How to Properly Calculate an Annualized Return

The S&P 500 index has increased from 1,549.38 at the height of the market in October, 2007 to 2,091.48 at the present. What is the annualized rate of return since then? Its been over eight years, that’s long enough that we should earn around what Wall Street considers the “average” return, right? I’ve seen some meme’s on Facebook showing how obama has grown the market. It must be pretty high, right?

Do you know how to calculate the return on an investment (ROI)? I think most people know how to calculate an ROI. Its pretty simple math. A recent conversation (debate) with a real estate investor on BiggerPockets.com prompted me to write this article.

This investor told me that his stock market holdings were up 50% since 2007. He sounded excited about this, almost bragging. 50% over 8 years doesn’t sound like a great return to me. Especially with the risk of losing 30-40% in any given year. After all, what does this mean on an Annual basis. So what does this look like when you convert it to an annualized rate of return. Note the terminology. It makes a big difference.

The ROI is easy to calculate. Where I see most people making mistakes is when they try to convert their nominal return to an annual number.

What is the Annualized ROI if your investment increased 50% over 8 years? The Return is 50%. That much is simple. But we really need to convert it to an annual rate of growth to compare it to what we could have made in other investments. The financial experts on Wall Street tell us that the market appreciates at a long term average of 9%. What does that number mean? How did they calculate it?

Do you simply take 50% and divide it by 8 years to get an average? No.

Let’s put this into some real numbers so you can see what happens when you use incorrect calculations. Let’s say this investor had $1,000 in mutual funds and he turned it into $1,500 over 8 years. The $500 gain divided by his $1,000 investment is how we get the 50% return on investment. To get a simple average we divide 50% by 8 years for a 6.25% Average Return.

This number is useless and doesn’t tell you anything about the investment.

Here’s why:

If I stuck $1,000 into an investment that pays 6.25% annually, here’s what my account balance would look like each year:

See the problem with using a simple average? Look at the account value at the end of the 8th year. I made $624.17. So how did my investor friend make a 50% return and I made a 62.4% return if we’re both earning 6.25%?

 

Its simple.

He actually earned something less than 6.25%.

The average doesn’t take compounding into effect. Compounding is an incredibly powerful force and you can’t overlook it.

I don’t care about average returns. They’re meaningless to me. If you’ve kept up with my articles over time, you know that the Rule of 72 is my favorite rule in finance. If you divide 72 by the interest rate you are earning, the result is the number of years it will take to double your money.

If you are going to accurately employ the Rule of 72, then you need to know thetrue annual rate of return.

I want to know the annualized rate of return for my investor friend. What rate of return would result in $1,000 turning into $1,500 after 8 years?

To get the correct answer requires a little more sophisticated math. Compounding interest results in Exponential Growth. So we need to play with exponents.

If I am earning 10%, then I can calculate my expected return by multiplying my investment amount by 1+10%. For example,  $1000 investment * 1.1 = $1,100. $100 is a 10% return.

If I roll this investment over for a second year, its $1,100 * 1.1 = $1,210. See how 10% compounding growth for two years gives me a 21% Return? (or a meaningless 10.5% Average Annual Return)

This can also be written as: $1,000 * 1.1 * 1.1 = $1,210

So if I kept this investment rolling for 5 years, I could easily calculate my expected rate of return by just typing =1,000 * 1.1^5 into Excel. The caret symbol denotes an exponent. This is the same as 

This is all great for projecting forward into the future, but how do we work backwards to solve for the rate if we know that we ended with $1,500, started with $1,000 and were invested for 8 years?

Easy. You need to find the root. If I solve for the square root of

 

I get back to 1.1. If I subtract “1”, I get the 10% rate that I started with.

How do you do this on a calculator or in excel?

That’s fairly easy as long as you know that the root can also be expressed as the reciprocal of the exponent.

Huh?

In layman’s terms, if

is what you would get if you invested 10% for 8 years (the exponent is the number of years), then

would be the reciprocal (root). One-eighth is the reciprocal of eight. (It’s the 1/xbutton on your calculator.) So…

Here’s what it would look like in excel:

=(V-End/V-Beg)^(1/Yrs)-1

Where V = Value

So let’s figure out my friend’s Annualized Rate of Return:

Value at End = $1,500

Value at Beginning = $1,000

$1,500 ➗ $1,000 = 1.5

8 Years

5.2% is a hell of a lot different from 6.25% average isn’t it? It will take him 13.8 years to double his money at this rate of growth. He’s underperforming the long term average return of the market by 4%. Did Wall Street analysts calculate the annualized return or are they giving us the Average? Hmmm.

Does that 50% return sound exciting now?

Let’s check our math:

$1,000 * (1.052) * (1.052) * (1.052) * (1.052) * (1.052) * (1.052) * (1.052) * (1.052) = $1,500

These calculations really aren’t that hard to program into excel. Remember also that you can use any time period. It doesn’t have to be an integer. If you’re a real estate investor and you turned $10,000 into $15,000 in a quick 6 month flip, just run the same calculation for your TRUE annualized ROI. (Use 0.5 for “years”) Most people incorrectly just double their actual return to estimate their annualized ROI.

Homework Assignment:

Calculate the annualized rate of return on the S&P 500 using the numbers from the first paragraph. Post the answer in the comments below.

In my next article, we’ll use this math to debunk the ways that the media and politicians tend to mis-state data.

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