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 July 2011 The Real Concept of Risk It's Not Standard Deviation
You probably know what standard deviation is, the average deviation of returns about the mean. Statisticians like this because they define the odds of not achieving the mean return as "risk". The greater the average deviation about the mean, the greater the "risk". It's a handy way to quantify risk -- but it's not what most investors think of as risk. Here's why: Standard deviation gives equal weight to deviations on the upside as well as the downside. Most (if not all) investors aren't worried about risk on the upside. In fact, they seek it. That's why when stocks are on a tear to the upside, investors can't get enough risk. They want as much as possible. That's also why they get burned so badly when stocks finally revert to the mean. The point is, investors never complain when they get more return than they expected. That's a pleasant surprise. Not only that, while not nearly as pleased with returns above the mean, investors aren't too upset with annual returns below the mean, as long as they aren't negative. In other words, as long as they don't actually lose money, they're not too upset. But when they do suffer a loss they're concerned. The greater the loss, the greater the concern. In essence, investors perceive risk only on the downside. Not only is the magnitude of loss important, frequency is, too. This isn't captured by standard deviation and is why even the most carefully crafted portfolio will go unappreciated by investors if there's too much risk perceived on the downside. To truly meet their expectations, advisors must address the frequency and downside aspects of risk.
It's in the Statistics
To understand how investors view risk, you only have to see the stream of returns from one investment. Chart 1 shows what's statistically known as a "normal distribution". In this case, it comes from lining up all the annual returns of an investment along the horizontal axis and then showing the frequency of each on the vertical. The curve follows the frequency moving from left to right. Because this is an example of a normal distribution, the vertical bar at the middle of the curve represents both the average and median return. There are just as many returns to the right as there are to the left. More are clustered around the average/median in the middle, and fewer appear in the two tails of the curve. Still focusing on Chart 1, notice that 68.26% of the annual returns are right around the average/median: 34.13% are to the right and an equal number are to the left. Over 91% are within two standard deviations of the average/median, and fully 99% are within three standard deviations. Put a different way and using the Cumulative Percentile scale, you'll notice that the third and fourth standard deviation at the left of the chart represent only the first 2.3 percentile of the overall distribution of returns. The green-shaded area represents 2%. You'll see why this is important in a just a minute. Before proceeding, it's important to note that the examples we're about to see are all historical data from actual asset classes. As you've probably heard repeatedly, past performance is no guarantee of future results. The same goes for risk. Nevertheless, it's still informative to look at the actual historical results because they clearly illustrate the difference in perceived risk and standard deviation.
Now let's take a look at actual large cap stock returns as measured by Ibbotson Associates' Stocks, Bonds, Bills, and Inflation S&P 500 Extended Series. We've added the white line highlighting the pattern of the distribution. Although it's not a perfect normal distribution (they never are out in the real world), the pattern in Chart 1 is quite close to that of Chart 2. The annual return over the period was 11.88% and the standard deviation was 20.4%. That's a nice average return but is that risk too high? Would you be comfortable with it? Maybe with hindsight looking back over the past 85 years you might say it would be OK. But do you have a real feel for what that standard deviation really means? Probably not. Instead, consider this: The second percentile return, comparable to the green shaded area under the curve in Chart 1, was -24.1%. In other words, once in 50 years large caps lost about a quarter of their value. The distribution curve doesn't tell you when it will occur, only that it roughly happens in one year out of fifty. It could happen next year or any year for that matter. Would such a loss cause you to panic and possibly make some poor short term decisions? You probably have a better feel for this than simply knowing the S&P 500's standard deviation is 20.4%. As far as the frequency of losses, large caps had negative years 28% of the time or roughly three out of ten years. Keep in mind this is a measure of the frequency of losses and not their magnitude. In this particular example, they ranged from -0.4% to -43.3%. To put that in context, the -0.4% loss was barely negative while the -43.3% was essentially a 1-in-100 year occurrence. The result is similar as before when estimating the size of losses: Now you know the frequency (a loss every three years or so) but not the size. It could be virtually negligible or the century's largest. Even so, you still probably have a better understanding of this than standard deviation, don't you? Isn't this what investors talk about when they talk about risk?
Curves Comparison Consider, for example, the choice between a large cap or small cap equity investment. Chart 3 overlays chart 2 with the return distribution of small cap stocks (green outline) for the same period. As you would expect, small caps have a much wider dispersion of returns -- that's why they have a much higher standard deviation than large caps (32.6% vs. 20.4%, respectively). Someone focusing only on that would conclude small caps have historically been much riskier.
But have they? It depends on how you define risk. If you define it as an annual loss, there's virtually no difference between small caps and large caps. Recall large caps suffered a loss roughly every three years. The same goes for small caps. Nevertheless, small caps had an average return of 16.74% versus large caps' 11.88%. Chart 3 shows you why. Small caps have a wider range of returns at both the high and low extremes. In fact, Chart 3 consolidates some of the highest on the right into a category of 65% or more. Indeed, the highest annual small cap return over the period was 142.87% -- all the way off your screen. Although the lowest annual return was well below that of large caps (-58.01 vs. -43.34, respectively), many more small cap returns are on the right side of the chart. Statisticians call this a "positive skew". Nevertheless, risk averse investors are often more concerned about the size of potential losses than the size of potential gains. It's here that small caps earn their reputation for risk. Recall the 2% or 1-in-50 year loss for large caps was just over 24%. It's fifty percent higher, 36% for small caps. That's pretty sobering when you realize in a 1-in-80 year occurrence like 2008, you can lose a lot more. That really brings it home, doesn't it? That's something that can't be said for standard deviation. If investors were shown these statistics and ways to quantify risk, perhaps there wouldn't as many rude surprises as there were in 2008. Unlike standard deviation, they can relate to the 2% size of loss and annual chance of loss. It's not that the average investor can't understand standard deviation, for the most part, they probably do. What they can't do, however, is relate to it in a context that's meaningful. That's too bad. What's even worse is that the professionals who are allegedly there to help them, really don't. Search this site! Just enter you key word or words:
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ATIONAL INVESTORS ARE inherently risk-averse although they may not really know what "risk" is. If they ask their advisors, they'll more than likely get a definition of standard deviation. Most are more or less accurate, but there's a major problem: Although standard deviation is the way academics define risk, it's not the way investors think of it. It just isn't.
This chart shows the annual returns measured on the horizontal axis and the frequency on the vertical. Here both the average return and the median return is 0%. The scales below the curve show the Standard Deviations, the Cumulative Percentages, and the Return Percentiles. The shaded area under the curve shows the 2% return.
Source: Ibbotson Associates
Source: Ibbotson Associates
