Comparing Mean–EVaR and Other Portfolio Efficient Frontiers

Summary

The document asks how an efficient frontier based on expected return and Entropic Value-at-Risk (EVaR) should compare with frontiers built using variance or Conditional Value-at-Risk (CVaR). It notes that EVaR is an upper bound on VaR and CVaR, then asks whether this implies a predictable vertical placement for the mean–EVaR curve. The responses caution that frontier comparisons across risk measures depend on the space used to evaluate portfolios.

In mean–variance space, portfolios optimized under CVaR or EVaR need not be efficient, and the answer states that both corresponding frontiers are contained within the mean–variance frontier. Whether the mean–EVaR frontier falls between the mean–variance and mean–CVaR frontiers depends on the selected securities. A second response points toward research on entropic portfolio optimization, but supplies no derivation or empirical illustration. Thus the key lesson is that a pointwise bound between risk measures alone does not determine the relative geometry of efficient frontiers; portfolio universe and risk representation matter.

Key ideas

• EVaR is described as an upper bound on VaR and CVaR.
• Efficient frontiers built under different risk measures cannot be compared without specifying the evaluation space.
• In mean–variance space, the answer places mean–CVaR and mean–EVaR frontiers within the mean–variance frontier.
• The relative placement of mean–EVaR and mean–CVaR depends on the available securities.

Tags

# Mean-EVaR efficient frontier


# Mean-EVaR efficient frontier












Entropic Value-at-Risk (EVaR) is an alternative and more efficient risk measure than conditional Value-at-Risk (CVaR). EVaR serves as an upper bound to both VaR and CVaR.

Below is a graph of the mean-variance efficient frontier and the mean-CVaR efficient frontier

https://www.scipedia.com/wd/images/e/e9/Wang_2020a_2053_Figura3.png

What would the mean-EVaR efficient frontier look like compared to the two shown, given that EVaR is an upper bound to CVaR? How would its curve be placed. in-between the two frontiers shown, lower than both?

## Answer by Nipper (score 2)

https://quant.stackexchange.com/a/69853

It makes very little sense compare efficient frontiers across different risk measures, as per your attached picture. That is because efficient Mean-CVaR portfolios are always sub-optimal compared to the efficient Mean-Variance ones within the Mean-Variance space and vice-versa. Therefore, if the Mean-Variance space is considered both Mean-CVaR and Mean-EVaR efficient frontiers are always contained by the Mean-Variance one. Whether or not the Mean-EVaR efficient frontier lies in between the other frontiers depends on the considered subset of securities.

## Answer by Pedro (score 1)

https://quant.stackexchange.com/a/61397

Yes, you can check the paper Entropic Portfolio Optimization: a Disciplined Convex Programming Framework in SSRN.

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.