5 TH INTERNATIONAL SYMPOSIUM ON ROBOTICS AND AUTOMATION 2006, SAN MIGUEL REGLA HIDALGO, M ´ EXICO, AUGUST 25-28, 2006. 1 A Prediction-Based Portfolio Optimization Model F´ abio Daros de Freitas, Member, IEEE, Alberto Ferreira De Souza, Member, IEEE, and Ailson Rosetti de Almeida Abstract— This work presents a prediction-based portfolio optimization model that uses Normal prediction errors as risk measure. A new autoregressive neural network predictor is used to predict future stock returns and its prediction errors are used as risk measure. In this predictor, the differences between the values of the series of stock returns and a specified past value are the regression variables. A large set of experiments with real data from the Brazilian stock market was employed to evaluate our portfolio optimization model, which included the examination of the normality of the errors of prediction. Our results showed that it is possible to obtain Normal prediction errors with non-Normal series of stock returns, and that our prediction-based portfolio optimization model outperforms the Markowitz portfolio selection model showing better return for the same risk. Keywords— Neural Networks, Time Series Prediction, Portfolio Optimization. I. I NTRODUCTION Statistical Modeling can be used for providing uncertainty measures of realization of investments return. These uncer- tainty measures, named risk, make Statistical Modeling suita- ble for comparing investment alternatives. This feature is cen- tral to the Modern Portfolio Theory (MPT), first introduced by Harry M. Markowitz through his celebrated model of portfo- lio 1 selection [1]. In this model, the total risk of an investment in various stocks is minimized by the optimal selection of stocks with low joint risk, which provides a mechanism of loss compensation known as Efficient Diversification. The portfolio selection process, then, consists of finding, in a large collection of stocks, the participation (i.e. individual proportion) of each stock that minimizes the portfolio’s risk at a given portfolio return, or maximizes the portfolio’s return at a given risk. The model assumes that the historical series of returns of each stock follows a Normal distribution, uses the mean of the series as a prediction of the stock future returns, the variance as a measure of the risk of the stock, and the covariance of the stocks’ returns as a measure of joint risk (the standard deviation, semi-variance and absolute deviation may also be used as a measure of risk in the model). After the Markowitz model, many other models that use its basic assumptions (that the series of returns are Normal F. D. Freitas, Secretaria da Receita Federal, Programa de P´ os Graduac ¸˜ ao em Engenharia El´ etrica – UFES, e-mail freitas@computer.org A. F. De Souza, Programa de P´ os Graduac ¸˜ ao em Inform´ atica – UFES, e-mail albertodesouza@gmail.com A. R. Almeida, Programa de P´ os Graduac ¸˜ ao em Engenharia El´ etrica – UFES, e-mail ailson@ele.ufes.br Universidade Federal do Esp´ ırito Santo (UFES), Av. Fernando Ferrari, s/n, 29075-910, Vit´ oria, ES, Brazil 1 A portfolio is a collection of stocks, bonds, or other forms of investment. In this paper a portfolio is always a collection of stocks only. and that the moments of these series can be used as measure of future return and risk) appeared [2]. In all these models, known today as classic models, the portfolio’s expected return is given by the linear combination of the expected returns of its stocks (the mean returns) and the participations of each stock in the portfolio. The portfolio risk, in turn, varies, but it is often related with the moments about the mean of the joint Normal distribution of the series of returns of its stocks. Despite the wide adoption of the classical methods of port- folio selection, it is important to mention that the distributions of the series of returns often exhibit kurtosis and skewness [3], challenging the assumption of normality of the series of returns. In addition, the realization of mean returns tends to be verified only in the long term. This has stimulated the development of predictive models based on Time Series Analysis and other non-linear methods, like artificial neural networks, as a way of supporting the needs of investment on shorter horizons. This paper presents a prediction-based portfolio optimiza- tion model that uses Normal prediction errors as risk measure. We have used a new autoregressive neural network (ARNN) predictor to predict future stock returns, and the variance of its prediction errors as risk measure. In this predictor, the differences between the values of the series of returns and a determined past value are the regression variables, instead of simply the returns, as in the traditional prediction methods. Our experimental results clearly suggest that the prediction errors are Normal (an earlier version of this verification of normality appears in [4]). Our experiments have also shown that our portfolio selection model outperforms the Markowitz model, presenting better returns with the same risk while using the same stocks in the same periods of time. II. AUTOREGRESSIVE MOVING REFERENCE NEURAL NETWORK (AR-MRNN) PREDICTORS The one-period stock return in time t, r t , is defined as the difference between the price of the stock at time t and the price at time t - 1, divided by the price at time t - 1, as shown in Eq.1. r t = P t - P t−1 P t−1 (1) where, P t and P t−1 are the stock prices at times t and t - 1, respectively. The series of N past returns of a stock, r ′ , is defined as: r ′ =(r t−N+1 ,...,r t−1 ,r t ) (2) The time-series-based prediction of a future return of the stock can be defined as the process of using r ′ for obtaining an estimate of r t+l , where l ≥ 1. The value of l directly affects the choice of prediction method. For l =1, or one-period