Workshop Portfolio A Foolish Math Primer, Part 2
Where we deviate from the standard

By Todd Beaird (TMF Synchronicity)
March 9, 2000

Last week, as you'll remember if you stayed awake, we looked at CAGR (that's Compound Annual Growth Rate). CAGR is one of the of the most important numbers we look at when analyzing mechanical strategies. This week we'll look at two ways of assessing risk: Standard Deviation and its "cousin," Geometric Standard Deviation. We'll try and keep it relatively painless.

Standard Deviation ("SD") is a measure of the variance in a set of data. In non-math speak, you could have two groups of people, both of which have an average weight of 150 pounds. (Can you guess that I've been on a diet?) However, Group A could consist of very fit people who all weigh between 140 and 160 pounds. Group B could include many people who are overweight or underweight, as well as "average" people. Both groups might have the same average, but group B would show more variety and have a higher standard deviation.

Standard deviation is calculated by taking each data point (a person's weight, or an annual return) determining how far away it is from the average, and then plugging all those distances from the average into a formula. Standard deviation is important for a bunch of reasons, which were discussed in more detail in our articles about statistical significance.

There's just one problem with standard deviation -- it's intended for the kind of data that you can add together to get an average. But with CAGR we do this weird "multiply and take the nth root" thing to get our average. Very different. We also have a slight problem in that our returns can never go below zero (once you've lost it all, you can't lose any more) but they can go up a gazillion percent. (Look at the Rule-Breaker's portfolio returns with AOL -- maybe they will reach a gazillion). This imbalance caused by our old friend, the Compounding Clown, means that there is a greater chance of volatility on the positive side of the average than on the negative side.

Enter Geometric Standard Deviation ("GSD"). What is GSD? It's a method for calculating standard deviation for situations like CAGR, where you're multiplying instead of adding to get results. How is it calculated? Here's the tricky part. You have to use logarithms. Remember those from high school? I don't either. But I can muddle through with a spreadsheet.

For each value, you take the log. Then you take the log of the CAGR. You compare the log of each value to the log of the average, and plug those numbers into the formula for standard deviation.

Here's an example. Remember our example last week, where we had four years of returns at +20%, +10%, -15%, and +30%. For CAGR, the values are 1.2, 1.1, 0.85, and 1.3. For each of these values, we take the log:

Year One    1.20    log(1.2)=   0.1823
Year Two    1.10    log(1.1)=   0.0953
Year Three  0.85    log(0.85)= -0.1625
Year Four   1.30    log(1.30)=  0.2624

The average of those log values is 0.0944. Not so coincidentally, the anti-log of that number is 1.099, which is our CAGR. (If you're using Microsoft's Excel spreadsheet, just take EXP(0.0944), to get the anti-log, and say a silent prayer of thanks for spreadsheets.)

Then, we take the standard deviation of those log values. Plugging these into Excel, we get a regular standard deviation of the logs of 0.1844. To get a return one GSD above average, just take 0.1844 (one standard deviation) plus 0.0944 (the average), which gives you 0.2788. To convert the logs to everyday terms, take the anti-log (that EXP thing) of 0.2788, and you get 1.3215. Subtract one and convert to a percentage, and you get 32.15%, which is the GSD in everyday numbers Say a BIG thank you to spreadsheets.

Remember that in a "normal" distribution, about two-thirds of all values are within one GSD of the average, and about 95% are within two GSDs of the average. Market returns probably aren't "normal," but that gets into some very complicated math, and I suspect most of you are running out of coffee and No-Doze. Suffice it to say that extreme events (three GSDs or higher) probably happen more often in markets than you'd expect from a "normal" distribution.

So now you know how we calculate the two most important numbers for our screens, the CAGR and the GSD. In general, we want screens with a high CAGR and a low GSD indicating both high returns and low volatility. The closer the yearly returns cluster around the average return, the better.

Next week we'll talk about how NOT to invest. If you're interested in how TO invest, just check out our Workshop Message Board, or the other stock strategies here in Fooldom.

Speaking of other stock strategies, I hope that some of you signed up for David Gardner's Rule Breaker Seminar. If you did and are in either the Printing Press or Radio teams, you'll be seeing me as the seminar progresses. Otherwise, tune in next week, same Fool time, same Fool channel!