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There are three different commonly used Value at Risk (VaR) methods:

  • Historical method
  • Variance-Covariance Method
  • Monte Carlo

What is the difference between these approaches, and under what circumstances should each be used?

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Is "expected shortfall" ever used in practice? –  Richard Herron Jan 31 '11 at 22:42
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@richardh Yes it is. One simple way to see this is to look at the most popular vendor for these statistics: RiskMetrics. They market shortfall and scenario testing almost as much as VaR. –  Shane Feb 1 '11 at 0:42
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Expected Shortfall also has the benefit of being a "coherent" risk measure, unlike VaR. Most of the recent literature in portfolio construction and optimization tends to use Expected Shortfall (or equivalently Conditional Value at Risk) –  Quant Guy Aug 5 '11 at 13:38
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2 Answers

The Historical Method, which I would call Historical Simulation requires that you have a reasonably clean and accurate time series of data for the underlying asset. Essentially, you are using the past performance of the asset to model its likely behaviour over a time frame of typically 1 to 10 days. Choosing and updating your time series data set needs to be thought about carefully as your VaR number can be impacted significantly by extreme events in the time series used.

Where good time series data is not available, it would be appropriate to use the Variance-Covariance method. This is generally considered to be less accurate than Historical Simulation due to assumptions about the distribution of returns that do not hold perfectly true in real markets (fat tail distributions).

Monte Carlo simulation is computationally a lot more expensive than Historical Simulation or V-CV and requires that a large number of asset paths are calculated to get a statistically significant result.

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There are many advantages and flaws to each quoted method by Shane(presuming that I understand them properly), the first one has the big main advantage that it doesn't need any evaluation of probability law, it is just some kind of evolved scenario re-playing "as of" today using the history of (usually) one day market evolutions over one or two year.

So once you know how to evaluate your protfolio (no matter how complex it is) you have something that allows you to mechanically know your quantile and so your VaR. The problem with the method is that there are manny econometrics hypothesis that are actually hard to test for this kind of method to be trustworthy.

Second "Variance-Covariance" I think that Shane means by this that risk factors returns obey multinomial normally distributed laws (or log-normally), once again the upside of method is its simlpicity once you econometrically estimate your VarCovar matrix. The truth is that unfortunately one day returns exhibit fat tails and asymmetry so those kind of laws aren't the best, this can though be partially overcome by using some other multinomial laws that are elliptic. Another thing you should keep in mind is that when you have a VaR indicator and a profolio that isn't rebalanced over a day, and when market becomes excited, then you expect (and your management) your VaR to go up even though nothing much has happened in your econometric estimation, so what you really want is then not a VaR which is unconditionnal but some conditionnal VaR and then you resort to some kind of overweighing the recent observations versus the old one (like EWMA methods). Then you get some indicator that is really "alive" instead of some indicator that takes a long time to adjust to market conditions. Let me now explain why I used the elliptic laws (multinomial Gaussain are part of this class), this is because with this assumption (If I remember well) then VaR requalifies as a true Risk Measure as defined by Artzner et al. (which is not true for genral probability laws has it lacks the subadditivity axiom). Another point, is when you have nonlinear positions in your protfolio such as options then what is usually done (and can be really wrong if not given sufficient attention) is that those position are Taylor derived at order one with respect to risk factors and then Linear VaR is calculated. You can extended your approximation to superior orders but if you have a lot of risk factors then the second derivative estimations are already an achievement, so what you do is that you stay linear and estimate periodically what is the error when quadratic terms are taken into account (or resort to MC methods).

Finally, MC methods then they are efficient methods but very time consumming as you are trying to evaluate a Quantile id est a rare event and you then need to get a very large number of simulation to get something good, moreover it is not always clear which law is to be simulated. In particular in portfolios with derivatives, because it is quite tempting tu use Risk Neutral measure in your simulations but they have two differences over historical estimates, first the underlying model is usually calibrated to model risk factors over large period of times when you are only trying to get estimates over the next day so the underlying measures have different purposes.

And second as you may know in theory the difference between Historical and Risk Neutral measures are hidden "in the drift" of the risk factor dynamics (well this is not true unless complete market is assumed but let's go with it) and over a day you can discard this difference with respect to the diffusion term which should be the same for both measures, it happens that almost always Risk Neutral Volatility (i.e. marekt calibrated volatilities) are higher than historical one (or realized ones).

Well here are my two cents

Best Regards

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It is worth pointing out that if you calibrate a standard variance-covariance return matrix over, say an N-day historical period, and then take the mean and variance of the empirical historical returns over those same N days, you will get the same normal distribution and VaR levels from the two different calibrations. –  Brian B Feb 14 '11 at 17:02
    
For the sake of completeness, one advantage of the historical approach is that it is non-parametric (i.e. does not commit one to distribution assumptions such as normality other than the empirical distribution itself). Disadvantage is that past is not prologue. –  Quant Guy Feb 22 '12 at 2:53
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