The Mathematics of Financial Modelingand Investment Management

19-EquityPort Page 554 Friday, March 12, 2004 12:40 PM

554 The Mathematics of Financial Modeling and Investment Management

■ Step 3. Obtain the difference between the values found in Step 1 and

Step 2. The difference is referred to as the active return.

■ Step 4. Compute the standard deviation of the active returns. The

resulting value is the tracking error.

The tracking error measurement is in terms of the observation

period. So, if monthly returns are used, the tracking error is a monthly

tracking error. Typically, tracking error is computed using either weekly

or monthly data. Tracking error is annualized as follows:

Annual tracking error = Monthly tracking error × f

where f is 12 for monthly observations or 52 for weekly observations.

A portfolio created to match the benchmark index (i.e., an index

fund) that regularly has zero active returns (that is, always matches its

benchmark’s actual return) would have a tracking error of zero. But a

portfolio that is actively managed that takes positions substantially dif-

ferent from the benchmark would likely have large active returns, both

positive and negative, and thus would have an annual tracking error of,

say, 5% to 10%. A hybrid portfolio (e.g., enhanced index fund) that

combines an index portfolio with an active portfolio would typically

have a tracking error below 2%.

An enhanced index portfolio’s is simply a combination of an

indexed portfolio and an active portfolio. That is, the tracking error of

the enhanced index portfolio is simply the tracking error of the active

portion times its weight in the overall portfolio. For example, if the

active portion constitutes 10% of the enhanced index fund (the other

90% being indexed), and the tracking error of the active portion is 5%,

then the tracking error of the enhanced index fund is 0.5% (= 10% ×

5%). To see this, let r = return, w = weight, σ^2 (.) = variance, and ρ(.,.) =

correlation. Using the following notation for subscripts, b = benchmark,

I = indexed portfolio, a = active portfolio, p = enhanced index portfolio

(a combination of the indexed portfolio and the active portfolio), then

rp = wi ri + wara

since wi + wa = 1,

rp – rb = wi (ri – rb) + wa (ra – rb)

So, the tracking error variance of the enhanced index portfolio equals