Estimating Portfolio Expected Shortfall with Scenarios
Summary
The document considers how to estimate a portfolio’s return distribution for an expected shortfall constraint when only a limited history of asset returns is available. It questions whether a normal approximation based on the central limit theorem is suitable when the portfolio contains relatively few assets and weights are concentrated, and notes that normal or lognormal assumptions may not represent returns well.
The proposed practical direction is to represent the distribution with scenarios. Scenarios may use historical portfolio outcomes directly, or may be generated by estimating a model and simulating returns. The latter allows non-normal features and time-varying behavior to be incorporated, but leaves substantial modeling discretion. The replies also mention Cornish–Fisher expansions for expected shortfall and research on scenario generation that preserves non-normal marginal distributions while addressing arbitrage. These approaches are options rather than a tested recommendation; the thread does not compare their performance or specify a preferred estimation window, dependence model, or validation procedure.
Key ideas
- Expected shortfall optimization requires an estimate of the portfolio return distribution.
- A normal approximation may be questionable for concentrated portfolios with few positively weighted assets.
- Historical outcomes or simulated scenarios are common ways to represent portfolio return distributions.
- Simulation offers flexibility to model non-normality and time variation, while making results dependent on modeling choices.
- Scenario-generation research can preserve non-normal marginal returns and impose arbitrage restrictions.
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# Determining the portfolio return distribution to calculate CVaR/ES # Determining the portfolio return distribution to calculate CVaR/ES 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. ## Answer by John (score 5) https://quant.stackexchange.com/a/8429 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. ## Answer by Nick (score 2) https://quant.stackexchange.com/a/30888 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. References 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.
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