21-Bond Portfolio Man Page 658 Wednesday, February 4, 2004 1:12 PM
658 The Mathematics of Financial Modeling and Investment Managementvariances, it not the sum of these two risks that sum to the portfolio’s
tracking error, but rather the squares of these two tracking errors that
will equal the square of the portfolio’s tracking error. Or equivalently,
the square root of the square of the two tracking errors will equal the
portfolio’s tracking error (i.e., [(45.0)^2 + (26.1)^2 ]0.5 = 52.0). Adding of
variances assumes that there is zero correlation between the risk factors
(i.e., the risk factors are statistically independent).
The alternative calculation for subdividing the tracking error is shown
in the last two columns of Exhibit 21.3, the “Cumulative” calculation. In
the second column the cumulative tracking error is computed by introduc-
ing one group of risk factors at a time and computing the resulting change
in the tracking error. The analysis begins with the 36.3 basis point tracking
error due to the term structure risk. The value shown in the next row of
38.3 basis points is calculated by holding the risk factors constant except
for term structure risk and sector risk. The change in the cumulative track-
ing error from 36.3 to 38.3 basis points is shown in the last column for the
row corresponding to sector risk. The 2 basis point change is interpreted
as follows: given the exposure to yield curve risk, sector risk adds 2 basis
points to tracking error. By continuing to add the subcomponents of the
risk factors, the cumulative tracking error is determined. Because of the
way in which the calculations are performed, the cumulative tracking
error shown for all the systematic risk factors in the next-to-the last col-
umn is 45 basis points, the same as in the “isolated” calculation.
Exhibit 21.4 can be used to understand the difference between the
“isolated” and “cumulative” calculations. For purposes of the illustra-
tion, the exhibit shows a covariance matrix for just the following three
groups of risk factors: yield curve (Y), sector spreads (S), and quality
spreads (Q). How the covariance matrix is used to calculate the subcom-
ponents of the tracking error in the “isolated” case is shown in panel a.
The diagonal of the covariance matrix shows the elements of the matrix
that are used in the calculation for that subcomponent. The off-diagonal
terms of the matrix deal with the correlations among different sets of risk
factors. They are not used in calculating the tracking error and therefore
do not contribute to any of the partial tracking errors. The elements of
the covariance matrix used in the calculation of the “cumulative” track-
ing error at each stage of the calculation are shown in Panel b of Exhibit
21.4. The incremental tracking error due to sector risk takes into consid-
eration not only the S × S variance but also the cross terms S × Y and Y ×
S which represent the correlation between yield curve risk and sector risk.
Note that the incremental tracking error need not be positive. When the
correlation is negative, the increment will be negative. This can be seen in
the last column of Exhibit 21.3 which shows that the incremental risk
due to the MBS sector risk is –1.7 basis points.