Imagine packing for vacation and checking your favorite weather app only to find one piece of information, " This week, we’ll have an average temp of 75 degrees." Sounds pleasant, so you pack your favorite clothes for sunny and 75. When you get there, you get some more information and realize that the city is 45 degrees one day and 105 the next, and it has been that way all year. The forecast wasn’t wrong. The average really is 75. It’s pretty much useless because it doesn’t tell the full story and help you figure out what to wear.
What you actually need to know is how far the temperature tends to wander from that average. That is exactly what standard deviation measures, and it is one of the most used risk numbers in the financial world.
What is standard deviation?
Standard deviation is a measure of how spread out a set of numbers is around its average. A low standard deviation means the returns cluster tightly. A high standard deviation means there is more variation in returns.
Let’s revisit the two hypothetical portfolios from the last post. Portfolio A returned 8, 6, 10, 7, and 9 percent over five years. Portfolio B returned 25, -12, 30, -8, and 5 percent. Both averaged 8 percent. Portfolio A's standard deviation works out to about 1.6 percentage points. Portfolio B's works out to about 19 percentage points. That one number captures the difference between a smooth ride and a wild one, which the average completely hid.
These figures are hypothetical, but not impossible to find in the real world. From 2003 through 2007, the S&P 500 returned between 4.9 and 28.7 percent every single year, with a standard deviation of about 10. Over the very next five years, 2006 through 2010, the same index had a standard deviation of about 25, because 2008’s 37 percent loss sat in the middle of it. Aside from looking at the individual returns of each year, standard deviation is one of the best measures to help measure the investor experience over a period.
How it is calculated
If you don’t like math, skip ahead to the next section. Here are the steps, using Portfolio B.
First, find the average. For B, that is 8 percent.
Second, for each year, subtract the average and square the result. Year one: 25 minus 8 is 17, squared is 289. Year two: -12 minus 8 is -20, squared is 400. Squaring does two things: it turns every miss into a positive number, so the misses don’t cancel each other out, and it punishes big misses far more than small ones.
Third, average those squared numbers. In practice, analysts divide by one less than the number of observations when working from a sample, which is a technical adjustment you do not need to worry about. Either way, the result is called the variance.
Fourth, take the square root of the variance so the answer is back in the same units you started with, percentage points. That is the standard deviation.
Written out, the formula is: standard deviation equals the square root of the sum of (each return minus the average return) squared, divided by the number of returns minus one. For math folks, it looks like this:
s = √[ Σ(xᵢ − x̄)² / (n − 1) ]
The rule of thumb that makes it useful
Standard deviation is useful because of a pattern in how returns tend to spread out. If returns are roughly bell-shaped around their average, then about two-thirds of the time, a year's return will land within one standard deviation of the average. About 95 percent of the time, it will land within two.
So, if a portfolio has an average return of 7 percent and a standard deviation of 10 percentage points, the rule of thumb says that in a typical year you should expect something between -3 percent and +17 percent, and that a year outside the range of -13 to +27 percent would be rare.
This is what makes standard deviation so valuable to us. It turns "this fund is aggressive" into "in a bad year, this could reasonably be down 13 percent, and once in a long while worse than that." Often, investors aren’t able to really quantify what ‘aggressive’ or ‘conservative’ means, but we are able to better understand their tolerance for risk when we use metrics like standard deviation to give them real expectations backed by statistics.
What it misses
Standard deviation has two well-known blind spots, and the rest of this series exists largely because of them.
The first is that it treats an upside surprise exactly like a downside surprise. A year that comes in 20 points above average raises standard deviation by exactly as much as a year that comes in 20 points below. No investor feels those two years the same way. The next two posts deal with this.
The second is that real market returns are not perfectly bell-shaped. Extreme years, especially extreme bad years, happen more often than the bell curve predicts. The rule of thumb above says a three-standard-deviation year should happen about once every 370 years. Markets have produced them considerably more often than that. Standard deviation is a good place to start. It’s like the dough for a pizza. By itself, it’s not a pizza, and it doesn’t taste too great, but when you add the toppings, it becomes the most important ingredient and keeps everything together.
What this means for you
A portfolio's standard deviation over the last 20 years and over the last 3 years can be very different numbers, and the one that matters depends on the person. For someone who plans to draw on the account starting in two years, the recent, shorter-window figure and the worst single-year figure are what we focus on, because a long-run average will not pay next year's bills. For someone in their thirties adding to the account every month, we care more about the long-run figure and much less about any single year.
A standard deviation of 12 means little to most people. We try and do our best to communicate to clients what that means in plain English, "In a bad year, this portfolio could reasonably lose about $60,000 on a $500,000 balance, and you would still be within the normal range" is something any client can react to. If that sentence makes someone's stomach turn, we have learned something important before any money is at risk, and we adjust.
Standard deviation is the ruler. The next post introduces the first tool that uses that ruler to answer a better question: not just "how bumpy was it," but "was the bumpiness worth it?
Any opinions are those of Kirk McCarty and not necessarily those of Raymond James. Expressions of opinion are as of this date and are subject to change without notice. Raymond James and its advisors do not offer tax or legal advice. You should discuss any tax or legal matters with the appropriate professional. Investing involves risk and you may incur a profit or loss regardless of strategy selected, including diversification and asset allocation. Prior to making an investment decision, please consult with your financial advisor about your individual situation. Investing in oil or the energy sector involves special risks, including the potential adverse effects of state and federal regulation and may not be suitable for all investors.