Surface plots of the mean-variance efficient frontier

Question

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 1

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?".