19-EquityPort Page 577 Friday, March 12, 2004 12:40 PM
Equity Portfolio Management 577noisy as a large fraction of the cointegrated pairs will be spurious. In
practice, the number of cointegrated pairs has to be reduced.
■ Searching for cointegrated indexes. This is performed testing cointegra-
tion on existing, commercially available indexes. These indices typi-
cally reflect economic sectors or geographies. After determining that
cointegration among the indexes exists, one has to select stocks within
the index to reduce transaction costs.
■ Searching for common trends. This is a recent development in statisti-
cal arbitrage. It is based on approximate robust techniques for finding
factors using state space models. Factors, in this sense, are linear com-
binations of price processes not of returns.In summary, statistical arbitrage is a new methodology for managing
long-short equity portfolios based on finding stable trends that signal
profit opportunities. Trends might be determined with classical factor
models of returns. More recently, cointegration techniques are being used.APPLICATION OF MULTIFACTOR RISK MODELS
In the previous chapter, we explained how factors are determined. In
this section we will see how multifactor risk models are used. In our
illustration with use the Barra model described in the previous chapter.Risk Decomposition
The real usefulness of a linear multifactor model lies in the ease with
which the risk of a portfolio with several assets can be estimated. Con-
sider a portfolio with 100 assets. Risk is commonly defined as the vari-
ance of the portfolio’s returns. So, in this case, we need to find the
variance-covariance matrix of the 100 assets. That would require us to
estimate 100 variances (one for each of the 100 assets) and 4,950 covari-
ances among the 100 assets. That is, in all we need to estimate 5,050 val-
ues, a very difficult undertaking. Suppose, instead, that we use a 3 factor
model to estimate risk. Then, we need to estimate (1) the three factor
loadings for each of the 100 assets (i.e., 300 values); (2) the six values of
the factor variance-covariance matrix; and (3) the 100 residual variances
(one for each asset). That is, in all, we need to estimate only 406 values.
This represents a nearly 90% reduction from having to estimate 5,050
values, a huge improvement. Thus, with well-chosen factors, we can sub-
stantially reduce the work involved in estimating a portfolio’s risk. Note
that the ease of estimation of correlation parameters is another facet of
the fact that factor models capture the stable correlation information.