## Growth rates and their statistical fallacies

Have you run into some fund or investment vehicle where the seller advertises a tremendous growth rate?

Watch out, because you might be getting played for a sucker!

When you come in here, you look for the sucker. And if you can’t find him, then the sucker is you. –Mark Cuban, Shark Tank

Let’s imagine a very tiny index fund. It’s worth a measly \$1. What is the total growth rate if it climbs to \$2? 100%!!! Someone can legitimately tell you they have a fund sporting 100% growth. Of course, it only grew by \$1 total.

What if your fund was worth \$100,000,000 and increased by \$1,000,000? The growth rate would be a tiny 1%. But it still grew by \$1,000,000.

What this says is the percent and absolute dollar are BOTH important metrics.

## Financial Math II: Averages and standard deviations

This 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.

Let’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.

## Financial Math I: Rule of 72 and other fast math

When making financial decisions, it’s important to understand how financial math works. This is the first of a short series where I hope to cover some financial math basics.

Rule of 72

For starters, when we are faced with an opportunity, it’s good to be able to quickly assess if it’s worth spending more time investigating. We often call these “back of the envelope estimates”. Without a spreadsheet can we get a ballpark estimate of the value of something?

One of the most well known mathematical devices is the “Rule of 72”. The formula provides a way to see in what time frame and with what interest we can double our money.

Interest x Years = 72

For an example, imagine we had an account growing at 10% annual interest rate. Take the 10, multiply if by 7.2, and you get 72. This means that in 7.2 years (approximately), our money would double.

You can flip things around. If we wanted our money to double in 10 years, we would need at least a 7.2% interest rate.

If you wanted something to double in six years, you would need 12%. 3 years? 24%.

To generalize, any situation that involves the compound interest effect works here. It’s also important to recognize this is only an approximation, and the further you get away from the middle, the less accurate it is. Rule of 72 may say that 2% takes 36 years to double, but this is much less accurate than 7.2% and 10 years. (BTW, you can swap the two such that 7.2 years and 10%.)

This type of quick math can help you. If you hear some investor advertise that there is no risk in doubling your money in six years. In your head, you can quickly deduce he is suggesting 12% growth. Sorry, but that is 3x what people average when using mutual funds. In real estate, this is definitely doable, but it requires either a lot of cash to mitigate the risk, or it requires taking on debt. None of this is risk free.

Gains and Losses

I’m always hearing things like “The Dow is up 5%” or “The NASDAQ is down 10%”. People seem to think that growth and losses are arithmetic, i.e. pluses and minues. They aren’t. They are multiplicative.

If you had \$1000, to grow it by 10% means you multiply by 1.1, another way of saying (1 + 10%) or (1 + .10). \$1000 x 1.1 = \$1100.

To reduce by 25% mean you multiply by 0.75, another way of saying (1 – 25%) or (1-.25). \$1000 x 0.75 = \$750.

Why is this important to understand? Because combining multiple gains and losses together involves multiplying all of the factors together, not adding them. When you have a 10% gain following by a 25% loss, what is the result? You might think 10% – 25% = -15%, but that isn’t right.

Instead, let’s take what we did earlier (1.1 and 0.75) and multiply them together. 1.1 x 0.75 = 0.825, or -17.5%. That loss (-17.5%) is actually greater than you would get if you just subtracted one from the other other (-15%).

It’s important to know that getting 10% each year for three years doesn’t add up to 30% gain. Instead, it’s 1.1 x 1.1 x 1.1 = 1.331 or 33.1%. This is power of compounding. But when we suffer losses, the effect is equally drastic. And people often don’t realize just how hard losses can be on overall performance.

Understanding this basic tenet of gains and losses is critical to building spreadsheets that evaluate things that are more complex. And the more you use this mechanism, the more you notice how people like to quote “Your investment will average x%!” You will be able to size something up and ask, “Hmm, if I multiple by 1.xx every year, do I really get what they just advertised?”

That’s enough for starters. Tune in for future posts about more financial math.