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By Edwin Burmeister; Richard Roll; Stephen A. Ross; Edwin J. Elton; Martin J. Gruber; Richard Grinold and Ronald N. Kahn

This monograph provides the paintings of 3 teams of specialists addressing using single-factor versions to provide an explanation for safeguard returns: Edwin Burmeister, Richard Roll, and Stephen Ross clarify the fundamentals of Arbitrage Pricing concept and talk about the macroeconomic forces which are the underlying assets of possibility; Edwin J. Elton and Martin J. Gruber current multi-index types and supply counsel on their reliability and value; and Richard C. Grinold and Ronald N. Kahn deal with multiple-factor types for portfolio danger.

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Principal Component Analysis Principal component analysis is perhaps easier to understand than factor analysis. For returns on a defined set of stocks over a predetermined time span, compute an index (weighted average of that set of returns) that explains the maximum amount of the variation in the variance-covariance matrix of security returns. l4 Usually, the first principal component looks somewhat like a market index with all stocks entering with positive weights. Then, search for the index, constrained to be orthogonal (uncorrelated with the first index), that explains as much as possible of the unexplained portion of the variance-covariance matrix.

The returns are positi;ely correlated between all pairs of countries, and the highest correlation is between the two North American countries and the two European countries. Thus, two indexes will be needed: a general world index and an index that reflects whether the country is North American or European. TABLE 1. 03 We ran a principal components analysis on the variance-covariance matrix and obtained two explanatory factors. The return on one factor was given by where R = return, f = factor value, B = Belgium, C = Canada, U = United States, F = France, and t = time period.

In addition, many of the problems with the multi-index model, and the economic intuitions behind the solution to those problems, can be illustrated for the case of a single-index model. The single-index model is simply a way of decomposing return on an asset into two parts. The first part, the systematic part, is the portion of return affected by influences common to all assets. The second part is the unsystematic part, which is assumed to be unique to the asset. Thus, A Practitioner's Guide to Factor Models Asset where Rit = is the return on security i in time t, bil = is the sensitivity of security i to returns on Index 1, f1, = is the return on Index 1 in period t, ai = is the expected level of nonindex-related return for security i, and eit = is a random variable with mean of zero and variance 0:;; Note that the unique return is also split into two parts: its mean level, a , and its variability, ei, For the single-index model to be a reasonable description of reality, the unique part should be truly unique to the security in question and not related to another influence.

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