166 CHAPTER
6 Risk Capital Aggregation
potential outcomes that are produced by different risk models (e.g., credit, market, opera-
tional, business risk models), their practical application has to face remarkable problems
as far as calibration is concerned. Further research on the topic is surely warranted, but
at the same time it would not be easy for copulas to substitute the simpler normal/hybrid
CaR approach that is still largely used in practice despite its limitations. The pure and
the mixed multifactor approaches represent interesting compromises between theoretical
soundness and the need to feed a model with reasonable data. By modeling business unit
P&L versus common factors, moreover, some extra useful information could be obtained
by decision makers. The methodology might also be easier to explain to top managers
than a copula function, and this may help make the methodology accepted and
successful.
Yet the issue of data availability is important in any case, so the problem of time series
is stressed frequently in Table 6-6. Note also that when a sophisticated technique is fed
with a very short series of historical data (e.g., monthly earnings of certain business units),
model parameters may change substantially from one year to another, simply because of
the increase in the size of the historical sample. It would then be problematic to provide
top managers with a report that could contain major changes in overall and individual
CaR values, even in the absence of similar changes in business unit activity. Consequently,
it may be advisable to test how the results of different methods change when new data
are added, especially if the sample is not totally reliable. In general, as Pezier (2003)
points out, it is important to make clear to all of the potential users that single CaR
numbers are estimates and often very uncertain ones. Yet a potential suggestion when
designing aggregated risk capital reports for senior managers would be to provide a range
of values rather than a single and apparently precise number. Knowing that aggregated
capital is not equal, say, to 1823.27, but rather is somewhere in the range between 1780
and 1870 (or perhaps between 1700 and 1950) may be a better support for decision makers,
without giving a false sense of certainty, which might be dangerous. Moreover, making
the range explicit may also convince senior managers to make the bank invest more money
and effort in improving its risk aggregation techniques if the importance to achieve better
and sounder measures were perceived.
6.6 Summary
After measuring market, credit, operational, and business risk, a bank still has to derive
an aggregated capital measure in order both to support capital management decisions and
to better understand the size of diversifi cation benefi ts arising from its business mix. The
starting point is to harmonize the different risk measures for each risk in terms of time
horizon, con dence level, and their underlying notion of capital. There are different
aggregation methodologies that can be used and that should be selected taking into con-
sideration its conceptual soundness, its computational complexity, and the actual possibil-
ity of consistently estimating the parameters that the aggregation methodology requires.
While adopting the usual portfolio VaR formula applied by the variancecovariance
approach in a context of multivariate normally distributed returns is the easiest and
perhaps currently most common solution, alternative and more refi ned solutions should
be considered, such as copulas, the multifactor approach, and the mixed multifactor
approach proposed here. Even in the simpler solutions, however, estimating proper param-
eters to model the dependence among different risks is far from easy. We pointed out in
the chapter potential risks deriving from using short series of monthly earnings data
without considering their potential impact on correlation estimates. The chapter has also

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