A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient portfolio using only these 60 securities. How many estimates of expected returns, variances, and covariances are needed to optimize this portfolio? If one could safely assume that stock market returns closely resemble a single-index structure, how many estimates would be needed?

In a single index model:

ri - rf = α i + β i (r M - rf) + e i

Equivalently, using excess returns:

R i = α i + β i R M + e i

The variance of the rate of return can be decomposed into the components:

The variance due to the common market factor

Bi^2stdvm^2

The variance due to firm specific unanticipated events

STDV^2 (ei)

In this model

Cov (ri, rj) = BiBjSTDV

The number of parameter estimates is:

n = 60 estimates of the mean E (ri)

n = 60 estimates of the sensitivity coefficient β i

n = 60 estimates of the firm-specific variance σ2 (ei)

1 estimate of the market mean E (rM)

1 estimate of the market variance

Therefore, in total, 182 estimates.

The single index model reduces the total number of required estimates from 1,890 to 182. In general, the number of parameter estimates is reduced from:

(n^2 + 3n / 2) to (3n+2)