Financial Math II: Averages and standard deviations

GDvsS&P5001996to2012This Part II of my series on financial math. Previously we talked about some simple math tricks that can help you think faster on your feet.

In this post, I want to talk about some key statistics that get thrown around and how to parse them. I’m sure many of you have read this famous quotation:

There are three kinds of lies: lies, damned lies, and statistics. –Benjamin Disraeli (according to Mark Twain)

Statistics are what happen when we try to look at a whole batch of data points and spot some sort of trend, correlation, or conclusion. The reason they have to be looked at with a discerning eye is because people will either knowingly (or unknowingly) perform some sort of statistical calculation and then TELL us what it means. What they tell us and what the numbers actually mean can be very different.

rodin_thinkeLet’s introduce an example. Whenever you take a collection of data, such as amount of income earned by every person, and average it together, you can produce a couple different outputs. One is known as the mean. This is when you add all income and divide by every person. In these situations, it is easy for a small group of either very high earners or very low earners to skew the metrics one way or the other.

But if you instead take the entire collection of people and split them into two groups, right down the middle, and look at the mid-point, this is called the median. The median and the mean might be very close together, or they could be far apart.

By itself, these two different statistical values hold no bias. They simply show a slightly different perspective on the spread of income. But people can pick and choose which particular data set to show when making a point. They might choose the data set that better trumps their point of view.

Continuing with our current example, when people calculate such values, the purpose at hand is usually to deduce, where do I fit in? And that is why using the mean, which can be heavily skewed based on outliers, tends to not be as good of a statistic as the median when it comes to predicting things like that.

Another factor we want to know is how spread out is the data from the mean. To do so, we commonly use the standard deviation. If we tried to average the difference of each person from the mean, we would actually reach zero. That’s because half of the data points are greater and half are less than the mean, by definition. So to come up with something of value, we instead square the difference, average that, and take the square root. (In science, this is known as the root-mean-square).

Much research has been done that shows that anything with one standard deviation of the mean has about a 68% chance of success. Two standard deviations = 95%. Three standard deviations = 99%+.

Because standard deviation is so easy to calculate, you should always ask for it whenever someone, such as a financial planner or whomever, attempts to woo you with averages. “The average performance of this fund is 18%.” “What’s the standard deviation?” If they scramble from answering that, it’s a sign that you should probably run.

You see, the bigger the gain, the bigger the risk, and the probably the bigger the standard deviation.

You can see an example in a blog post I wrote for Dr. Dave. In it, I compare the average performance of the S&P 500 compared to an EIUL. To do an analysis, I figured that most people will have about 25 years to get serious about saving in either plan in order to “catch up” if they are late to the game. So, what if I looked at EVERY 25-year window of the S&P 500 going back to 1950, calculated it’s actual performance, and averaged them together? On top of that, let’s find out what the standard deviation?

Turns out, we have a 68% chance of landing somewhere between 4.77% and 9.53% in total growth. If we don’t do so well, we might barely be grazing past inflation. Or we might be well ahead of it. For something in which we only have one shot, I don’t really care for those odds.

Compare that to an EIUL, for which we must trade in a certain amount of highs to avoid certain lows. Turns out in that scenario, we have a 68% chance of landing somewhere between 7.52% and 8.84%. The 8.84% is certainly lower than 9.53% of the S&P 500. But in exchange we are almost three points higher than the low point, meaning our odds are pretty good of beating inflation, a key factor for investing in EIULs.

Neither of these stats factor in costs or how much cash you can put away. Remember, you can always clobber returns rates by putting away more money. The key is that means and standard deviations are important statistics you need to understand if you plan to take an active role in investing.


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