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I've been a researching minimum variance portfolios (from this link) and find that by building MVPs adding constraints on portfolio weights and a few other tweaks to the methods outlined I get generally positive returns over a six-month to one year time scale.

I am looking to build some portfolios that are low risk, but have good long term (yearly) expected returns. MVP (as in minimum variance NOT mean variance) seems promising from backtests but I don't have a good intuition for why this works.

I understand the optimization procedure is primarily looking to optimize for reducing variance, and I see that this works in the backtest (very low standard deviation of returns).

What I don't have an intuitive feel for is why optimizing variance alone (with no regards to optimizing returns, i.e. no mean in the optimization as in traditional mean-variance optimization) gives generally positive returns. Any explanations?

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What number of stocks is practical for building MVP? I suspect few stocks or huge number will not work. – user2296 Apr 15 '12 at 3:20

12 Answers 12

up vote 29 down vote accepted

The minimum variance solution loads up on securities that have low variances and co-variances. Theoretically you are correct that this should have a low expected return profile.

However, it turns out - in contradiction to modern portfolio theory - that securities that have low-volatility or low-beta experience higher returns than high-volatility or high-beta stocks. This is well-documented in the literature as the low-volatility anomaly. As a result, many funds and ETFs have been launched in recent years to exploit this phenomenon.

There are a couple arguments as to why the anomaly exists. The paper I cite above argues that institutional investor objectives and constraints create the anomaly:

Over the past 41 years, high volatility and high beta stocks have substantially underperformed low volatility and low beta stocks in U.S.markets. We propose an explanation that combines the average investor's preference for risk and the typical institutional investor’s mandate to maximize the ratio of excess returns and tracking error relative to a fixed benchmark (the information ratio) without resorting to leverage. Models of delegated asset management show that such mandates discourage arbitrage activity in both high alpha, low beta stocks and low alpha, high beta stocks. This explanation is consistent with several aspects of the low volatility anomaly including why it has strengthened in recent years even as institutional investors have become more dominant.

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The following papers may help.

A New Look at Minimum Variance Investing by Bernd Scherer

Minimum Variance Portfolio Composition by Clarke, De Silva & Thorley

Under a multifactor risk-based model, if the global minimum variance portfolio dominates the market portfolio, the implication is that the market portfolio is not multifactor efficient and that the global minimum variance portfolio is picking up some risk-based anomaly. In this case, it shows up at beta / idiosyncratic volatility - however, that still needs to be connected with some underlying characteristics of the assets. (if low market beta beats high market beta stocks with less risk and more return, that's fishy)

You may try these articles, both by John Cochrane to get some idea about how financial economists think about these issues. Particularly, see figure no. 2 of the 4th paper and the associated explanation.

New facts in finance by John H. Cochrane

Portfolio advice for a multifactor world by John H. Cochrane

Though they do not talk about your problem directly, you may get an idea about why the global min. variance portfolio may show the sort of behavior it does. (of course, this implies that you want to work with the mental model of mean-variance, CAPM, multifactor model, of some version of a risk-based explanation of returns - which you implicitly are, if you are concerned with the global min. var anyway)

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Cochrane has good stuff. Fama and French's three factor model is a great starting point. It will also explain growth versus value. – MathAttack Feb 4 '12 at 7:29

Unlike the tangency portfolio on the efficient frontier (which represents the most efficient portfolio in terms of max expected sharp ratio), min var portfolios have no ex ante theory that suggests it should outperform a cap weighted market portfolio. The same can be said about other risk-weighted portfolio construction schemes, including equal risk contribution portfolios or max diversity portfolios.

In practice, however, expected returns are notoriously difficult to accurately forecast (the expected covar matrix is as well, but to a lesser extent). This often leads to poor out-of-sample performance. Risk-weighted portfolios ignore potentially error prone expected return forecasts.

Additionally, there is reason to believe high beta / high vol names get bid up as certain investors seek short-term large gains ('lottery ticket' effect in the behavioral finance literature). As others have alluded to, this tends to result in 'cheaper' offers for lower beta names (beta puzzle, or low vol anomalies). The unconstrained min var portfolio is (heavily) tilted towards lower vol names, and thus does not buy into rich high vol names.

Min var portfolios tend to be more resilient in market downturns. As the business cycle eventually recovers, a min var portfolio tend to start compounding performance at a far higher capital base than a similar mkt cap weighted portfolio.

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The minimum variance optimization framework does not guarantee positive return whatsoever.

As a matter of fact what you are trying to do is something close to the following:

$$\underset{w}{\arg \min} \quad w' Q w \quad \text{s.t} \quad Aw \leq b,\quad \sum_i w_i=1$$

The fact that you get positive return is a nice result that you get from your backtest (i.e a coincidence in a sense), but it should not be the reason why you choose this asset-allocation technique.

One of the main advantages of minimum variance is that it actually removes expected returns from the optimization (which are difficult to handle, see this post for more details).

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Any explanations? Yes. Within each asset category we find that stocks may be:

  1. Unattractively underperforming the category norm
  2. Attractive as they meet the expected norm
  3. Unsustainable as their returns exceed the category norm and may suffer mean reversion

By focusing on low variance, we exclude type (3) stocks that damage portfolio performance through high variance and so we achieve index-beating returns.

I have posted a paper that recommends portfolio optimisation through holding only stocks seen as attractive: type (2). The paper includes an investment coefficient for ranking stocks according to strength of EMA less the price premium to the EMA. This ranking serves as a value for expected returns.

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Minimizing risk alone would not imply a positive expected return, except for the following:

The assets that are being included have positive expected returns. If you took a portfolio of assets that had a negative expected return, and minimized their risks, you would probably still end up with a portfolio that has a negative expected return. Most of these portfolios start by including equities and bonds, which each have embedded risk premium.

Why should minimum variance outperform other portfolios, such as 60/40 or mean-variance? Asness, Frazzini, and Pedersen (2011) have laid out an arguement that minimum variance portfolios will typically require using leverage, and this introduces a leverage aversion risk premium.

We fill what we believe is a hole in the current arguments in favor of Risk Parity investing by adding a theoretical justification based on investors’ aversion to leverage and by providing broad empirical evidence across and within countries and asset classes.

Minimum variance and risk parity have the benefit of not requiring an estimation of expected return. It turns out that volatility is one of the most predictable return characteristics, while expected return is inherently very difficult to study and unstable.

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First, you find the (global) minimum variance portfolio, without any restriction on the expected return. Second, you trace the efficient frontier above and to the right of the (global) minimum variance portfolio by finding the (local) minimum variance portfolio for each expected return above the expected return corresponding to the (global) minimum variance portfolio. You do not consider portfolios with an expected return under that of the (global) minimum variance portfolio because each such portfolio is dominated by a portfolio above the (global) minimum variance portfolio for which the variance is the same as that of that portfolio under the (global) minimum variance portfolio. Third, you trace a half-line from the height corresponding to the risk free rate on the y axis through the point of tangency of that half line to the efficient frontier and choose a point on this line according to your aversion for risk. Since you are presumed averse to risk, you want to get paid to endure it, so you will choose a point on this half line such that the expected return of the corresponding portfolio is higher than the risk free rate, thus ensuring a risk premium for yourself on average. Since the risk free rate is usually positive, the expected return of the chosen portfolio will also be positive. QED

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If one performs a simulation of the historical performance of the minimum variance portfolio in U.S. and global stocks, there is no statistically significant alpha remaining after projecting the portfolio's excess returns onto common risk factors (e.g. Fama French). See Tzee-man Chow, Jason Hsu, Vitali Kalesnik, and Bryce Little. "A Survey of Alternative Equity Index Strategies", Financial Analysts Journal, September/October 2011, Vol. 67, No. 5: 37–57 for the results.

What this suggests is minimizing the variance of a portfolio of stocks systematically provides positive loadings on size and value factors. That explains the lion's share of why a minimum variance portfolio performs well in stocks. Note my explanation does not touch on the "low-beta" anomaly which controls for size and value, however, I omit that because it is less relevant to the explanation of the returns of the minimum variance portfolio.

Interestingly, a minimum variance portfolio in Treasury bills, for example, doesn't provide an analogous result. In this asset class, picking government securities with the lowest volatility will almost surely decrease your returns. It is an odd feature of equities that we see such desirable factor loadings in stocks with low sensitivity to the overall market.

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A minimum variance portfolio is going to be the most diversified portfolio you can achieve given your portfolio weighting constraints. A diversified portfolio will nearly always achieve the best risk adjusted returns over the long term into an uncertain and unpredictable future. A max Sharpe portfolio by contrast is not only trying to overcome the uncertainty with the prediction of returns but it is also risk concentrated into it's precarious forecasts and therefore while a max Sharpe portfolio may out perform in certain periods of the short term it is most likely to under perform a min var portfolio over the medium to long term on a risk-adjusted basis.

Therefore if you select the min var portfolio weights, it is possible to leverage these portfolios to achieve one's desired level of portfolio risk or absolute returns, so why would you not choose the more diversified approach, with other positive attributes such a low maximum drawdown etc.?

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I don't agree: Minimum variance is something different than maximum diversification. – vonjd Mar 19 '13 at 15:23
It is as near as makes no difference max diversification, especially if you run real money like i do. – Sanj Mar 19 '13 at 16:26
@Sanj I disagree also. They are not that close. – SRKX Mar 27 '13 at 13:28

Simple, if you're investing in Equities, you have a negative skew, decrease the variance and that will decrease the number of hits you take. Returns are on a log normal scale so, if you have two equal but opposite moves you will still be down money. E.g. If I have 100 dollars and I lose ten percent and then make ten percent, I will still be down one percent.

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Assumptions on Equities skewness or any statistical properties is at the discretion of the investor right? Technically risk should be rewarded by expected return, if that's not the case, then why don't you simply invest in the risk-free rate? – SRKX Oct 17 '14 at 3:25

Most portfolios offer positive returns, and minimum variance portfolios are not exceptions to this rule. But by offering "minimum variance," they also offer the lowest possibility of a negative deviation large enough to pull the actual return (expected return minus deviation), into negative territory.

This property is attractive, which is to say that (all other things being equal), investors will be willing to "pay up" for it, until the anticipated return falls below some other market-based threshold

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One of the more prominent proponents has a current paper on the reasons - and why this anomaly "will persist":

Why the Low Volatility Anomaly Will Persist
by Eric G. Falkenstein, March 2015


Common explanations of the low volatility anomaly involve biases or frictions that cause investors to overpay for high volatility assets, giving them a negative alpha within the CAPM model, yet currently all such mechanisms are either heuristic or partial equilibrium. This paper shows that leverage constraints of Frazzini and Pedersen (2014) alone cannot explain this result if there also exist rational investors. If 3 non-standard assumptions are added — hybrid relative utility, delusional subset of investors, residual systematic risk across beta — then we can capture several facts existing models cannot simultaneously capture: a positive return to the market, positive holdings by rational investors to negative CAPM-alpha stocks, and a negative Security Market Line. New data relevant to these assumptions are presented.

And from the conclusion:

In sum, you need three assumption to generate the low volatility anomaly: beta estimation risk so that the high beta assets are not perfect substitutes to the market portfolio, some relative preferences that prevent rational agents from avoiding high beta stocks systematically when they offer below-CAPM returns, and lastly you need some sort delusion that causes some investors to reach for the high beta assets outside of the standard model.

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