By Edwin Burmeister; Richard Roll; Stephen A. Ross; Edwin J. Elton; Martin J. Gruber; Richard Grinold and Ronald N. Kahn
This monograph offers the paintings of 3 teams of specialists addressing using single-factor types to provide an explanation for safety returns: Edwin Burmeister, Richard Roll, and Stephen Ross clarify the fundamentals of Arbitrage Pricing thought and speak about the macroeconomic forces which are the underlying assets of threat; Edwin J. Elton and Martin J. Gruber current multi-index types and supply tips on their reliability and value; and Richard C. Grinold and Ronald N. Kahn deal with multiple-factor types for portfolio threat.
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In addition, the structure should be parsimonious; that is, returns can be described in terms of a limited number of indexes. Finally, having the indexes represent separate influences would be desirable. Two statistical techniques accomplish these goals: factor analysis and The most common technique is factor analysis. ~ Factor analysis was devised to define a set of indexes mathematically so that the covariance between security returns is minimized after the indexes have been removed. This assures that cov(e,ej) is as close to zero as it can be.
For any hypothesized number of factors, factor analysis finds the indexes and the loadings on each index for each security to make the covariance between the unique returns as small as possible. Although the indexes produced by factor analysis need not be orthogonal to l5 In addition, researchers often want to construct the indexes from a larger sample of stocks or to have other properties such as being widely diversified. See, for example, Lehrnann and Modest (1988). l6 For example, in a single-index model, beta has been shown to be positively related to residual variance.
Additional principal components are then extracted until the user decides that they are picking up random influences in the data rather than real information. Of course, a prior estimate of the number of relevant influences will narrow the choice of how many principal components to extract. To obtain a multi-index model from a principal component solution some adjustments are usually performed. First, the indexes obtained from a principal components solution have decreasing standard deviations as additional influ- l4 Many researchers choose to perform principal component or factor analysis on the correlation matrix rather than the variance-covariance matrix.