To assess potential return, investors rely on historical averages. Variations from those averages are called risk. But the greater risk may lie in their assumptions—namely that history repeats itself.

That’s because investors often rely on a normal distribution of returns, commonly called the bell curve. Here’s how it works. Assume a portfolio has a mean return of 10%, with a risk of 19%. That’s approximately how the S&P 500 has performed from 1926 until now.

In a normal distribution, two thirds of the time an investor will receive 10%, plus or minus 19%. Thus, an annual return could vary between -9% and +29%. That may be a reasonable risk assumption. But as investors have learned this past decade, the range of annual returns has varied more widely—5% of the time, the portfolio will return between -28% and +48%. Why?

First, risk is defined by standard deviation, a measure of variance. It is the engine behind mean-variance optimization — first proposed by Harry Markowitz in his doctoral dissertation in the 1950s and now part of the foundation of modern portfolio theory.

Most investors are satisfied with returns that fall within one standard deviation of the historical result. For years, institutions have modelled the possibility of loss as the range of outcomes that could occur 95% of the time—two standard deviations—or 19 out of every 20 days, in a process known as Value at Risk.

## Inaccurate assumptions

What makes this problematic, says Morningstar research, is extreme stock returns occur much more frequently—in fact, 10 times more frequently than expected if returns followed the path of a bell curve.

Thomas Idzorek, chief investment officer and director of research at Ibbotson Associates, has written, “Asset-class return distributions are not normally distributed, but the typical Markowitz MVO framework that has dominated the asset-allocation process for more than 50 years relies on only the first two moments of the return distribution.”

He also found investors are “particularly concerned about downside risk, which is a function of skewness and kurtosis”.

Skewness describes the bunching up of returns on either side of the mean. Kurtosis refers to how peaked the curve is: whether it’s steeper (positive kurtosis) or flatter (negative kurtosis). For stock market returns, the devil is in the tails. Most of the time, stock market returns show negative skewness. That is, assuming a mean of 10%, there are far more years where the return is less than 10%, and far fewer where the return is greater than 10%, than a normal distribution would imply.

That suggests buying and holding: ignoring the years of less-than-average returns, expecting that a few good years will bump up the portfolio’s value. That’s what lies behind presentations that illustrate the perils of market-timing—the investor will miss out on the best days that compensate for the dozens more that are below average.

Skewness introduces asymmetry into the normal curve. With excess kurtosis—fatter tails, meaning more outsize returns on the downside or upside, or both—the concern is how these asymmetrical returns bunch up. Are they only slightly less than 10%, or are they significantly less (lots of 20% losses, for instance)? (See “An arithmetic mean”.)

## The numbers lie

The non-normality of stock returns has been known for decades, part of Eugene Fama’s doctoral dissertation in 1964. Fama, who pioneered the efficient markets hypothesis, is a Board member of Dimensional Fund Advisors.

**AN ARITHMETIC MEAN** CAN HIDE A ROLLERCOASTER OF RETURNS

**Consider a distribution of returns that is virtually flat.** Out of 100 returns, the curve could show the majority essentially cancel each other out, but still produce a mean return of 10% because just two were 10%. That’s simply the arithmetic mean. It’s also an example of excess negative kurtosis (platykurtosis).

Take a return series of -40%, +40%, +20%, +20%. The arithmetic mean is still 10%. But it’s been quite a rollercoaster along the way. The -40% return and the 40% return are fat tails, and they occur more frequently than one would expect.

Looking at the geometric mean, the picture can be quite dire, depending on the sequence of returns.

In the sequence just outlined:

That’s not quite what the investor expected of a 10% annualized return: he thought he’d get $1,464.10 ($1,000*[1.1]4).

“Distributions of daily and monthly stock returns are rather symmetric about their means, but the tails are fatter (i.e., there are more outliers) than would be expected with normal distributions,” he writes on DFA’s website. “The message for investors is: expect extreme returns; negative as well as positive.”

If the normal distribution doesn’t quite fit stock-market results, why is it so widespread? “First, it’s easy to understand; the average person can instantly grasp the bell curve,” says Campbell Harvey, professor of finance at Duke University’s Fuqua School of Business.

“Second, it’s simple to work with. You don’t need much information; you just need the variances and the co-variances and you’re done.” Harvey also argues some tail risk can be diversified. Called the central-limit theorem, it’s when you put several non-normal assets together so the portfolio is approximately normal.

“Think of one asset with a big left tail and another with a big right tail. Put them together and you’re diversifying the tails.” He warns, “theory is only theory. Diversification does not guarantee the left-tail risk is eliminated.”

George Athanassakos, a professor of finance at University of Western Ontario’s Ivey School of Business, takes a less benign view of diversification. “John Maynard Keynes concluded not all risks can be measured,” he says.

“However, Thomas Bayes [an 18th century mathematician] said we can quantify risk in the same way we place bets on roulette. That is, we observe how many times the numbers come up. If we can plot how many times the numbers came up, we can come up with a [normal] probability distribution. What he described was easy to understand.”

“It’s nice if you play roulette,” says Athanassakos. “But not when you play with life or the markets. In roulette, the odds are fixed.The markets are like playing poker: your decisions are influenced by the behaviour around you. The problem with normal distributions is they assume players play no role in the risk.”

## Unnatural, non-normal

Actuarial tables work for modelling fire rates or mortality, but “financial markets are not natural phenomena; they are man-made,” points out Athanassakos. “As we manage risk, the environment and the risk changes. The normal distribution cannot handle this, because it assumes we know everything. In real life, the distribution has fat tails: there’s a lot of stuff at the ends we don’t know.”

Pierre Saint-Laurent, president of AssetCounsel Inc., says people are becoming more aware of fat tails. “[But] several of the models are complex to build.”

So what’s the solution? “Tail-risk measures are not hard to calculate,” says Harvey. “You can calculate the skewness and kurtosis [using spreadsheet software]. Software can also optimize your portfolio by incorporating an investor’s aversion to left-tail [bad outcome] risk.”

He says knowing the behaviour of the asset is key. “If you’re just in a plain-vanilla portfolio of equity and fixed income, maybe you don’t need to worry about this. If you’re invested in derivative-based strategies, alternatives, or funky fixed income, like distressed debt, junk, European sovereigns they will spill out of the normal distribution.”

Athanassakos says, “Risk is not a signal that we can derive from the normal distribution.The margin of safety helps to manage the downside and protect capital better than diversification or any other modelling based on the normal distribution.”

Saint-Laurent acknowledges that risk management is more art than science. “We need to use human judgment. At some point, we need to move away from magic numbers.”

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Scot Blythe is a Toronto-based financial writer.
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