whereepandemare the factor exposures of the portfolio and the market,
andhpandhmare their respective holdings vectors. We are thus able to eval-
uate the optimal hedge ratio.
Application: Tracking Basket Design
A tracking basket is a basket of stocks that tracks an index. If the basket is
the same as the index, then the prices of both will be the same at all times.
However, if the tracking basket is composed of fewer stocks than the index,
then there is likely to be tracking error; that is, the returns for the tracking
basket are not exactly the same as the returns for the index. The discrepancy
in the returns is expressed in terms of tracking error.
Tracking error may be defined as the standard deviation of the differ-
ence in the return between the tracking basket and the index. In the defini-
tion of tracking error, the mean value of the difference is assumed to be zero.
To see this more clearly, consider a long–short portfolio where we are long
the index and short the tracking basket. The expectation is that the return on
the index and tracking basket is the same. Therefore, a profit made on one
leg of the portfolio is likely to be neutralized by an equal loss on the
other leg. The expected value of total return on the portfolio is therefore
zero. However, while the expected return is zero, it is possible for the actual
return value to be a nonzero value. The extent of this variation from zero is
captured by the standard deviation of returns of the long–short portfolio
and forms a measure of the tracking error.
A natural deduction from the preceding discussion is that the design of
a tracking basket involves designing a portfolio such that it minimizes the
tracking error. Writing out the equations for the variance of the error in re-
turns, we have
(3.26)
whererpis the return on the tracking basket and rmis the return on the mar-
ket. Expanding the terms using APT constructs, we have
(3.27)
Bear in mind that there are some constraints on the values of hp; that is,
some of them are forced to have zero values even though they are part of the
index.
−+ (^2) [h XVX hm T pT hmp∆h]
Min:h XVX hm T mT++hmm∆hT h XVX hp T pT+−h hppT
Min:Er()mp−r = ()rm+ ()rp− ()r rmp
2
var var 2 cov
Factor Models 49