Explaining the facts with adaptive
agents: The case of mutual fund flows
A financial market with adaptive agents
Mutual fund flows and GA learning
報告人 : 范志萍
The paper studies portfolio decisions of boundedly
rational agents in a financial market.
The data set set uses aggregate flows instead of flows
of individual mutual funds.
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-short
Two version of the model are studied. The first models a
population of agents whose investment portfolio
converges to a common value over the lifetime of the
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.
The models with adaptive agents provide a viable
alternative to conventional rational models in explaining
observed behavior of participants in financial markets.
In 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.
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 speculator behave under certain
parameter value like price chaser.
A financial market with adaptive
The genetic algorithm
The GA implementation
Agent’s utility is U(w)=- exp(-r w)
p0 be the price pf asset and exogenously
How many units s of the risky asset she should
Constant absolute risk
The utility of the payoff is evaluated every period
so that there is no dynamic link between periods.
Under full rationality the optimal portfolio is linear
function of mean value of the asset and the
current price p0,
with optimal and
To calculate her optimal portfolio the agent must
know all the relevant parameters: the distribution of
the asset and her own utility function.
pvs αα −=
Agents observe the current market outcome and revise
their next periods portfolio using observation.
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
This paper assume a weaker notion of noise trading in
that investors use a simple learning method to improve
Investors use GA to update their portfolio decisions.
The genetic algorithm(1)
The initial population is generated randomly.
GA first randomly selects copies of strings in the current
[Proportionate Selection]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
The GA then introduces new strings through crossover and
mutation that alter some strings of the population.
Did not use the election operator? Why?
It supports the development of compact building
blocks in the population.
A compact building blocks 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.
Crossover guarantees that all building blocks of the
search space are sampled at a rate proportional to
Mutation simply flips a 0 to a 1 and vice
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.
A strategy or decision rule maps the observable
parameters to the demand for the asset. In this model,
the observable parameters are p0 and and the asset
demand decision can be written as s(p0, ).
This model choose a parsimonious encoding of the
demand function. I assume that p0 and are
constant and known to the agents. Hence, the agent’s
problem is reduced to finding a scalar in the linear
)0( pvs tt −=α
The GA implementation(1)
Let T represent the lifespan of each agent.
Let J be the (constant) number of agents in each period.
Each string of length L represents a value for a parameter in
the demand function of one agent.
The range of values for the parameter is the interval
Let denote the ith bit of a string in period t. then
the value of parameter in the demand function decoded by
the string is given by
)....( ,,1 tLtt www =
The GA implementation(2)
In each period t the agents face S portfolio decisions, so that
each agent makes a total number of T*S decisions in her
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.
Each decision rule is assigned a probability of being copied to
the next generation based on its performance. The probability
be the cumulative utility
after S drawings.
∑ == S
j jii wUV 1 )(
Let the number of strings in the population is
J=30,and each string is of length L=20, the
probability of crossover is 0.4, The initial mutation
rate is set to MUT=0.08. The strings decode possible
parameter values between MIN=-4 and MAX=4.
Each simulation is repeated 25 times for a given set
of parameter value.
The variance of the asset value and the coefficient of
absolute risk aversion are set to unity.
S 相同， T 較
小 bias 較大
當 s ，與
Simulation results (1)
Mutual fund flows and GA learning
Mutual fund flows: Some empirical evidence
Financial flows in models with rational agents
A learning model with entry and exit
Mutual fund flows
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.
The four fund groups are
aggressive growth funds(AG), growth funds(GR), growth
AG, GR, GI, BP
AG seek to maximize capital gains thus
focusing on risky stocks.
GR invest in common stocks of well-
GI invest in stocks of companies with a solid
record of paying high dividends.
BP have a portfolio mix of bonds, preferred
stocks, and common stocks.
Statistics of return and fund flows
1.highest average asset value
2.highest average flows
3.lowest average return
4.lowest standard deviation of
1. Lowest risk
flow in % of
1. highest risk
2. highest average return
3. highest standard deviation
4. highest standard deviation
of asset flows
AG’s returns and flows
The correlation coefficient of two series is 0.73 .
Whenever the is positive, flows into aggressive growth funds tend to be positive.
At 1987/10 , AG funds a return of about –25%, AG funds lost about 10% of its
assets net the capital loss.
GR ’s returns and flows
The positive tend in
flows of growth funds
1988~1992 年 .
coefficient of two
series is 0.54
GI ’s returns and flows
coefficient of two
series is 0.24
BP’s returns and flows
The correlation coefficient of two series is 0.10 .
1985~1986 年， BP funds experienced very high inflow.
若不看 1985~1986 年的資料，則 BP 落在 GI的範圍 .
Summarize the behavior of mutual funds
Flows into mutual funds are positively correlated with returns.
Flows are more sensitive to negative returns than to positive ones.
Evidence is stronger for riskier mutual funds.
Financial flows in models with
Markowitz (1952, 1959) and Sharpe (1964) is also
not able to explain why mutual investors are
changing the portfolio composition after observing
the return of their investment portfolio.
Wang (1993) shows the uninformed agents behave
like trend chasers. This leads to a positive correlation
between changes in the holdings of the risky asset in
the portfolio of the uninformed investors and price
A learning model with entry and exit
A population of adaptive agents will not be
able to reproduce the behavior of the mutual
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
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.
The coefficient of absolute risk aversion is set to
To let the population settle down, it evolve for 5000
periods before starting with the analysis.
The model with GA agents exhibit the same asymmetry after
positive and negative returns realizations while the
coefficients β+and β- are both bigger than for the AG funds
As investors in mutual fund, the GA population of adaptive
agents adjusts its portfolio after each market outcome.
The GA population exhibits the same asymmetry after
positive and negative returns realizations.
Plots the change of the portfolio of the GA population while the 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 returns.
Fig. 5B demonstrates this overreaction of the GA agents when
compared to investors into aggressive growth funds (Fig. 5A )
Mutual fund investors change their portfolio composition
after observing marker outcomes. The correlation
between flows into mutual funds and returns is positive
and large, at least for riskier funds.
Mutual fund investors exhibit an asymmetric response
after positive and negative returns. They tend to react
more heavily after negative market outcomes than after
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.
They tend to hold a portfolio which is too risky
compared to the portfolio of a rational agent who
maximizes expected utility.
A second model with a varying set of agents is
introduced. The behavior of the population of
adaptive agents is consistent with the mutual fund
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
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
Not using the election operator. Why?
Role of S: Why is the performance so much
dependent on S?
When S is large:
When S is small:
Trade-off between S and T:
GA Learner and Bayesian:
Population Allowing for Entry and Exit:
Read the last line of p. 1126 to the end of first
paragraph of p.1127.
Read the first paragraph of p. 1132 with
Figures 2.A and 2.B very carefully.
This pattern of behavior is a very typical one
for the GA agent. In what sense is it typical?
When S is large..
Lettaul (1997) provides an analysis of the
Using Monte-Carol Simulation, Lettaul (1997)
When S is small..
Since is chosen to be positive, this
tendency is more pronounced for weights
close to MAX.
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
This explains why the GA selects portfolio
weights that are too high as long as S is low.
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.
Under survival pressure, agents tend to behave
as if they are less risk-averse than otherwise.
This result should be compared to Szpiro
S, T and ST
The second contribution of Lettau is that he
actually made an economic analysis of the
optimal combination of T and S given a fixed
amount of resource to be spent in search.
But, usually S is set exogenously by the
institution, and T has the sky as the limit.
Modeling Survival Pressure with
This is a research topic open for students who
are interested in doing research on the relation
among S, T and ST.
Fundamental Failure of Conventional
Financial Modeling based on Bayesian
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 unreasonable large amount of noise
Intuitively, the reason is that in dynamic
setting, Bayes’ rule puts little weight on
recent information and relies more on the
Cognitive Features of Bayesian
Thus, any new information does not change
beliefs too much and thus does not cause
substantial portfolio updates.
In other words, Bayesians tend to have a long
window of memory, and normally the window
size is fixed during the adaptation process is
Exit and Entry
While Lettau (1997) considered the population
which allowed for exit and entry, it is not clear
how this added mechanism can help match the
empirical features of the mutual fund markets.
The most interesting part of the empirical
behavior is the asymmetric response to
positive returns and negative returns.
But, to replicate the pattern of asymmetric
response, it seems enough to have a small S,
as the author also well noticed: ``The reason
for this asymmetry is the risk-taking bias as
discussed in Section 4.2.’’ (p. 1142)
Therefore, it has nothing to do with the added
entry and exit mechanism.
While S plays the major role in Section 3, the
author did not carefully make this distinction
by only saying ``Agents update their decision
rule after each market outcome, i.e, S=1.’’ (p.
My conjecture is that if the S is enlarged to
1000, there should be no discernible pattern
between the adjustment of portfolio and the
Therefore, Lettau’s experiment suggested an
empirical size of S, which seems to be small,
while not necessary one.
This study provides probably the only empirical
evidence on the length of evaluation cycle.
Since the evidence is in favor of a small S, there
is no ground for the choice of a large S in the
GA simulation as many existing studies did.
Also see LeBaron (1999) for a different and
related study on the choice of time horizon.
There is no natural economic interpretation of
entry and exit used in this paper. Dawid and
Kopel (1998) is probably the only paper
considering entry and exit as a part of decision
variable to determined by GA.