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P&L Implications of VaR/Economic Capital

Please note, as *ART Consulting/Research* is a fee based
service, in the following the results have been "sanitised" to
disguise the specific markets, trading factors, strategy parameters and many
other essentials. Of course, all of the analyses is based on real market
conditions and real world trading considerations (trans cost, funding, etc). For access to the
"un-sanitised" results, and for analysis tailored to your needs, please
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The *ARBLab* relies on
PaR analysis and the Pr/rO ®
software (see other examples in the *ARBLab*
Samples section, and
the
TG2RM1st
- Chapter 12
provides a detailed introduction to PaR analysis).

The basic notion of setting risk limits
or funding/provisioning requirements based on a probability weighted
expectation of likely portfolio or P&L behaviour is the corner stone of
almost all trading activity. Though the practical implementation of
this idea is rather tricky, and can lead to unexpected costs/losses, or
sometimes hidden losses and complaisance.

Over the past decade or two,
Value-at-Risk (VaR) methodologies have emerged as one of the primary
techniques for implementing this idea in the context of risk limits.
A variation of the VaR methodology, called Economic Capital (ECap),
is used to derive expected funding and provisioning levels for credit and
default expectations, and shares much of the same costs/benefits as VaR
(see
TG2RM1st
for a detailed introduction
Request
More Information).

The basic idea behind VaR/ECap
methods is to analyse the statistical properties of position or P&L
behaviour, and estimate "tomorrow's" P&L distribution. Based
on that distribution, one may examine "how low the P&L can go" with some
statistical confidence level (usually taken as the 5% loss level). The image to
the right (click to enlarge) shows such a P&L distribution. The
cross-hatched bar depicts the chosen loss threshold at 5%, and so the implied
P&L loss value can be read along the horizontal axis. This "VaR
number" can then be "managed"
by altering positions etc (e.g. if the VaR is too small or too large).

Similarly, ECap loss provisions and
funding levels may be estimated this way by assessing the level of defaults
or other credit implications that a distribution implies (requires a distribution calculation reflecting P&Ls based on
defaults, recoveries, etc).

VaR methodologies are available in
several "flavours"; CVaR, HVaR, MCVaR, and so forth (see
TG2RM1st
for a detailed introduction). Roughly
speaking each flavour offers a trade-off between cost of
implementation/usage, and reliability/integrity. CVaR is by far
the least "expensive^{1}" flavour (which may account for its
popularity).

All VaR methods ignore important real
world effects, such as hedging/rebalancing. In addition, such
methods may take "shortcuts" by making various (technical) simplifying
assumptions regarding the "shape" of the distribution and other dynamics.
Importantly, CVaR methods rely on the assumptions that all distributions
are Gaussian (i.e. Normally distributed). This is very convenient from
a parameter estimation, distribution^{2}, and VaR calculation
perspective. This is part of the reason for CVaR's "low cost".

Unfortunately, these assumptions can
also lead to very considerable unreliability and lack of integrity,
impacting the firm's (real) P&L. Notably, the firm's P&L may be
adversely affected whether the VaR/ECap values are over- or
under-estimated. Over-estimation may result in artificially
constrained trading/business thereby leading to an under utilisation of
the firm's assets (i.e. low RoA). Under-estimation results in
carrying positions that are riskier than otherwise thought, leading to a
lower than expected risk-adjusted returns (or large losses).

One "experiment" back-tests the
validity or reliability of VaR/ECap to assess whether (on average) the
CVaR-like results are indeed within 5%. That is, using a long history
of data, start at the beginning to perform the VaR/ECap calculations as
usual (typically using 100 days of data). Then compare the VaR to the
next day's actual P&L from the actual historical data. Repeat this
process over the entire history/dataset, tracking the expected (i.e. VaR)
vs. actual (real) P&L.

Analysis of various individual and
combined positions shows that some instruments and indices do follow the
Gaussian/multiples-rule assumption reasonable well. However, many positions
are very far away from the “Gaussian ideal”. Using many years of daily data
for VaR/ECap analysis, some positions were “outside” the 5%-ntile many more
times than the Gaussian ideal permits, and biased to "below" or "above" the
threshold.
More sophisticated methods can relate the
tracking error to the "equivalent VaR/RCap".
For example, one position had such a
high frequency of being “outside” that it equated to a VaR/ECap limit of
2.46%-ntile. In other case, the “equivalent measure” was 8.22%-nitle.

The first case implies a much riskier
position than that assumed by the CVaR analysis, or that the ECap would
provision too little capital. The latter case has the opposite effect, in
that it over states the risk and overstates the capital requirements.
Notice also that while a discrepancy of 8.22-5 = 3.22% does not sound like
much, in fact, it implies 64.4% error in funding levels. That is a very
substantial error, and will have a very substantial effect on P&L. Notice
also that overstating the risk limits can be equally bad, since it may cause
management to incorrectly restrict the traders from doing as much business
as they should do, again with potentially materially effect on P&L.

Notably, this is the effect of
just the CVaR distribution assumption. Imagine the impact when other
reality issues are accounted for (particularly hedging/rebalancing,
"drifting conflict", etc).

As usual, caution is required.
The analysis here, though including thousands of trades, and incorporating
many real world factors cannot be taken as any perfect predictor of the
future, and additional specific analysis may be required for your due
diligence.

For detailed research results on this issue please
Request
More Information and please feel free to indicate specifics of
interest to you.

________________________

^{1} Here,
"expensive" is an all encompassing matter and covers cost of
purchase/building, implementation, usage, interpretation, and "popularity".
Popularity is included, since sometimes less effective, but market
convention, methods are less expensive to implement (e.g. to obtain
regulatory or shareholder approval etc).

^{2} In fact,
with CVaR, the distribution is not actually "calculated" since it is assumed
to be Gaussian, and then there is often not even a direct quantile
calculation, but may rely on the application of so-called "multiples rules"
(e.g. VaR = Today'sP&L - *n* ∑, where *n* is
the multiples rule (e.g. ~1.68 for 95%ntile) and ∑ is the position's standard
deviation).

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