# Portfolio analysis statistical considerations

Portfolio analysis is essentially a statistical technique. However, because the ‘‘true’’ population parameters for the input data (expected returns, variances, and covariances) are unobservable, sample statistics must be estimated. Thus, the efficient portfolios generated by portfolio analysis are no better than the

statistical input dataon which they are based.

Three methods are used for generating the statistical inputs:

- The parameters may be directly estimated on an asset-by-asset basis without assuming any return-generating process
- A one-factor
**return-generating process**can be used to estimate the parameters - A multiple-factor return-generating process can be assumed to exist, from which the parameters may be estimated

A key assumption of the *single-index market model* is that the covariance between the error terms of any two securities is zero, thus *the covariance of returns between any two securities arises primarily through their relationship with a common market factor*. However, if some relevant omitted variables exist, the covariance of returns between any two securities arises through their common relationships with more than a single common factor: for instance the growth rate in the index of the nation’s industrial production. That is, security returns could be assumed to be related to both a market factor and an industrial growth factor:

*The two-index model* can be constructed so that the two independent variables, r and g , are orthogonal by removing the market effect from the growth rate in the index of industrial production. In such case, their covariance is zero. Furthermore, such indexes are assumed to be uncorrelated with the error term. The orthogonalization procedure for removing each factor’s effect on the other factor can be extended to any number of indexes, and essentially consists of the substitution of each index with the residuals of its regression against the other indexes because, by assumption, such residuals are uncorrelated with the other indexes.

As with individual securities, the variance of returns from a portfolio can be decomposed

The unsystematic risk will approach zero as the number of individual securities in the portfolio, n, increases. Thus, the total risk of a well-diversified portfolio converges to its systematic risk for large n.

Efficient portfolios can be derived from any return-generating processes by minimizing a Lagrangian objective function:

where α’s elements are the intercept terms of the single-index model; β’s elements are beta coefficients for the securities:

and the matrix:

is a variance-covariance matrix of the variances for the error terms and the market. However, the covariance matrix must be a *positive definite covariance matrix* for the first-order conditions to be necessary and sufficient for a **global optimum**. Below I briefly explain why: