Choosing Historical Simulation or Copulas for Portfolio VaR

# Do I need a copula to accurately estimate the VaR of a portfolio of risky assets?


# Do I need a copula to accurately estimate the VaR of a portfolio of risky assets?












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?

## Answer by Alexey Kalmykov (score 6, accepted)

https://quant.stackexchange.com/a/3299

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.

## Answer by Bob Jansen (score 3)

https://quant.stackexchange.com/a/3298

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.

## Answer by tagoma (score -1)

https://quant.stackexchange.com/a/3300

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).

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).

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).

(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).

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