Portfolio Frontier Visualizations Beyond Volatility and Return

Summary

The document considers what a third dimension could add to a mean-variance efficient-frontier visualization, whose familiar axes are portfolio volatility and expected return. It explains that these two statistics can describe the trade-off when returns are normally distributed, but volatility may miss important features of non-normal portfolio outcomes.

Suggested alternatives for representing risk include value at risk, conditional value at risk or expected shortfall, maximum loss, and downside semi-variance. A third axis could instead encode portfolio attributes such as sector, country, or style exposure, or other investor priorities. More broadly, a utility function can combine several dimensions to express how an investor evaluates a portfolio. The discussion offers candidate measures and a conceptual framework, but no plotted example, empirical comparison, or recommendation that one measure is universally best; the useful choice depends on the investor’s objectives and the distribution of returns.

Key ideas

• A mean-variance frontier maps expected return against portfolio volatility.
• Volatility may be inadequate as a risk proxy when portfolio returns are non-normal.
• Alternative risk dimensions include expected shortfall, maximum loss, and downside semi-variance.
• A visualization can also show exposures or investor priorities beyond conventional risk and return.
• Utility functions can combine multiple dimensions, with choices depending on investor objectives.

Tags

# Surface plots of the mean-variance efficient frontier


# Surface plots of the mean-variance efficient frontier












3d surface plots contain an X, Y and Z axis. For the mean-variance efficient frontier:

- X axis is portfolio volatility ($\sigma_p$)

- Y axis is portfolio expected return ($\mu_p$)

any ideas for what could be used for the Z axis? I know the higher moments skewness and kurtosis can be of interest (any sources that look at these for surface plot?), but what else can be used for Z in the context of portfolios?

## Answer by Steinwolfe (score 1)

https://quant.stackexchange.com/a/50099

The concept behind showing volatility vs expected return is that a risk averse investor will wish to minimise risk, and maximise return.

However, how good a proxy is volatility for risk? Given a normal distribution, risk and return (sigma and mu) alone will suffice. But especially for non-normally distributed returns (as alluded to by the mention of higher moments) we might be interested in other risk sensitivities, such as VaR (Value at Risk) or CVaR (Conditional Value at Risk or Expected Shortfall), maximum loss, negative semi-variance (downside only risk), or possibly something else such as how much weighting the portfolio gives to various sectors, countries, socially responsible stocks, growth or value assets, etc., or any arbitrary weightings of what we consider important other than simply risk and return.

More generally, a Utility function could arbitrarily weight various other dimensions depending upon our value system, and makes a good additional dimension on which to consider "how optimal is our portfolio?".

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.