The Mathematics of Financial Modelingand Investment Management

19-EquityPort Page 578 Friday, March 12, 2004 12:40 PM

578 The Mathematics of Financial Modeling and Investment Management

Multifactor risk models allow a manager and a client to decompose

risk in order to assess the potential performance of a portfolio to the

risk factors and to assess the potential performance of a portfolio rela-

tive to a benchmark. This is the portfolio construction and risk control

application of the model. Also, the actual performance of a portfolio

relative to a benchmark can be assessed. This is the performance attri-

bution analysis application of the model.

Barra suggests that there are various ways that a portfolio’s total risk

can be decomposed when employing a multifactor risk model.^16 Each

decomposition approach can be useful to managers depending on the

equity portfolio management that they pursue. The four approaches are

(1) total risk decomposition; (2) systematic-residual risk decomposition;

(3) active risk decomposition; and (4) active systematic-active residual

risk decomposition. We describe each below and explain how managers

pursuing different management strategies (i.e., active versus passive) will

find the decomposition helpful in portfolio construction and evaluation.

In all of these approaches to risk decomposition, the total return is first

divided into the risk-free return and the total excess return. The total excess

return is the difference between the actual return realized by the portfolio

and the risk-free return. The risk associated with the total excess return,

called total excess risk, is what is further partitioned in the four approaches.

Total Risk Decomposition

There are managers who seek to minimize total risk. For example, a

manager pursuing a long-short or market neutral strategy, as discussed

later in this chapter, seek to construct a portfolio that minimizes total

risk. For such managers, total risk decomposition which breaks down

the total excess risk into two components—common factor risks (e.g.,

capitalization and industry exposures) and specific risk—is useful. This

decomposition is shown in Exhibit 19.5. There is no provision for mar-

ket risk, only risk attributed to the common factor risks and company-

specific influences (i.e., risk unique to a particular company and there-

fore uncorrelated with the specific risk of other companies). Thus, the

market portfolio is not a risk factor considered in this decomposition.

Systematic-Residual Risk Decomposition

There are managers who seek to time the market or who intentionally

make bets to create a different exposure than that of a market portfolio.

Such managers would find it useful to decompose total excess risk into

systematic risk and residual risk as shown in Exhibit 19.6. Unlike in the

(^16) See Chapter 4 in Barra, Risk Model Handbook United States Equity: Version 3.
The discussion to follow in this section follows that in the Barra publication.