Fool.com: Calculating the Growth Rate [The Fool Ratio]

http://www.fool.com/school/calculatinggrowth.htm

We've already mentioned that when we talk about growth rate, we're talking primarily about the future earnings per share (EPS) growth rate. The most relevant numbers -- earnings estimates -- are the ones that haven't been reported yet. A whole industry has sprung up to guesstimate those numbers and provide them to investors.

The future EPS growth rate for a single year is quite simple to calculate. Let's say Addict Your Toddlers Toys has just reported EPS of $0.60 this year, and analysts estimate that it will earn $0.90 EPS for the upcoming year. The one-year growth rate, a straight percentage gain calculation, is 50%.

How did we calculate that? By using a simple formula, we can figure out the single year future EPS growth rate:

(Earnings Est. for upcoming year -Current Earnings)  x 100                      Current Earnings 

This is really easy math -- really! -- and we'll walk you through each step. All you have to do is plug in the numbers and subtract, divide, and multiply. A piece of cake!

So let's figure out the one-year growth rate in earnings per share. First we need our earnings numbers. Remember, the earnings estimate for the upcoming year is $0.90 per share and current earnings are $0.60 per share.

Plugging those numbers into the above formula, we get a one-year growth rate of 50%:

($0.90 - $0.60)  x 100  =  50%       $0.60 

Let's go through this formula step-by-step. First take the earnings estimate for the upcoming year ($0.90) and subtract what Addict Your Toddlers Toys earned over the past year ($0.60). This gives us the expected increase in EPS ($0.30). We want to compare the expected increase in EPS to current earnings. To do this, we divide the difference in earnings ($0.30) by the current year's earnings ($0.60). The result is 0.5. Multiply 0.5 by 100 to express the growth rate as a percentage. Voila! We get 50%.

    Estimated Earnings    $0.90      -   Current Earnings   $0.60      Increase in Earnings  $0.30         Divided by Current Earnings: $0.30/$0.60 = 0.5     Convert to Percentage: 0.5 times 100 = 50% 

So what does this really mean? In this case, the company is currently earning $0.60 per share and we expect that to grow by 50% over the coming year to $0.90 per share.

But hold on a sec here, Harry, because the more Foolish among us generally try to look forward at least two years when calculating annual growth rates. This makes the math a teensy bit more involved. Let's check out Addict Your Toddlers Toys' projected earnings for the next two years, then work up a two-year annualized growth rate:

  1996 (current)   $0.60   1997 (estimated)   $0.90e   1998 (estimated)   $1.35e 

To perform the calculation, you first need to find the total percentage growth from 1996 to 1998. We use the same formula we used above for the one-year growth, except that we use the earnings estimate for the most distant year -- in this case, from 1998. The key thing to remember here is that since we are looking at growth over two years, we also need to annualize our result to determine the average growth rate per year. Investors tend to talk about growth rates on an annual basis unless otherwise specified. To calculate the total expected two-year growth rate:

     Estimated Earnings    $1.35       -   Current Earnings   $0.60     Increase in Earnings   $0.75         Divided by Current Earnings: $0.75/$0.60 = 1.25     Convert to Percentage: 1.25 times 100 = 125% 

Is this right? Let's check it. If we multiply the current earnings ($0.60) by the total growth (1.25 +1=2.25) we should get the estimated earnings. So $0.60 x 2.25 = $1.35. Ok, you're wondering why we added 1 to 1.25 to get 2.25 for total growth. Good question. When multiplying by a percentage, we add 1, or 100%, so that the original amount is included in the answer. (It is just like adding in tax. If tax is 5%, you can multiply by 5%, which is the same as .05, and then add that to the original amount to find the total cost... or you can just multiply by 1.05. Try it!). For this example it would be: $0.60 x 1.25 (which is the same as 125%) = $0.75 + $0.60 = $1.35.

Now, to make sense of this total growth of 125% and to be able to compare it to other companies' growth rates, we need to annualize it. Careful, it gets tricky here. First, you can't just divide the total growth by two. Let's try that and see what happens. 125% divided by 2 = 62.5%. Is that correct? Well, let's apply it to the earnings and see what happens.

Year 0 EPS = $.60 x 1.625 = $0.975  Year 1 EPS = $0.975 x 1.625 = $1.58  Year 2 EPS = $1.58 -- I don't THINK so! 

Now, you might have noticed that what we are doing is multiplying each year's EPS by the same number -- the presumed average annual growth rate. Another way to write that would be 0.60 x 1.625 x 1.625 = 1.58. Even though the rate we are using gives us the wrong answer, it also gives us a clue to how to find the right answer...

If you are mathematically inclined, you will have noticed that 1.625 x 1.625 is the same as 1.625 squared (we write that as 1.625^2) and 1.625 is the square root of 1.625^2. (Not so obvious? Try 2 x 2 is the same as 2 squared, and 2 is the square root of 2^2). But when we used the 62.5% figure we didn't arrive at the correct earnings estimate. Eureka! The correct annual growth rate must be the square root of whatever number, multiplied by the current EPS, will give us the estimated EPS two years from now.

And we already know what number we multiply the current EPS by to get the estimated EPS -- the total growth! (Actually, the total growth plus one, so that the original EPS is carried into the answer.)

$0.60 x 2.25 = $1.35 

So the average annual growth rate must be the square root of 2.25! Let's try it. Power up your voice-activated calculator and command it to take the square root of 2.25 (or enter 2.25 and hit that little square root symbol thingy.) Voila! The square root of 2.25 is 1.5, or 50% annual growth (remember, we have to subtract 1). Let's check and see if it works:

Year 0 EPS = $0.60 x 1.5 = $0.90  Year 1 EPS = $0.90 x 1.5 = $1.35  Year 2 EPS = $1.35 -- Bingo! 

Now, it should be obvious (yeah, right) that if you are lucky enough to have earnings estimates that go out 3 years, the average annual growth rate would be the cube root of the total growth -- for 5 years it would be the fifth root. (If it's not obvious, just take our word for it.) But what if you are in the middle of the company's fiscal year so that it's not a nice clean two years until the estimate? Can you say fractional roots?

In many cases the time period between your current earnings ($0.60 in this example) and your estimated earnings ($1.35 above) will not be exactly two years. Remember, for ANY time period, the root you take to annualize the growth rate is exactly the number of years from present value to future value. (Please read that last sentence again, fully digest it, and commit it to memory in preparation for our 6:30 a.m. quiz tomorrow morning.) Thus, if we were dealing with three years, you'd use the 3rd (or cube) root; five years, the fifth root, and for one and a half years we'd use the 1.5 root, etc.

(NOTE: When you have a simple situation like our example above where you have only 2 FULL years, you can estimate the average annual growth rate by simply averaging the individual years' rates. But when you are dealing with fractional years or estimates several years out, that method is not very accurate.)

You can figure out the fractional roots by finding out exactly how many quarters it will be to the furthest-out estimate. If we are using annual earnings that include the first quarter of fiscal 1996 and we have estimates for fiscal 1997, then there will be seven quarters between estimates. You just divide the number of quarters by 4 (the number of quarters in a year) to convert to a fraction. In this case: 7/4 (seven-fourths).

Now, most calculators won't take a 7/4 root, but many calculators and all spreadsheets will take something to any power. To find the x root of a number, you simply raise it to 1/x powers (the reciprocal). (Just ask any high school kid.) So the square root of 4 would be written as 4^(1/2) meaning 4 to the power of one half. (The reciprocal of a fraction is just the fraction reversed.) The 7/4 root of 2.25 would be written 2.25^(4/7). (You put parentheses around the fraction so that you don't accidentally tell your calculator to take 2.25 to the fourth power and then divide the answer by 7 -- you get points off for that.)

Whew!

We realize that most people don't find fractional roots all that exciting, so we have created a little program know as the PEG Calculator (a.k.a. PEGulator) that takes all the work (we mean "fun") out of finding the PEG. All you need is the current price, the current 12-month trailing earnings, the estimated earnings, and the number of fiscal quarters between the last actual earnings and the date of the earnings estimate. You plug the numbers into the PEGulator and out pops the PEG. Click here to check it out.

Returning to a plane of reality on which most of us feel more comfortable, you've just gained a basic understanding of how to calculate the EPS Growth Rate -- more than the average investor possesses. Let's load these two bits of ammo into our giant Nerf dartgun (not available in FoolMart), and start looking for some stocks.