It can be difficult for investors to understand how their professional managers are really performing. A few statistical tools can help illuminate whether investors are getting value. 

The numbers below illustrate performance for two sets of investment managers. How can we compare them?

		Manager A		Manager B
Year 1		14%		10%
Year 2		15%		15%
Year 3		16%		20%

The best way to start is by asking a simple question: What would you expect each manager to earn next year if these were the past three years of returns? Well, unless you have a working crystal ball, there is no way to know what the managers’ returns will be in the future. Luckily, we can make good guesses: one of the best ways to get a sense of what will happen in the future is to examine what just happened.

In other words, statistically speaking, the words “expected” and “average” are interchangeable. What we expect in the future is the average of the historical data. So, all we have to do is take the average return for each manager.

So which manager should you hire? I get a lot of great and different answers when I pose this question to CFA candidates. Some say Manager B because he had the highest return, 20% in Year 3. Many say Manager A because his returns are all grouped around one number; he may be off a little each year, but not by much. Is that enough to justify choosing him over Manager B?

Let’s take a look at what investors can expect on average. We can do this adding up the items that we have and then dividing that sum by the number of things we added.

		Manager A		Manager B
Year 1		14%		10%
Year 2		15%		15%
Year 3		16%		20%
Expected		15%		15%

What can we do now? It looks like they have the same average (or expected) returns. That doesn’t help us determine who to hire.  It looks like, just from eyeballing the figures, that Manager A does indeed have a tighter grouping of his returns.

What if we tried to quantify this just to be sure? How close or how far apart are these two managers from their expected return every year? I guess we could compare each individual year’s return with the expected return to see by how much it missed and call that the “error.” Let’s do that. Would you look at that! They have the same average, or expected, returns. That doesn’t help us decide who to hire.

	Manager A	A Error		Manager B	B Error
Year 1	14%	1		10%	5
Year 2	15%	0		15%	0
Year 3	16%	–1		20%	–5
Average	15%			15%

Now all we have to do is take the average error and see which one is bigger to mathematically prove who has the wider swings in performance. Just add up the errors and divide by three. Simple as pi!

The errors for Manager A add up to zero. And zero divided by three is zero, or indeterminate. Statistically speaking, it’s meaningless. It’s the same for Manager B.

It turns out that another definition of average is the point that minimizes, or sets equal to zero, the distances on the number line between all of the observations. So, for any set of numbers, you take an average of the differences between each number and the average will always sum up to zero.

Okay, so the problem is the errors always add up to zero. How can we get around this mathematical stumbling block?

One thing we could do might be a bit surprising at first. We can just forget about the positive and negative signs, and add the numbers up, and  then divide by three. That would be called the “mean absolute deviation.”

Hold on, are you confused? Oh, yes. “Mean” is a wonky statistics word that means “average” or “expected.” There certainly are a lot of words for that concept. It’s a very important one.

But see here; ignoring positive and negative signs doesn’t seem completely sane. If I were to offer you a job that paid $1 million per year, you’d be delighted. Negative $1 million would be a whole different story. So, we’re still stuck….

The problem is that we can’t take the average of the errors because they always sum to zero, so what if before we added up the errors, we square them and then add them up? This operation has a nifty name: “sum of the square of the errors”! That’s a mouthful, so let’s just say SSE from now on, okay? Let’s try it and see!

	A Error	A Error Squared	B Error	B Error Squared
Year 1	1	1	5	25
Year 2	0	0	0	0
Year 3	–1	1	–5	25
SSE		2		50
MSE		0.6667		16.6667
STDEV		+/– 0.8165		+/– 4.0825

Quick, what is the square root of 100? You’re smart, and almost before your eyes finished reading the answer, “10” popped into your brain, didn’t it?

But what if I told you that was not right? You’d be indignant. What if I told you it was incomplete? You’d be contrite after quickly realizing 10 is not the only correct answer to that question. Although 10 is correct, NEGATIVE 10 is also the square root of 100. That, in fact, is why standard deviation is by definition both positive and negative. We had to jump back through the mathematical hoop from when we originally squared the numbers so they wouldn’t add up to zero.

Manager A has an SSE that is much lower than Manager B’s SSE, but we haven’t taken the average or mean yet. When we divide the SSE by three we get the average squared difference between what the manager actually got and what we expected him to get. Let’s call this the “mean squared error,” or MSE for short.

At last, we have some meaningful descriptive statistics, and we didn’t violate any mathematical principles to get there. The MSE is also called the “variance” by statisticians, and the square root of the variance is the “standard deviation” (STDEV). Now you know.

To add some additional color, we divided by three because we were taking the average of all of the observations of returns, or what is called the “population.” If we were just taking a partial set of observations, say for instance the PE ratio of 100 out of the 500 stocks in the S&P 500 Index, we would have to give ourselves an added cushion or room for error by dividing the SSE by one less than the number of observations we have, or n – 1.

This is called a “degree of freedom,” and we do it because we are making an estimate of the population (PE for the S&P 500) from our sample of 100 stocks that we looked at. This gives us a bigger range than if we divided by n. For Manager B, if this were a sample of his returns, we would divide SSE = 50 by n – 1 = 2, which would give us an MSE or variance of 25, which is definitely a bigger number than 16.6667, and this would of course yield a standard deviation of +/– 5, which is bigger than +/– 4.0825. The bigger range from –5 to +5 gives a wider margin of error because we are using a sample that may be skewed or biased rather than using all of the data.

So, that’s how you calculate variance and standard deviation. And just to be complete, the formula for the standard deviation of a sample is at right.

Notice all the Greek letters in the formula? This is where the expression “It’s all Greek to me” actually comes from. I hope my explanation made it clearer and easier to remember than that looks!

(Extra Credit: How would the above formula be different if it were for a population? Let me know on Twitter at @SconsetCapital or in the comments below.)

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