The Mathematics of Financial Modelingand Investment Management

21-Bond Portfolio Man Page 659 Wednesday, February 4, 2004 1:12 PM

Bond Portfolio Management 659

EXHIBIT 21.4 Illustration of “Isolated” and “Cumulative” Calculations of

Tracking Error Subcomponentsa

a. Isolated Calculation of Tracking Error Components

Y × Y Y × S Y × Q

S × Y S × S S × Q

Q × Y Q × S Q × Q

b. Cumulative Calculation of Tracking Error Components

Y × Y Y × S Y × Q

S × Y S × S S × Q

Q × Y Q × S Q × Q

a Y – Yield curve risk factors; S – Sector spread risk factors; Q – Credit Quality

spread risk factors.

Source: Exhibit 12 in Lev Dynkin, Jay Hyman, and Wei Wu, “Multi-Factor Risk

Models and Their Applications,” in Frank J. Fabozzi (ed.), Professional Perspec-

tives on Fixed Income Portfolio Management: Volume 2 (New Hope, PA: Frank

J. Fabozzi Associates, 2001).

The “isolated” calculation helps a portfolio manager identify the

relative magnitude of each subcomponent of the tracking error. The

advantage of the “cumulative” calculation is that it takes into consider-

ation the correlations among the subcomponents of the risk factors and

the sum of the tracking error components is equal to the total tracking

error. The drawback of the “cumulative” calculation is that it is depen-

dent upon the order in which the risk factors are introduced.

Another portfolio risk measure provided in Exhibit 21.3 is the vola-

tility of returns. That is, the standard deviation of the return for each

systematic risk factor and the standard deviation for the portfolio return

can be computed. Similarly, the standard deviation of the benchmark

return can be calculated. Note the difference between tracking error and

standard deviation of returns. The former is computed by using the his-

torical differences in return between the portfolio and the benchmark.

The latter only considers the historical returns. As was computed for

tracking error, there are systematic return and nonsystematic return

components. The last panel in Exhibit 21.3 reports the total standard

deviation for the portfolio and the benchmark and the composition of

each in terms of systematic and nonsystematic risk factors. Notice that

the portfolio’s standard deviation (430 basis points) is greater than that

of the benchmark (417 basis points).