# Are minimum-risk and minimum-variance portfolios equivalent?

When reading a paper by DeMiguel and Nogales (2007; http://papers.ssrn.com/sol3/papers.cfm?abstract_id=911596), I came across the following formulation:

Comparing the proposed minimum-risk portfolios to the traditional minimum-variance portfolios, we observe that the proposed portfolios have more stable weights than the traditional minimum-variance portfolios,...

And I am left wondering: is a minimum-risk and minimum-variance portfolio the same thing? Usually, risk is measured by standard deviation, if I'm not mistaken, which is not equal to variance; however, a larger variance means larger risk and vice versa. Did the authors just want to avoid repeating themselves?

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## 3 Answers

The problem lies in the definition of risk.

It seems that in the cited paper, the authors treat risk as a concept connected with the uncertainty of the out-of-sample performance of the portfolio. In that way portfolios constructed using the proposed robust estimators would be what they call minimum-risk portfolios. Contrasted with minimum-variance portfolios, which aim to minimize the in-sample variance, they may differ significantly. So, at least in my perception, that's where the difference comes from.

In a general context you may see minimum-risk and minimum-variance portfolios used interchangeably as synonyms. Indeed, standard deviation (square root of variance) is most commonly used as a means to express risk (especially in the Markowitz's optimization context) so that may be the cause of your confusion.

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Risk is a broader concept than variance. That paper is specifically focused on robust estimators (i.e., estimators that are less sensitive to outliers) of dispersion. A robust estimator of dispersion is not the same thing as variance (which may be a dispersion parameter for some classes of distributions). Nevertheless, these robust estimators could be used to estimate portfolio risk. Other approaches to minimizing risk could include minimizing conditional Value at Risk/Expected Shortfall.

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Answering "No" to the title question, I'll mention that variance is a rather poor measure of risk, even if convinient and nicely behaving.

Variance is not even a risk measure, with the standard deviation eventully being a deviation risk measure, while not necessarily for downside risk (see David Nawrocki-"A Brief History of Downside Risk Measures" for historical alternatives)

Nowadays the expected shortfall seams to be the most fashionable (tail) risk measure for portfolios (ex: Turan G. Bali, K. Ozgur Demirtas and Haim Levy: "Is There a Relation Between Downside Risk and Expected Stock Returns?")

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