The Theory and Practice of Investment Management: Asset Allocation, Valuation, Portfolio Construction, and Strategies, Second Edition

RISK DECOMPOSITION

The real usefulness of a linear multifactor model lies in the ease with which the risk of a portfolio with several assets can be estimated. Consider a portfolio with 100 assets. Risk is commonly defined as the variance of the portfolio’s returns. So, in this case, we need to find the variance–covariance matrix of the 100 assets. That would require us to estimate 100 variances (one for each of the 100 assets) and 4,950 covariances among the 100 assets. That is, in all we need to estimate 5,050 values, a very difficult undertaking. Suppose, instead, that we use a three-factor model to estimate risk. Then, we need to estimate (1) the three factor loadings for each of the 100 assets (i.e., 300 values), (2) the six values of the factor variance–covariance matrix, and (3) the 100 residual variances (one for each asset). That is, we need to estimate only 406 values in all. This represents a nearly 90% reduction from having to estimate 5,050 values, a huge improvement. Thus, with well-chosen factors, we can substantially reduce the work involved in estimating a portfolio’s risk.

Multifactor risk models allow a manager and a client to decompose risk in order to assess the exposure of a portfolio to the risk factors and to assess the potential performance of a portfolio relative to a benchmark. This is the portfolio construction and risk control application of the model. Also, the actual performance of a portfolio relative to a benchmark can be assessed. This is the ...