148 CHAPTER
6 Risk Capital Aggregation
it to the target rating the bank is willing to maintain) is very important and, on the other
hand, a certain amount of discussion about the rescaling methodology might emerge (from
a purely pragmatic point of view, rescaling from 99% to 99.97% as if the distribution
were normal has at least the advantage that it does not change the relative ranking of
market-risk-sensitive business units).
Yet a third relevant problem, which is usually neglected, is whether VaR measures for
different risks are measuring losses with reference to the same concept of capital. Market
risk VaR is measured in mark-to-market terms and hence also potentially identi es the
reduction in market capitalization subsequent to adverse market changes; book capital at
risk would be the same too, provided that all assets are evaluated at fair value. In the case
of credit risk, in contrast, credit portfolio models may value loans either at book value or
in mark-to-market terms. For both operational and business risk, different numbers might
be obtained, depending on whether the focus is on the one-year accounting loss that opera-
tional and business risk can generate or on the market capitalization impact of those
losses.
This potential lack of consistency between capital-at-risk measures could be solved
differently, depending on the objective. If the goal is to measure risk-adjusted performance
(RAP) measures to support capital allocation consistently, then the market cap value of
capital is likely to be the correct solution, since it is the most consistent with the share-
holders view of the bank.
If the purpose is to support decisions about the bank’s optimal capital structure policy,
then in theory two different aggregated measures (based on BCaR and MCaR, respec-
tively) could be created and compared with actual book capital and market capitalization
as measures of available capital. In practice, however, the main emphasis would be attrib-
uted to book capital at risk.
This approach clearly differs from the common practice of assuming that all albeit
different measures of capital at risk have to be compared against a single measure of
available capital. Clearly, it may be costly to maintain different measures of capital at risk
according to different criteria and to explain the differences to senior management
members who have to make key decisions. Yet recalling this difference which is
stronger for some risks, such as business risk, than for others is relevant, since the
choice of either a BCaR or an MCaR approach may impact on the relative share of the
bank’s capital allocated to different businesses and then on how return targets for those
businesses should be assigned (see Chapter 9). Note that from now on we use CaR rather
than VaR to remind us that aggregation should be based on measures of capital at risk
calculated or harmonized over a yearly horizon, at the same con dence level, and after
harmonizing to the maximum possible extent the concept of capital behind each risk
measure. In this way CaR values could differ from VaR values used for day to day risk
monitoring and control.
6.2 Risk Aggregation Techniques
After harmonizing capital-at-risk measures as much as possible, aggregating those mea-
sures into a single number for the whole institution requires tackling three different issues:
(1) identifying the components that have to be aggregated, (2) identifying the aggregation
technique or algorithm to be used, and (3) calibrating the parameters (e.g., correlation
coeffi cients) needed to derive the single risk measure. We approach these issues in this
sequence. As a necessary premise, risk aggregation represents a very important area, but
it is still not as developed as individual risk modeling.

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