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enter image description here

The mean-variance efficient frontier holds the minimum variance portfolio, but in the graph above it shows that the minimum VaR (Value-at-Risk) and minimum ES (CVaR) portfolios (expected shortfall/conditional VaR) lie on and share the same frontier as the minimum variance portfolio.

I thought though (and have seen in articles) that there are frontiers unique to the mean-VaR and mean-ES efficient frontiers? Which is right?

Source

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When returns follow an elliptical distribution (e.g. the Gaussian distribution), then minimising VaR and ES is equivalent to minimising variance. See https://people.math.ethz.ch/~embrecht/ftp/pitfalls.pdf. Then, the frontiers will be the same.

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    $\begingroup$ Interesting. Are there a distribution that the efficient frontiers would be different? $\endgroup$ – Preston Lui Oct 30 at 14:12
  • $\begingroup$ @PrestonLui, anything nonelliptical should do. I think one can be constructed using nonelliptical copulas (there are many) even if the marginals are elliptical, and even more easily if they are not. $\endgroup$ – Richard Hardy Oct 30 at 16:45
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This is a result of the two fund separation theorem or mutual fund separation theorem. Any (optimal) portfolio choice will take place on the efficient frontier. In a Markowitzian world, the asset universe is fully characterised by first and second (co-)moments. Hence, for any performance metric, you would always be able to obtain "more return at a given risk" or "less risk at a given return" by simply moving your portfolio towards the efficient frontier. The VaR and ES metrics are (simply) combinations of portfolio mean and risk: Hence, they can be improved by "moving left/up" as well.

What may be observed, though, is a different diagram depicting mean-return vs VaR, or mean-return vs ES. They may look different.

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