a genuinely stochastic s
latter becomes even more difficult if the chaotic system is occasionally hit
random shocks: this is known as ‘noisy chaos’. Tests for chaotic systems requ
amount of data, if inferences are to be reliable (e.g. in excess of 20000 data p
hence with the ‘length’ of most economic data this becomes an acute problem
in the variables one can always ‘add on’ stochastic elements to represent the r
in human behaviour. Hence as a first step we need to examine the dynamics p
non-linear systems. If asset prices appear random and returns are largely un
we must at least entertain the possibility that these results might be generated
systems.
We have so far spoken in rather general terms about chaos. We now briefly
explicit chaotic system and outline how an economic model based on noise
smart money is capable of generating a chaotic system.

The Logistic Equation
About the simplest representation of a system capable of chaotic behaviour i
linear) logistic equation:
Yt+l = m ( 1 - Y r )
The steady state Y* is given when Yt+l = Y , = Y,-1 . . .
Y * = AY*(1 - Y * )

and the two solutions are:
Y* = 1 - l / h
Y* = 0

Not all non-linear systems give rise to chaotic behaviour: it depends on the
values and initial conditions. For some values of h the system is globally stabl
any starting value Y o , the system will converge to one of the steady state soluti
other values of h the solution is a limit cycle whereby the time series eventuall
(for ever) between two values U; and Y ; (where Y r # Y * , i = 1,2): this is k
two-cycle (Figure 8.8). The ‘solution’ to the system i therefore a differenti
s
and is known as a bifurcation. Again for different values of A the series can a
two-cycle to a 4, 8, 16 cyclic pattern. Finally, for a range of values of h the tim
Y , appears random and chaotic behaviour occurs (Figure 8.9). In this case if
value is altered from Y O = 0.3 to 0.30001, the ‘random’ time path differs afte
time periods, demonstrating the sensitivity to very slight parameter changes.
The dynamics of the non-linear logistic system in the single variable Y
solved by simulation rather than analytically. The latter usually applies a fortio
complex single variable non-linear equations and to a system of non-linear
where variables X , Y , 2, say, interact with each other. A wide variety of v
patterns which are seemingly random or irregular can arise in such models.

Noise Traders and Smart Money
It is not difficult to set up ad-hoc models of the interaction of noise traders (NT
money (SM) that are non-linear in the variables and hence that may exhibit ch
we say ad hoc we imply that the NT and SM need not necessarily maximise
defined function (e.g. utility of end of period wealth). De Grauwe et a1 (1993)
0.6
"Q˜ooowooow
0.4




L
I
I
I
40.0
30.0
20.0
10.0

Generation Number

Figure 8.8 Period 2: Limit Cycle. Source: De Grauwe (1993). Reproduced by pe
Blackwell Publishers

1.o


0.8


0.6

Yfr)
0.4


0.2


0.0
20.0 30.0
10.0
0.0 40.0

Generation Number

Figure 8.9 Chaotic Regime. Start Value Yo = 0.3 (Solid Line), Start Value Y O= 0.30
Line). Source: De Grauwe (1993). Reproduced by permission of Blackwell Publishers

interesting model of this interaction where NT exhibit extrapolative behaviou
feedback) and the SM have negative feedback, since they sell when the pric
fundamentals. The model they develop is for the exchange rate St, although
can replace this by the price of any speculative asset, without loss of genera
aspect of the model is the heterogeneity of expectations of different traders. T
where E,S,+l is the market expectation at time t. The dating of the variable
needs some comment. The time period of the model is short. Agents are assum
information for time t - 1 and they take positions in the market at time t, bas
+
forecasts for period t 1. This is required because S, is the market solution of
and is not observable by the NT and SM when they take their positions in t
The expectations of NT are extrapolative so that

.&NI
E;[S,+1ISr-11 = f[S,-l, Sr-27 **



where f is a non-linear function. A simple form of extrapolative predicto
De Grauwe et a1 is based on chartists' behaviour, who predict that the pric
in the future if the short moving average SMA crosses the long moving ave
from below. (Hence at point A in Figure 8.10, chartists buy the asset since the
price rise.) A simplified representation of this NT extrapolative behaviour is:




In contrast, the SM have regressive expectations, relative to the long-run e
value S:-l given by fundamentals:


and adjust their expectations at the rate given by the parameter a. Note that
not have RE since they do not take into account the behaviour of the NT and




Moving Average




Time

Figure 8.10 Moving Average Chart. A, Chartists Buy Foreign Exchange; B, Chartists
Exchange. Source: De Grauwe (1993). Reproduced by permission of Blackwell Publi
De Long et a1 (1990) the relative weight given to NT depends on the profi
NT relative to those of the SM. The model of De Grauwe et a1 is rather aki
De Long et a1 in that the relative weight of NT and SM varies over time de
economic conditions and it provides the key non-linearity in the model. The
NT in the market m, is given by:


The SM are assumed to have different views about the equilibrium value S
views are normally distributed around St. Hence if the actual market price
the true equilibrium rate then 50 percent of the SM think the equilibrium rate
and 50 percent think it is too high. If we assume all the SM have the same
risk aversion and initial wealth then they will exert equal and opposite pres
market rate. Hence when Sr-1 = SF-l the SM as a whole do not influence
price and the latter is entirely determined by the NT, that is m, in (8.32) e
and the weight of the SM = (1 - m,)is zero (Figure 8.11). On the other ha
falls below the true market rate then more of the SM will believe that in the
equilibrium market rate is above the actual rate and as a group they begin t
the market rate, hence m in (8.32) falls and the weight of the SM = 1 - m,
t
Finally, note that /3 measures the degree of confidence about the true
market rate held by the SM as a whole. As # increases then mf+ 0 and (1 -
I
Hence the larger is B the greater the degree of homogeneity in the SMs’ view
the true equilibrium rate lies. In this case a small deviation of S,-1 from S:-l
to a strong influence of SM in the market. Conversely if /3 is small there i
dispersion of views of the SM about where the true equilibrium market rate l
weight of the SM in the market (1 - m,)increases relatively slowly (Figure 8
To close the model we need an equation to link the markets’ non-linear e
formation equation:




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