A portfolio management organization analyzes 64 stocks and constructs a mean-variance efficient portfolio using only these 64 securities.
a. How many estimates of expected returns, variances, and covariances are needed to optimize this portfolio?
b. If one could safely assume that stock market returns closely resemble a single-index structure, how many estimates would be needed?

Explanation:

A) To optimize this portfolio one would need:

n = 64 estimates of means

n = 64 estimates of variances

n^2-n / 2 =

(64^2-64)/2

(4096-64)/2

4032/2

= 2016 estimates of covariances

To get estimates of expected

n^2+3n / 2

(64^2+3(64))/2

(4096+192)/2

4288/2

= 2144 estimates

B) in a single index model:

ri - rf = α i + β i (r M - rf ) + e i

using excess returns equivalently we have:

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 = 64 estimates of the mean E(ri )

n = 64 estimates of the sensitivity coefficient β i

n = 64 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, 194 estimates.

The single index model reduces the total number of required estimates from 2144 to 194.

In conclusion, the number of parameter estimates is reduced from:

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

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