The Mathematics of Financial Modelingand Investment Management

19-EquityPort Page 555 Friday, March 12, 2004 12:40 PM

Equity Portfolio Management 555

σ^2 (rp – rb) = σ^2 {wi(ri – rb)} + σ^2 {wa(ra – rb)}

+ 2 wiwaρ(ri – rb, ra – rb)σ(ri – rb)σ(ra – rb)

But, the variance and the standard deviation of the indexed portfo-

lio relative to the benchmark would be zero. So, the first and the last

terms in the above equation vanish, leaving.

σ^2 (rp – rb) = σ^2 {wa (ra – rb)}

Taking the square root on both sides, we have

σ(rp – rb) = σ(wa(ra – rb))

Since, σ(wa(ra – rb)) is the tracking error of the active portion, the track-

ing error of the enhanced index portfolio is the product of weight of the

active portfolio and the tracking error of the active portfolio.

Backward-Looking versus Forward-Looking Tracking Error

We have just described how to calculate tracking error based on the

actual active returns observed for a portfolio. Calculations computed

for a portfolio based on a portfolio’s actual active returns reflect the

portfolio manager’s decisions during the observation period with

respect to the factors that affect tracking error. We call tracking error

calculated from observed active returns for a portfolio backward-looking

tracking error, ex post tracking error, or actual tracking error.

A problem with using backward-looking tracking error in portfolio

management is that it does not reflect the effect of current decisions by

the portfolio manager on the future active returns and hence the future

tracking error that may be realized. If, for example, the manager signifi-

cantly changes the portfolio’s exposure to risk factors during the obser-

vation period, then the backward-looking tracking error, which is

calculated using data from prior periods would not accurately reflect the

current portfolio risks going forward. That is, the backward-looking

tracking error will have little predictive value and can be misleading

regarding portfolio risks going forward.

The portfolio manager needs a forward-looking estimate of tracking

error to reflect the portfolio risk going forward. The way this is done in

practice is by constructing a multifactor risk model using as the market

index the portfolio manager’s benchmark. Given a manager’s current

portfolio holdings, the portfolio’s current exposure to the various risk

factors can be calculated and compared to the benchmark’s exposures to