Value at Risk and Bank Capital Management

VaR delta are described. Section 7.5 concludes the chapter by analyzing autonomy and

pricing limits for credit risk.

7.1 Defi ning VaR-Based Limits for Market Risk: Identifying

Risk-Taking Centers

When defi ning a risk policy for business units in the trading business, which would be

exposed mainly to either market risks (e.g., in the case of equity trading) or the credit

risk of traded fi nancial instruments (e.g., corporate bonds, credit derivatives), the fi rst

topic to be considered is identifying which units should be given a limit. This in turn

requires (1) defi ning the highest level in the hierarchy of business units and desks to which

VaR limits should be set and (2) checking whether the boundaries between the tasks of

different units allow one to defi ne VaR limits effi ciently by reducing risk overlaps as much

as possible.

The fi rst problem is related, for instance, to whether VaR limits should be set only

up to the equity trading business unit or also to the level of the main desks composing

the unit (e.g., cash, futures, plain-vanilla options, exotic option trading) or even to

the level of the individual trader. A more detailed disaggregation of VaR limits might

help if VaR is considered an effi cient way to set operational risk limits for the single entity

(i.e., business unit, desk, or individual trader) and if the bank is willing to measure

the performances based not only on expost utilized VaR, but also on the amount of

capital assigned to the unit, which depends precisely on its VaR limit. As we see later on,

this may also be used to create incentives to use more intensively the allocated capital for

each entity, i.e., to reduce the percentage of idle, unutilized capital. In theory, then, a more

pervasive attribution of VaR limits would be desirable.

At the same time, there may be reasons to keep VaR limits at a higher hierarchical

level. One case is when VaR limits may control only partially the real risk of the exposures

of a desk. A bank adapting a variance–covariance approach could hardly use it to control

the day-to-day risk of its equity option desk. Of course, a more sophisticated VaR model

could be more helpful, but a bank with a small option desk may still consider that invest-

ing in a better VaR model is not worth the extra cost and effort. In this case, VaR limits

should be set at a higher level and the desk should simply be given limits based only on

option greeks such as delta and vega (i.e., sensitivities to changes in the underlying asset

price and implied volatility). These simpler limits are also widely used in more sophisti-

cated banks, since they help in capturing the risk profi le of the option portfolio, even on

an intraday basis, when VaR measurement with a more sophisticated model could be too

diffi cult.

More generally, it is useful to recall that VaR is usually calculated based on the trader’s

portfolio after the mapping process has been performed. A poorly mapped portfolio in

which some of the real risks “disappear” during the mapping phase (e.g., an option posi-

tion that is transformed into a linear equivalent through a delta-only approximation, an

undiversifi ed stock portfolio whose VaR is approximated as beta times the stock index

VaR) can hardly be controlled through VaR limits. Failing to model existing risks properly

could be considered one of the sources of operational risk in the trading business, but a

careful check of existing limitations is important when defi ning a VaR limit system (see

also Box 7-1).

DEFINING VaR-BASED LIMITS FOR MARKET RISK 171