Why Mean-Variance Utility Penalizes Variance Rather Than Standard Deviation
Summary
The document asks why portfolio utility commonly subtracts a variance penalty instead of a standard deviation penalty. For a single risky asset, a position scales expected return linearly and variance quadratically, which can produce a finite optimal holding; standard deviation also scales linearly, so this simple setup may not yield an interior optimum under a fixed risk-aversion coefficient.
The answer distinguishes known parameters from estimated ones. With known parameters, variance and squared standard deviation rank choices identically, including in a mean-risk tradeoff. With estimated parameters, taking the square root of an unbiased variance estimate generally introduces bias, and the choice of functional form can affect calculations. Variance is also the second central moment and, when a moment-generating function exists, contributes to a moment sequence that characterizes a distribution. The discussion is conceptual: it does not establish that variance is universally preferable, and it notes that estimation assumptions matter.
Key ideas
- Variance grows quadratically with position, while standard deviation grows linearly in the single-asset setup described.
- With known parameters, variance and squared standard deviation lead to the same risk ordering.
- Taking the square root of an unbiased variance estimator generally does not give an unbiased standard deviation estimator.
- When parameters are estimated, the risk measure and estimation framework can change the resulting calculations.
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Full text
# Why does risk aversion use variance instead of standard deviation?
# Why does risk aversion use variance instead of standard deviation?
The risk-aversion component of a portfolio utility function is expressed as the variance of the portfolio. Why the variance, instead of standard deviation, is used in here?
I'm asking this question because of the following calculation: suppose I only have a single stock and also a fixed risk-aversion parameter. Then I use mean-variance tradeoff to determine the optimal amount of the stock to hold. If the variance is used in the utility function, then I can get an optimal position because the variance is quadratic of the position. However, if standard deviation is used, then both the expected return and risk are linear in the position, hence we cannot get an optimal position in this setup. On the other hand, if indeed standard deviation is also a reasonable choice of expression of risk aversion, then it seems the "optimal position" obtained using the variance is purely an artifact of the functional form selected.
## Answer by Dave Harris (score 2)
https://quant.stackexchange.com/a/46339
Let us start with some underlying math. First, $\sigma=\sqrt{\sigma^2}$, but the minimum variance unbiased estimator (MVUE) for standard deviation is not the square root of the MVUE of the variance, $\hat{\sigma}\ne\sqrt{\hat{\sigma^2}}.$ Taking the square root of the unbiased sample estimator of the variance introduces bias because it is a non-linear function. See derivation of MVUE of SD If the parameters are known, then it doesn't matter which way you do it.
Standard Frequentist stochastic calculus assumes the parameters are known, even though they are not. Nonetheless, there is an advantage, if you believe people are going to create estimates from the formula to using the variance in that it is the second central moment of a distribution. If a moment generating function exists, then the moment generating function uniquely defines the distribution.
If the parameters are known, then minimizing the variance and minimizing the square of the standard deviation are the same thing. It would also be true in a mean-variance tradeoff situation. If the parameters are not known, and partly depending on your estimation assumptions such as Frequentist, Likelihoodist or Bayesian, then your functional form affects calculations. As economics is primarily a Frequentist discipline, mostly by default, the distinction matters, except in the exceedingly rare case where the parameters were actually known.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.