19-EquityPort Page 556 Friday, March 12, 2004 12:40 PM
556 The Mathematics of Financial Modeling and Investment Managementthe factors. Using the differential factor exposures and the risks of the
factors, a forward-looking tracking error for the portfolio can be com-
puted. This tracking error is also referred to as predicted tracking error
and ex ante tracking error. Given a forward-looking tracking error, a
range for the future possible portfolio active return can be calculated
assuming that the active returns are normally distributed.
It should be noted that there is no guarantee that the forward-look-
ing tracking error at the start of, say, a year would exactly match the
backward-looking tracking error calculated at the end of the year. There
are two reasons for this. The first is that as the year progresses and
changes are made to the composition of the portfolio, the forward-look-
ing tracking error estimate would change to reflect the new exposure to
risk factors. The second is that the accuracy of the forward-looking
tracking error at the beginning of the year depends on the extent of the
stability in the variances and correlations used in the statistical model to
estimate forward-looking tracking error. These problems notwithstand-
ing, the average of forward looking tracking error estimates obtained at
different times during the year will be reasonably close to the backward-
looking tracking error estimate obtained at the end of the year.
The forward-looking tracking error is a useful in risk control and
portfolio construction. The manager can immediately see the likely
effect on tracking error of any intended change in the portfolio. Thus,
scenario analysis can be performed by a portfolio manager to assess
proposed portfolio strategies and eliminate those that would result in
tracking error beyond a specified tolerance for risk. We will illustrate
the use of multifactor risk models and tracking error later in this chap-
ter and in bond portfolio management in Chapter 21.The Impact of Portfolio Size, Benchmark Volatility, and Portfolio
Beta on Tracking Error^4
There are have been several empirical studies that have investigated the
relationship between a portfolio’s variance and number of stocks. These
studies have found that between 15–20 names are needed to eliminate
most of the unsystematic risk in a portfolio. These studies focus on the
standard deviation of returns of a portfolio relative to a benchmark, not
on tracking error.
Tracking error decreases as the portfolio progressively includes
more of the stocks that are in the benchmark index. This effect is illus-
trated in Exhibit 19.2 which shows the effect of portfolio size for a large(^4) This discussion draws from Raman Vardharaj, Frank J. Fabozzi, and Frank J.
Jones, “Determinants of Tracking Errors for Equity Portfolios,” unpublished manu-
script, October 2003.