JOURNAL OF
Journal
ELSEVIER
Economic
DpalnicS
& Control
of Economic Dynamics and Control
21 (1997) 1117-1147
Explaining the facts with adaptive agents:
The case of mutual fund flows
Martin Lettau
CentER j6r Economic Research and Department of Finance, Tilburg Universily,
P. 0. Box 90153, 5000 LE Tilburg, Netherlands
Received
1I April 1995; accepted 4 December 1996
Abstract
This paper studies portfolio decisions of boundedly rational agents in a financial market.
Learning is modeled via a genetic algorithm. Learning as modeled in this paper leads
agents to hold too much risk as compared to the optimal portfolio of rational investors.
Moreover, adaptive agents exhibit an asymmetric response after positive and negative
returns where the portfolio adjustment is more pronounced after negative returns. It is
demonstrated that investors in mutual funds show the same investment patterns as the
adaptive agents in the model. A model with entry and exit of agents is able to match the
mutual fund data closely.
Keywords: Genetic algorithm; Mutual fund flows; Learning;
1EL clussiJcution:
D83; Gl I
Bounded rationality
I. Introduction
Standard models of financial markets imply that rational utility maximizing
agents should not trade for speculative reasons (see e.g. Milgrom and Stokey,
I thank John Campbell, David Easley, Dan Friedman,
Kai-Uwe Kiihn, Blake LeBaron, Georg
NGldeke, Ariel Rubinstein, Harald Uhlig, Timothy Van Zandt, and two anonymous referees for
helpful comments. Seminar participants
at Carnegie Mellon, Carlos 111 (Madrid), EUI (Florence),
IAE (Barcelona),
IAS (Vienna), Ohio State, Pompeu Fabra, Princeton, the Santa Fe Institute and
Tilburg provided many constructive remarks. I am also grateful for the hospitality of the Santa Fe
Institute where part of this research was done. The Investment Company Institute kindly provided
the mutual fund flow data. An earlier version of this paper was distributed under the title “Adaptive
Learning in a Financial Market.” All remaining errors are mine.
0165-l 889/97/$17.00 8 1997 Elsevier
PII SO165-1889(97)00046-S
Science B.V. All rights reserved
1118
M. LetraulJournal
of Economic Dynamics and Control 21 (1997)
1117-1147
1982). These results are at odds with high trading volume in many financial
markets. Since these no-trade theorems rely on the strong assumption that it is
common knowledge that all market participants are rational, it remains unclear
which kind of trading patterns emerge once the common knowledge assumption
is relaxed. In this spirit, this paper studies portfolio decision of boundedly rational
agents. Agents are boundedly rational in the sense that they cannot compute the
calculations required for expected utility maximization. Hence, they have to learn
from observed outcomes of their investment decision. This learning process is
modeled via a genetic algorithm as developed by Holland (1989). One feature of
adaptive agents is that they adjust their portfolio composition after observing the
return of past investment decisions. This behavior leads to interesting deviations
from rational decisions. First, the adaptive agents tend to take on too much risk
compared to rational agents. The magnitude of this risk taking bias depends on
the number of market observations that the agents use before they update their
investment portfolio. Under certain conditions this bias does not vanish as their
lifetime increases. Second, the risk taking bias leads to an asymmetric response
after positive and negative returns: the portfolio adjustment after negative returns
is larger in absolute value than after positive returns.
Adaptive behavior that leads to systematic deviations from full rationality might
be able to shed light on some of the puzzles in the finance literature. I use
data on flows into mutual funds to present evidence that investment strategies
of mutual fund investors cannot easily be explained by models with rational
agents. However, the mutual fund flows are consistent with adaptive investment
behavior as exhibited by the adaptive agents as modeled in this paper. The data
set used in this study differs from most other studies on mutual fund flows in
that it uses aggregate flows instead of flows of individual mutual funds. This
data set contains monthly net monetary flows into aggregate groups of funds.
In other words, individual funds with similar investment objectives are aggregated
together into classes such as aggressive growth funds or growth and income funds.
Most previous studies on mutual fund flows focused on the monetary movements
across individual funds, see e.g. Ippolito (1992), Lakonishok et al. (1992) and
Sin-i and Tufano ( 1993). A common result is that flows into individual funds are
positively correlated with its past performance. This might be expected because
the performance of a fund might reveal information about the quality of the fund
manager which in turn will lead rational agents to move money into funds which
performed well in the past. The advantage of a methodology using aggregate fimd
flows is that the information about fund managers will cancel out and therefore
cannot lead to flows across fund classes. Hence, any imperfection in information
about fund managers can be ignored and the focus of this study will be how
investors change their portfolio between mutual funds with different investment
objectives and hence different risk structures. The most important findings are
as follows. Mutual fund flows are highly positively correlated with returns of
the funds. After a positive return, investors move funds into mutual funds while
M. LetrauIJournal
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1119
they decrease their holdings in mutual funds after a negative market return. The
correlation is higher for riskier funds than for low risk mutual funds. Moreover,
the flows after a negative return are larger than after a positive return. Both
of these features are present in the investment behavior of the learning agents.
Indeed, when confronted with actual stock returns the adaptive agents produce
patterns which are very close to the mutual fund flow data.
How do these results square with conventional financial theory? In the standard
model following Markowitz (1952, 1959) and Sharpe (1964) the optimal portfolio
strategy is a passive one, i.e. investment flows should not follow any pattern
and can only be due to liquidity needs. The no-trade theorems of Milgrom and
Stokey ( 1982) show that rational agents should not trade with each other even
when there is asymmetric information. Wang (1993) presents a model which is
partially consistent with the mutual fund flow patterns. In his dynamic model with
asymmetric information
and noise trading, the uninformed
speculators behave
under certain parameter values like price chaser: buy when prices go up and
sell when they go down. This is just the investment pattern of mutual fund
investors. However, this result obtains only under very specific parameter values;
for example, it requires a very high amount of asymmetric information and strong
mean reversion in the process of noise trader’s demand. Moreover, the correlation
between the payoff and the portfolio adjustment in his model is much smaller than
in the data, even with the most favorable parameter choices. Since the optimal
investment policy is symmetric with respect to positive and negative returns, this
model is also inconsistent with the asymmetry found in the data. While the Wang
model is at least partially consistent with the mutual fund flow data, traditional
models are not able to create a substantial amount of investor’s responsiveness
to
current payoffs. Models with boundedly rational agents are a promising alternative
to explain trading volume and flow data. ’
In this spirit, I study how boundedly rational agents learn about their investment behavior in a very simple financial market. Agents are assumed to be unable
to perform any kind of maximization
as required in standard settings with rational
agents. They learn solely from outcomes of past investment decisions. The model
is kept as simple as possible to isolate the effects of learning compared to rational
behavior. Agents have to decide how much to invest in a single risky asset. Their
investment horizon is one-period so that learning takes place as repeated one-shot
investment decisions. Two version of the model are studied. The first model models a population of agents whose investment portfolio converges to a common
value over the lifetime of the agents. The second model consists of a population
of agents with new agents coming into the market and some of the existing agents
leaving the market. In this way I can compare the behavior of the adaptive agents
to the mutual fund flow data. The second model is able to replicate the behavior
’ Strictly speaking, the Wang (1993)
models since they assume completely
model and many others using noise trading
irrational noise traders to break the no-trade
are not ‘rational’
theorems.
1120
M. Lettoul
Journal
of Economic
Dynamics
and Control
21 (1997)
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of mutual fund investors more closely than any available model with rational
agents. The adaptive agents produce a reasonably high correlation of returns and
portfolio adjustment and they also exhibit the asymmetric reaction after positive
and negative returns. I therefore conclude that models with adaptive agents provide a viable alternative to conventional rational models in explaining observed
behavior of participants in financial markets. It remains to be seen whether it is
possible to explain a wider array of puzzles with boundedly rational agents. 2
The remainder of the paper is organized as follows. Section 2 defines the
model and describes how agents learn. The results for the single agent model
are presented. Section 3 gives the empirical results of the mutual fund flow data,
presents the multi agents model and compares the results of the adaptive agents
to the behaviour of the mutual fund investors. Section 4 concludes.
2. A financial market with adaptive agents
2.1. The model
As mentioned in the introduction, the model is taken to be a very simple one.
In order to obtain an analytical solution I make assumptions so that optimal portfolio decisions are linear in the expected excess return of the asset. Specifically,
I assume that there is a single asset whose value is normally distributed (with
mean 0 and variance cr,“), and that the agent’s utility of net payoff w has constant
absolute risk aversion with coefficient y:
U(w) = - exp( -yw).
(1)
In order to maximize expected utility the agent has to decide how many units
s of the risky asset she should buy. Let pe be the price of the asset. I assume
that the price of the risky asset is determined exogenously and is not influenced
by the adaptive agents. This is a restrictive assumption but can be defended on
multiple grounds. First, the focus of this paper is on how adaptive agents change
their portfolio over time and not on the patterns in prices generated by boundedly
rational investors. Given the available data set on mutual fund flows, a model with
exogenous prices appears to be a reasonable simplification. Of course, studying
the price formation in this market might also be in interesting extension. Second,
orders to change holdings in mutual funds are carried out only with a fairly
long lag, usually half a day, often only at the end of a trading day. Even if
a fund investor tried to react to current information, such as prices, his investment
decision will not impact prices until some later period. Hence, demand of fund
investors cannot affect current prices of assets. If there is a link from demand
2 Timmerman
returns.
(1993) uses
a learning model to explain excess volatility
and predictability
of stock
M. LettauIJournal
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to current prices, it has to be lagged. Hence, it seems a reasonable modeling
approach to assume that prices are given to mutual fund investors and that they
can only react with a lag to observed prices. The real shortcut is that the delayed
reaction is ignored in terms of their impact on prices.3
The agent’s net payoff given s and the realization of u is given by w = ~(a--PO).
The utility of the payoff is evaluated every period so that there is no dynamic
link between periods. The risk free interest rate is assumed to be zero. Under the
full rationality paradigm the optimal solution is relatively easy to calculate. The
optimal portfolio is a linear function of the mean value of the asset E and the
current price PO,
with optimal coefficients CL;and cl;:
1
LX;= CL;= --Yj’
P”
Note that for the agent to calculate her optimal portfolio she must know all
the relevant parameters: the distribution of the asset and her own utility fimction. Moreover, she must be able to perform the necessary calculations to obtain
the above equations. None of these requirements is necessary when we assume
that the agent is learning the portfolio decisions through an inductive learning
process. Of course, there are many ways to model learning in the context of
portfolio decision making. In this paper I model learning in a very simple way.
Agents observe the current market outcome and revise their next periods portfolio
using this observation. This is obviously a very stylized model of learning and
restricts agents severely in the way they can form portfolios. For example, they
are assumed not to use any statistical model and use observations to estimate it.
Despite these restrictions, the learning model captures the way many investors
behave in real markets such as chartists or technical traders.
Mutual fund investors appear to be a logical starting place to model boundedly rational behavior in financial markets. Apart from arguments made by the
popular press that mutual fund investors are the least informed participants in
financial markets, there are other theoretical arguments as well. The decision of
mutual fund investors to consciously put their money into the hands of fund
managers can be viewed as a decision in terms of an optimal allocation of time.
Researching the financial market is time-intensive and hence it might be optimal
3 Of course, this simplification
leaves a number of interesting questions unanswered.
For example,
it should be interesting to study how more rational investors could exploit such behavior. It is not
clear, however, that investors with this type of boundedly rational behavior should be driven out of
the market by more rational investors, see e.g. Palomino (1996). It is also not obvious how to write
down a model including GA investors and rational investors. Because the GA is highly nonlinear
algorithm, no closed-form solution for the behavior of the rational agents exists.
1122
M. Lettaut Journal of Economic Dynamics and Control 21 (1997)
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to allocate one’s time into other activities. Instead, mutual fund investors choose
to use simple ways to adjust their portfolio without spending too much time on
specific investment decisions. Often noise-trading models have been motivated by
these arguments. In this paper, I assume a weaker notion of noise trading in that
investors use a simple learning method to improve their investment decisions. The
GA described below allows investors to learn from past experience. However, the
learning mechanism is not optimal in the Bayesian sense. Investors use a mechanistic algorithm to update their portfolio decisions. The GA is a useful tool in
this context, because on the one hand more successful investment policies are
more likely to be used again in the future, and on the other hand, new investment policies are introduced by combining existing ones. These features appear
to be sensible ingredients when one aims at modeling adaptive behavior in financial markets. In that sense, modeling the behavior of mutual fund investors
in this very simple way appears to be a reasonable framework. Of course, the
results depend on the learning model and different learning models will most
1ikeIy produce different results.
2.2. The genetic algorithm
GAS were introduced by Holland (1989) to study learning, adaption, and optimization in complex systems. The basic idea of GAS is taken from processes
of evolutionary dynamics. A GA consists of a set of operations which manipulate a given population which is defined as a finite number of strings. In most
applications a string is a list of binary numbers, i.e. bits. The interpretation of
a string depends on the application. When applied to game theory each string
might represent a strategy of an agent. For the purpose of this paper a string
will be interpreted as a set of coefficients in an agent’s demand function. The
population of binary strings represents a set of economic agents whose behavior
is characterized by the strings. Each string is assigned a measure of performance.
In economic applications this fitness criterion might be a payoff or a utility level.
The task of the GA is to manipulate the strings to improve their performance
according to the fitness criterion. In other words, the population evolves over time
creating new strings which are better adapted to the environment. The key task
of the GA is to produce new populations based on the existing ones.4 Usually
the initial population is generated randomly.
To perform this transformation, the GA first selects copies of strings in the
current population. This is done by random draws from the existing population.
The probability that a given string is copied to the new population is based on
its performance or fitness: a string that did well according to the fitness measure will be more likely to be copied than a string with a lower fitness. The
4 The GA used in this paper is a fairly standard version. Buckles and Petry (I 992) and Goldberg
(1989) describe different implementations of GAS.
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GA then introduces new strings through two genetic operators that alter some
strings of the population. These two operations are called crossover and mutation. Both operations retain important characteristics of the ‘better’ strings in the
existing population. Crossover is the most important genetic manipulation.
In real
organisms, chromosomes rearrange their genetic code through a similar process.
Specifically, two strings are chosen randomly from the parent population. They
are lined up against each other, a point is selected randomly and the portions to
the right of the point are swapped between the two strings. To give an example,
suppose the strings ( 101101) and (011000) are selected. Suppose the crossover
point is chosen to be three. After crossover we obtain (101000) and (011101).
Both new strings replace the parent strings in the new population. The key feature
of the crossover operator is that it supports the development of compact building
blocks in the population. A compact building block is a block of genes that are
located next to each other. These building blocks tend to survive crossover and
in the long run only ‘successful’ blocks will prevail. It turns out that crossover
enables GAS to focus their attention on the most promising parts of the search
space. More precisely, crossover guarantees that all building blocks of the search
space are sampled at a rate proportional to their fitness (see Holland, 1989 for
a proof). The second genetic operator is mutation. Mutation simply flips a 0 to
a 1 and vice versa. Each chromosome undergoes mutation with a given (small)
probability.
The purpose of mutation is to introduce new genetic material and
to avoid the development
of a uniform population which will be incapable of
further evolution over time.
Without getting too deeply into the theory of GAS, I will illustrate briefly
why GAS are a powerful tool for simulating evolutionary
processes, including
economic learning. GAS exhibit implicit parallelism. In effect, each binary string
covers many different regions of the search space. For example, the string (10010)
belongs to the region of all strings whose first chromosome
is a 1. But it also
belongs to the region of strings which end with 10. This feature allows the GA
to cover many different regions with relatively few strings. This property is very
important when the GA is faced with nonlinear problems. Moreover GAS are also
able to balance the trade-off between exploration and exploitation.
Exploitation
means that an algorithm exploits actions which are currently successful. However,
if exploitation
is overemphasized
the system bears the potential cost that the
current action might not be the globally optimal one. To avoid this dilemma, the
system has to explore new parts of the search space. It has to try new things;
most of them will fail, of course, so too much exploration
will hurt overall
performance. Striking a healthy balance between exploration and exploitation is an
important component of adapting and learning systems. It can be shown (see e.g.
Holland, 1989) that GAS exhibit a very efficient trade-off between exploration
and exploitation. 5
’ The proof of this statement
is closely linked to the well-known
multi-armed
bandit theorems
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M. Lettaul Journal of Economic Dynamics and Control 21 (1997) 1117-1147
Another positive feature of GAS is that they can be employed in a very broad
class of models. In other words, regardless of the economic model the adaptive
agents endowed with a GA always use the same set of operations. Of course,
one can always find more elaborate rules, e.g. Bayesian learning, but this requires
learning methods that can vary considerably from one situation to another. For
example, a Bayesian learning rule is usually strongly affected by the statistical
distribution of what is unknown. A set of learning rules that can be used in more
general situation seems to be preferable for modeling learning in an economic
context.
Examples of GA applications in economics include Arifovic (1992, 1994, 1995,
1996), Miller (1986) and Miller and Andreoni (1990). Some of these papers
compare the behavior of the GA with experimental evidence and conclude that
the GA tends to produce results which are consistent with experiments. When
compared to other learning models such as Bayesian learning or least squares
learning, the GA tends to do a better job in reproducing the experimental data.
2.3. Decision rules
A strategy or decision rule maps the observable parameters to the demand for
the asset. In this model, the observable parameters are pa and fi and in general the
asset demand decision can be written as s(pe, fi). There might be unobservable
parameters or uncertainty (noise) that is resolved only after the agent makes her
decision. In this model, the unobservables are the realization of the asset value.
A priori the decision rule could be any function of the observables. However, it
is impossible to completely encode this infinite dimensional space. Although it is
possible to get an arbitrarily close approximation of the space (e.g., using polynomials), this is not desirable either because (i) the computational requirements
are larger the longer are the encodings of the strategies, and (ii) although allowing a wider range of functions makes it more likely that the optimal decision
rule can be approximated by an encodable decision rule, this also slows down
learning (convergence). Therefore, I choose a parsimonious encoding of the demand function. I assume that i! and po are constant and known to the agents.
Furthermore, I specify the simplest class of demand functions which includes the
optimal one. Hence, the agent’s problem is reduced to finding a scalar a, in the
linear function 6
s, = a,(6 - PO).
6 An earlier working paper version included results using more general timctional forms and variable
parameters 6 and po, e.g. s,(po,l)=q,
+ al,,po + CQ,,I?.The only important difference compared
to the simple case (4) is a slower convergence speed because agents have to learn more parameters.
All other conclusions are essentially unchanged.
M. LertaulJournal
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1125
2.4. The GA implementation
I implement the GA algorithm as follows. Let T represent the lifespan of each
agent. In each period t = 1,. . . , T the decisions of an agent are represented by
a binary string. Let J be the (constant) number of agents in each period t. Each
string of length L represents a value for a parameter in the demand function of
one agent. The strings are decoded in the following way. The range of values
for the parameter is the interval [MIN, MAX] 7 and let oi,* E (0, l} denote the
ith bit of a string in period t. Then the value of parameter CQin the demand
function decoded by the string wI = (~1,~. . . a~,,) is given by
a,=MINf(MAX-MM)~i=Z~~r:i-‘.
(5)
Eq. (5) is the binary decoding, scaled appropriately. Of course, this implies that
the parameter value MIN is encoded by the string (0.. . 0), while (1 . . .1) represents MAX. All other strings are between these two values. To give an example,
let MIN = - 1, MAX = 1 and L = 5. The string (11010) encodes the parameter
value a = - 0.290322.
Before the first period, the initial population of strings is chosen randomly.
Each bit is independently set to 0 or 1, each with equal probability. In each
period t the agents face S portfolio decisions, so that each agent makes a total
number of T * S decisions in her lifespan. The population of decision rules is
constant for all S drawings in a period t. After S realizations the population is
updated using crossover and mutation as described in the previous section. The
performance of each rule is measured by the sum of utilities generated by the
S drawings of the asset. It turns out that the number of decisions per period, S,
is crucial for the behavior of the GA agent (see simulation results below). Each
decision rule is assigned a probability of being copied to the next generation
based on its performance. A rule with a high cumulative utility receives a higher
probability than a rule with a low cumulative utility. The drawings from the
pool of rules is done with replacement. Specifically, let K = CT=, Vi(Wj) be the
cumulative utility after S drawings. Then string i is assigned the probability
(6)
of being copied in each drawing. Note that this transformation keeps the relative
strengths in proportion, i.e. a string with cumulative utility of -1 has a four
times higher probability than a string with cumulative utility of -4.
After J rules are drawn, the strings are randomly lined up in pairs of two.
Each pair undergoes crossover with a constant probability CROSS. The mutation
operator is then applied to each bit in the population with probability MUT. MUT
’ MIN and MAX are exogenously
chosen constants.
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is exponentially decreasing over time with half-life T* to guarantee a fixed limit
as T grows. This completes the formation of the new population.
The GA parameters used in the simulations are as follows. The number of
strings in the population is J = 30, and each string is of length L = 20. The initial mutation rate is set to MUT = 0.08. MUT decays exponentially with half-life
T’ = T/2, so that the mutation rate in the last period is 2%. Pairs of strings undergo crossover with a constant probability of 40%. The strings decode possible
parameter values between MIN = - 4 and MAX = 4. It should be noted that the
performance of the algorithm is not very sensitive to changes in these parameters.
If the mutation rate falls too rapidly the population might get stuck in a suboptimal bitstring. Recall that the crossover operator does not change two equal
bitstrings so that only mutation is able to alter a completely homogeneous population. Changing the crossover rate has mostly an influence on the convergence
speed.
2.5. Simulation results
Tables 1A and 1B report the simulation results for various values of the parameters. Table 1A reports the results for the simplest case where all observables
are constant. The price of the asset po is set to zero while the values for all other
parameters are given in the table. A ‘*’ denotes the theoretical optimal value for
the parameters. Each simulation is repeated 25 times for a given set of parameter values. Recall that in each period t = 1,. . . , T there are S subperiods. The
population of strings is constant for S periods before the crossover and mutation
operators are applied. To get a good measure of the performance of the GA the
following statistics are reported. First, I compute the average parameter values
of the population in each of the last 25 periods, T - 24,. . . , T, thus allowing the
algorithm to settle down. Then I compute the average and the variance of these
25 values for the 25 runs for each set of parameter values. The tables report
these averages with the variances in parentheses.
The first row shows the results for a benchmark case. The variance of the
asset value and the coefficient of absolute risk aversion are set to unity. Hence
the optimal value for parameter at is 1 as well. The total number of periods
T for the benchmark simulation is 500 with S = 150 portfolio decisions in each
period. The average value of coefficient (rt of the demand function of the GA
agent is 1.0233 which is about 2% higher than the optimal value. The effect that
the GA agent holds ‘too much’ of the risky asset can be observed in almost
all simulations. I will return to this feature in detail later. The low variance of
0.0088 shows that this result is very precise across repetitions. Note that the
average utility generated by the GA agent is extremely close to the utility under
full rationality. The maximal (average) utility is -0.6074 while the utility using
GA decision parameter is -0.6099, this amounts to a loss in utility of less than
0.5%. This is remarkable since mutation of a single bit in one of the last periods
M. LettauIJournal
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Table IA
Simple GA model
T
s
u
Y
u;
a;
I
500
150
I .oo
1.00
1.oo
1.oo
2
500
150
I .oo
2.00
1.00
0.50
3
500
150
1.oo
1.oo
2.00
0.50
4
500
150
1.oo
1.00
4.00
0.25
5
500
100
1.00
1.oo
1.00
1.oo
Simulation
6
500
50
I .oo
1.oo
1.00
1.oo
7
500
25
I .oo
I .oo
1.oo
1.00
8
500
10
1.oo
1.oo
1.oo
I .oo
9
100
100
1.00
1.oo
1.oo
1.oo
10
100
50
1.oo
1.oo
1.oo
1.oo
II
100
25
1.00
1.oo
I .oo
1.oo
12
100
10
1.oo
1.oo
1.oo
1.00
Now
Average
parameter
Table 1B
E[arg max L/J depending
values for 25 runs, variances
1.0233
(0.0088)
0.5230
(0.0038)
0.5 132
(0.0006)
0.2581
(0.0025)
1.0502
(0.0024)
1.1020
(0.0140)
1.1257
(0.1276)
1.3064
(0.0552)
1.0891
(0.0054)
1.1005
(0.0359)
I .3201
(0.0741)
1.4876
(0.0846)
in parentheses.
on S
CG =
1 a’ = 1
Eiarg max U,]
G=l
6 = 0.5
I
2
5
IO
25
50
100
500
3.3704
3.0779
2.4986
2.0177
2.7656
2.4097
1000
5000
S
1.2027
1.0969
1.8879
1.5466
1.2089
1.0897
1.0424
1.0205
1.0113
1.0103
I .0003
1.0058
1.0003
1.4187
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can cause a parameter in the population to change dramatically. Of course, this
string will most likely not survive to the next population but it will affect average
utility negatively.
Simulations 2, 3 and 4 change the parameters of the economy while keeping
the parameters of the algorithm constant. The algorithm produces stable results
for different coefficients of absolute risk aversion and variances of the asset value.
The following simulations reduce the number of total runs T and the number of
drawings S per period. Simulations 5-8 keep the lifespan T at 500 periods, but
reduce the number of drawings per period, S. As might be expected, the portfolio
chosen by the GA agent moves further away from the optimal one. Note that
the GA agent holds more of the risky asset as S decreases. I will discuss this
phenomenon later in more detail. Note, that the variance is increasing as well.
The subsequent simulations use a gradually decreasing lifespan while keeping
the same pattern for S. For a given S a lower T implies in most cases a bigger
deviation from the optimal portfolio. But the effect of decreasing S is much
stronger than of lowering T. If S is high enough, the algorithm produces results
close to the optimum even with a very low T. For example, simulation 10 uses
T = 100 and S = 50. The GA agent holds 1.1005 units of the risky asset which
is about 10% above the optimal holding of one unit. On the other hand, for
T = 500 and S = 10 (simulation 8) the portfolio of the GA agent consists of
1.3064 units of v which is more than 30% above the optimum. Even though both
simulations have 5000 realization, the case with higher S yields a portfolio which
is closer to the optimal one. Thus it is clear that the number of decisions before
the population of strings is updated, S, is much more important for the behavior
of the algorithm than the lifespan of the agent, T.
Why is the performance so much dependent on S? To develop an intuition,
consider the following experiment. Start with a set of portfolio weights, at I . . . <
a~. Recall that the rational agent maximizes expected utility, thus the optimal
portfolio weight is given by a* = arg max, E[ U,(W)]. Now fix S. Suppose that the
agent has to choose an a after observing only S drawings. Naturally, she would
choose the portfolio weight that induced the highest cumulative utility after S
drawings. Hence, she picks a** = arg max,, XT=, Uol,(wj). Note that in general
Era**] # a*. Thus the agent’s portfolio weight will in expectation deviate from
the optimal one, at least for finite S. I was not able to obtain a closed form
for Era**] except for the trivial case S = 1. In this case a** is either al, if the
only realization of u is negative, or aN, if the realization of v is positive. Hence,
E[a’*] = Pr(u ~0) al + Pr(v > 0) aN. This expression might be smaller or bigger
than a* depending on the set of a’s and the distribution of V. For a1 = 0, a,V = 4,
5 = 1, and 0,” = 1, we get E[a”] = 3.3704 >a*. Table 1B shows Monte Carlo
results for different S using the same values. For small S the bias is positive,
as S increases the bias is converging to 0.
Of course, the mechanism of the GA is not that simple. The set of portfolio
weights is not constant as only a part of the old population is carried over to the
M. Lettau I Journal of Economic Dynamics and Control 21 (1997)
1 I 17-1147
1129
new population and new strings are invented. Nevertheless,
the above analysis
is helpful in understanding
the behavior of the GA agent. After S drawings the
GA copies new strings from the old population according to their cumulative
utility. For small S the strings close to the minimal and maximal values of
the parameter space (i.e. weights close to MIN or MAX) tend to generate above
average fitness. Since V is chosen to be positive, this tendency is more pronounced
for weights close to MAX, for negative i! the effect is reversed. In other words,
since there are more ‘good’ states of the world with a positive payoff than ‘bad’
states with negative outcomes, the riskier strategies are more successful than
safer ones in more than 50% of market realizations. This explains why the GA
selects portfolio weights that are too high as long as S is low. As S increases
this effect becomes smaller since more realizations of u are observed. The whole
range of possible realizations will be covered and hence parameters closer to a+
will on average receive a higher utility than parameters close to the endpoints
of the parameter range. The fact that E[argmax U,] + argmax E[U,] as S + 00
illustrates this behavior. In other words, the agent must have a lot of information
about the distribution of the asset in order to learn how to behave optimally. If
she observes only a few realization of the uncertain asset, she selects portfolios
with too much risk since she does not take rare negative events correctly into
account.
Table 1C demonstrates the effect of S and fi in the simulations. The first three
rows show that the bias decreases when ij is lowered. Even for relatively high
values of S this effect is noticeable. Next, consider the extreme case when the
population is updated after each realization of the risky asset (rows 4-6). Recall
that the maximum value of a is set to 4. For fi = 1, the GA converges to a value
very close to MAX. When V is lowered the bias goes down even for S = 1. These
simulations reflect the theoretical intuition developed in the last paragraph. The
last four rows demonstrate that when S is set to a very large number, the bias
towards risky strategies vanishes. This suggests a trade-off between S and 7’.
Suppose that the agent has a fixed total of T t S decisions to make. What is the
optimal choice of S to maximize average utility? Fig. 1 illustrates this trade-off
for S * T = 1000 (other parameter values from simulation
1 in Table 1A). The
average utility for very low S is very small for the same reasons explained above.
The agent holds too much risky asset implying a low average utility. When S
is high, the demand parameter is updated rarely resulting again in lower utility.
The optimal S is about 20 with corresponding
T of 50. In other words, the
intermediate values of S guarantee a good trade-off between improving existing
strategies and producing suboptimal ones when the updating is too fast.
Figs. 2A and B illustrates the evolution of the parameter in the demand function
of the GA agent. Fig. 2A shows the average parameter value while Fig. 2B
presents the variance in each period t. Since the initial population
is chosen
randomly, the mean is close to 0 for the first periods. Of course, the variance is
very large initially. The GA agent is able to recognize quickly that the negative
M. Lettaul Journal of Economic Dynamics and Control 21 (1997)
1130
1117-1147
Table 1C
Simple GA model: effect of S and a
T
Simulation
1
S
100
100
v
Y
2
cu
a;
1.00
1.00
1.00
1.00
aI
1.0671
(0.0080)
100
2
3
100
4
100
5
100
0.50
1.00
1.oo
1.0250
(0.0082)
0.9927
(0.005 1)
0.25
1.oo
1.oo
I .oo
1
1.oo
1.oo
1.oo
1.oo
100
1
0.50
1.oo
1.00
1.00
6
100
1
0.25
I .oo
1.00
I .oo
7
100
1000
1.00
1.oo
1.oo
1.oo
8
100
5000
1.00
1.oo
1.oo
1.00
10
100
10000
1.oo
1.oo
I .oo
I .oo
11
100
Note: Average
parameter
100
1.00
1.oo
25000
I .oo
values for 25 runs, variances
1.00
3.9786
(0.0018)
3.5574
(0.0282
3.2576
(0.0876)
1.0014
(0.0106)
0.9937
(0.0085)
1.0010
(0.0059)
0.9974
(0.0110)
1.oo
in parentheses.
d
I
9
i”
7
h,”
.e
._;
=ul
,
S?
,o
u 9
z
‘:
21
s
I
rl
i
’ 0
100
200
300
400
500
600
700
800
900
1000
s
Fig. 1. Average lifetime utility of the agents
S l T = 1000 for different values of S.
for a fixed number
of lifetime
portfolio
decisions
M.
x
‘0
Letraul
50
Journal
100
of Economic
150
Dynamics
200
and Control
250
JO0
21 (1997)
350
1117-1147
1131
I
4w
450
500
ulo
450
500
period
iaa
Fig. 2. GA population
following parameters:
150
200
250
300
350
average and variance for each period t, respectively,
T = 500, S = 150, B = 1, y = 1, ut = I.
for a simulation
with the
parameter values are not successful and concentrates more on positive parameter
values. The variance is still fairly large indicating that the algorithm has not yet
pinned down a certain value. Nevertheless, the mean is slowly increasing towards
the optimal value while the variance is going down. After about 300 periods
1132
M. LettaulJournal
of Economic Dynamics and Control 21 (1997)
1117-1147
the mean is fairly close to one, the optimal value. It is slightly above one for
the same reasons as explained before. Nevertheless, the crossover and mutation
operators still alter the strings, hence the variance is still fairly large. After about
300 periods, the variance becomes very small, as indicated by the almost solid
line on the horizontal axis. The dots above the horizontal axis indicate that the
mutation operator still alters some strings. Since most of these mutations are not
successful they are discarded in the next period. Hence, the variance is large
for just one period. Since the mutation rate decreases over time, the number
of mutations is falling. The variance is almost always zero, only an occasional
mutation drives it up. The system settles down close to the optimal value. This
pattern of behavior is a very typical one for the GA agent.
To summarize, the behavioral pattern, the adaptive agents tends to hold too
much of the risky asset because there are more positive than negative events. In
other words, the agents are not able to take rare events correctly into account.
This bias tends to zero in the limit as the agents observe many asset realizations
before she updates her set of decision rules. Thus it is by no means trivial that
learning always leads to optimal portfolio decisions even in very simple static
settings as the one considered in this section.
3. Mutual fund flows and GA learning
The last section identified some consistent deviation of the learning agents
compared to how rational agents are behaving. This section is concerned with the
question whether there is any empirical evidence that learning as studied in this
paper is present in real financial data. To this end, I present next some evidence
that investment patterns of mutual fund investors are inconsistent with standard
financial markets theory but are consistent with the learning model on numerous
counts. Moreover, a slightly adapted version of the model in the previous section
is able to match the mutual fund data quite closely.
3.1. Mutual fund POWS: Some empirical evidence
This section contains an empirical investigation of flows into and out of commonly held mutual funds. The data set consists of monthly data starting in
February 1985 and ending in December 1992. 8 Instead of looking at individual mutual funds the data set focuses on aggregate flows of different mutual fund
categories thus summarizing the flows of funds with comparable investment objectives. The four fund groups are (in order of decreasing risk) aggressive growth
funds (AG), growth funds (GR), growth and income funds (GI), and funds with
balanced portfolio (BP). The investment objectives of these fund categories differ
* The data set was provided by the Investment Company Institute.
M. Lertau I Journal of Economic Dynamics and Control 21 f I997)
Table 2
Descriptive
statistics
Number
of individual
Average
Std. dev.
Risk
Average
Average
Average
Std. dev.
return
of return
of mutual fund data (monthly
funds
asset value (in $1000)
flow (in $1000)
flow in % of asset value
of asset flows
Note: Risk measure
is OLS coefficient
1117-l 147
1133
data 1984-1992)
AG
GR
GI
BP
326
547
428
160
0.013
0.052
1.063
37182
271
0.455
0.021
0.011
0.047
0.947
64330
559
0.606
0.012
0.009
0.040
0.859
93030
843
0.858
0.009
0.010
0.036
0.623
11924
202
in regression
on CRSP-VW
1.440
0.020
returns.
considerably. Aggressive growth funds seek to maximize capital gains thus focusing on risky stocks. Growth funds invest in common stocks of well-established
companies. Growth and income funds invest in stocks of companies with a solid
record of paying high dividends. Balanced funds have a portfolio mix of bonds,
preferred stocks, and common stocks. The data set contains the monthly flow into
the mutual funds measured in percentages of total asset value and the monthly
return of the fund category. Table 2 presents some descriptive statistics of returns and fund flows. Row 1 indicates the number of individual funds belonging
to each of the groups as of 12/92. There are 547 individual growth funds in the
data set while the number of funds with balanced portfolio consists of only 160
funds. The other two fund groups contain 326 funds (aggressive growth) and
428 funds (growth and income). Consider next the returns. For comparison, note
that the CRSP value-weighted index had an average monthly return of 1.2% with
a standard deviation of 4.6% during the sample period. The aggressive growth
funds (AG) yield a return of 1.3% with a standard deviation of 5.2%. Compared
with the CRSP-VW index, the return is 0.1 percentage points higher but at cost
of a higher standard deviation of 0.6 percentage points. This higher risk is also
indicated by a coefficient of 1.063 in an OLS regression of AG returns on CRSPVW returns. Growth funds (GR) are slightly less risky than the CRSP-VW index
and also have a lower average return. For the growth and income funds (GI) and
funds with balanced portfolio (BP) it is interesting to note that the BP funds
have a higher average return and a lower risk than the GI funds. Thus GI funds
did perform very poorly over the sample period.
Consider next the flow statistics. The total asset value for the fund categories
ranges from about 93 billion US dollars for growth and income funds to 12
billion dollars for BP funds. Note that the worst performing group (GI) has
the highest total asset value. The average monthly flows into the funds are all
positive thus indicating that mutual funds attracted investment over the sample
period. GI funds had the highest average monthly inflow of 843 million dollars
1134
hf. LettaulJournal
of Economic Dynamics and Control 21 (1997)
1117-1147
which translates to an average inflow of 0.86% of the total asset value. Only BP
funds had a higher relative inflow of 1.44% of the asset value. Aggressive growth
funds along with BP experience the highest variation in flows. GR and GI funds
have somewhat smaller standard deviations in asset flows. The high average flow
and standard deviation number for the BP funds deserve an additional comment.
BP funds experienced very high inflows early in the sample period, 1985 and
1986. After that period the average inflows are much lower. After deleting the
1985 and 1986 data from the sample, the number for the BP funds are in the
range of the GI funds. The subsequent analysis uses the whole sample period,
however. None of the time series except GR exhibit a significant time trend. The
t-value for a linear trend in an OLS regression of the GR data is almost 4. To
correct for this trend all the following regressions for the growth fund time series
include a linear trend.
Next, I study the relationship between mutual find flows and mutual fund
returns. Figs. 3A-D show plots of both time series for each fund category. Consider first the case of aggressive growth funds in Fig. 3A. It is clear from the
picture that both series exhibit an substantial amount of positive contemporaneous
correlation (the correlation coefficient is 0.73). Whenever the return is positive,
flows into aggressive growth funds tend also to be positive. On the other hand,
investors in aggressive growth fund tend to withdraw money in months with negative returns. Take as an example October 1987, the month of the stock market
crash. Aggressive growth funds yielded a return of about -25%. In the same
month AG funds lost about 10% of its assets net of the capital loss. The lower
panel in Fig. 3B illustrates the positive trend in the flows of growth mnds especially between 1988 and 1992. The top panel shows that the GR returns are less
volatile than the AG as is to be expected, of course. The correlation coefficient
of returns and flow of GR funds is 0.54 which is somewhat smaller than for
AG funds. Figs. 3B and C suggest that the correlation of returns and flows are
weaker for GI and BP funds. The visual evidence is supported by corresponding
OLS regressions which are presented in Table 3.
Row 1 of Table 3 shows the correlation coefficient of mutual funds flows S
and returns r for the four fund categories. Funds with higher risk tend to have a
higher correlation. The riskiest fund category, aggressive growth funds, exhibit the
highest correlation with returns with a value of 0.73. All correlation coefficients
are positive. The fairly large correlation coefficients also translate into significant
coefficients in OLS regressions of fund flows on fund return, at least for the
riskier funds. For aggressive growth funds the regression coefficient is 0.30 with
a highly significant t-value of 10.52. Moreover, retums explain more than 50%
of the variation in flows. The numbers for the less risky growth funds are a bit
smaller but the coefficient of returns is still highly significant. The R2 is 0.44. The
evidence for the less risky funds is less clear. The coefficient in the BP regression
is not significant at conventional significance levels while the GI coefficients is
significant at the 1.7% level. The R2’s are low with 0.01 and 0.04, respectively.
M. LeitaulJournal
of Economic Dynamics and Control 21 (1997)
1117-1147
1135
n
FIOWS
Fig. 3A. Top Panel shows the return of aggressive growth funds; bottom panel shows the monetary
flows into the class of aggressive growth funds, monthly data, 2/85-12/92.
II36
M. LelraulJournol
of Economic Dynamics and Control 21 (1997)
1117-1147
:
P IS
I
67
w
89
90
91
B-2
(b)
Fig. 3B. Top panel shows the return of growth !?mds; bottom panel shows the monetary flows into
the class of growth funds, monthly data, 2/85-12/92.
M. LerraulJournd
of Economic Dynamics
and Conrrof 21 (1997) i117-(147
I137
FIOWS
.
8
1
.
I
9
063
6.6
6T
69
6Q
90
91
92
a3
Cc)
Fig. 3C. Top panel shows the return of growth and income funds; bottom panel shows the monetary
flows into the class of growth and income funds, monthly data, Z/85-12/92.
1138
n
:
hf. LerraulJournal
of Economic Dynamics and Control 21 (1997)
1117-1147
_
P
0
‘.
P 95
w
87
WJ
89
90
91
92
(4
Fig. 3D. Top panel shows the return of balanced portfolio timds; bottom panel shows the monetary
flows into the class of balanced portfolio fknds, monthly data, 2/85-12/92.
M. LettaulJournal
of Economic Dynamics and Control 21 (1997)
Table 3
Mutual fund flows f and returns r (OLS regression:
Corr(/,,rz)
P
f-statistic
R2
8’
B-
fi = a + pr,, monthly
1117-1147
data 1985-1992)
AG
GR
Gl
0.73
0.30
10.52
0.54
0.20
0.34
0.54
0.14
1.34
0.44
0.13
0.14
0.24
0.05
2.42
0.04
0.02
0.11
Note: B+ (fl- ) are OLS coefficients
of f after a positive
(negative)
1139
BP
0.10
0.06
0.98
0.01
-0.06
0.19
return
Thus, there is substantial evidence, at least for the riskier funds, that returns are a
major explanatory variable for flows into and out of mutual funds. The evidence
is very strong for the riskier aggressive growth and growth funds and weaker for
the less risky growth and income funds and funds with balanced portfolio.
However, there is another interesting fact in the data. Consider the responses
of mutual fund investors to positive or negative returns. To check whether there
is any asymmetry in investor’s responses, I split up the data set in months with
positive returns and negative returns and perform regressions on both subsets. The
respective regression coefficients are shown in the bottom row of Table 3. Note,
that the regression coefficients for the subset of months with negative returns are
larger than the respective coefficients for positive months in all four cases. The
difference is substantial for all fund categories but growth funds. Thus, it appears
that mutual fund investors are reacting more heavily after negative returns than
after positive ones. To summarize the most important facts of the behavior of
mutual fund investors, (i) flows into mutual funds are positively correlated with
returns, (ii) flows are more sensitive to negative returns than to positive ones,
and (iii) evidence is stronger for riskier mutual funds.
3.2. Financial flows in models with rational agents
These findings pose a substantial challenge for standard financial theory with
rational agents. Recall that the data used here is on aggregate fund classes and not
on individual funds. The positive correlation of returns and fund flows could be
expected with individual fund flows since a high return on a specific fund might
reveal some information about the quality of the fund manager. This in turn might
lead investors to update their prior of the quality of that manager and increase
their investment in this fund. However, this information story does not apply to
aggregate fund flows used here since investors would move money to a fund in
the same class. Since only flows across different fund classes are reflected in the
data, the flows due to new information would show up in this data set.
The standard market model following Markowitz (1952, 1959) and Sharpe
(1964) is also not able to explain why mutual investors are changing the
1140
M.
Leftaul Journal of Economic Dynamics and Control 21 (1997)
1117-1147
portfolio composition after observing the return of their investment portfolio. In
their model with complete information investors should follow a passive investment strategy ignoring any market outcomes. More generally, it seems unlikely
that any model with complete information is able to explain the reactiveness of
investors. Wang (1993) presents a model of incomplete information in which
returns provide information about the state of the world, thus making the optimal
portfolio strategy dependent on observed market outcomes. In his dynamic model,
agents have asymmetric information about the future growth rates of dividends.
Informed and uninformed agents trade against serially correlated noise demand.
Apart from the noise traders, all agents in the model are rational in that they use
Bayes’ rule in updating their guesses about the noise supply and dividend growth
rate. He shows that for certain parameter values, the uninformed agents behave
like trend chasers, that is, they buy when the price goes up and sell when it goes
down. This leads to a positive correlation between changes in the holdings of
the risky asset in the portfolio of the uninformed investors and price changes. In
that sense the uninformed investors in Wang’s model behave like mutual fund
investors as shown in the preceding section. However, this result is only true
for very specific parameter values in the Wang model. A very high degree of
asymmetric information and a substantial amount of serial correlation in the noise
trader demand are needed. Moreover, the degree of positive correlation between
asset holding changes and price changes is rather small even for the most favorite
parameter settings. This makes Wang’s model more consistent with mutual fund
investors in less risky funds than with high risk mutual fund investors. The optimal portfolio policy of Wang’s traders is also symmetric with respect to positive
and negative price changes which is inconsistent with the mutual fund data.
In general, it seems unlikely that models with agents who are using Bayes’ rule
to update their beliefs are able to generate substantial trading volume in steady
state without relying on unreasonably large amounts of noise trading. Intuitively,
the reason is that in dynamic settings, Bayes’ rule puts relatively little weight on
recent information and relies more on the past. Thus, any new information does
not change beliefs too much and thus does not cause substantial portfolio updates.
Another rational explanation of positive correlation between returns and portfolio changes might be portfolio insurance. If the risk tolerance of an investor
is steeply decreasing with wealth it might be optimal for her to sell stock when
the price goes down and increase her holdings in risky asset when the price goes
up (see e.g. Black, 1988; Leland and Rubinstein, 1988). However, it seems unreasonable that all or most investor in mutual funds followed portfolio insurance
strategies.
3.3. A learning model with entry and exit
The model in Section 2 focused on how a fixed set of agents is learning over
time. The agents learn via the crossover and mutation operators. The algorithm
M.
Lettaul
Journal
of Economic
Dynamics
and Control
21 (1997)
II 17-1147
1141
was designed so that the portfolio of the agents converges to a fixed amount
of the risky asset which is the same for each agent in the population.
In the
long run, the agents will not adjust their portfolios after observing the payoff.
The declining mutation rate and the crossover operator are responsible for this
behavior. Thus, it is clear that a model with a constant population of adaptive
agents will not be able to reproduce the behavior of the mutual fund investors as
shown in Section 2. The simplest way to achieve a heterogeneous population even
in the long run is to insert new random strings over time and to delete some of
the existing strings from the population. These new strings introduce new genetic
material which again will be subject to crossover and mutation. However, the
population will not converge to a homogeneous one as it is the case with only
crossover and mutation due to the constant inflow of new strings. In terms of the
economic interpretation of this new model, some of the investors are leaving the
market and some new ones are entering it. The new agents start with random
strings. In the simulations below, the exiting agents are chosen randomly with
equal probability. Other methods such as basing the probabilities
on the fitness
of the strings are also possible. I experimented with different methods but the
results were not affected much.
The following setup was used in the simulations. The population consists of
60 agents. In each period t = 1,. . . , T three agents, chosen randomly, exit the
market and are replaced with new agents who start with a random strategy.
Agents update their decision rule after each market outcome, i.e. S = 1. To make
the simulations
as close to existing markets as possible, I set the distribution
parameters of the asset payoff, V and oV, so that they match the values of the
first two moments of the returns of the four mutual fund groups assuming that
returns are indeed normal. The coefficient of absolute risk aversion is set to unity.
To let the population settle down, I let it evolve for 5000 periods before starting
with the analysis. The figures and tables use periods t = 5001,. . . ,5 100 because
there are also about 100 observation in the mutual fund data set.
Consider Table 4 and the corresponding
Fig. 4A which shows a plot of 100
market realizations and the portfolio adjustments of the GA population for the
case where the asset payoffs are drawn from a normal distribution with the first
two moments matching the aggressive growth fund returns. Just as investors in
mutual fund, the population of adaptive agents adjusts its portfolio after each
market outcome. The agents become more bullish after positive realizations and
hence increase their holdings of the risky asset and vice versa. The correlation
between returns and portfolio adjustments is 0.76, the OLS regression coefficient
is 0.38. While the regression coefficient is larger than the respective number
for the mutual fund investors, the correlation coefficient is virtually the same
as for the flows into the aggressive growth fund. But this aspect is not the
only similarity with mutual fund investors. The GA population exhibits the same
asymmetry
after positive and negative realizations.
Again, separating the data
into reactions after positive and negative returns reveals the same pattern as the
1142
M. LettaulJournal
of Economic Dynamics and Control 21 (1997)
Table 4
Entry and exit GA model (portfolio
adjustment
fi, return rz, OLS regressions:
1117-1147
fi = c( + /&, )
Asset return r
Corr(fr, rr )
B
f-statistic
R2
P+
B-
N(0.013,0.052)
N(O.O1 1,0.047)
N(0.009,0.040)
N(0.010,0.036)
0.76
0.69
0.67
0.59
0.38
0.29
0.23
0.15
11.23
10.83
9.48
7.03
0.64
0.52
0.50
0.38
0.31
0.26
0.20
0.12
0.48
0.34
0.29
0.21
AG
GR
GI
BP
0.70
0.60
0.54
0.45
0.32
0.25
0.21
0.13
10.38
9.72
8.99
5.65
0.63
0.47
0.46
0.34
0.26
0.22
0.19
0.10
0.41
0.30
0.26
0.18
returns
returns
returns
returns
Note: p+ (p- ) are OLS coefficients
after a positive
(negative)
return.
mutual flow data. The downward reaction after a negative return is more extreme
than the corresponding portfolio revision after a positive outcome. The regression
coefficients are p- = 0.3 1 and /I+ = 0.48, respectively. This is also evident from
Fig. 4A. The reason for this asymmetry is the risk-taking bias as discussed in
Section 4.2. With the distribution moments t7= 0.013 and 0” = 0.052, about 65%
of the market outcomes are positive. As seen in Section 4.2 this asymmetry in
the asset distribution leads the adaptive agents to be too bullish on average. In
this model, the mechanism is very similar. Since the adaptive agents hold too
much of the risky asset, the utility after a negative payoff is very small causing
the agents to suddenly become very bearish. After a series of positive outcomes,
the adaptive agents increase their holdings of the risky asset more and more.
Thus, the following downward revision after a negative return is very extreme
since the level of utility will be very low.
The following three rows of Table 4 present the simulation results for returns
matching the first and second moments of GR, GI and BP mutual fund returns. As
might be expected the values for correlation of returns and portfolio adjustment
and regression coefficients are smaller for the returns with a lower risk. Note that
the numbers are all slightly larger than the numbers for actual mutual fund flows
in Table 3. This indicates that the adaptive agents in the model are more reactive
than mutual fund investors. However, the general pattern of investment behavior
is very similar.
Finally, I confront the GA agents with the actual mutual fund returns and
compare the portfolio reaction to the mutual fund flows. Again, the population
evolves 5000 periods using a normal distribution with matched first two moments
as payoff before feeding the mutual fund returns. Fig. 4B plots the change of
the portfolio of the GA population while the lower panel of Table 4 presents
the corresponding correlation and OLS regression results. The results are very
similar to the case with normal distributions. The portfolio adjustment is more
extreme for high risk mutual fund returns. As before, the numbers are larger
M. LettaulJournal
of Economic Dynamics and Control 21 (1997)
1117-1147
1143
period
3020
5060
5040
5060
5100
period
Fig. 4. Top panel shows the normally distributed asset payoff and the change in the holdings of the
risky asset of the GA population. The bottom panel shows the changes of the asset holdings when
the return of aggressive growth funds are used as input.
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-0.10
-0.05
AC
-0.00
0.05
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0.10
0.t
return
:!
d
I
-0.25
-0.20
-0.0
-0.10
-0.05
-(
0
0.05
0.10
5
ffi return
Fig. 5. Top panel shows the flows of aggressive growth funds plotted against the return of the AG
funds. Bottom panel shows the changes of the asset holding of the GA agents plotted against the
return of the AG funds.
M. LertaulJournal
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than for the mutual fund flows in Table 3. The model with GA agents exhibit
the same asymmetry after positive and negative returns while the coefficients p+
and /?- are both bigger than for the AG funds. While the investment patterns of
the GA agents and the mutual fund investors are very similar in many aspects,
the GA population appears to be even more sensitive to market outcomes than
investors in risky growth fimds. The scatter plot in Fig. 5B demonstrates this overreaction of the GA agents when compared to investors into aggressive growth
funds (Fig. 5A) very clearly.
4. Conclusion
The empirical part of this paper investigates data on how investors in mutual
funds move money into and out of four different groups of mutual funds. The
paper shows that mutual fund investors change their portfolio composition after
observing market outcomes. The correlation between flows into mutual funds and
returns is positive and large, at least for riskier funds. OLS regressions show that
returns alone explain up to 54% of mutual fund flows. The regression coefficient is significant for the riskier fimd categories. The second interesting finding
is that mutual fund investors exhibit an asymmetric response after positive and
negative returns. They tend to react more heavily after negative market outcomes than after positive ones. This effect is also more pronounced for riskier
funds.
These findings pose a substantial challenge for conventional theory. Most standard models suggest that rational agents should follow a passive investment strategy. In these models there is usually little trading volume. Wang’s (1993) model
with asymmetric information
is partially consistent with the mutual fund flow
data. For certain parameter values, some of the agents in his model rationally
follow a price-chasing
investment strategy. However, this is only true for very
specific parameter values and the correlation of portfolio adjustment and returns
is lower than in the mutual fund flow data even in the most favorable case. In
general, it seems unlikely that a rational model is capable of explaining the facts
about mutual fund investors in a satisfying way. Instead, this paper suggests an
adaptive learning approach.
The paper presents a simple model in which agents use a genetic algorithm
to update their portfolio decisions. In the first version of the model the adaptive
algorithm is implemented
so that the agent’s investment strategy converges to
a fixed portfolio as time grows. During the learning process the agents look at
market outcomes and revise their portfolio after observing a certain number of
observations.
It turns out that the adaptive agents exhibit a positive risk taking
bias that depends how many market outcomes are observed before the agents
update her portfolio. Thus, they tend to hold a portfolio which is too risky compared to the portfolio of a rational agent who maximizes expected utility. Next,
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a second model with a varying set of agents is introduced. In this model a constant inflow of new agents and outflow of old ones keeps the market going. The
behavior of the population of adaptive agents is consistent with the mutual fund
flow data. They adjust their portfolio after observing the market outcome just
like mutual fund investors. The GA population also reacts asymmetrically
after
positive and negative returns just as the mutual fund investors. The regression
coefficients for the GA agents tend to be larger than the corresponding
numbers in the mutual fund flows. In summary, the learning approach as used in
this paper produces behavioral patterns which are consistent with most aspects of
mutual investor’s strategies in real financial markets and thus provide a simpler
and better fitting theory of observed behavior than standard theories with rational
agents.
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