Being an investor is tough enough with all of the investment choices available today (that’s why we’re here!). But some investors make the challenges even greater by misunderstanding some of the most important numbers used in making decisions and keeping score. Here’s a detailed look at two of these pivotal metrics.

#### Total return

Here’s a quiz. Imagine your investment account generated the following returns over the past five years: 18%, 1%, -12%, 5%, and 8%. What was your total return?

Did you come up with 20%? Imagine the sound of a loud, annoying game-show buzzer and get a picture in your mind of Alex Trebek frowning at you.

To get an accurate read on your portfolio’s performance, you can’t just add up the rates of return. That’s because each year’s return changes the amount of money that will be invested the following year.

Here’s a quick example. If you invest \$100 and earn 10% in your first year, your portfolio would grow to \$110. If you also earn 10% in year two, you’d have an \$11 gain, pushing your account total to \$121. As you can see, our example didn’t result in a 20% total return—the total return was 21%, showing that simply adding the returns doesn’t work. Here’s the accurate way to calculate total return:

Step 1: Convert the returns to decimal format. A 10% return is expressed .10, an 11% return is expressed as .11, and so on.

Step 2: Add “1″ to each rate of return, so that, for example, a return of 10% is now 1.10.

Step 3: Multiply the rates together. Using our quick example of two years of 10% gains, you get 1.10 x 1.10 = 1.21.

Step 4: Subtract 1, leaving .21.

Step 5: Convert back to a percentage format by multiplying by 100. In our example, this would give you a 21% return.

That may sound complicated, but it’s not once you’ve done it a few times. Here’s the full longhand for the five-year example we looked at initially:

Steps 1 and 2: 18%=1.18, 1%=1.01, -12%=0.88, 5%=1.05, and 8%=1.08.

Step 3: 1.18 x 1.01 x .88 x 1.05 x 1.08 = 1.1893

Step 4: 1.1893 – 1 = .1893

Step 5: .1893 x 100 = 18.93%

#### Average annual return

Now, let’s build on that idea. Using the example above, what’s the average annual return? If you answered 4%, Alex Trebek is still frowning at you.

It seems logical to add up the five annual returns and divide by 5. But that leads to what is known as the arithmetic average. That’s the right approach for figuring out your average golf score over your last 10 rounds, but it doesn’t work when figuring out annualized investment returns.

The difference is that golf scores (and most other things you average) are independent events. Unfortunately, that miracle round you posted last weekend has no bearing on your next round, other than providing a bit of false confidence.

When it comes to your investment returns, this year’s performance is not an independent event. As mentioned earlier, each year’s return changes the amount of money that will be multiplied by the next return in the sequence.

To accurately calculate average annual investment returns, you need to use the geometric average. Here’s the formula using our earlier example. First, go through the initial three steps we used when calculating the total return. This gets us to 1.1893.

The next step is more complex. You have to raise 1.1893 to the power of 1 divided by the number of years. In this case, that’s 1 divided by 5, or 0.2. Fortunately, Google can take you the rest of the way. Just enter “1.1893 to the power of 0.2″ in the search field and it’ll give you the answer, which is 1.0353. (And you thought Google was just for finding web sites.)

Last, to convert the answer to a percentage, go through steps 4 and five 5 above—subtracting 1 and multiplying by 100. That leaves us with 3.53%. While using the geometric average didn’t give us as attractive a return as when we used the arithmetic average, it does have the advantage of being correct!

How did you do on this investment math quiz? Knowing how the numbers really work is an important part of being a wise investor.

### By Matt Bell

Matt Bell is Sound Mind Investing’s Associate Editor. He is the author of three personal finance books published by NavPress, leads workshops at churches and universities throughout the country, and has been quoted in USA TODAY, U.S. News & World Report, and many other media outlets.

• http://www.kgaction.com/ Mary Kaplan

Good post and important information for any investor. Even though the geometric average is a much small return, at least it is accurate and you can move forward with the right information.

• lsteens

GREAT(!) information…thanks for sharing

• Good \$ense guy

A related “wake up call” is to ask yourself the question, “Would I rather have a 4 year run of returns of +50%; -33%; +50%; -33% or 4 years of a straight 5% return?” Putting aside risk aversion tolerance for a moment, at first glance it looks like the first example would be the way to go. Wrong again! If you had invested \$100, the first example leaves you with \$107 after 4 years; the second leaves you with \$121 and some change.

• Good \$ense guy

Ooops – just realized i mistyped… the 3rd year return should have been +60% in the first example.

• http://www.soundmindinvesting.com/ Matt Bell

It’s sort of like golf. The players who grip it and rip it are fun to watch, but it’s usually the ones who hit the most fairways and greens who come out on top. Maybe an even better example could be seen in Moneyball, where a true understanding of the numbers led to a winning strategy.

• Merritt

These formulations are probably best performed by computer. Not even a powerful hand held calculator (except by means of the cloud) easily holds and operates on data in the manner described. NOW with the information provided by this short tutorial, I may add programming to my investment spreadsheets that will utilize the historical data contained therein to display the yields of my investments as they are reported. Thanks.