System Dynamics and stock markets
Martin Timothy Rafferty
Abstract
This paper presents an analysis of the behaviour of a stock market; the London stock exchange main
market as expressed in the FTSE 100 index. The paper examines the main features of the literature
relating to the academic and practitioner views on market behaviour. One of the main pillars of
current academic understanding of stock markets, the efficient market hypothesis, is examined and
tested. A novel variation on a known flaw in the efficient markets hypothesis is examined; the sub-
Monday variation on the Monday effect. Using actual data this variation is tested and found to be in
violation of the efficient markets hypothesis. The paper describes two index/market designs; one
with actual data and one with hypothetical data. Limitations of system dynamics models in data rich
environments are illustrated. The models presented here are put forward as prototypes for the
development of further stock market simulations. This paper presents a proof of concept that
cyclical behaviours exist in stock markets and that stock markets are therefore amenable to analysis
using the system dynamics paradigm.
Keywords
Stockmarket, “Stock market”, “FTSE 100”, “Efficient markets hypothesis”, “EMH”, “Integration
error”, “sub-Monday effect”, “Monday effect”
Introduction*
The behaviour of financial markets occupies much intellectual effort from both the academic and
practitioner communities where a fairly sharp difference of opinion is in operation. However, it is
also a field where the academic and the practical approaches occasionally unite to create crossover
systems (Black and Scholes, 1973).
From the academic point of view the field has been dominated for many decades by the idea that
markets behave in a near random fashion. A milestone in the development of this paradigm came
when Eugene F. Fama gave full voice to the efficient markets hypothesis (Fama, 1970). Fama
acknowledged that the hypothesis was and remains a work in progress or as he concludes “we
certainly do not want to leave the impression that all issues are closed” (page 416). To supply some
context to the analyses that follow the three categories that Fama (1970) provided as degrees of the
efficiency of the market with some basic implications for practitioners are outlined.
Strong form efficiency would mean that market participants have access to and act on all relevant
and knowable information regarding price formation. Any market that was efficient at this level
would preclude earning of any excess returns other than by chance. No analysis of the market would
consistently provide excess returns. Semi-strong form efficiency means that market participants
have access to and act on all publicly available information. No analysis of the market, that did not
involve private information, would yield better than average returns other than by chance. Weak
form efficiency means that market participants have access to and act on historic price information.
No technical analysis of past price performance would yield better than returns produced by chance.
Underpinning all of these degrees of efficiency is the idea that the current price of an asset reflects,
to one of the degrees noted above or some shade thereof, the available information about that
asset. Hence, as new information arises prices will move to reflect that new information. As new
information arises in a near random fashion, causes are numerous and opaque, asset price
behaviour will reflect this near random information flow. This is the random walk (Malkiel, 2003,
page 3). In recognition of the observation that markets tend to rise over time the random walk idea
is generally supplemented with a component for this gentle rise upwards. The basic model of market
behaviour then becomes the random walk with drift.
* This is not a paper about system dynamics per se it is a paper about the value of system dynamics in
modelling stock markets. This research makes use of the system dynamics paradigm to describe an empirical
phenomenon. Hence this paper is to be read as a proof of concept rather than a definitive exercise in market
modelling.
If the efficient markets hypothesis holds true the implications are profound. There is little incentive
for anyone to carry out research on stock markets and little room for system dynamics type
simulations to uncover any feedback processes. Any attempt to model a market which was strong
form efficient would be a simple random process with a degree of drift. Any attempt to model a
market which was semi-strong form efficient could model only the strong version with additional
feedbacks limited to the effects of private information. Any attempt to model a market that was
weak form efficient could only model the semi-strong form with additional feedbacks limited to the
effects of publicly available information that was not already incorporated in the share price.
Adhering to this philosophy would mean that models of market behaviour can still be valuable but
are likely to be limited to special situations or to testing the degree of efficiency that is in operation.
Examples of these types of analysis are provided by Dubow and Monteiro (2006) and Monteiro et al
(2007) which both focus on insider trading.
However, the efficient markets hypothesis is not universally accepted and even in those areas where
it is accepted this acceptance can be conditional on so called exceptions or anomalies. Malkiel (2003)
identifies several calendar based market anomalies including the January effect, Monday effect, turn
of the month and holiday effects. In addition to asserting that these effects are not dependable
Malkiel further dismisses them as follows “these nonrandom effects (even if they were dependable)
are very small relative to the transaction costs involved in trying to exploit them” (page, 64). The
results of his analysis are a kind of acceptance, albeit grudging, that the markets Malkiel examined
are not entirely efficient at any level since all of the effects mentioned above can be isolated by
examination of historic price series. If a market is not efficient at the lowest level it cannot be
efficient in any higher or stricter sense.
In addition to the admission that markets are less than efficient from an academic viewpoint the
practitioner viewpoint can differ radically. A reasonably typical, if there is such a thing amongst such
a diverse population, practitioner view is expressed by George Soros (Soros, 1994). When discussing
the idea of a random walk in stock price behaviour, a foundation of the efficient markets hypothesis,
he asserts that “the theory is manifestly false” with the justification that “I have disproved it by
consistently outperforming the averages over a period of twelve years” (page 47). In a wider sense
Soros’ approach can be summed up as follows: investor behaviour and stock prices as well as
behaviours and prices in other asset classes, are interlinked in such a way that patterns are
occasionally observable due to the existence of a feedback process. Soros refers to this feedback
process as reflexivity. It is the ability to discern and take action on these patterns that illuminates the
feedback mechanism which produces further actionable patterns or investment opportunities.
4
Soros’ approach is at the more philosophical end of the spectrum of processes that rule in the world
of practitioners. Nonetheless the central core of Soros’ thinking appears to be that there are
patterns in the data which are exploitable by those with sufficient insight to see them; this is the
general thrust of much of the literature and other material, broadcast media for example, on
practical investing (Rafferty, 2012).
There is therefore some uncertainty about whether a system dynamics based investigation of stock
market behaviour or the efficient markets hypothesis would be valuable. Further, there is no point in
modelling the supply and demand relations of the participants if prices are apparently determined
by supply and demand though in reality are completely determined by a near random information
flow.
To date, explicit attempts to model stock market dynamics in the system dynamics literature are
sparse. Getmansky (2003) implicitly examined market efficiency/inefficiency when examining
arbitrage opportunities available to hedge funds. Chiarella and Gao (2002) modelled differences in
the dynamic behaviour revealed by comparing noise with fundamental trends on the Standard and
Poor’s 500 index. They also supply a critique of attempts to model the market pricing mechanism
using traditional regression models. Provenzano (2002) created a market model with the focus on
behaviours that arose as part of the trading style of the participants. Askar et al (2007) modelled the
Egyptian stock market with the focus on the implications for a stock index of differing investor
behaviours. Sterman (2000) discusses the business cycle and economic long wave which hint at a link
with stock market behaviour. Most of these analyses focus on supply and demand with delays and
feedbacks produced by the participants as price determining mechanisms. As noted above this is
unlikely to result in a meaningful analysis if prices, via supply and demand, are fundamentally
determined by the near random information flow.
From a system dynamics viewpoint we can postulate a link, much as Soros does, between the
information flow and the actions of the market participants; each influencing the other. At a
fundamental level demand is expressed as a positive price reinforcing loop and supply acting as a
balancing loop to counter the positive thrust of the demand. We could frame such a scenario from
the viewpoint of a simple price responder “prices are rising so buy, prices are falling so sell” which
also gives the basis for unwarranted boom and bust cycles. We could further characterise boom or
bust behaviour as just a significant divergence of current valuation from true value; though there is
no universally accepted method for determining true value and therefore no certain way to tell if
boom or bust behaviour is occurring other than with hindsight. In this model value and price are
detached. The up phase of this type of cycle was characterised by Alan Greenspan as “irrational
5
exuberance” (Greenspan, 1996) and we could venture to characterise the down phase as irrational
pessimism.
That there are feedback effects within stock markets is undeniable. The price yield figure below
shows the relationship between share price and yield on the FTSE 100 index (yield has been
multiplied by 700 for ease of comparison and is an average of 3.76% of price over the period shown).
Index FTSE 100 price and yield
May 1984 to Sept 2012
FTSE Close
FTSE 100 yield (restated)
1984 1987 1990 1993 1996 1999 2002 2005 2008 2011
Figure 1”
The key finding from the initial analysis is that, according to the efficient markets hypothesis the part
played by market participants as buyers and sellers is minimal in terms of determining initial prices
then subsequently in determining price changes. Rather starkly, this perspective views buyers and
sellers of stocks as mere conduits for the implementation of the information flow which is the
essential price setting mechanism. In contrast the practitioner view is largely the opposite of the
academic view; that the markets are inefficient and hence there is scope for uncovering exploitable
opportunities. In pursuance of this theme the analysis presented here focuses on investigating the
random nature of the information flow as revealed and expressed in market prices. A consequence
of revealing any significant non-random behaviour would be that the system dynamics paradigm
could be meaningfully employed in modelling stock market behaviour.
Whilst the apparently free actions of the market participants are in this view anything but free such
a view is not unknown throughout the breadth of academic thought. Consider, for example, Richard
Dawkins view that humans and all other creatures that pass on their inheritance genetically are
* Data sourced from Rafferty (2012, page 30).
unwittingly acting as vessels for the selfish gene that is the true determinant of their behaviour
(Dawkins, 2006).
Focus of this analysis
If the information flow were random with positive drift the first day of any trading period after a
weekend break would theoretically yield the greatest returns as it has two or more days of
information to incorporate in the share price; this is the Calendar time hypothesis (French, 1980). If
any other result were systematically obtained then the information flow could not be random given
that in the EMH information flows are the key price determinants. Previous research has suggested
that the returns on Mondays may be inconsistent with a random flow of information and the
calendar time hypothesis (French, 1980). This phenomenon is known as the Monday effect and it
generally states that in certain markets share prices are lower on Mondays than on any other day of
the week.
For this paper it is a variant on the Monday effect’ that will form the focus of attention. The
variation that is used here is to invest, via the index, on a Monday when that Monday is below the
price of the previous Monday — when there was a Monday in the previous five trading days. This
variant, together with others, is introduced and described in Rafferty (2012, pages 70-71).
French (1980) investigated the basic Monday effect in some detail over the period 1953 to 1977 on
the Standard and Poor’s composite index. French concludes both that the “persistently negative
returns for Monday appear to be evidence of market inefficiency” and that “an active trading
strategy based on the negative expected returns would not have been profitable because of
transaction costs” (page 68). In other words the effect had/has statistical significance but was/is of
limited practical value. French goes on to suggest that the greatest value that the effect had was to
allow an investor who was going to invest anyway to delay those purchases until Monday when
prices were relatively depressed or reschedule sales to sometime other than Monday. By induction
this would lead to outperformance and a weakening of the status of the efficient markets
hypothesis. In large part this investigation focuses on French’s observation that the value of this
anomaly lies solely in applying it to a passive rather than an active trading strategy. It is noted that
the passive investing technique is not the property of the efficient markets hypothesis it is merely
the most efficient way of accessing the underlying paradigm. Those who do not accept the efficient
markets hypothesis as valid may still make use of passive investing.
“The Monday effect is also known as the weekend effect or the (Monday) day of the week effect and the
variant is described in Rafferty (2012) as the sub-Monday effect.
7
Draper and Krishna (2002) analyse the anomalous behaviour of Monday returns in the UK with
reference to a set of explanatory variables. They confirm the existence of the Monday effect and
note that it can be largely explained with reference to the explanatory variables. They have sought
to add insight into cause to assist in explaining the effect. The addition of cause to effect would not
reduce the significance of the Monday effect or the implications of this effect for other explanations
of more general market behaviour including the efficient markets hypothesis. A significant element
of the explanatory power they describe is attributed to their observation that UK dividends tended
to be paid on Mondays and in particular they suggest that nearly 94% of ex-dividend days in their
sample are Mondays. Dividends paid by FTSE 100 constituents are now normally paid on
Wednesdays and have been since 2001; prior to 2001 ex-dividend days on the London Stock
exchange were Mondays (McStravick, 2000a). A timetable is produced by the exchange each year
detailing the acceptable ex-dividend dates for the forthcoming year. Any index constituent is not
bound to follow the timetable but there are additional responsibilities for those that do not. A
cursory examination of adherence to the timetable by FTSE 100 companies was undertaken via an
analysis of ex-dividend days for the six months to 28" September 2012; Mondays were significantly
underrepresented as ex-dividend days. Announcement dates, record dates and payment dates in the
same period were also examined and again Mondays were significantly underrepresented. The move
to Wednesdays occurred on Monday 5" February 2001 and is briefly documented and confirmed in
(McStravick, 2000b). At the same time as the change from Monday to Wednesday ex-dividend days
was introduced, changes to the settlement period, shortened from 5 to 3 days, were introduced;
record dates remain as Fridays.
Although the majority of academic writing concerning the Monday effect finds that it persists
Connolly (1991) provides an abstract statistical analysis of the Monday effect which finds that it is
weak or easily explained away. This is, Connolly suggests, due to the perceived unreliability of
abstract statistical analysis who states that “inferences about the DOW and weekend effects may
reflect a deficiency of classical statistical methods rather than systematic (and anomalous) return
behavior”, (page 94) the analysis provided here is as straightforward as possible; thereby avoiding
the criticism levelled by Connolly. Support for Connolly’s general position on the Monday effect is
provided by Schwert writing in Constantinides et al (2002) who offers the observation that “the
weekend effect seems to have disappeared, or at least substantially attenuated, since it was first
documented in 1980” (page 945).
Model design*
Figure 2 below shows one interpretation of the EMH in graphical format.
The EMH interpretation
Price Information Converted
determinants converters information
Conversion
layer
Available Share price
information behaviour
Figure 2
Figure 2 presents an interpretation of the prevailing academic view. Figure 2 shows the
transformation of information into share price behaviour. The available information includes all
information of all types that may affect share prices. The conversion layer contains buyers and
sellers, brokers, market makers, stock exchanges and so forth that convert the information into a
form that can be implemented as share price behaviour. The conversion layer is a kind of clear lens.
In the EMH this lens is highly efficient and transparent in its effects — information is rapidly and
accurately transformed into share price behaviour.
We may now turn to the practitioner view. A version of the practitioner view is expressed graphically
in figure 3 as the non-EMH interpretation.
* As this research is primarily aimed at providing a proof of concept the models provided with the paper are
primarily illustrative only.
The non-EMH interpretation
Price Information Converted
determinants converters information
Available Conversion
information layer
= 2 Share price
. behaviour
rE ?
Figure 3
Figure 3 shows the transformation of information into share price behaviour in a less efficient
manner. As before, the available information includes all information of all types that may affect
share prices. Explicitly including market inspired information and behavioural investing patterns that
are to some extent predictable; the documented Monday and sub-Monday effect for example. In
this interpretation the conversion layer is a kind of translucent lens. In the EMH the conversion layer
is highly efficient and transparent in its effects — information is rapidly and accurately transformed
into share price behaviour. In this version the conversion layer is less efficient, containing frictions
(bid-ask spreads, information asymmetries...etc) and is somewhat opaque in its effects. The track of
the information through the conversion layer is neither straightforward nor exact. Figure 3 differs
substantively from figure 2 in that it includes a looped structure from the share price behaviour to
the available information. This subset of available information has been labelled market inspired
information and includes, for example, the dividend yield information/relationship described in
figure 1. For the purposes of this analysis, differentiating market inspired information from other
information is a process that is complex to the point of impossibility due to the number of links and
feedback processes in operation within the set of all information.
Examples of the subset of all available information labelled cyclical behavioural information are that
such as mood/weather related information (Garrett et al, 2004) and temperature information (Cao
and Wei, 2005) which have been postulated as partial determinants of price behaviour and are
cyclical in nature with the progression of the seasons.°
5 Note that these two pieces of research, and similar, are noted as unpersuasive for a number of reasons
primarily due to the data being tied to physical market location which may differ radically from market
10
Two index designs are presented for consideration. The first uses actual closing price data for the
FTSE 100 for each trading day in the period expressed in index points. This design covers the period
1 November 2007 to 31 October 2012. Data has been sourced from commercial suppliers. The
second design uses artificial data constructed from theory and calibrated with observed values.
In addition to the closing price data an indication of whether the closing price belongs to a sub-
Monday or not is required. This takes the form of a binary signal which reads 1 if this is a sub-
Monday and a 0 otherwise.
The data series contains 1262 trading days of which 235 (18.62%) are Mondays and 104 (8.24%) are
sub-Monday variants. Figure 4 illustrates the overall shape of the data.
FTSE 100 closing prices,
value 1st November 2007 to 31st October 2012
00
11/2007 05/2008 11/2008 05/2009 11/2008 05/2010 11/2010 05/2011 12011 05/2012
Figure 4
Conceptually the model adopts a measure that is designed to give an indication of normal market
behaviour; a benchmark, the behaviour of any variant can then be measured against this
benchmark. In this case the benchmark is the ‘buy all rule’ described in Rafferty (2012, pages 57-60).
The buy all rule is an implementation of the three questions (a) what did it cost? (b) What is it worth
now? And (c) what is the difference? The three questions are applied to a process that just buys into
the index every available trading day and can be expressed as the equation below:
participant location. It is acknowledged that some attempt has been made to address this issue “[the effect]
remains strong even after controlling for the geographical dispersion of investors relative to the city where the
stock exchange resides” (Cao and Wei 2004, page 1561). Other issues are with inter-year data comparisons
(are returns increased in very cold years?) or between large scale differences in ambient environment (are
returns lessened in areas where light levels are generally lower?). Overall, even where these questions are
considered insufficient detail on these is provided to reach a conclusion of the validity of the research.
11
t t
Benchmark return = (» *Pr -¥ °) i> P «100
i
i
Where; n is number of transactions (all buys), P is the index price, iis the initial time and t is the
current time. The result of this formulation for any given day is the percentage of profit that the
benchmark is returning. Capital purchase costs are included in the formulation but transaction costs
are not. The benchmark figure is equivalent to the excess of the current value of all purchases over
the costs of buying the index every available trading day over the time period chosen.
Constructing the return figure for buying on a sub-Monday is the same as the benchmark except that
all purchases are conditional on the day being a sub-Monday. The calculation of returns from buying
on a sub-Monday compared to the benchmark is shown in the equation below. Symbols are as per
the benchmark return except that P’ is the price on a sub-Monday, RR is the net running return and
BR is the benchmark return specified above.
t t
RR = (nn -Yryye *100-—BR
7 7
The formulation shown above underestimates the discrete return on the anomaly due to the
inclusion of the anomalous returns in the benchmark return.
Findings
First we discuss the simulation based on actual data.
Initially the model was run with a time step of 1 to calibrate the model exactly to the number of days
that comprise the data series. The output of the model as expressed by the net running return figure
is illustrated below.
® Transaction costs are assumed to be the same for a passive investing strategy using the efficient markets
hypothesis or the timed, investing on sub-Mondays, passive investing strategy illustrated here.
12
% Efficacy of sub-Mondays v benchmark
(dt =1)
35
3.0
25 \
2.0 | N,
15 4
1.0
os
0.0
1 101-2012 301.402. S027, 80S ss901s100Ss 10S 1201
Time
Figure 5
The model output shown in figure 5 accords exactly with the expected performance of the differing
buying strategies, that is, the buying on sub-Mondays option outperforms the benchmark
consistently. In this case the sub-Monday option outperforms the benchmark 100% of the time.
Technically it outperforms 100% - 1 observation as on the initial pass both figures are zero.
The model is then reset to have a time step of 0.25 and is run again. The outputs are shown in figure
6.
13
% Efficacy of sub-Mondays v benchmark
(dt = 0.25)
1.0 \
os
\!
0.0
0 100 200 «300» 400s S00. s«60D.—s—«s700-S-«800.-S «900s ««1000« 1100-1200
Time
Figure 6
The outcome of this run is that the overall behaviour remains unchanged, sub-Monday outperforms
buy all, though the detail of the series is different. To highlight these differences between the two
series the values from figure 6 have been subtracted from those underlying figure 5 and are
displayed as figure 7.
One specific aspect of behaviour that arises from this simulation is an inconsistency regarding the
different buy signals. With timestep set to less than 1 any change in signal will result in a gradual
change from 1 to 0, and vice versa, according to the size of the timestep. When mutually exclusive
buying regimes are tested this can then result in buying on signals which are in reality mutually
exclusive processes. The differences that arise in this fashion do not change the overall shape of the
data here but they do present a small, though fundamental, inconsistency in the ability of the model
to mimic reality. Such differences are in the nature of continuous simulations and are exacerbated
with the particular model design selected here. To accept this type of simulation we must be willing
to write this off as good enough for simulated behaviour. It would not be acceptable for modelling
actual behaviour.
14
% Differences in reported efficacy
(dt 1 - dt 0.25)
08
0.6
0.4
0.0
oO 100-200-300 400 500. 600, 700» 800 900 1000 1100 1200
ime
Figure 7”
This section discusses a simulation with hypothetical index data calibrated from real values.
Part of the strength of system dynamics is that it presents the opportunity to distil the essence of
systemic behaviour and use that to model structural factors that give an insight into similar but
different systems. For example, in this case it would be appropriate to see this similar but different
system as the behaviour of the FTSE 100 over the coming 5 years or some other index over the same
period.
To create a hypothetical simulation for any stock market we must look to the existing theory and
data to guide us in replacing the actual index data. As we have discussed above the EMH asserts that
the market is a random process with some degree of drift. We can therefore look to this basic
paradigm to provide the framework on which to construct the model.
Choosing a random number from 0 or 1 would provide the basic model of the market where the
switch was the determinant of the direction of market movements; say 1 for a move upwards and 0
for a move downwards. Looking to the data would provide us with some refinement on this basic
structure. In particular if we look at the number of moves upward/downward in our calibration
period these reveal the drift in terms of how many moves have been observed in each direction. If
we then extend this analysis to uncover the size of the moves in either direction we can further
refine the model.
The method outlined above is not the only way to model the market. The figure below shows the
distribution of the data for the period 1 November 2007 to 31* October 2012. With so much data
available we are spoilt for choice regarding how to calibrate our model(s).
7 As the two series are of different lengths, there are four times as many data points for the series with dt =
0.25, the values at the end of each time period, corresponding to one day, only are included in the figure.
15
Sount, Categorical distribution of FTSE 100 daily moves
of moves
450 1st November 2007 to 31st October 2012
400
350 |
300 +
263
247
250
200
150
104
50 29
1 2 0 7 6 9 9 6 2 1 0 3 1
-7.9% -7.0% -6.0% -5.1% -4.2% -3.2% -2.3% -1.4% -0.4% 0.5% 1.4% 2.4% 3.3% 4.2% 5.2% 6.1% 7.0% 8.0% 8.9% 9.8%
Upper category limit (size of moves)
Figure 8
A visual inspection of the data in the figure above reveals that it is for all practical purposes normally
distributed in this period. So rather than using simple binary switches we could simply distribute our
index moves directly using the data above.
A detailed examination of figure 8 shows there is a slight bias to the downside. The actual respective
starting and ending index values are 6586.10 and 5782.7 in this period so the index did indeed drift
lower. Bearing in mind that the stock market, for many investors, is a long term investment vehicle
we might look to a slightly longer period for a more definitive dataset. Figure 9 below shows the
distribution of moves over the period 1* May 1984 to 28" September 2012.
Count
of moves Categorical distribution of FTSE 100 moves
3500 1st May 1984 to 28th September 2012
3089
3000
2500 2395
2000
1500
1000 ”
542
1 0 0 2 1 4 17 29 321430220020 3
-11.1% -10.0% -8.9% -7.8% -6.7% -5.6% -4.5% -3.4% -2.3% -1.2% -0.1% 1.0% 2.1% 3.2% 4.3% 5.4% 6.5% 7.6% 8.7% 9.8%
Upper category limit (size of moves)
Figure 9
16
From figure 9 we can clearly see the expected slight positive drift in the data that was absent from
figure 8.
Analysing the data over these two time periods gives us a set of figures that we can use to calibrate
the simulation — which really just amounts to replacing the actual index data with simulation
generated data based on whatever set of observations we use to calibrate it.
The figures in the table® below show two sets of calibration data.
Calibration period
Description 1* Nov 2007 to | 1* May 1984 to
31° Oct 2012 | 28” Sep 2012
Size of move upwards 0.010657205 0.007927
Size of move -0.01061137 -0.00806
Size of sub Monday move -0.00702 -0.00515
Number of moves upwards 629 3741
Number of moves 630 3421
Number of sub-Monday moves | 104 559
Percent moves upwards 0.498415 0.521176
Percent moves 0.499208 0.476595
Percent sub-Monday moves 0.082409 0.077877
Table 1
We can now calibrate our artificial index with something more subtle than the basic 1 for up and 0
for down with drift process then run that simulation for a number of iterations to give us an idea of
the accuracy of the model. The results of 10 iterations of the model, without adjustment for the sub-
Monday variant, are described in the figure below.
Index Maximum, minimum and average run values
od calibrated with data from 1st November 2007 to 31st October 2012
Thousands
0) 100 200 300 400 500 600 Fime 700 800 900 1000 1100 1200
Figure 10
® There are also a small number of occasions when the closing price of the index showed no change from one
day to the next these are 3 (0.24%) and 16 (0.22%) for the shorter and longer periods respectively. These no-
change days have been left out of the model for simplicity.
17
Figure 10 shows the maximum values of all ten iterations of the model, the top of the shaded area,
the lowest values from all ten iterations of the model, the lower limit of the shaded area, and the
average of these two values, the grey line along the centre of the shaded area.
As the model contains unseeded pseudo random values that determine when the moves up or down
occur no additional sensitivity analysis has been carried out; the model will naturally vary as the
pseudo random values change with each iteration. The outcomes illustrated in figure 10 are not
intended to be definitive and have been produced with the sub-Monday variant switched off. After
extensive testing with the sub-Monday variant behaviour included the model tends to
underestimate the actual behaviour of the index to a greater degree than that shown in figure 8.
We can now take this analysis a step forward and compare the average value from figure 10 against
the actual data from the period used to calibrate the model. This is illustrated in figure 11 below.
index Average simulated data v actual data
1st November 2007 to 31st October 2012
Simulated
Thousands
Actual
11/2007 04/2008 09/2008 02/2009 07/2009 12/2009 05/2010 10/2010 03/2011 08/2011 01/2012 06/2012
Figure 11
How accurate is the simulation in figure 11? Visually the relationship is close for much of the time
though as the simulated data is an average it would not be expected to follow the peaks and troughs
of the actual series and it obliges in this regard. Statistically the relationship is remote; the basic
correlation is 0.31 (31%) and the coefficient of determination is 0.1 (10%) between the two series.
This index design can present an outperformance by the sub-Monday rule, when that behaviour is
enabled, given that a bias has been built into this version of the index.
To project the relationship forward in time or switch it to another index the initial value of the index
in the simulation needs to be set to the value of the known data series e.g. a five year forward
projection for figure 11 needs the initial index value to be set at 5782.7 which is the closing value of
the FTSE 100 on 31* October 2012. The model can be calibrated with either data set shown in table
1 or any other that the user chooses.
18
Review and discussion
The model reveals the advantages of buying according to the sub-Monday variation compared to
buying according to the benchmark. The efficient markets theory precludes these findings and the
findings are therefore anomalous with the efficient markets hypothesis.
The strength of the outperformance revealed in figures 5 and 6 is total (noting the exception for the
initial observation given above). Given the complete dominance of the buying sub-Mondays strategy
it is not possible to dismiss these observations as statistically significant but practically insignificant
as French (1980) does with similar variants. Over the period illustrated the sub-Monday strategy
outperforms the buy all rule, in terms of returns, by 20.44% over the five years shown. The practical
significance of this result cannot be overestimated. It would leave the investor who achieved it in a
position to outperform both the index and almost the entire body of professional investors.
The model design using actual data, without structural modification, can be varied to model any five
year period in the history of the market shown. This involves replacing the data set (closing index
prices), the buy sub-Monday signal and possibly the run length.
Either model design can simulate any five year period from any other market without further
modification simply by replacing the data sets and varying the run length.
With some straightforward structural modification the model can be varied to model varying time
periods and variants from the EMH. A selection of variations that outperform the FTSE 100 index and
the S&P 500 index over different time periods is shown in Rafferty (2012).
It is a feature of financial analyses that they tend to be data rich. This is both a blessing and a curse
when considering a system dynamics approach to modelling stock markets. This is a positive feature
in that any model can be calibrated exactly using the available data. It is a negative feature in that
the availability of the data exposes any variations, however slight, between the simulated behaviour
and the actual behaviour. The latter can be due either to integration errors associated with the
method of incorporating flows into stocks (Euler’s method has been used here) or simply software
limitations. Figure 7 highlights these differences and prompts the recommendation that such
simulations are used to model periods or indices where real data is unavailable.
Differences between simulated results are an issue when exact data values are known or expected.
They are not as substantive an issue when the model is being used to create forecasts or indications
of general behaviour where exact, point by point, results are not the prime requirement.
It is not entirely clear why the system dynamics paradigm has largely ignored financial systems
behaviour. However, the specific issues introduced by integration errors together with the weight of
intellectual consensus in favour of the EMH, which would make system dynamics largely redundant
if valid, may partly explain the lack of system dynamics work in this vast and lucrative field of
research.
Philosophically the research presented here demonstrates, by reference to empirical observation,
the power of inductive research over well regarded theory in the form of the EMH; the sub-Monday
anomaly was discovered accidentally when searching the data for other patterns.
19
Conclusions
The concept that was being tested by this research, that there are significant cyclical patterns in
market behaviour, is found to be valid. Therefore, useful simulations of stock market behaviour can
be developed in standard system dynamics software and two such designs have been demonstrated
here.
The exact nature, causal relations, of the feedbacks that are in operation is yet to be uncovered.
The simulation reveals that the efficient markets hypothesis is flawed in regard to the sub-Monday
buying behaviour shown. This flaw goes beyond any simple statistical imperfection.
Care needs to be exercised when simulating behaviour with varying time steps if exact results are
required.
20
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