Skip to content
All library documents

Why Mean-CVaR Portfolios Use Expected Return Instead of Median Return

Article Quant Q&A · Author: Eiffelbear
Read the original

Summary

The document asks why portfolio optimization commonly combines expected return with conditional value at risk (CVaR), rather than using median return as the performance input. The answer suggests that mean-CVaR optimization is familiar partly because mean-variance optimization is an established framework, while a parallel median-based model has not become conventional.

It distinguishes the mean, an algebraic statistical quantity with convenient properties, from the median, which is defined by ordering observations. The discussion also notes that mean-return estimates can be unreliable and points to robust statistics and resistant regression as examples where median-based estimators and absolute-loss methods can be useful. It does not present a median-CVaR model, empirical comparison, or proof that such a model is absent or inferior. Whether median inputs improve portfolio outcomes enough to justify a distinct approach remains an open research question in the text.

Key ideas

  • Mean-CVaR optimization pairs expected return with a tail-loss measure as an alternative to mean-variance optimization.
  • The answer attributes the lack of a common median-CVaR framework partly to the established use of mean-based portfolio models.
  • Median-based estimators and absolute-loss methods appear in robust statistics and resistant regression.
  • The document gives no empirical evidence on whether median return inputs improve portfolio solutions.

Tags

Portfolio constructionRisk managementStatistics
Full text
# Why no median-CVaR optimization for portfolios?


# Why no median-CVaR optimization for portfolios?












#### Question

- Since CVaR is a concept that can be applied to all probability distribution, even if they do not follow normal distribution, I thought CVaR should be more concerned with median, not the mean of return of any asset.

- However, I haven't heard any about 'Median - CVaR portfolio optimization technique', whereas we have a mean-CVaR portfolio optimization technique. You can read research papers like 'CVaR Robust Mean-CVaR Portfolio Optimization' to get more information about 'mean-CVaR portfolio'

- I am wondering why there is no such thigs as 'median - CVaR portfolio optimization'.

## Answer by develarist (score 1, accepted)

https://quant.stackexchange.com/a/49544

Mean-CVaR portfolio optimization is an alternative to the more widely known and simpler mean-variance model. Since there doesn't seem to be any median-variance model out there, the familiarity surrounding the traditional model stuck.

The mean is an actual equation with convenient properties in statistics, whereas the median is obtained through a counting procedure. The use of median asset returns is absent in finance, even though estimates of mean asset returns have been shown to be unreliable.

Does calculating median asset returns as an input drastically improve the portfolio solution enough from the usual mean asset returns to write a whole paper on the topic?

Robust statistics and resistant regression models often use estimators based on the median instead of the mean, and the absolute loss function instead of the widely used squared loss function, so you have a good point anyway.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.