# Growth Rates & ETF Performance Math

Compounded annual growth rates versus mean annual growth rate.

My secret shame is that despite a lifetime career in finance, I'm not really a math guy. When I went to business school, I had to do a summer course in investment math and statistics as a condition for being accepted. Words, I get. Math? Sometimes it's hard.

Which is why I felt enormous sympathy for this request when it came in to my mailbox:

"What is the different significance of the CAGR [compounded annual growth rate] vs. the AAGR or MAGR [average or mean annual growth rate]. I know if you have a 37 percent loss in your first year of analysis vs. a 37 percent loss in your last year of analysis, it's a big difference—but how exactly is it different?"

Two Ways To Measure Performance

As an ETF investor, one of the things you probably look at on a regular basis is, "How'd this ETF do?" So to compare how the SPDR S&P 500 ETF (SPY | A-99) did over the last little while versus the Guggenheim S&P 500 Equal Weight ETF (RSP | A-80) is entirely rational.

If you pop back and forth between the two fund reports, you'll see these numbers:

 1 Month YTD 1 Year 3 Year SPY -3.10% 8.73% 12.57% 19.86% RSP -2.55% 9.04% 13.05% 21.45%

A few things about how to interpret a simple table like this.

The first thing is that it's very important to make sure you're looking at total return performance; that is, performance that takes into account reinvesting any dividends the ETFs may have paid. After all, if you don't include those dividends, and just look at the price for the ETF over time, you're acting as if any dividends you received don't count!

The good news is that here at ETF.com, and at most brokerage sites and fund issuer websites, you'll see total return in tables (although, confusingly, sometimes not in charts).

So, assuming we're looking at total return, the first three columns are easy to understand—they're just your actual return as an investor in those periods. Year-to-date, your \$10,000 in SPY is now worth \$10,873.

The Pain of Annualization

Where it gets confusing, and where it gets to the question asked above, is in the "3-Year" column. If you look on our fund pages—or, again, at any industry-standard source for performance—it will say "All returns over 1 year are annualized."

That means the 19.86 percent figure quoted for three years for SPY is saying "If you put in \$10,000 three years ago, you've had the equivalent of 19.86 percent returns a year for three years in a row. So in the first year, your \$10,000 became \$11,986, in the second year, that became \$14,366, and at the end of three years, \$17,219. Not too shabby!

But your returns were not exactly that much each yea. That's just the compound annual growth rate—the single best way we have of expressing "how something did" over more than a year.

How is this different than average? Consider the following two examples:

 Scenario 1 Scenario 2 Investment Return Investment Return \$10,000 \$10,000 Year 1 \$8,000 -20.00% \$10,375 3.75% Year 2 \$8,400 5.00% \$10,764 3.75% Year 3 \$9,240 10.00% \$11,168 3.75% Year 4 \$11,088 20.00% \$11,587 3.75% Total Return: 10.88% 15.87% Average Annual Return 3.75% 3.75% CAGR 2.62% 3.75%

In scenario 1, we have a wild and wooly ride, starting off with a huge loss, and three years of increasing gains.

At the end of the four years, we have \$11,088 from our \$10,000 initial investment, for a total return of 10.88 percent. If you take the simple average of those four years individual returns, it's 3.75 percent.

CAGR

But if we really wanted to be able to reconstruct the end result—that 10.88 percent return after four years, we need to compute a CAGR—the number you apply to \$10,000 for four years in a row to get to the \$11,088. (The actual formula is very math-y, and you can go study it here, or you can plug numbers into your financial calculator or Excel and just trust it.)

To prove how different it is, scenario 2 just takes that \$10,000 and grows it by the average rate of return—3.75 percent—for four years. As you can see, in this case, you make a lot more money—a total of 15.87 percent, and because your returns are exactly the same every year, your average growth rate and your CAGR are the same.

In almost any case where you're evaluating returns for any period over a single year, CAGR is going to give you a much more accurate picture of how a given investment performed.

The Pain Of Losing Early?

There's a second part of our emailer's question, however, which is actually more pernicious.

There's a common belief that losing—or winning—early in your investment career is far more important than later on. That's not strictly true. Consider the following two scenarios with extremes on both sides:

 Scenario 1 Scenario 2 Investment Return Investment Return \$10,000 \$10,000 Year 1 \$8,000 -20.00% \$12,000 20.00% Year 2 \$8,400 5.00% \$13,200 10.00% Year 3 \$9,240 10.00% \$13,860 5.00% Year 4 \$11,088 20.00% \$11,088 -20.00% Total Return: 10.88% 10.88% Average Annual Return 3.75% 3.75% CAGR 2.62% 2.62%

The first scenario is the one we just looked at—a terrible first year, and then slowly increasing returns. The second scenario inverts the pattern—a fantastic first year and a slow slide into a dismal final year.

The end result is exactly the same, and the CAGR is exactly the same. There's nothing tricky about the math.

However, there are some kernels of truth to this belief about the early periods mattering more. Obviously if in scenario 1 you can avoid that early loss altogether, you're vastly better off—you'd go into year 2 with \$10,000 intact, not that \$8,000 from the loss.

It's axiomatic—avoiding a loss is always better than experiencing a loss. And if you have a surefire method of doing that while still getting exposure to gains, well, you should probably open a hedge fund.

One last thing—if you're thinking in investment math, every percent down is worth more than a percent up. If you have \$10,000 and lose half your money, how much does your remaining \$5,000 have to grow to recover? 100 percent. It has to double. If your \$10,000 goes up 50 percent to \$15,000, how big a loss will it take to wipe out your gains? 33 percent.

Most investors intuitively feel this—that losing hurts far more than winning helps. But understanding the math is the key to making rational decisions.

At the time this article was written, the author held no positions in the securities mentioned. You can reach Dave Nadig at [email protected], or on Twitter @DaveNadig.