Most of the popular models for VaR calculations seem to assume a multivariate normal distribution for the risk factors. This seems quite unrealistic to me. Some alternatives have been put forward in the research literature. Does anyone use them?Or is the state of the art to use quadratically approximated valuations of scenarios sampled from a normal distribution?

Some practicioners don't bother unless the real-world distribution is significantly non-normal (like options?). VaR is supposed to be an easy-to-implement-and-understand risk measure. It is not meant to be so precise as to estimate the probable loss to the last unit of currency. Just as long as the VaR methodology quantifies risk without any bias, to make comparisson of risk between different assets possible, the risk measure is alredy useful.

VaR is usually calculated using a hist sim type approach, so that no assumption about the distribution of risk factors is required (other than that the past distribution is a good approximation to the future distribution....) and correlations are easily captured.... If you are going to assume a distribution I think it is probably more standard to use a Pareto distribution as research shows that beyond a certain cut off point the tail follows a pareto type distribution - the key being to estimate the cut-off point and parameters without introducing too much error....

You misunderstand what the Normal distribution is used for.It happens to be mathematically convenient to simulate drawings from a multivariate Normal distribution with a given covariance matrix. But once you do this, you can transform the variables to any distribution you like. Looking at it from the other end, you could always define your risk factors in such a way that they follow a Normal distribution. You don't assume every security log return has a Normal distribution.For large, diverse portfolios, the type that VaR was designed to monitor, the marginal distribution of individual securities is not important for VaR, it is the tail correlation structure that matters. There isn't much data on tail correlations, at what there is suggests they are not neither simple nor stable.As jomni says, precision is not the issue. In practice, you calibrate your VaR measure by back-testing, not by getting everything analytically correct from first principles. The important points are financial control and aggregation across markets.Historical simulation is much less common than variance-covariance VaR. It is suitable only for portfolios whose risk comes mostly from a single risk factor.People do use more complicated approaches for VaR, but generally because they want to use it for something different that it was originally designed (or, too often, because they have no idea what people use it for).

QuoteOriginally posted by: Aaron The important points are financial control and aggregation across markets.Historical simulation is much less common than variance-covariance VaR. It is suitable only for portfolios whose risk comes mostly from a single risk factor. I am confused by this, as from my view view the real advantages of hist sim are it is can easily tackle mutiple risk factors, and allows easy aggregation accross markets??

I am a bit confused about the single risk factor point aswell - I wasn't quite sure what is meant by" b]mostly a single risk factor" Can you elaborate a little by what you mean with mostly a single risk factor or give an example of an asset with multiple risk factors where it would be suitable to estimate HS VaR using a single one - do you mean like an option with underlying as main source of risk?

You're correct that historical simulation can easily tackle multiple risk factors, but it can't tackle them very well.Suppose the portfolio depends mostly on the general level of interest rates. The recent ups and downs of interest rates can be reasonably used as a sample of potential future movements. If we use 500 days, for example, we will have one or two hundred examples of each of up days, down days and small change days.Suppose instead the portfolio depends on 20 different market factors: 1 year interest rates, 10 year interest rates, USD/CHF, USD/JPY, oil prices, gold prices and so on. I don't just mean that these each affect the portfolio, I mean each has a strong effect and they are not additive. For example, the portfolio might do well if the 1 year interest rate increases more than the 10 year interest rate and USD/CHF goes up; but if the 10 year interest rate increases more than the 1 year interest rate, the portfolio does better when USD/CHF goes down.There are 3^20 = 3,486,784,401 different combinations of up/down/small change for these 20 market factors. Only a tiny fraction of them will be represented among the last 500 days, in fact only a tiny fraction will ever have occurred in history. While some combinations are improbable, even the most probable combinations are unlikely to be represented. A parametric model will give some weight to all these combinations, historical simulation will not.

QuoteOriginally posted by: AaronYou're correct that historical simulation can easily tackle multiple risk factors, but it can't tackle them very well.There are 3^20 = 3,486,784,401 different combinations of up/down/small change for these 20 market factors. Only a tiny fraction of them will be represented among the last 500 days, in fact only a tiny fraction will ever have occurred in history. While some combinations are improbable, even the most probable combinations are unlikely to be represented. A parametric model will give some weight to all these combinations, historical simulation will not.I can agree that historical VaR cannot capture the combinations that Aaron has mentioned. However, from my limited experience, a parametric model weights a limited section of past data to make a forecast. I'm sorry, but I can't see how it can give some weight to all combinations. I would have thought that only Monte Carlo could capture the numerous combinations for multiple risk factors.

QuoteI can agree that historical VaR cannot capture the combinations that Aaron has mentioned. However, from my limited experience, a parametric model weights a limited section of past data to make a forecast. I'm sorry, but I can't see how it can give some weight to all combinations. I would have thought that only Monte Carlo could capture the numerous combinations for multiple risk factors.A parametric model can give weight to all possible outcomes within the model. That's not saying much, tractable models have severely restricted outcome sets and inaccurate weightings.For example, suppose I compute the delta and gamma of my portfolio with respect to 20 risk factors. If I assume the factors follow a multivariate Normal random walk with known mean vector and covariance matrix, I can compute the exact probability of any outcome, taking into account all possible values of all 20 factors from negative infinity to infinity.You might object that more than 20 factors influence the value of the portfolio, and that the delta/gamma approximation is not valid for all values of the factors, and the deltas, gammas, means and covariances are measured with substantial error, and the distribution is not constant, let alone Normal, and that it's not a random walk. All true, but I have accounted for every possible set of parameter values.In practice, I can do better than delta/gamma/Normal, but however fancy I get, the results are not very good. But you have to base decisions on something, and a well-designed parametric model, interpreted carefully, is a reasonable guide.

Aaron,I could not agree more on the keep it simple approach: simple -but not simplistic- model, well understood limitations so that you could eventually stress test the parameters, bottom line: you know what's going on as opposed to do a lot of number crounching that adds little value but a lot of unecessary complexity.This is why the simple multivariate normal distribution for the risk factors is very popular although everybody knows that in practice this doesn't hold (a bit like Black&Scholes assumption). Of course, to do your due diligence, you will have to supplement those results with stress tests (historical, forward-looking, etc...) and eventually with some extreme worst case loss using EVT with a very high confidence interval.I have however one practical issue here: how do model/integrate spread risk in your VaR calculation? It looks to me that the assumption that spread movements are normally distributed is a bit too much (which is the case in quite a few of risk engines that uses gov rates, swap rates, credit bond rates as risk factors as one normally distributed minus another normally distributed variable is also normally distributed). Any view on this one?Thanks in advanceJM

Spreads are a challenge to this approach. Something like credit spreads are relatively easy, you have the treasury curve, the AAA to treasury spread, the AA to AAA; and so on. Each spread can be a Normal random walk with a cap and a floor, or else you transform the spread in some manner to make it always positive.That doesn't work for say, Heating Oil to Jet Fuel. You can't let each of the underlyings propagate randomly, even with correlation, because you will get unreasonable spreads. So you need some kind of tree model, starting with a general energy price and branching into the finest gradations of traded products.For some applications, it's enough to model individual prices as linear combinations of factors, with the last factor being idiosyncratic. That works for equities pretty well, and has some value in commodities.

QuoteOriginally posted by: AaronA parametric model can give weight to all possible outcomes within the model. That's not saying much, tractable models have severely restricted outcome sets and inaccurate weightings.Aaron, thanks very much for the explanation. Looking forward to your book on Poker

thanks Aaron