Minimum-Variance Portfolios and Broader Definitions of Risk
Summary
The post asks whether minimum-risk and minimum-variance portfolios are equivalent, prompted by a paper that distinguishes proposed minimum-risk portfolios from traditional minimum-variance portfolios. The answers explain that the distinction depends on how risk is defined. In Markowitz-style optimization, risk is often represented by variance or its square root, standard deviation, so the terms may be used loosely as synonyms.
In the cited discussion, however, minimum risk can refer to out-of-sample performance uncertainty or to portfolio construction using robust dispersion estimators, while minimum variance minimizes in-sample variance. These objectives may produce different portfolios. Risk can also be measured using downside-focused approaches such as conditional value at risk or expected shortfall. The discussion is conceptual rather than an empirical comparison: it gives no portfolio data or performance results, and it does not establish that one risk measure is universally preferable. The useful takeaway is to inspect the measure and estimation method behind a portfolio’s risk label before comparing it with a variance-minimizing alternative.
Key ideas
- Minimum variance is one specific way to define and estimate portfolio risk.
- Minimum risk may refer to robust dispersion estimates or uncertainty in out-of-sample performance.
- Standard deviation is the square root of variance and is commonly used to express risk.
- Alternative portfolio risk objectives include downside measures such as expected shortfall.
- Portfolio labels alone do not reveal whether two optimization objectives are equivalent.
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Full text
# Are minimum-risk and minimum-variance portfolios equivalent? # 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? ## Answer by Karol J. Piczak (score 4, accepted) https://quant.stackexchange.com/a/8221 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. ## Answer by John (score 3) https://quant.stackexchange.com/a/8220 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. ## Answer by user7056 (score 2) https://quant.stackexchange.com/a/8260 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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