Apr
9

## Value at Risk Estimation with Copula

Value at Risk is widely used to measure the downside risk, and Copula is a generalized dependence structure instead of linear correlation to model dependence, especially for lower tail dependence, therefore the combination of VaR with Copula is fantastic in terms of accurately capturing the true risk embedded.

I read roughly a working paper

For example, below are two simulated return series, one is under Gaussian copula and the other one is under Student t copula, as you can easily see, although both have the same marginal distribution, Gaussian copula has much smaller upper and lower tail dependence than Student t copula, which eventually underestimates the Value at Risk and other risk measures.

I would stay away Gaussian Copula if I were a risk manager, and you? Download Copula toolbox and other code files at Copula if interested.

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I read roughly a working paper

*Value at Risk – MATLAB Application of Copulas on US and Indian Markets*, where the authors calculate the Value at Risk (VaR) using the bivariate Gaussian Copula distribution implemented in MATLAB for the Dow-Jones index and the National Stock Exchange index. It is good to use Gaussian Copula together with some rank correlation (like Spearman's rho or Kendall tau) to model the dependence, however, we must be very careful as basically Gaussian Copula assumes the joint dependence structure normally distributed and as a result, no matter which marginal distribution you choose, the upper and lower tail dependence approach to zero when the significant level limits to one and zero, in other words, in a bivariate case, the probability that X2 exceeds its q-quantile, given that X1 exceeds its q-quantile (upper tail dependence) when q->1, and the probability that X2 is below its q-quantile, given that X1 is below its q-quantile (lower tail dependence) when q->0 are zero.For example, below are two simulated return series, one is under Gaussian copula and the other one is under Student t copula, as you can easily see, although both have the same marginal distribution, Gaussian copula has much smaller upper and lower tail dependence than Student t copula, which eventually underestimates the Value at Risk and other risk measures.

I would stay away Gaussian Copula if I were a risk manager, and you? Download Copula toolbox and other code files at Copula if interested.

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