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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 submit an email via  Request More Information.

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 "expensive¹" 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, distribution², 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.

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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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