I'm trying to do a portfolio optimization with an expected shortfall constraint. For this, it is necessary to know the distribution of expected portfolio returns. When doing this empirically, my plan is to assume a small investment universe in which all assets' daily returns are known. If the portfolio is optimized with a 100-day moving horizon, then on day 101, I have 100 past returns for each asset, which would also allow me to get an estimate of each of their distributions. But if I now put them together in a portfolio, what is the way to determine the portfolio return distribution?

I thought about an application of the central limit theorem and then assume a normal distribution accordingly, but I will not have enough assets for this, and they will not be evenly weighted in the portfolio (in fact, the portfolio will be robust, therefore only quite few assets will have positive weights).

Alternatively, is it possible to take some defined distribution that is known to approximate portfolio returns more or less well, and then take this as a basis for the ES-calculation? I guess that normal and lognormal distribution both don't fit very well, so is there an alternative?

I'm planning to use stock indices as the assets in my universe.


2 Answers 2


There is a formula for calculating ES from a normal distribution. There is also a formula for ES of arbitrary distributions using a Cornish-Fisher expansions (easy for univariate processes but frustrating for multivariate).

However, the most common approach is a scenario representation of the distribution. This could include using the historical distribution or it could include estimating parameters to a distribution and simulating many times from it. The second approach is very arbitrary and gives you a lot of freedom to incorporate whatever non-normalities or time-varying properties you feel are important.


In the paper (Klaassen, 2002) the author propose a scenario generation method as well as a rule in order to precludes arbitrage opportunities.

In the paper (Davari-Ardakani, et al. 2016) authors propose a new scenario generation method, it preserves marginal (non-normal) distributions of asset returns and it precludes arbitrage opportunities.


Klaassen, P. (2002) Comment on ‘‘Generating scenario trees for multistage decision problems’’. Management Science 48 (11), 1512–1516.

Hamed Davari-Ardakani, Majid Aminnayeri and Abbas Seifi (2016) Multistage portfolio optimization with stocks and options. International Transactions in Operational Research, 23, 593–622.

  • $\begingroup$ You should also include links to your references, it would enhance your answer. $\endgroup$
    – SRKX
    Nov 7, 2016 at 5:42
  • $\begingroup$ if scenario generation means "simulating many times", how can it be possible to preserve marginal distributions of asset returns? what does preserve mean here $\endgroup$
    – develarist
    Dec 4, 2020 at 1:58

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