The Mathematics of Financial Modelingand Investment Management

19-EquityPort Page 584 Friday, March 12, 2004 12:40 PM

584 The Mathematics of Financial Modeling and Investment Management

EXHIBIT 19.11 Portfolio ABC’s Risk-Return Decomposition

Number of Assets 202     Total Shares 62,648,570

Average Share Price $59.27

Portfolio Beta 1.20 Portfolio Value $3,713,372,229.96

Risk Decomposition Variance Standard Deviation (%)

Active Specific Risk 15.33 3.92

Active Common Factor

Risk Indices 44.25 6.65

Industries 17.82 4.22

Covariance 19.27

Total Active Common Factor Riska 81.34 9.02

Total Activeb 96.67 9.83

Benchmark 247.65 15.74

Total Risk 441.63 21.02

a Equal to Risk Indices + Industries + Covariances

b Equal to Active Specific Risk + Total Active Common Factor Risk

Source: Exhibit 13.8 in Frank J. Fabozzi, Frank J. Jones, and Raman Vardharaj,

“Multi-Factor Risk Models,” Chapter 13 in Frank J. Fabozzi and Harry M.

Markowitz (eds.), The Theory and Practice of Investment Management (Hobo-

ken, NJ: John Wiley & Sons, 2002).

the difference between the portfolio exposure and the benchmark expo-

sure. The exposures to the risk index factors are measured in units of stan-

dard deviation, while the exposures to the industry factors are measured

in percentages. The portfolio has a high active exposure to the momentum

risk index factor. That is, the stocks held in the portfolio have significant

momentum. The portfolio’s stocks were smaller than the benchmark aver-

age in terms of market cap. The industry factor exposures reveal that the

portfolio had an exceptionally high active exposure to the semiconductor

industry and electronic equipment industry. Exhibit 19.12b combines the

industry exposures to obtain sector exposures. It shows that Portfolio

ABC had a very high active exposure to the Technology sector. Such large

bets can expose the portfolio to large swings in returns.

An important use of such risk reports is the identification of portfo-

lio bets, both explicit and implicit. If, for example, the manager of Port-

folio ABC did not want to place such a large Technology sector bet or

momentum risk index bet, then she (or he) can rebalance the portfolio

to minimize any such bets.