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I need to estimate the daily VaR of a portfolio of various exposures in $n$ risky assets (say equity futures).

The simplest approach, I think, would be to just estimate VaR from a multivariate normal distribution using historical daily return covariances. Would this approach significantly underestimate the risk because of fat tails? Or is it still a reasonable estimate?

If I were to use a copula, what would be the simplest approach? Would a Gaussian copula suffer from the same problems as simply estimating the multivariate normal distribution as above?

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  • $\begingroup$ both these models are inherently flawed. It is well know and eaisly searchable on google especially with the gaussian copula $\endgroup$ – pyCthon Apr 20 '12 at 3:11
  • $\begingroup$ Well no model is perfect. I am aware of the criticism that people have put too much trust in VaR and gaussian copula, but that doesn't mean they have no use. What would you suggest instead? $\endgroup$ – user2303 Apr 20 '12 at 3:25
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    $\begingroup$ VaR and Copula are two different things $\endgroup$ – tagoma Apr 20 '12 at 14:26
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In general you don't need copulas to calculate VaR on portfolio. You can use historical method if you have time series of returns for the assets in your portfolio. If you have sufficiently enough data this will allow you to take into account correlation risk, non-normality of returns.

Example of code in R for equally weighted portfolio without assuming any copula or distribution (using RMetrics package and the LPP indices data provided with this packages):

library(fPortfolio)
lppData  <-  100*LPP2005.RET[,1:6]
eqWSpec  <- portfolioSpec();
nAssets <- ncol(lppData)
setWeights(eqWSpec)  <- rep(1/nAssets, times = nAssets)
setAlpha(eqWSpec)  <- 0.05
ewPorfolio  <- feasiblePortfolio (data = lppData, spec=eqWSpec);
print(ewPorfolio)

Output:

Target Return and Risks:
  mean     mu    Cov  Sigma   CVaR    VaR 
0.0431 0.0431 0.3198 0.3198 0.7771 0.4472

(Note that this is most probably not the best way to calculate VaR of portfolio in R)

It worth mentioning that if you need to use copulas, you will have to do Monte Carlo VaR calculation (i.e. sample copula and calculate VaR on that data), as there are no closed form solutions available for VaR for most of the copula classes.

And yes, Gaussian copula would suffer the same problems as estimating the multivariate normal distribution. Instead of Gaussian copula you can try elliptical t-copula (but note that it's symmetric) or empirical copula. Yes, Gaussian copula and other normality assumptions are highly criticized in many papers for underestimating the tail risks.

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  • $\begingroup$ Thanks for your reply, but if I were to use historical VaR on 20 assets (or even 6 as in your example), wouldn't it suffer from the curse of dimensionality? I thought that's why I needed to make distributional assumptions. $\endgroup$ – user2303 Apr 20 '12 at 10:58
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    $\begingroup$ It won't. Historical VaR is by far most used VaR approach in banks (I would say 90% of banks are doing it for really large portfolios). $\endgroup$ – Alexey Kalmykov Apr 20 '12 at 11:12
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    $\begingroup$ Thanks, one last question. You say that a Gaussian copula suffers the same tail risk problems as a multivariate normal, but what about a Gaussian copula with non-Gaussian marginals? Couldn't that fit fat tails? $\endgroup$ – user2303 Apr 20 '12 at 11:26
  • $\begingroup$ Historical VaR will not measure events that "have not already happened" in your data set. Hence, you will get a more general result if you do some distributional assumptions. $\endgroup$ – AdAbsurdum Apr 20 '12 at 14:33
  • $\begingroup$ @AdAbsurdum There always exists a model risk trade-off $\endgroup$ – Alexey Kalmykov Apr 20 '12 at 14:46
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It depends on the assets which copula is best and other methods may still be better and comparable in complexity.

If you want to use copula's for equities you can take a look at Clayton copula. While the Gaussian copula is symmetric the Clayton copula has asymmetric tail dependency. This makes modeling the increase in correlation during a crisis possible.

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I would start by studying the distribution of the returns. Normal or not? (probably not).

Then, study the behavior of your stocks one against the others. Other actors to take into account are your data, and your requirements in terms computational time.

All that will allow you to decide whether VaR or copula, and which particuliar methodology to follow.

(Last time I recommended a book I had got a penalty, pero bueno ..)

You will find hints in Frequently Asked Questions in Quantitative Finance (P.Wilmott) and Market Risk Analysis Vol.3 (C.Alexander).

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