# 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?

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.

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.

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?")