ñòð. 21 
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 longrun view of the
as does the smart money (i.e. p* = 0) and that there are no â€˜surprisesâ€™ (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
noisetrader 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 â€™.
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 â€˜basic mispricingâ€™ level. In this case the model would give a moveme
prices which was â€˜excessively volatileâ€™ (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 â€˜fadsâ€™ and â€˜fashionsâ€™. 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 investorsâ€™ best guess (optimal forecast) of p* is
value. However, as â€˜newsâ€™ (a,) arrives noise traders alter their views about p
â€˜change in perceptionsâ€™ 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 â€˜bull and bearâ€™ 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 â€˜
rateâ€™ 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 50Year 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 noisetrader 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 adhoc 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 â€˜oldâ€™ 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 noisetrader 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 noisetrader risk. Changes in noisetrader sentiment (i.e. in p* and p,
cause the discounts to vary over time and as noisetrader risk is systematic, d
most funds are expected to move together.
In the noisetrader model a number of closed end funds should also tend to
at the same time, namely when noisetrader 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 oneâ€™s
to the possibility of changes in noisetrader sentiment, which are persistent.
â€˜persistenceâ€™ 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 longsh
(Rr) 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 noisetra
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 noisetrader 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 noisetrader model explains the stylised
still leaves us a long way from a formal test of the noisetrader model in the bo
ShortTermism
In a world of only smart money, the fact that some of these investors take a â€˜
view of returns should not lead to a deviation of price from fundamentals. The
chain of shortterm â€˜rational fundamentalâ€™ investors performs the same calcul
investor with an infinite horizon.
With a finite investment horizon and the presence of noise traders the abov
doesnâ€™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 â€˜longâ€™ but finite horizon there is
resale risk. The share in the total return from dividend payments over a â€˜lon
period is large but there is still some risk present from uncertainty about p
â€˜final periodâ€™.
We note from the noisetrader model that if a firm can make its equity
subject to noisetrader 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 firmâ€™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 ModiglianiMiller hypothesis).
Mispricing and ShortTermisrn: Shleifer Vishny Model
The underpricing of an individual firmâ€™s stock is not a direct result of the fo
trader model of De Long et a1 since the formal model requires noisetrader 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
ShleiferVishny 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 firmâ€™s managers might ensue, that is they choose less profi
term physical investment projects rather than longterm 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 noisetrader 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 longhorizon 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, arbitrageursâ€™ rational behaviour results
current mispricing of â€˜long assetsâ€™ when the mispricing is revealed at long ho
terms â€˜longâ€™ and â€˜shortâ€™ 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 longterm 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 â€˜expen
the better. Hence it is the â€˜carrying costâ€™ or per period costs of borrowed fu
important in the model. The demand for the longterm mispriced asset is low
for the shortterm mispriced asset and hence the current price of the longterm
asset is lower than that for the shortterm 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 shorttermism 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 longterm physical investment proj
A hostile acquirer can abandon the longterm investment project and hen
shortterm 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 firmâ€™s assets. The above scenario implies that some profitable (in D
longterm investment projects are sacrificed because of (the rational) short
arbitrageurs who face â€˜highâ€™ 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 longterm 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 shorttermism by considering variants on the RV
fiveyear horizon the RVF for the price of equity of the jth company at time
+ +
where djr+i = 1/(1 r,,t+j rpj)â€™ and rt,r+i is the riskfree rate at horizon t
is the risk premium for company j which is assumed to be constant for ea
Shorttermism could involve a discount factor that is â€˜too highâ€™ 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, â€˜shorttermism
+
each time period t i. Another form of shorttermism 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+
Shorttermism 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 firmâ€™s beta, p j , and the firmâ€™s level of gearing,
Miles uses a crosssection of 477 UK nonfinancial companies, with Pjf set fo
He then invokes RE and â€˜replacesâ€™ the expectations terms in dividends and t
price by their known outturn values in 19851989 (and uses instrumental v
estimation). The rr,f+jare measured by the yield to maturity on UK governm
for maturities 15 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 crosssection
which is nonlinear in the unknown parameters a1 and a2 (which appear in all t
and in the unknown shorttermism 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 â€˜centralâ€™ tax case). All the measures (see
Table 8.2) used indicate that shorttermism 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. â€˜Year t + 5 â€™ 7.4
discount rates high (4.7) (3.8)
(6.3)
+ 13.2
h = 0.52 0.10
4. â€˜Year t 5â€™ cash
flows pessimistic (3.7)
(6.1) (5.2)
Source: Miles (1993).
(.) = t statistic.
flows five years hence are â€˜undervaluedâ€™ 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 710 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 shortterm
although the robustness of these results requires further investigation there is
evidence of shorttermism.
8.2.4 Noise Waders and Contagion
We now discuss a noise trader model based on Kirman (1993). Kirmanâ€™s mo
different to that of DeLong et a1 in that it explicitly deals with the interactio
individuals, the rate at which individualsâ€™ opinions are altered by recruitment an
phenomena of â€˜herdingâ€™ and â€˜epidemicsâ€™. The basic phenomenon of â€˜herdingâ€™
by entomologists. It was noted that ants, when â€˜placedâ€™ 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 â€˜flipâ€™ 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 5050 between the bridges
of recruitment process in an ant colony is â€˜tandem recruitingâ€™ 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
lineâ€™ (queue). Note that there may be externalities in being â€˜part of the crow
loosely tied to â€˜fundamentalsâ€™. The parallel with the behaviour of the ants
A model that explains â€˜recruitmentâ€™, and results in a concentration at one so
considerable time period and then a possibility of a â€˜flipâ€™, 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).
Kirmanâ€™s stochastic model of recruitment has the following assumptions:
There are two views of the world, â€˜blackâ€™ and â€˜whiteâ€™, 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 â€˜blackâ€™ view of the world.
ñòð. 21 