The Mathematics of Financial Modelingand Investment Management

19-EquityPort Page 558 Friday, March 12, 2004 12:40 PM

558 The Mathematics of Financial Modeling and Investment Management

capitalization portfolio benchmarked to the S&P 500, a mid-cap portfo-

lio benchmarked to the S&P 400, and a small cap portfolio bench-

marked to the S&P 600.^5 Notice that an optimally chosen portfolio of

just 50 stocks can track the S&P 500 within 2.3%. For mid cap and

small cap stocks, the corresponding tracking errors are 3.5% and 4.3%,

respectively. In contrast, tracking error increases as the portfolio pro-

gressively includes more stocks that are not in the benchmark. This

effect is illustrated in Exhibit 19.3. In this case, the benchmark index is

the S&P 100 and the portfolio progressively includes more and more

stocks from the S&P 500 that are not in S&P 100. The result is that the

tracking error with respect to the S&P 100 rises.

The impact of benchmark volatility is as follows. Managed portfo-

lios generally hold only a fraction of the assets in their benchmark.

Given this, a highly volatile benchmark index (as measured in terms of

standard deviation) would be harder to track closely than a generally

less volatile benchmark index.

This can be seen by using the market model:

rp = βrm + e

EXHIBIT 19.3 Tracking Error versus the Number of Nonbenchmark Stocks in the

Portfolio

Source: Exhibit 7.3 in Raman Vardharaj, Frank J. Jones, and Frank J. Fabozzi,

“Tracking Error and Common Stock Portfolio Management,” Chapter 7 in Frank

J. Fabozzi and Harry M. Markowitz (eds.), The Theory and Practice of Invest-

ment Management (New York: John Wiley & Sons, Inc., 2002), p. 171.

(^5) The tracking errors for the various portfolios were obtained from Barra Aegis soft-
ware. These are forward-looking tracking errors rather than backward-looking track-
ing errors. Also, the portfolios were optimally constructed to minimize tracking error.