Why Mean-Variance Plots Use Standard Deviation

Summary

The note explains why portfolio theory commonly plots expected return against standard deviation rather than variance. Its example combines a risk-free asset with a risky asset: scaling the risky position by one half scales standard deviation by one half, while variance falls to one quarter. Consequently, the portfolio appears halfway between the two assets on a standard deviation axis, but not on a variance axis.

This makes the risk-return relationship for combinations of a risk-free and risky asset a straight line when risk is measured by standard deviation, and a curve when it is measured by variance. The geometric simplicity is the practical reason for the usual convention. The explanation is limited to this setting; it does not argue that variance is invalid or that standard deviation is always the best risk measure. Both forms of graph can be used.

Key ideas

• Standard deviation scales linearly when exposure to a risky asset is scaled against a risk-free asset.
• Variance changes quadratically with that exposure.
• The standard deviation and return plot makes such portfolio combinations linear.
• Variance and return plots remain possible but produce a curved locus in this example.

Tags

# Mean-Variance Portfolio Axis Description


# Mean-Variance Portfolio Axis Description












I'm currently looking into the mean-variance approach to portfolio theory and I wonder, why the standard deviation $\sigma$ is graphed on the x-axis and not the variance $\sigma^2$ as a measure of volatility (as the name would indicate). Does anybody know the reason for that?

## Answer by nbbo2 (score 5, accepted)

https://quant.stackexchange.com/a/67989

Suppose you have a risk-free security R and a risky security B. A portfolio with a 0.50, 0.50 combination will have a standard deviation of $0.5 \sigma_B$, but a variance of $0.25 \sigma_B^2$. So if you draw it in Standard Deviation space it will be half way between R and B, in Variance space it won't be. This linearity is the reason it is more convenient to draw (std dev, return) space rather than (variance, return) space.

The locus of all combinations of R and B will be a straight line in the (std dev, ret) diagram, it will be a curve in the other diagram. Hence the former is usually preferred (although the other is also sometimes used).

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