Adaptive Portfolio Managers in Stock Market: An Approach Using Genetic Algorithms

1
Adaptive Portfolio Managers in Stock Market An
Approach Using Genetic Algorithms

• K.Y. Szeto

2

• Introduction
• Data Preprocessing
• Results of GA as a Forecasting Tool
• Portfolio Management
• Summary

3
Introduction

• Complex System
• The Analysis of Stock Market
• Modify the Traditional Economic Models
• Model the Individual Investor
• Forecast Financial Time Series

4
Complex System

• Stock market is an ideal complex system for
  investigations by financial analysts.
• The laws underlying the dynamics are not even
  proven to exist.
• Even if the underlying laws of economics trends
  are known, there is no way to predict the elusive
  human behavior.
• As the complex system evolves, the underlying
  laws should also evolve along, albeit at a
  slower time scale.

5
The Analysis of Stock Market

• Computer Programs
• A given set of investment rules, extracted from
  historical data of the market.
• Rules based on fundamental analysis or news
  obtained from the inner circle of the trade.
• Statistical Results

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Modify the Traditional Economic Models

• To incorporate a certain level of communication
  among traders.
• (human interactions)
• Mean Field Theory
• Heterogeneous agents
• (psychological response)
• different rules of investment
• different human characters

7
Mean Field Theory

• Each trader will interact with the average trader
  of the market, who is representative of the
  general atmosphere of investment at the time.

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General Atmosphere

• Model the general atmosphere of the market
• We do not deduce it from a model of microscopic
  interaction between agents, but rather by a
  source of random news that serves as a kind of
  external, uncontrollable stimulus to the market.
• Quantitative Parameters to measure individual
  characteristics of the trader, so that the
  response to the general atmosphere of the market
  is activated according to these parameters.

9
The Individual Investor

• A rule of investment
• Supplement the agent with specific value and
  character, representing the human psychology of a
  particular subset of investors.
• Endow him/her with a particular skill of
  technical analysis.

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Forecast Financial Time Series

• Max (The rate of correct prediction)
• s.t. (A set of constrains)
• set by past patterns
• Prediction is transformed into a problem of
  pattern recognition.
• The data can be preprocessed using standard
  signal processing techniques.

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Data Preprocessing

• The signal processing techniques used here is
  mainly for the purpose of noise reduction and not
  for prediction, and no attempt is made on the
  theory behind the trend.
• Transform a time series of rational numbers into
  one of alphabets or integers.
• Divide the data with N points into two parts.

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Transform a Time Series

• Use a vector quantization technique to encode the
  time series as a sequence of integers
  corresponding to q classes.
• For a given q, the original data set of N points
  will be divided into q sets, with N1, ,Nq
  members respectively.
• For q2
• Large fluctuation as class 1
• Small fluctuation as class 0
• For q2m1

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Fluctuation

• The fluctuation of the input X'(t) is computed
  as the fractional change in each interval. (The
  daily rate of return)

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• For q2m1, one can put down 2m boundary values
  y-m, , y0, ym for q levels of fluctuations,
  such that
• y(t)? y0 , X(t)0 ??
• y0? y(t) ? y1 , X(t)1
• y-1? y(t) ? y0 , X(t) -1
• we convert a time series into an integer sequence
  of data X(t) defined on the alphabets
  A?-m,,0,,m
• Rename the q classes with the alphabets
• A?0, 1, , q-1

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The Choice of the Boundary Values

• Maximize the signal to noise ratio.
• Set N1N2 Nq
• This criterion imposes a strict constraint on the
  boundary values, but the results will ensure a
  more precise comparison on the performance of the
  prediction tools on each class.
• Adopt those normally used by traders on the daily
  rate of return to achieve immediate application.

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Divide the Data

• Training SetM-L strings of length equal to L
  digits, along with the known associated action
  unit of each string.
• Test Set(N-M-L) sets of strings of length L, but
  the action unit should be used for performance
  evaluation.
• The choice of L is important and one method is to
  use information entropy.

17
Results of GA as a Forecasting Tool

• Generation of Time Series
• Forecasting of Artificial Time Series
• Forecasting of Real Financial Time Series
• Self Organizing Behavior in GA

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Generation of Time Series

• Inverse Whitening Transformation
• ?diagonalizable nxn matrix
• ?diag?1,,?n
• eigenvector matrix ??1,,?n
• ????
• Using T ?1/2 ?T, the correlation matrix of YTX
  can be shown to be the identity matrix if the
  covariance matrix of the random variable X is ?.

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• Our first step is to generate an independent,
  normally distributed random variable Y with zero
  mean and unit variance.
• Then we define the covariance matrix ?, which is
  a Toeplitz matrix with entries CijCjiC(i-j).
• It is a real symmetric matrix with unit diagonal
  and the function C(i-j) is related to the
  correlation function with given memory structure.

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• If the required time series has a short memory,
  one assumes an exponentially decaying function
  for these elements CnC(n)exp(-n/?).
• ? is the range of correlation
• n is the number of days in the past
• The final result is a random variable xT-1Y
  with correlation given by the covariance matrix ?.

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Forecasting of Artificial Time Series

• Three sets of short memory time series with 2000
  data points are produced, with correlation
  functions C(n) with ?5, 10, 15.
• Training Setfirst 1000 data points
• cutoff values are used to put data points into
  five categories
• Test Setnext 1000 data points

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• We want to maximize the correct percentage as
  well as the guessing percentage.
• Specific rulesincrease the correct percentage
• General rulesincrease the guessing percentage

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• In general the ratio of correct guess/total
  number of guess on the test set is around 50 to
  60.
• For benchmark comparison with random guess, it
  gets a maximum of 20 since there are 5 equally
  likely classes.
• Another benchmark is random walk. It uses the
  value of previous time unit to predict the
  present unit. This method gives 25 of correct
  predictions. (See Table 1).

???
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Table 1Results of Prediction

• genetic algorithm performs best, especially in
  time series with long memory (larger time
  constant ?).

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Forecasting of Real Financial Time Series

• Experiment Setting
• Two Benchmark Tests
• Stock Price (Hang Seng Index)
• Performance of Genetic Optimizer
• The Performance of the Predictor
• Probability of Correct Prediction

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Experiment Setting

• Training set1712 data points
• Test set190 data points
• Each experiment is started with different seed of
  random numbers.
• 2000 generations
• Repeated 15 times

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Two Benchmark Tests

• The first one is the random guess of two classes
  stock prices go up and down with equal
  probability.
• The second one is the random guess of two classes
  based on past statistics, in which case the
  probability of choosing 1 is 0.5144 and choosing
  0 is 0.4856.

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Stock Price (Hang Seng Index)
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Performance of Self-Organized Genetic Optimizer
662.6942.3lt1712sometimes no guess is made
C0the correct guess minus the wrong guess
G0the total number of guess made
C1? G1?
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The Performance of the Predictor

• The average number of correct guess minus the
  wrong guess, which is the sum of C0 and C1.
• Ideal ResultC0110, C180, Sum190
• Worst CaseC0-80, C1-110, Sum-190

??
??
Genetic Optimizer C0C18.2
Random Guess with Equal Prob C0C1 - 4.1
Random Guess Using Prob(1)0.5144, Prob(0)0.4856 C0C1 -5.6
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Probability of Correct Prediction

• Pktest and Pktrain are the probability of
  correctly predicting the class k for the testing
  set and training set.
• P0test0.59, P1test0.45, P0train0.60,
  P1train0.59
• The sum of probability of making a correct guess
  for both classes in the test set is
  0.590.451.03, greater than one, an upper limit
  of any scheme of random guess.

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Self Organizing Behavior in GA

• A more important observation is the
  self-organizing behavior of the genetic optimizer
  without the Lagrange multipliers.

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• Suppose that the penalty of having too many or
  too few don't care bits compared to a chosen
  frequency (for example, 0.3) in the rules is not
  controlled by a Lagrange multiplier ?.
• The exponent ? in the fitness controls the
  penalty, so that the fitness measure is modified
  by a factor of

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• Similarly, assume that the penalty of having a
  guessing frequency very different from the
  frequency of occurrence in the training set is
  not controlled by a Lagrange multiplier ?.
• This exponent ? is to modify the fitness measure
  by a factor of

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• Then, these two factors will modify the
  expression of fitness by

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• Hk is independent of the rule index i so that it
  is a positive constant for the population of
  rules.
• A new variable
• If 1? is positive, then fik is a monotonic
  function of .
• Thus we can forget about Hk and use for our new
  fitness.

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Portfolio Management

• The level of confidence reflects the probability
  of change of the original strategy of investment,
  which is based on hard work on past data.
• The degree of greed reflects the relative portion
  of each asset involved in each transaction, which
  definitely affects the final outcome of the
  investment.
• Response to News and Level of Confidence
• Level of Greed
• Portfolio Management in the Presence of News

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Response to News

• News
• A randomly generated time series
• Take some kind of average of many real series of
  news.
• An internally generated series that reflect the
  dynamics of interacting agents
• For an agent who had originally forecasted a drop
  in the stock price tomorrow and planned to sell
  the stock at today's price, may change his plan
  after the arrival of the good news, and halt
  his selling decision or even convert selling into
  buying.

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Four Scenarios
re-evaluate

• News is good and he plans to sell.
• News is good and he plans to buy.
• News is bad and he plans to sell.
• News is bad and he plans to buy.

re-evaluate
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Level of Confidence

• fthe level of fear
• 1-fthe level of confidence
• If f is 0.9, then the agent has 90 chance of
  changing his decision when news arrives that
  contradicts his original decision.

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• Choose a random number p.
• If p gt f, he will maintain his prediction,
  otherwise he reverses his prediction from 1 to 0
  or from 0 to 1.
• The bigger the value of f, the smaller the chance
  the random number p will be greater than f,
  implying that the smaller the chance he will
  maintain his original prediction.

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Level of Greed

• For a greedy investor, he may be very aggressive
  in all his investment, while a prudent investor
  will be more conservative in his action.
• gcharacterize the percentage of asset allocation
  in following a decision to buy or sell.
• If g is 0.9, it means that the agent will invest
  90 of his asset in trading.
• g can be interpreted as a measure of greed.

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Portfolio Management in the Presence of News

• Training set
• 800 points
• Extract a rule using standard GAs.
• Test set
• 100 points
• Evaluate the performance of the set of rules
  obtained after training.
• News set
• 1100 points
• Investigate the performance of investors with
  different degree of greed and confidence.

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• x(t)the daily rate of return of a chosen stock
• x(t) is a function of the value at x(t-1),
  x(t-2),..., x(t-k). Here k is set to 8.
• Min MSE (to find a set of ?i)
• x(t)?1, ?i?1, i1,,k

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• gt0, the agent predicts an increase of the
  value of the stock.
• ?0, the agent predicts either an unchanged
  stock price or a decrease.
• Count the guess as a correct one if the sign of
  the guess value is the same as the actual value,
  otherwise the guess is wrong.
• If the actual value is zero, it is not counted.

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• Performance index Pc
• PcNc/(NcNw)
• Nc is the number of correct guess
• Nw is the number of wrong guess
• While most investors make hard decision on buy
  and sell, the amount of asset involved can be a
  soft decision.

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Fitness
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• The final set of agents, all with the same
  chromosome (or rule), but with different
  parameters of greed g and fear f.
• Initial Asset
• Cash10,000 USD
• Shares100 (at 99 a share)
• The value of f and g ranged from 0 to 0.96 in
  increment of 0.04 will be used to define a set of
  25x25625 different agents.

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Final Net Asset Values in Cash
Greedy and confident investors perform better.
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Summary

• We construct a learning classifier system based
  on genetic algorithm.
• Transform the problem of forecasting time series
  into a pattern recognition problem.
• It performs better than both the random guess and
  random walk method on artificial data as well as
  real data.
• This is superior to the use of Lagrange method
  for implementing constraints in the statistical
  properties of the rules.

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• The problem of interacting adaptive portfolio
  managers.
• Heterogeneous agent
• With different set of human characters (level of
  confidence and greed).
• The agent maybe change the technical prediction
  when confronted with breaking news.

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Discussion

• Use of the idea mean-field to agent-based
  artificial stock markets. This is sort of a
  bridge to connect two different approaches to
  large interacting heterogeneous agents in
  artificial markets.