19-EquityPort Page 587 Friday, March 12, 2004 12:40 PM
Equity Portfolio Management 587Risk Control Against a Stock Market Index
The objective in equity indexing is to match the performance of some
specified stock market index with little tracking error. To do this, the
risk profile of the indexed portfolio must match the risk profile of the
designated stock market index. Put in other terms, the factor risk expo-
sure of the indexed portfolio must match as closely as possible the expo-
sure of the designated stock market index to the same factors. Any
differences in the factor risk exposures result in tracking error. Identifi-
cation of any differences allows the indexer to rebalance the portfolio to
reduce tracking error.
To illustrate this, suppose that an index manager has constructed a
portfolio of 50 stocks to match the S&P 500. Exhibit 19.13 shows out-
put of the exposure to the Barra risk indices and industry groups of the
50-stock portfolio and the S&P 500. The last column in the exhibit
shows the difference in the exposure. The differences are very small
except for the exposures to the size factor and one industry (equity
REIT). That is, the 50-stock portfolio has more exposure to the size risk
index and equity REIT industry.
The illustration in Exhibit 19.13 uses price data as of December 31,- It demonstrates how a multifactor risk model can be combined
with an optimization model to construct an indexed portfolio when a
given number of holdings is sought. Specifically, the portfolio analyzed
in Exhibit 19.13 is the result of an application in which the manager
wants a portfolio constructed that matches the S&P 500 with only 50
stocks and that minimizes tracking error. Not only is the 50-stock port-
folio constructed, but the optimization model combined with the factor
model indicates that the tracking error is only 2.19%. Since this is the
optimal 50-stock portfolio to replicate the S&P 500 that minimizes
tracking error risk, this tells the index manager that if he or she seeks a
lower tracking error, more stocks must be held. Note, however, that the
optimal portfolio changes as time passes and prices move.
Tilting a Portfolio
Now let’s look at how an active manager can construct a portfolio to
make intentional bets. Suppose that a portfolio manager seeks to con-
struct a portfolio that generates superior returns relative to the S&P 500
by tilting it toward low P/E stocks. At the same time, the manager does
not want to increase tracking error significantly. An obvious approach
may seem to be to identify all the stocks in the universe that have a
lower than average P/E. The problem with this approach is that it intro-
duces unintentional bets with respect to the other risk indices.