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"Average Annual Returns"
What the numbers really mean

by Ann Coleman (TMF AnnC@aol.com)

Reston, VA (March 9, 1999) -- We love to quote investment returns. The Foolish Four has an average annual return of 22.59% per year over the last 10 years. The Foolish Four has an average annual return of 24.55% per year over the last 25 years. The Foolish Four has an average annual return of 19.56% per year over the last 38 years.

As we asked yesterday, though, what do those numbers really mean?

First, they are not the average of the returns for all those years. (Wait a minute, didn't she just say...?)

"Average annual return" is a somewhat misleading name. I used it because it has been adopted by the mutual fund industry and it makes sense to let people know that we are comparing apples to apples. But if you think you can add up the yearly returns from 1974 to 1998, divide by 25, and get 24.55%, you would find you were wrong. We are not using what most people think of as the average, or "mean." (We aren't even using the median or mode!)

What we, and the mutual fund industry, are using is the "geometric mean," also known as the Compound Annual Growth Rate, or CAGR, to its friends. The CAGR is the rate at which a lump sum, invested in the strategy at the beginning with no additions, would have grown each year. And, more importantly, it is a number you can extrapolate into the future, given, of course, that your investment continues to perform in the future as it has in the past. The mean (or arithmetic average) doesn't do that very well.

Here's an example.

Initial amount: \$4,000 Account balance
Year 1 return: 20% \$4,800.00
Year 2 return: 12% \$5,376.00
Year 3 return: 2% \$5,483.52
Year 4 return: 39% \$7,622.09
Year 5 return: 25% \$9,527.62

Now, if you "average" each of those yearly returns (add them up and divide by 5) you get 19.6%. But if you try multiplying your starting balance by 19.6% per year for 5 years, you get \$9,788.49. Whoa. You've overstated your actual, real-life balance by more than \$250 bucks. Imagine that error compounded for 25 years. (OK, don't imagine it: After 25 years, you would be overstating your account balance by \$44,349.12. There go your plans for the SUV with the dual TVs.)

That's why we don't use the regular old garden variety average. It's just too high. And that's why the mutual fund industry gave up quoting it (with some reluctance, perhaps).

The CAGR avoids that trap by working backwards. The real question is, what do you multiply \$4,000 by to reach a balance of \$9,527.62 after 5 years?

\$4000 * N * N * N * N * N = \$9,527.62
N is the mystery number, the real average annual return.

Again working backwards, we want one number (N) that you can multiply by itself 5 times, which you can then multiply by \$4,000 to get \$9,525.62.

Hang on now: N * N * N * N * N = N to the fifth (N^5), right? Just like 2 * 2 = 2 squared (2^2) or 2 * 2 * 2 = 2 cubed (2^3).

Let's find N^5 first, because it's easy. It's just whatever number you multiply \$4,000 by to get \$9,527.62, right? \$4,000 x N^5 = \$9,527.62

All we need is fifth grade math. A * B = C and C / A = B.

So 9,527.62 / 4,000 = 2.381905 = N^5

Now you just need the fifth root of 2.381905, which is 1.189555987. (You might have to trust me on this -- most calculators don't do anything tougher than square roots. Don't worry, I'm not doing it in my head.)

Does it work? (This part anyone can verify.)

\$4,000 * 1.189555987 * 1.189555987 * 1.189555987 * 1.189555987 * 1.189555987 = \$9527.60 (you lose a couple of cents to rounding)

Of course, 1.18955 etc., doesn't look much like a growth rate. The common way to quote it would be to subtract one and change to a percentage. Voila! 18.95%.

(The digit 1 is there so you don't "lose" your original value each time you multiply. If you want to increase something, you multiply by 1 plus the percentage. 18 * 0.15 = 2.7, which is 15% of 18. To get 18 plus 15% you go: 18 * 1.15 = 20.7. You could just add the 2.7 to 18, but multiplying by 1 is more efficient. Ask any waitress.)

Other than calculating tips quickly, knowing what the "average annual return" really means can keep you from being deceived when comparing investment returns. Anyone using the arithmetic average, or mean, is probably overstating the real average growth rate.

And it makes for delightful dinner party conversation. You are going to explain this to all your friends, aren't you?

Oh, yes -- go, J.P. Morgan (NYSE: JPM)! That stuffy ol' investment bank is really on a roll. Check out Today's Numbers.

Fool on and prosper!

 ```Stock Change Last -------------------- CAT - 13/16 48.81 JPM + 13/16 116.38 MMM + 15/16 77.81 IP - 1/4 40.81 ```
 ``` Day Month Year History FOOL-4 -0.11% 3.63% 4.68% 6.24% DJIA -0.35% 4.16% 5.73% 5.31% S&P 500 -0.23% 3.35% 4.44% 4.69% NASDAQ -0.20% 4.59% 9.13% 10.63% Rec'd # Security In At Now Change 12/24/98 24 Caterpillar 43.08 48.81 13.31% 12/24/98 9 JP Morgan 105.51 116.38 10.30% 12/24/98 14 3M 73.57 77.81 5.77% 12/24/98 22 Int'l Paper 43.55 40.81 -6.29% Rec'd # Security In At Value Change 12/24/98 24 Caterpillar 1034.00 1171.50 \$137.50 12/24/98 9 JP Morgan 949.62 1047.38 \$97.76 12/24/98 14 3M 1030.00 1089.38 \$59.38 12/24/98 22 Int'l Paper 958.12 897.88 -\$60.25 Dividends Received \$15.04 Cash \$28.26 TOTAL \$4249.43 ```