Genetic Control Applied to Asset Managements

This paper addresses the problem of investment optimization using genetic control. Time series for stock values are obtained from data available on the www and asset prices are predicted using adaptive algorithms. A portfolio is optimized with the genetic algorithm based on a recursive model of portfolio composition obtained on-the-fly using genetic programming. These two steps are integrated into an automatic system the final result is a real-time system for updating portfolio composition for each asset. Introduction. IBM and other companies are undertaking massive studies on the application of advanced computing technologies to stock brokerage and obtaining better results than the New York stock market’s sharpest traders [1]. Every investor knows that there is a trade off between risk and reward: to obtain a greater than expected return on investment one must be willing to take on a greater risk [2]. Portfolio optimization theory assumes that for a given level of risk, investors prefer higher returns to lower returns. Similarly, for a given level of expected return, investors prefer less risk to more risk. It is standard to measure risk in terms of the variance, or standard deviation, of return. The portfolio optimisation problem consists of obtaining the biggest return on investment with the least risk exposure necessary under the prevailing market dynamics. These dynamics are unpredictable due to both exogenous factors (such as government actions, market rumors, unexpected events, etc.) and endogenous factors (such as company and stock fundamentals). Mahfoud and Mani [17] developed a rule-based system for managing each individual asset where the rules are of the form: IF (price < limit and EPS > value) THEN buy, where price is the asset price, limit is the buy threshold, and EPS is the earning per share. Chang et al [18] used the genetic algorithm to find the portfolio of assets with different risk exposures (called the efficient frontier). Usually, this problem is solved with quadratic programming, but, for practical purposes it is desirable to limit the number of assets in a portfolio, as well as the proportion of the portfolio devoted to any particular asset. Kivine and Warmouth [19] proposed that the portfolio vector itself encapsulate the necessary information from the previous price relatives. Thus, at the start of day t, the algorithm computes its new portfolio vector w as a function of w and the just observed price relatives x, using a linear regression. Helmbold et al [14] select a more complex function and Parkes and Huberman [16] generalize the idea for investment group model for the portfolio selection problem, adjusting their portfolio as they observe movements of the market over time and communicate to each other their current portfolio and its recent performance. Investors can choose to switch to any portfolio performing better than their own. In this work the goal is to obtain a recursive mathematical law using genetic programming and the genetic algorithm that establishes a relation between the available information and the percentage of various assets to be held in the portfolio. The general framework, genetic control Werner [4], is represented in fig. 1. It uses data from experimental setup (the simulated market in this case) to feed genetic programming for the purpose of building a model of the market. Later, the genetic algorithm adapts real values to obtain the optimal percentages of the assets in the portfolio that will feed genetic programming, closing the loop. Fig. 1. Genetic control: obtaining the structure of the solution with genetic programming and adapting its parameters with genetic algorithm. The genetic algorithm. The genetic algorithm (GA) mimic the evolution and improvement of life through reproduction, where each individual contributes its own genetic information to the building of new ones adapted to the environment with higher chances of survival. This is the basis of genetic algorithms and genetic programming ([5], [6], and [7]). Specialized Markov Chains underline the theoretical bases of this algorithms change of states and searching procedures. Each ‘individual’ of a generation represents a feasible solution to the problem, coding distinct algorithms/parameters to be evaluated by a fitness function. GA operators are mutation (the change of a randomly chosen bit of the chromosome) and crossover (the exchange of randomly chosen slices of the chromosome). The best individuals are continuously being selected, and crossover and mutation take place. Following a number of generations (Fig. 2), the population converges to the solution that performs better. Fig. 2. Genetic algorithm: the sequence of operators and evaluation of each individual. A generalization of the Genetic Algorithm is Genetic Programming (GP) (Holland [5] and Goldberg [6]) where each ‘individual’ in a generation represents, with its chromosome, a feasible solution to the problem; in our case, a mathematical function to be evaluated by a fitness function. There are two kinds of information defined for the GP algorithm: terminals (variable values and random numbers) and functions (mathematical functions used in the generated model). The virtual market. The first problem when studying market investment is in what environment to test the concepts and how to obtain time series of assets, stocks and currency, and market index for a period of 2 years at least. To solve this problem we extracted time series values from the history graphics available in the Yahoo Finance site [3]. We built a database contains the following information [9] between July 1999 and July 2001: FTSE100 stock quotes, trade volume and quotes by sector; Fix interest; European and American indices; Stock exchanges indices around the world: Argentina, Brazil, Canada, Chile, Peru, Venezuela, Australia, China, Hong Kong, India, Indonesia, Malaysia, New Zealand, Pakistan, Philippines, Singapura, South Korea, Sri Lanka, Thailand, Taiwan, Austria, Tchec Republic, Finland, Greece, Nederland, Portugal, Russia, Slovakia, Spain, Swiss, Turkey, Egypt, Israel; Commodities: Gold, silver, paladio; Currency convergence to pound: USD, Australia, Canada, Argentina, Brazil, Euro, Franca, Germany, Hong Kong, Japan, Mexico, Russia and Swiss. The benchmark for portfolio return. The reference for return evaluation is the stock exchange index. In the case of London this is the FTSE 100 [10]. The investment operation would build a portfolio with reflects the same constitution as the FTSE 100 index, and its performance is the same as the market. Asset forecast. To develop a forecast of assets price (or any other time series value) there are two necessaries definitions: the mathematical function to be adjusted (termed filter) and the adaptation algorithm (responsible for calculate the parameter values of the filter following temporal changes of the series). The FIR (finite impulse response) filter of N dimension is a filter with trivial poles (z=0) in its transference function: W(z) = w0 + w1.z -1 + w2.z -2 + .... + wn-1.z -N+1 (1) Let W the filter coefficient vector, and Xk last N inputs to the filter in k instant: W = [ w0 w1 w2 w3 .... wN-1] T Xk = [x(k) x(k-1) x(k-2) x(k-2) .... x(k-N+1)] T (2) Filter output is defined as: W X ) i k ( x w ) k ( y k 1 N 0 i i ⋅ = − ⋅ = ∑ − = (3) Any stock/asset contains in its price two components: one depending of its fundamentals and other completely random, modeled by the Brownian model. The intrinsic component would be adapted by Least Means Square LMS (see [11]), which cancels the random component. Let us define the performance function ξ = ξ(W) (4) a quadratic function with one minimum point. For any initial condition W, evaluate new values of W into the contrary direction of the performance hyper surface gradient (with indicate the maximum direction). Following the contrary direction, certainly will hit the minimum. To evaluate the gradient of ξ is necessary do some approximations. Let ξ = E[e(k)] ≅ e(k) (5) Where E is the average of stochastic variable e, the error between the real value d and the adapted by fitting y. Then: