Estimation of Empirical Expected Shortfall of a heavy tailed distribution

Question

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!

Answer 1

Glassermann et al. have published an approach where the loss distribution is approximated by a quadratic function in the risk factors. Based on this estimation they can apply importance sampling and stratified sampling to reduce the variance of the monte carlo estimate. I have not implemented their technique, but their numerical results look very good.

You can find the paper here: "Variance reduction techniques for estimating Value-at-Risk", R. Glassermann, P. Heidelberger, and P. Shahabuddin, Management Science, 46(10), p. 1349-1364, 2000.

Answer 2

The best approach depends very much on your specific requirements and constraints. For example if you know a importance sampling distribution, you can estimate ES but you need to weight the observations in the empirical distribution much as you do when using the MC-estimator for the expectation see these slides. I never really found importance sampling in practice useful though.

I did successfully implement the approach considered here. The "reduced model" which is necessary for this can be derived from a delta-gamma model, i.e. quadratic approximation model.