Table of Contents
Extrapolating Expectations: An Explanation for Excess
Volatility and Overreaction
Mila Getmansky
PhD Candidate in System Dynamics
Sloan School of Management
30 Wadsworth Street, Bld E53, Room 358
Cambridge, MA 02139
Phone: 617-253-8094
Fax: 617-468-8132
mgetman@ mit.edu
Jannette Papastaikoudi
PhD Candidate in Finance
Sloan School of Management
30 Wadsworth Street, Bld E52, Room 442
Cambridge MA, 02139
Phone: 617-253-3637
jpapa@ mit.edu
Eamings announcements have always been in the spotlight of the financial world.
Whether as a predictor for future price movements, or for the formation of the
contemporaneous rational stock price, earnings have been used by analysts, investors and
academics. Within the framework of a rational valuation model (Gordon growth model)
future earnings determine today’s stock price. On the other hand, news on earnings
induce a so called post-earnings-announcement drift, as first noted by Ball and Brown
(1968), thereby affecting the future stock price.
The last phenomenon has been documented ever since, and today it is mostly
known as momentum. Momentum in general refers to the tendency of “winner stocks”,
i.e. stocks that had positive abnormal holding period returns for 3-12 months relative to a
benchmark, to continue to outperform for the subsequent 12 months, while “loser
stocks”, i.e. stocks that had negative abnormal holding period retums for 3-12 months
relative to a benchmark, to continue to underperform for the subsequent 12 months.
Several explanations have been proposed to account for the phenomenon, among
them transaction costs, delayed stock reaction to market risk factors and underreaction, or
even overreaction. Overreaction appears as a quite popular explanation in the recent
finance literature, indicating investors being overly optimistic about future stock retums.
They tend to extrapolate past performance into the future, resulting into positive feedback
trading and hence into self fulfilling expectations. Shiller (1988) and Frankel and Froot
(1988) present interesting evidence to support overreaction.
Momentum strategies are applied in the financial markets, for investors to capture
abnormal retums, as high as 25% per year. They are indeed a viable and applicable
trading strategy and not a theoretical artifact. In fact, several mutual funds follow those
strategies to capture excess retums, in their effort to beat the market.
Yet, the existence of momentum itself, violates one of the cornerstones of modern
financial theory, the Efficient Market Hypothesis (EMH), introduced by Eugene Fama
(1970). According to the EMH, an efficient market fully reflects all available
information, therefore price changes must be unforecastable if they are properly
anticipated, i.e. if they fully incorporate the expectations and the information of all
market participants. Momentum is in contrast to the EMH, since it implies the opposite,
namely the partial incorporation of information into stock prices.
In addition, a counterargument against the EMH is the excess volatility exhibited
in financial markets. Under excess volatility we refer to the volatility of stock retums that
cannot be explained through the variation in fundamentals (earnings). The debate was
first initiated by LeRoy and Porter (1981) and Shiller (1981), who first made the claim
that prices move too much to be rational forecasts of future earnings discounted at a
constant rate. Hence, excess volatility is equivalent to predictability in stock prices,
which is a direct violation of the EMH.
In this paper, we try to explain excess market volatility by means of momentum.
Price setting mechanisms are introduced based on demand/supply balance as well as
based on mechanical trading strategies of investors. Two types of investors are
introduced: fundamental investors who believe that the marketed assets have some
intrinsic value and make their trades based on the relative value of the current price
relative to that intrinsic value; and momentum traders, who extrapolate past stock
performance into the future and act accordingly by buying when the stock price rises and
by selling when the price falls. Momentum investors should not be characterized as
irrational; they do not completely ignore news about eamings, which are equivalent to
fundamentals in this model. Their trades partially reflect the fact that they do take
eamings announcements into account while submitting their demand. Incorporating news
on earings in their demand has a direct impact on the market clearing price, making it
move volatile, regardless of the nature of the news (good or bad).
An additional feature of the proposed model it that it can actually explain post
eamings announcement drift. The presence of momentum traders pushes the stock price
further up (down) in the presence of good (bad) news, hence leading to the actual
realization of their extrapolated expectations: it is a self fulfilling prophecy. Stock returns
seem to be moving even more in the anticipated direction (up or down) therefore
producing the observed momentum effect.
The efficient markets hypothesis (EMH) has been the central proposition of
finance for nearly thirty years. In his classic statement of this hypothesis, Fama (1970)
defined an efficient financial market as one in which security prices always fully reflect
the available information. If the theory holds, the market truly knows best and investors
are better off holding passive market portfolio. Investors should forget active money
management altogether. However, excess volatility, especially recent volatility in
NASDAQ average, cannot be fully explained by EMH.
Amongst the literature of most relevance to the whole volatility issue is Robert
Shiller's “Market Volatility” (Shiller, 1990). Shiller proposes that investor reactions, due
to psychological or sociological beliefs, exert a greater influence on the market than good
economic sense arguments. Shiller does not totally disregard the work of economists
before him who proposed the Efficient Market Hypothesis (EMH). In fact, he admits that
the EMH can be substantiated by statistical data, but he believes that investor attitudes
are of great importance in determining price levels. His book provides statistical
evidence that excess volatility exists in the stock market and therefore volatility cannot be
totally explained by the EMH. Excess volatility is the level of volatility over and above
that which is predicted by efficient market theorists.
The model strives to explain market volatility by introducing price setting
mechanism based on demand/supply balance as well as introduce psychological factors
of investors who trade based on their believes about the market. Fundamental investors
are value investors. They believe that the market (or an asset) has some intrinsic price.
Fundamental investors make their trades based on the relative value of a current price to
this intrinsic value. Momentum investors try to chase a trend by buying when the price
rises and selling when a price falls. Note, the model assumes only two types of assets:
risk-free (cash or Treasury bills) that is not traded on the equity market and equity that is
traded on the market. In this model, volatility of the market is the same as the volatility
of the risky asset - equity.
A. Time Horizon
The model runs for 2 years. There are exactly 253 trading days per year.
Therefore, there are 506 trading days per two years. To round up, the model is run for
500 days. Trading days are chosen instead of calendar days because equities are traded
only on the trading days. Earnings are exogenous to the model. They are reported
quarterly. Therefore, the minimum time considered was a quarter - 63 days. However, it
takes time for fundamental investors to obtain quarterly eamings information and make
trading decisions based on it. Therefore, the model should be run for at least a year,
given delays in the model. Delays as well as combined behavior of fundamental and
momentum investors lead to oscillations in price, volatility, and other variables.
However, over time the price comes to equilibrium fundamental value, assuming no
further surprises in eamings. Therefore, the model is allowed to be mun for 2 years.
B. Core Structure
Model Boundary
The model boundary depends on the model purpose (Sterman, 2000). The
purpose of the model is to formulate a dynamics model that explains why markets are
excessively volatile. Non-risky and risky assets are modeled. Everything that is not
invested in a risky asset (“Equity Invested”) is assumed to be invested in a non-risky
asset (“Cash”). “Cash” and “Equity Invested” are modeled in a way to track how much
equity was bought and sold. Income and Consumption are introduced in the model;
however, they are not explicitly modeled. They are constant. Financial institutions and
instruments such as short sales and margin purchases play a role in explaining excess
market volatility. However, they are not modeled in order to attain as simple model as
possible.
Assumptions
¢ Eamings are assumed to be reported continuously. In reality, eamings are
reported quarterly.
¢ The model only has two types of investors: fundamental and momentum. In
reality, more types of investors exist. For example, Shleifer (2000) introduced
noise as well as arbitrage investors. However, even among fundamental and
momentum investors, differences in risk preference, age, and family situation
exist. The model presents average representative groups of fundamental and
momentum investors.
* The relative balance of fundamental and momentum investors can only be
changed exogenously. During the run of the model, investors cannot change their
preferences and move between two groups.
* Total number of shares in the market is constant due to the absense of IPOs and
share buybacks.
¢ Fraction of earnings reinvested in more capital is zero.
* Two types of assets exist: Risk-free and risky.
Investor Equity and Cash Holdings
Figure 1 presents stock and flow structure for “Cash” and “Equity Invested”.
“Cash” can be increased by “Income” and “Cash Increase,” increase in cash due to
selling stocks. Note that the model has an array structure built in. Each stock is modeled
both for fundamental and momentum investors (see model documentation). “Actual
Equity Weight” for each type of the investors is calculated in order to compare it with the
desired equity weight and make further decisions whether to buy or sell a stock. In the
question, I was asked to set up the model in such a way that relative balance of
fundamental investors and trend chasers can be changed. “Fundamental Specification
Mix” in the model serves this purpose.
Figure 1. Investor Equity and Cash Holdings
Fundamental Specification Mix
Initial Total Cash
SB a = pi
Income Cash Consumption
Cash Increase Cash Decrease
®
<Equity Purchase —_ .
fo Equity Purchased> Total Wealth ‘Actual Equity
<Price> <Shares> Pee ay i Weight
~~ Overall
Average Cost Wealth
" Market Value of
Equity Equity Invested
Equity Purchased invested Equity Sold <Shares>
<Total ab ate> Initial <Total Sell Rate> <Price>
eh Duy NOUS. <Initial Jo),
Price> <Shares>
Shares Balance and Calculation of Equity Buy and Sell Fractions
Figure 2 has two purposes. First, based on “Desired Equity Weight,” “Actual
Equity Weight,” and “Normal Roll-Over Fraction,” each type of investors decides
whether to buy or sell and how much to trade. The second part tracks “Shares.” The
total amount of shares should be conserved. Everything that is sold by one type of
investor should be purchased by another type.
Figure 2: Shares Balance
<Trading Volume>
InitialShares
rading Volume>
Total Desired Sell Rate Total Desired Buy Rate
~~ a <Cash>
ieaien San Trading Volume
Desired Buy Rate
<Price>
ee Hee
Desired Sell Fraction
Desired Buy Fractioin
of
Fundamental -Momentinn
weight Normal Portfolio
r Rebalancing Time
Desired
es «_——
Desired Portfolio ioe
Rebalancing for Sell
Desited Portfolio
ope Rebalancing for Buy
RollOver
Frachga Effect of Equity Effect of Equity
Weight on Buy “+—_ Weight on
site a Buy/Sell Table
Weight on S
Fundamental Investor Decision
Figure 3 presents the stock and flow diagram for the fundamental investor
decision. Fundamental investors buy a stock if they think that a stock is undervalued.
They compare current price with the intrinsic value of a stock. The intrinsic value of a
stock is calculated based on the forecast of earnings and the inverse of the cost of equity
minus the growth in earnings. They sell if they perceive that the stock is overvalued.
Figure 3: Fundamental Investor Decision
<Price> 7
~—™ Price/Value pcre
ue
Table for Desired ae Time to
Equity WeightF—_, pV og pargpive
watts
amin
ie Unga fect eg on
“S~ indicated fundamental
value
“Effect of keg ratio on
indicated fundamental valu
Eamings Forecast table
i a ks
Perceived |—- Trend in Eamings
Change in Earnings
Perceived Egmings 7 ‘ rs Cost of Equity
Normal k-g”
Historical
Time to Perceive Change in Historic L_Eamings
Eamings Eamings Riskless Rate Risk Premium
Duration Over Which to
Calculate Eamings Trend
Momentum Investor Decision
Figure 4 depicts the stock and flow structure of the decision taken by momentum
investors. Momentum investors only care about the trend of the price in making their
decisions. They buy on increasing trends and sell on decreasing trends.
Figure 4: Momentum Investor Decision
Table for Desired
Equity Weight M
Bese
Weight Normal Trend ir
Price
<Price>
Perceived
(han e i Price
Timeto —P *preiye
<initial Price> Trend in Price
Historical
fig ein Price
Du Hon over
alu ate ee
c
‘rend?
10
Pricing
Figure 5 depicts pricing structure. Price is determined by demand/supply balance
and expected price.
Figure 5: Pricing
Initial Price
<Total
D egited Buy
Expected
Price
pace \
mand <Total,
ae Depinrg sell
——
Time to Adjust }
expected price’ Price
erate ,
Demand Supp!
Effect of
aeuatstpty L_ Balance J” "cangen
demand supply
rice
E _ balance
Time to
Demand Supply table de pee ty
balance
C. Flow Equations and Decision Rules
The model equations are carefully documented in Question3.mdl and are attached on
adisk. The most important relationships and decision points are listed in this section.
Volatility
Figure 6 describes a stock and flow structure of the way volatility is measured in
the model. It should be noted that it is measured in the simplest possible way in order to
strive for elegancy and simplicity of the model. However, at the same time the
formulation captures the time-varying nature of volatility. “Moving Average of Return”
measures a weighted average of the current return and past returns with recent retums
weighted most heavily. It is assumed that a time constant over which the moving average
is calculated is 5 days. An exponential average possesses this quality and is easy to
11
represent in a simulation model (Forrester 1961, p. 406-411). Volatility is calculated as
the squared difference between a current retum and the moving average of return.
Figure 6: Volatility
Time to Change
Past Price
> ee Past Price
Change in Past
Price wee
\ Ret
La
<Price>
<Initial Price>
Duration Over Which
Retum is Calculated
Moving
Increase in Moving Awetene of
Average of Retumna+
Time to Update Moving Volatility
Average of Return Measure
In the model volatility exhibits a time-varying nature. The time varying nature of
asset retum volatility was first proposed by Fischer Black (Black, 1976). Robert F. Engle
proposed to use autoregressive conditional heteroskedasticity (ARCH) to calculate stock
volatilities (Engle, 1982). According to ARCH, a natural way to update a variance
forecast is to average it with the most recent squared “surprise.” The squared “surprise”
can be calculated as the squared deviation of the rate of return from its mean.
Engle also pioneered GARCH, generalized autoregressive conditional
heteroskedasticity model. (Engle, 1982). GARCH allows greater flexibility in the
specification of how volatility evolves over time compared to ARCH approach.
According to GARCH approach, the updated estimate of market-retumm variance in each
period depends on both the previous estimate of variance and the most recent squared
residual return on the market. Even though GARCH is the most used and more
appropriate to estimate volatility of retums, this model is going to strive for simplicity
12
and use squared difference between an asset return and a moving average of asset returns
with more emphasis on the most recent return as an approximation for stock volatility.
In a discrete case, volatility can be measured as a mean squared error (MSE) of
asset returns:
where s° is the volatility
x, is an asset return
X is an average asset retumn
nis the number of observations
Note that this measure of volatility is unbiased only in the case of returns distributed
normally.
Bounded Rationality of Investors
Decision rules for both fundamental and momentum investors are formulated
separately. Each type of investors decides on his own rules that are not altered during the
simulation. However, two types of investors interact in the market and are both faced
with the same price. The behavior of each type of investors is bounded rational
(Morecroft, 1983). Decision rules and bounded rationality of investors lead to the overall
oscillations in prices in the market and excess volatility as presented in Part D.
Equity Valuation Model and Fundamental Investors Decision
Fundamental investors make their trading decisions by comparing current price
with expected earnings. “Earnings Forecast” is modeled based on TREND function
(Sterman, 2000). The function is explicitly modeled (see attached model). The model
assumes that fundamental investors make their decisions based on a dividend discount
model, pioneered by Myron J. Gordon. According to the model:
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D,
k-g
0
where V, is an intrinsic price of a stock
D, is the value of dividends in next period (t=1)
k is the cost of equity
g_ is the growth rate of dividends
This dividend discount formula relates the P/E multiple to the cost of equity k
and the real eaming growth rate g. Note, in the derivation of this formula it is assumed
that the fraction of earnings reinvested in more capital is zero. Therefore,
Ey _ E
k-g r, +EMRP —g
P, =
where P, is the intrinsic stock price
E, is the value of earnings next period (t=1)
r, is the risk-free rate
EMRP is the market risk premium
g_ is the real earnings growth rate
In the model, risk-less rate is assumed to be 3.8%/year and EMRP is 7.4%/year
(Salomon Smith Bamey, 1999).
Momentum Investors Decision
Momentum investors trade based on the trend in price. If a trend is positive, they
buy. If itis negative, they sell. The model is formulated in such a way that the desired
equity weight for momentum investors is a function of “Trend in Price” to “Normal
Trend in Price.” Initially, the model had “Price Forecast” and compared it to the current
price; however, according to Shleifer (2000), momentum traders place a market order
today in response to a past price change. They do not formulate any price forecast unlike
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fundamental traders who formulate earnings forecast. Therefore, the model was changed
to represent this rule. “Trend in Price” is calculated according to a TREND function
(Sterman, 2000) that is explicitly modeled (see attached model).
Pricing
Price setting mechanism is modeled by using an anchoring and adjustment
process (Sterman, 2000). The anchor is the “Expected Price” and the cue is the effect of
the Demand/Supply balance. If demand exceeds supply, then the price is adjusted
upward. In a reverse case, the price is adjusted downward.
D. Model Analysis
Test 1: Fundamental Investors Decision
Earnings were increased by 100% leading to the increase in the desired equity
weight by fundamental investors.
Graph for Desired Equity Weight F
1
0.5
0
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Desired Equity Weight F : run2 fraction
Desired Equity Weight F : 1Unh n-ne fraction
muni: Earnings = 11.2/252 ($/Share/Day)
run2; Earnings = 2* 11.2/252 ($/Share/Day)
The behavior is as expected.
15
Test 2: Momentum Investors Decision
A pulse in price was introduced.
Price = 18.97+PULSE(2, 5) where Initial Price is 18.97 $/Share
Graph for Desired Equity Weight M
0.6
0.5 rr
0.4
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Desired Equity Weight M : run2 fraction
Desired Equity Weight M : runt fraction
runi: Price = Initial Price
run2: Price = Initial Price + PULSE(2, 5 )
As can be seen from the graph, the trend of price increased and then decreased as
expected.
16
Experiment 1: Different Earnings Inputs
Graph for Volatility Measure
0.06
0.03
0
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Volatility Measure : test2 1/(Day* Day)
Volatility Measure : test1-
- 1/(Day*Day)
17
Graph for Price
40
20
0
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Price : test2 $/share
Piceties!) = $/share
Test 1: Earnings = 1/63+RAMP(0.0001,2,25) ($/Share/Day)
Test 2: Earnings = 1/63 ($/Share/Day)
Note, when earnings are growing constantly for 23 days, behavior is more oscillatory
with higher amplitude of oscillations compared to the run where earnings are
constant.
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Experiment 2: Change in Fundamental Specification Mix
Graph for Shares
100,000
50,000
0
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Shares[ Fundamental] : test
Shares[Fundamental] : test2 --
Shares[Momentum] : testl --.
Shares[Momentum] : test2 ---
Graph for Volatility Measure
0 50 100 150 200 250 300 350 “400° 450° 500
Time (Day)
Volatility Measure : test1 1/(Day* Day)
Volatility Measure : test2 dvesdtmronsinbaninntsinebeettintensahnbnsainsiseusineenteeitiins 1/(Day*Day)
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Test 1: Fundamental Specification Mix =0.1
Test 2: Fundamental Specification Mix = 0.8
It is interesting and will be mentioned in Section E, that when most of the traders are
fundamental investors (80%), they do not immediately drive out momentum investors.
The number of momentum investors slowly decreases. However, when most of traders
are irrational - momentum traders (90%), the majority is driven out by fundamental
traders, but it takes time before momentum traders are driven out. However, the fraction
of momentum traders does not reach 0 even if the simulation is run for a longer period of
time. Momentum or irrational traders survive. As can be seen from test 1, the volatility
is higher when initially there are more momentum traders than fundamental traders.
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Experiment 3: Aggressiveness of Momentum Traders
In this experiment, aggressiveness of momentum traders was changed by
manipulating “Normal Trend in Price” value.
Graph for Price
40
20
UH
0
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Price : test2 $/share
Price : testl ~~ ~~ $/share
21
Graph for Volatility Measure
0.2
0.1
0 hn
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Volatility Measure : test2 1/(Day* Day)
Volatility Measure : test —~-------------rrrr 1/(Day* Day)
Test 1: Normal Trend in Price = 0.015 (1/Day)
Test 2: Normal Trend in Price = 0.027 (1/Day)
According to Test 2, momentum investors do not try to buy all shares of the asset as soon
as the price trend increases a little bit or sell everything in the opposite case. They wait
until the trend is bigger. This behavior actually leads to the equilibrium behavior as
depicted by the Price graph above. The price stabilizes and the volatility becomes 0 as
expected.
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Experiment 4: Perception Time By Momentum Investors
In this experiment, “Time to Perceive Price” by irrational investors is varied.
Graph for Price
20
10
0
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Price : test2 $/share
PIU ese L sisttitstnisetcearistrinreittcireanteereamn oir $/share
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Graph for Volatility Measure
0.06
0.03
0 Nr Oe ne
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Volatility Measure : test2 1/(Day* Day)
Volatility Measure : test —~-------------rrrr 1/(Day* Day)
Test 1: Time to Perceive Price = 1 (Day)
Test 2: Time to Perceive Price = 2 (Day)
In test 2, oscillations decrease in amplitude and frequency and die out in approximately
400 days. Volatility goes to 0. Due to the increase in the delay, momentum investors are
less likely to be in time to execute a buy (sell) when a price is increasing (decreasing).
Therefore, they are less likely to exacerbate increases or decreases in price that lead to an
increase in volatility.
E. Discussion
According to the results of the model, the equilibrium price is not reached
instantaneously. Trading of fundamental and momentum investors lead to oscillations in
prices, thus, an excess of volatility. This excess of volatility cannot be explained by the
EMH. The excess of volatility is due to bounded rational behavior of the traders.
The model shows that trend chasers are not quickly forced out of the market by the
fundamental investors. Indeed, it is possible to reach price equilibrium with a small
fraction of momentum traders left. To illustrate these two points, the graphs for total buy
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and sell rates and perceived demand/supply balance are provided below. As it can be
seen, when momentum investors are at the peak to buy(sell), the fundamental investors
are at the peak to sell(buy). The balance/supply balance oscillates until it reaches 1. As
long as there is an imbalance and there are buy and sell orders from two types of
investors, momentum investors are not driven out.
Potentially it will be interesting to include a third type of an investor: arbitrageurs
who maximize utility as a function of the last period consumption. As it was shown by
Shleifer (2000), the model with fundamental traders, momentum investors and
arbitrageurs lead to more excess volatility than a model with only fundamental and
momentum investors.
Graph for Total Buy Rate
200
100
0
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Total Buy Rate[Fundamental] : test; -——————— shares/Day
Total Buy Rate[Momentum] : testl-------~- shares/Day
25
Graph for Perceived Demand Supply Balance
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Perceived Demand Supply Balance : test} fraction
Graph for Total Sell Rate
200
100
0 Sm ney NRVC VCD BLA B Lb ee Ort
0 50 100 150 200 250 300 350 400 450 500
Time (Day)
Total Sell Rate[Fundamental] : test shares/Day
Total Sell Rate[ Momentum] : testl—-~-----------n shares/Day
26
References
1. Black, Fischer, “Studies in Stock Price Volatility Changes,” Proceedings of the
1976 Business Meeting of the Business and Economic Statistics Sections,
American Statistical Association, pp. 177-81.
2. Engle, Robert F., “Autoregressive Conditional Heteroskedasticity with Estimates
of the Variance of U.K. Inflation,” Econometrica 50 (1982), pp. 987-1008.
3. Fama, E. (1970). “Efficient Capital Markets: A Review of Theory and Empirical
Work,” Journal of Finance, 25, pp. 383-417.
4. Forrester, J. W., Industrial Dynamics. 1961. Cambridge, MA: Productivity
Press.
5. Morecroft, J.D.W., System Dynamics: Portraying Bounded Rationality. Omega,
1983. 11(2): p. 131-142.
6. Salomon Smith Barney, Financial Strategy Group, “The Industry Cost of Equity,”
1999.
7. Shiller, R. J. (1990) Market Volatility - pages 1-4, 71-76, 197- 214, MIT Press,
London.
8. Shleifer, Andrei, Inefficient Markets: An Introduction to Behavioral Finance.
2000, Oxford University Press.
9. Sterman, J.D., Business Dynamics: Systems Thinking and Modeling for a
Complex World. 2000, Chicago: Irwin-McGraw Hill.
Appendix: Model Equations
For model equations, please, contact Mila Getmansky at mgetman@ mit.edu
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