Note that in Equation 3.27, the variance of the market portfolio is not
at all affected by changing the composition of the tracking basket. The
only two terms that are affected when we change the tracking basket com-
position is the tracking basket variance and the covariance term. Therefore,
minimizing the tracking error is equivalent to minimizing the sum of the
two terms.
The error variance may also be viewed as a sum of two components;
namely, a common factor component and a specific component. Now, if we
were to design the tracking basket such that the factor exposures of the bas-
ket match the factor exposures of the index exactly (even though their con-
tents may not be identical), then the common factor component goes to
zero. We are now left only with the specific components of the variance. Re-
call from our earlier discussion that the contributions to the total variance
from the specific components are a lot less than they would be from the com-
mon factor component. Furthermore, if the two portfolios are highly diver-
sified, the expected value of the specific returns on the portfolios is zero.
Hence, the tracking error contribution in this case is solely due to the dif-
ferent specific returns in the portfolios. It is now easy to appreciate that a
good starting point to the design of tracking baskets will ensure that the fac-
tor exposures match as closely as possible. Hence, APT constructs may be
used to design tracking baskets.
Sensitivity Analysis
In our discussion on the covariance matrix we talked about the mining syn-
drome, the idea being that the estimation of the covariance matrix is biased
to the past and may not hold going forward into the future. This apprehen-
sion may be objectively examined by studying the stability of the covariance
matrix.
One approach would be to estimate a sequence of covariance matrices
and study the variations between two consecutive ones. The extent to which
the variations affect the particular situation—say, the evaluation of beta, the
measurement of risk, or the design of tracking baskets—may be gauged by
perturbing the current covariance matrix with the sequence of observed
changes and running the calculations with the perturbed matrix.
This results in a set of values for the estimated parameter. We can then
treat the set of values as realizations from a probability distribution and get
an idea of the error in our estimates using the current covariance matrix to
help us quantify the extent of uncertainty due to the mining syndrome.
50 BACKGROUND MATERIAL