Читать книгу Alternative Investments - Black Keith H. - Страница 33

The problems with covariance estimation include the potentially large scale of the inputs required. This is typically not an issue when working at the asset class level, at which the investor may consider 10 asset classes for inclusion in the portfolio. However, optimizing an equity portfolio selected from a universe of 500 stocks has very large data requirements. A 500-asset optimization problem requires estimates of covariance between each pair of the 500 assets. Not only does this problem require n(n − 1)/2, or 124,750 covariance estimates, but it is also difficult to be confident in these estimates, especially when there are too many to analyze individually. Also, notice that to estimate 124,750 covariance terms, we need more than 124,750 observations, or more than 10,000 years of monthly data or more than 340 years of daily data.

The problem of needing to calculate thousands of covariance estimates can be reduced with factor models. Rather than estimating the relationships between each pair of stocks in a 500-stock universe, it can be easier to estimate the relationship between each stock and a limited number of factors. While some investors simply choose to estimate the single-factor market model beta of each stock, others use multifactor models. To see how a factor model can reduce the data requirement, suppose the return on each asset can be expressed as a function of one common factor and some random noise:

(1.23)

where F is the common factor and a_i, b_i are the estimated parameters. It is assumed that for two different assets, the error terms ϵi and ϵj are uncorrelated with each other. Under this assumption, the covariance between two assets is given by:

(1.24)

This means that to estimate the covariance matrix of 500 assets, we need 500 estimates of b_i and one estimate of Var[F].