Capital Asset Pricing Model Using Regression Methods for Interval-Valued Variables

It is known that the Capital Asset Pricing Model (CAPM) provides an expression which relates the expected return of an asset to its systematic risk. In a decision making problem involving financial data however, we have to take in account the uncertainty given by the imprecision and the incompleteness of the information. Uncertainty in the data may be treated by considering, rather than a single value, the interval of values in which the data may fall. The extension of the CAPM to the case in which the returns of any considered asset are interval-valued variables (IntervalCAPM) has been introduced by Gioia (2009). The methodology makes use of the interval regression method Iregr presented in Gioia and Lauro (2005), but other different methods are already present in the literature and are described in the present work. A contribution of this work is the comparison of those interval regression methods on sets of real data. The algorithms of the considered methodologies have been implemented in MATLAB and the numerical results are compared to one another highlighting the good advantages for the method Iregr. Interval regression methods showed to be useful in the application of CAPM to interval financial data. However, as shown in this work, the interval regression method Iregr is more suitable in the framework of the IntervalCAPM, with respect to the other interval regression methods presented in the present work.