REVISTA INVESTIGACIÓN OPERACIONAL Vol. 27, No. 2, 129-142, 2006 REGRESSION EQUATION FITTING AS AN APPROACH TO MODELLING FINANCIAL DATA Carlos N. Bouza Herrera and Sira Allende Alonso, Universidad de La Habana Daniel C. Chen, Smith and King College Josefina Martinez Barbeito, Universidade La Coruña ABSTRACT Many financial models deal with concepts that are linked to regression. The unpopularity of its use in application is due to the fact that the residuals distribution is not normal. A key example is that of the study of the risk-adjusted return of the portfolio. The general equation is regarded as r - Rf = α + β ( Km - Rf ) (1) Where r is the fund's return rate, Rf is the risk-free return rate, and Km is the return of the index. This can be regarded as the usual equation for CAPM excepting the existence of α. β is the ´beta´ derived from the classic Sharpe’s representation of equilibrium prices. When fitting a regression we include an error term ε and α represents how much better the fund did than the predicted CAPM. We revise this problem considering that the residuals are distributed according to Stable distributions, not necessarily a normal. Some related financial problems are considered in a similar fashion. Monte Carlo experiments are developed for comparing different methods for estimating the so-called beta-coefficients. Key words: CAPM, robust regression, beta, Monte Carlo experiments, outliers. MSC: 62P05. RESUMEN Muchos modelos financieros se ocupan de conceptos asociados a la regresión. La impopularidad de su uso en las aplicaciones se debe al hecho de que los residuos tienen una distribución no normal. Un ejemplo clave es el del estudio del retorno de un portafolio ajustado al riesgo. La ecuación general es planteada como: r - Rf = α + β ( Km - Rf ) (1) Donde r es la tasa de retorno del fondo, Rf es la tasa de retorno libre de riesgo, y Km es el índice de retorno. Esto puede considerarse es la usual ecuación para el CAPM exceptuando la existencia de α. β es el ´beta´ derivado de la representación clásica de Sharpe para los precios de equilibrio. Al ajustar la regresión incluimos un termino de error ε y α representa cuan mejor está el fondo que el predicho por el CAPM. Revisamos este problema considerando que los residuos se distribuyen de acuerdo a una distribución estable , no necesariamente una normal. Algunos problemas financieros relacionados son considerados en forma similar. Experimentos de Monte Carlo se desarrollan para comparar diferentes métodos para estimar los llamados beta-coeficientes. 1. INTRODUCTION It is evident that financial data is generally generated by a heavy-tailed distribution or by a mixture of distribution functions. Give a look to the graphs of common series of equity returns data and you will note the no normality of the series. That equity plays a crucial role in finance is well known, see Carleton (1985) for example, and its behaviour is typical for financial data. Portfolio theory gives models for analysing the risk present in stocks by incorporating it into an assessment of securities. It is based in the following assumptions 1. Investors are risk averse (i.e. less risk is preferred to more) 2. Investors seek to maximise their wealth (i.e. more wealth is preferred to less) 3. The wealth maximisation is represented by an expected rate of return 4. The expected rate of return can be first estimated using past rates of return. There are two types of risk: unsystematic and systematic. The unsystematic risk can be eliminated by the diversification the portfolio. The systematic risk is undiversificable. For any particular security, the unsystematic component is greater than the systematic one. But, when a security is combined with other securities, the unsystematic risk of each is offset, as long as the securities are not correlated by the other securities in the 129