### What We Talk About When We Talk About Volatility.

When we as investors talk about volatility, we’re usually talking about variability in price returns. If an investment goes up and down 5% to 10% per day, that’s high volatility; if it goes up and down 0.05% to 0.1% per day, that’s low volatility. It’s a relatively simple concept, and is traditionally measured using standard deviation.

But when we compare investments to each other, we start talking not only about variability in price returns, but also about *beta*. And the implicit assumption is that beta measures something very different from variability. Beta is market-related risk; variability is unrelated to the market. For example, one writer, whom I won’t name because of my respect for him, wrote recently, “Beta is magnificent in theory but unstable and erratic in practice. . . . Imagine a debt-heavy gold mine whose stock soars in a dull market because of news of a big find. Beta is low or maybe negative, because the move is out of sync with the market, but risk is still massive!”

There’s one major problem with this kind of thinking. Even if the gold mine’s stock was almost completely uncorrelated to the market, its beta would be lower if its price variability was lower. This article will explain why.

### Why Beta and Variability Are Interdependent.

It can be observed empirically that the lower the beta of a stock’s returns is, the lower is the standard deviation of its price returns. Below is the three-year beta of the S&P 500 graphed on the *x*-axis and the three-year percentage deviation of the weekly price returns on the *y*-axis. (All figures are obtained from Portfolio123; beta is here calculated by comparing the weekly returns over the last 150 weeks to those of the S&P 500 without subtracting a risk-free rate from either one and percentage deviation is calculated by taking the standard deviation of the weekly returns over the last 150 weeks.) The relationship is very clear.

The reason for this is simple. Beta is the slope of the linear regression of a stock’s returns to a benchmark’s returns. The equation of the slope of a linear regression is

or, in plain English, the correlation of returns times the standard deviation of the stock’s returns divided by the standard deviation of the benchmark’s returns. Since the standard deviation of the benchmark’s returns is fixed (it’s the same for all stocks), the slope has a largely positive relationship with both the correlation and the standard deviation of the stock’s returns.

As beta goes to zero, two things could be happening: either the correlation goes to zero, or the standard deviation of the returns goes to zero. A stock whose price never changes has a beta of zero, just like a stock whose price changes in a manner that’s completely uncorrelated to the benchmark.

Now notice there’s a huge amount of white space at the bottom right of the graph. Why, for instance, are there no stock returns with a standard deviation of 5% and a beta of 2?

It’s because *r _{x,y}* (the correlation between the market’s returns and the stock’s returns) cannot be greater than one. So how small can

*σ*(the standard deviation of a stock’s returns) go? If we rearrange the above equation, we get

_{y}The higher the correlation, the lower *σ _{y }*can get, but the lowest it can ever get is

*σ*, and since

_{x}β*σ*= 2.56% (the percentage deviation of the S&P 500), a stock with a beta of 2 can never have a standard deviation of less than 5.12%.

_{x}### What About Negative Betas?

If we introduce short ETFs to the picture, we will get negative betas, and that’s because we’ll have negative correlations. With negative correlations, the higher the standard deviation of the ETF, the *lower *the beta. In other words, the relationship between beta and standard deviation is *V*-shaped with the lowest point at zero. After all, standard deviation, being the sum of squares, can never be negative.

This chart shows a sample of the betas and percentage deviations, measured exactly like in the above S&P 500 chart, of 1,600 ETFs, many of them leveraged and short.

Note the slopes of the edge of the plots, which are ± 2.56%—the percentage deviation of the S&P 500.

### What Does All This Imply for Low-Volatility Investing?

Judging from empirical data, low-beta and low-variability equity portfolios have both historically outperformed high-beta and high-variability equity portfolios. Portfolios based on beta and on variability are quite similar in constitution because of the interrelationship of these factors. The scale of their outperformance is consequently similar. With monthly rebalancing and no slippage costs, the lowest 20% of the Russell 3000 in terms of variability outperformed the highest 20% by 616.32% over the last twenty years (10.35% annualized); the lowest 20% in terms of beta outperformed the highest 20% by 540.28% (9.73% annualized). There’s not much difference there. The numbers for the S&P 500 are similar: 9.34% annual outperformance for low variability; 10.57% annual outperformance for low beta. Many studies have gone back close to a hundred years and have also looked at non-US markets. The results are strong.

There is a mathematical reason why low-beta equity portfolios outperform: as I have proven, in markets in which returns tend to be positive, alpha and beta are inversely correlated by their very nature. As far as I can tell, there is no mathematical reason why low-variability equity portfolios would outperform high-variability ones.

This outperformance goes against modern portfolio theory, which posits that investors demand greater return for greater risk. By any measure, low-volatility portfolios are less risky than high-volatility portfolios. Why should they outperform?

One theory is that investors tend to overpay for “exciting” stocks and ignore “boring” stocks, and “exciting” stocks clearly have more price variability than “boring” ones. But there seems to be no correlation between value stocks and low-volatility stocks, and this theory depends on one. There are a number of other theories out there, including a “stocks as lotteries” hypothesis. I haven’t found any of them very convincing—certainly not convincing enough to explain outperformance of such magnitude.

It’s possible that low-variability equity portfolios outperform *simply because they tend to have low betas*. But in that case, the outperformance of low-beta portfolios should be more apparent than the outperformance of low-variability portfolios, and that is not what we see. So there’s probably another reason as well. It merits further investigation.

CAGR since 1/1/16: 23%.

My top ten holdings right now: RHHBY, ACIU, ADRNY, ECOM, ESALY, III, CO, PCTI, ELMD, PCOM.

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