Speaking Mathanese: Compound Annual Growth Rate

Welcome back to another edition of "Speaking Mathanese," our Motley Fool series that tackles financial math myths and deconstructs the computations that make the biggest difference to your bottom line.

This week, our quest to make you smarter than a fifth-grader takes us from the tantalizing topic of total return to the confounding confines of the compound annual growth rate.

The myth
Sadly, there are no reader questions to kick off this week's lesson (come on, people!), so let's get right to it by reliving last week's lesson.

Remember, each time your investment doubles, you've gained 100%. Compound annual growth rate, or CAGR, goes further by calculating your average return per year that you hold the investment.

So, if I asked you what the CAGR was for these stocks, what would you say?


Total Return


Boeing (NYSE:BA)



Caterpillar (NYSE:CAT)



Goldman Sachs (NYSE:GS)



Hewlett-Packard (NYSE:HPQ)



Research In Motion (NASDAQ:RIMM)



Source: Capital IQ, a division of Standard & Poor's.

Just divide the total return by 3, right? Right?!?

The math
Not exactly. CAGR isn't a perfect average. Heck, it's not even a real number. Here's how Investopedia defines it:

CAGR isn't the actual return in reality. It's an imaginary number that describes the rate at which an investment would have grown if it grew at a steady rate. You can think of CAGR as a way to smooth out the returns.

To do that, you'll need the math, which is:

 ((Ending value / beginning value) ^ (1 / years)) - 1              

Still confused? Believe me, I understand. CAGR gets complicated because, instead of averaging, we're using exponents, raising the initial result of our total return calculation to the one-third power.

Let's run through an example. Say your $10,000 invested in Goliath Extra-Large Cell Phone Company (TICKER: THUD) has grown to $15,000 in three years. Dividing 15,000 by 10,000 equals 1.5.

Now, if you have a scientific calculator, or Excel, you can punch in 1.5 ^ 0.3333, which equals the one-third root. Hang on while I join you. The answer is ... (key-punching sounds) ... 1.1446.

Subtracting 1 from that gives us .1446, which, when expressed as a percentage, is 14.46%. Voila! I give you the compound annual growth rate, 14.46%.

Questions? Submit them here. (Please!) And see you next week for more mathanese.

For more money ideas, get 30 days of free access to Motley Fool Green Light right now. There's no obligation to subscribe.

Fool contributor Tim Beyers writes weekly about personal finance and investing basics. Have a Foolish money tip? Tell him. Tim didn't own shares in any of the companies mentioned in this article at the time of publication. Tim's portfolio holdings can be found at his Fool profile. His thoughts on personal finance, Foolishness, and investing in general may be found in his blog. The Motley Fool's disclosure policy is lobbying its local school district for a course in beginning mathanese.

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