Comparing risk measurements and P&L

Holding period
For regulatory purposes, the maximum loss over a 10-business-day period at the
99% confidence level must be calculated. This measurement assumes a static
portfolio over the holding period. In a realistic trading environment, however, portfolios
usually change significantly over 10 days, so a comparison of 10-day P&L with
market risk would be of questionable value. A confidence level of 99% and a holding
period of 10 days means that one exception would be expected in 1000 business
days (about 4 years). If exceptions are so infrequent, a very long run of data has to be
observed to obtain a statistically significant conclusion about the risk measurement
model. Because of this, regulators require a holding period of one day to be used for
backtesting. This gives an expected 2.5 events per year where actual loss exceeds
the market risk figure. Figure 9.2 shows simulated backtesting results. Even with
this number of expected events, the simple number of exceptions in one year has
only limited power to distinguish between an accurate risk measurement model and
an inaccurate one.
As noted above, risk figures are often calculated for a holding period of 10 days.
For backtesting, risks should ideally be recalculated using a 1-day holding period.

For the most accurate possible calculation, this would use extreme moves of risk
factors and correlations based on 1-day historical moves rather than 10-day moves.
Then the risk figures would be recalculated. The simplest possible approach is simply
to scale risk figures by the square root of 10. The effectiveness of a simple scaling
approach depends on whether the values of the portfolios in question depend almost
linearly on the underlying risk factors. For instance, portfolios of bonds or equities
depend almost linearly on interest rates or equity prices respectively. If the portfolio
has a significant non-linear component (significant gamma risk), the scaling would
be inaccurate. For example, the value of a portfolio of equity index options would
typically not depend linearly on the value of the underlying equity index. Also, if the
underlying risk factors are strongly mean reverting (e.g. spreads between prices of
two grades of crude oil, or natural gas prices), 10-day moves and 1-day moves would
not be related by the square root of time. In practice, the simple scaling approach is
often used. At the whole bank level, this is likely to be reasonably accurate, as
typically the majority of the risk of a whole bank is not in options portfolios. Clearly,
this would not be so for specialist businesses such as derivative product subsidiaries,
or banks with extensive derivative portfolios.