Answer: 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)
