# Estimation of Empirical Expected Shortfall of a heavy tailed distribution

Assume that you have a portfolio for which you have estimated a parametric model to the underlying instruments, but the distribution of the portfolio as a whole is too complicated to compute explicitly. Now you want to determine the expected shortfall by Monte Carlo simulations.

We know that for our r.v. $Y$ the empirical cdf can be estimated by $$\hat{F}_Y(y)=\frac{1}{n}\sum\limits_{i=1}^n I(Y_i \leq y)$$ and the quantiles can be estimated by $$\hat{y}_q=\text{inf}[y:\hat{F}_Y(y)\ge q] =\Upsilon_{[nq]+1}$$ where $\Upsilon_i$ is the i:th order statistic. Thus the ES can be estimated by $$\widehat{ES}_p(Y) = \frac{1}{p}\left(\sum\limits_{i=1}^{[np]}\frac{\Upsilon_i}{n}+\left(p-\frac{[np]}{n}\right)\Upsilon_{[np]+1}\right)$$

However, as we will see for this numerical approximation is that it converges very slow for increasing sample size N! This is illustrated with an example where the random variable Y is standard normal (the x axis is N/100)

Maybe you could naively repeat the simulation for fixed N (sufficiently large, eg. ~200*100) and then take the mean. But isn't there any other techniques that deal with this problem (especially in the case of heavy tails)? I've managed to find several different methods, for example using control variates, importance sampling, delta-gamma approximation etc. But none of these doesn't apply to the case of empirical ES.

All comments, including references to articles, are welcome!

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I've been confused by this question since when I estimate the empirical ES/CVaR, I normally don't simulate anything. For instance, for a portfolio of equities, I would use the current portfolio (with some assumption about rebalancing) with the historical returns and then follow the above formulas to get the ES. Are you concerned with securities that wouldn't have historical returns in the same way, like exotic options? (so then you simulate the returns for those conditional on everything else, and calculate the distribution of ES) –  John Jan 24 '14 at 17:04
This may be too obvious but have you tried bootstrapping? –  user25064 Jan 27 '14 at 19:29