What is the difference between the methods for calculating VaR?

• 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
@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
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) – Ram Ahluwalia Aug 5 '11 at 13:38

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.

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. – Ram Ahluwalia Feb 22 '12 at 2:53

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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