# Equivalence of Standard Deviation and Variance as a risk measure - WRONG?

In Modern Portfolio Theory, I often see that people seem to view Standard Deviation and Variance as equivalent. Example from Markowitz himself: "Thus far I have used the standard deviation (or equivalently, the variance) of return as a measure of the risk involved in the portfolio” (p.50, see reference below).

I have the following thoughts about this (in one case it is harmless/correct, but wrong in the second case):

1. If we do ordinary portfolio optimization, i.e. maximize the expected return for a given variance. Then variance and Standard Deviation is equivalent since there is a one-to-one correspondence between them. So it does not matter which measure of risk I use in this situation since we can go from one of the measures to the other in a unique way.

2. If we want to understand risk, I believe Variance and Standard Deviation is NOT Equivalent. This because Standard Deviation is in general sub-additive (i.e. Diversification never leaves you worse off) whereas Variance is in general NOT sub-additive (i.e. Diversification can leave you worse off).

So as a conclusion: Standard Deviation and Variance is NOT equivalent.

Am I overlooking something?

Source: [Markowitz, H. M. (1976). Markowitz revisited.Financial Analysts Journal,32(5):47–52]

• They are not equivalent for every purpose, certainly. But for minimization, they are. I.e. $\min \sigma$ subject to such and such conditions is equivalent to $\min \sigma^2$ subject to the same conditions. Put differently, one is a monotonic transformation of the other, and this can be exploited in optimization problems such as Markowitz was concerned with. But in other situations they have to be kept clearly distinct and separate, yes. – noob2 Mar 9 at 15:47
• They are different in terms of estimation - for example it is non-trivial or at least non-obvious how one gets an unbiased estimator of the standard deviation, whereas it is fairly easy to get an unbiased estimate of the variance. This is due to Jensen's inequality. – rubikscube09 Mar 10 at 14:17

## 1 Answer

I would disagree more with the initial premise (ie the intellectual "straw man" you then kill) than with any of your perfectly cogent arguments that follow.

I trust we both accept that stdev and var are monotonic in nature, each being the square (root) of the other?!?! The basic premise of Markowitz is thus indeed gloriously indifferent to which measure is used... IF the objective to maximise returns for any given level of acceptable stdev or var; or to minimise these for any required level of (targeted) return. Markowitz indifference between stdev versus var is thus - to me - perfectly defensible...

The issue then becomes whether one should attempt to maximise returns with respect to volatility (ie MaxSharpe) or with respect to variance (ie Kelly-betting)...

The former is defensible in the sense that it maximises the probability of a positive outcome. The latter is defensible, in the sense that it maximises logarithmic wealth/growth.

Where I suspect the false distinction/dichotomy lies is in exactly your point about only one being additive... If you only cared about 12m (or n-month) returns, then volatility/sigma matters. But change your time horizons, and you will get a different "answer" using a Sharpe measure that is predicated on a different stdev (that is root time vol). So MaxSharpe only really makes sense if one chooses to specify a specific time-horizon. Choose any different horizon, and the maximum probability of profit will differ.

That is, the portfolios with the max p(profit) over the next 1d, 1w, 1m, 3m, 12, 3y, or 5y can/will/might look very different. Precisely because (log)returns are time-additive but volatilities are not (precisely as you say).

Yes, this temporal mismatch disappears using variances instead of vols... but then that takes you to other strange places you might not wish to visit... like you might believe that stocks should return 2% over riskless for 14% vol... causing your "optimal" positioning to suggest levering up 10x stockmarkets at historically uber-pricy valuations ;-)

To summarise: appreciating that vol and var are not identical is indeed useful; but, equally, playing them off against each other rarely helps in any pragmatic sense. I suppose the challenge here is to identify if and how using one versus the other could really bggr-up versus save one's portfolio. That I find a toughie... but stand VERY ready to get corrected ;-)

thank you very much for the food for thought, much appreciated. DEM