to do exactly that. Consider a portfolio with long one share of A and short
gshares of B. The return of the portfolio for a given time period is given as
[]log(pp ppAti++)−log(tA) −−γ[]log(tiB ) log( )tB. (5.5)Overview 81
COINTEGRATION AND TRACKING ERROR
Tracking error is an idea that arises naturally in the context of an in-
vestment technique called indexing. The basic premise for indexing is
the notion that it is extremely hard to time the market. Therefore, the
strategy adopted is to passively invest in the market index. Alternately,
to reduce trading costs, the investment is made in portfolios designed
to mimic index returns. These portfolios are sometimes referred to as
tracking baskets. The ability of the tracking basket to mimic the mar-
ket returns is characterized by its tracking error. The tracking error
may therefore be thought of as a measure of discrepancy or margin of
error that one can expect in the tracking process.
Implicit in the definition of tracking error is the time interval for
which the returns are measured. It is typically one year (assumed to
have 252 trading days). If the holding period is different from one
year, then the tracking error needs to be scaled accordingly. The con-
vention is to assume that the tracking error is a random walk series
and scale the tracking error using the formula below. If Tis the hold-
ing period for the portfolio, then(5.6)
The tricky part here is in the random walk assumption for tracking
error in the process of scaling. In the case of long–short portfolios con-
sisting of cointegrated stocks, the tracking error is not a random walk
series. As a matter of fact, it is a stationary time series. The variance or
standard deviation in this case is independent of the holding period. In
other words, the tracking error remains the same regardless of the
holding period, and scaling formulas are not required.
We therefore caution against making the random walk assump-
tion blindly, particularly in indexing situations, because they tend to
distort the tracking error.err
err(1 year)
()TT=
252