Pop quiz: Pretend your investment portfolio looked like one of these three graphs. Which of these represents an 8% average annualized return each year?

The answer:

All of them!

###### An Introduction to the 8% Return:

One of the most widely quoted and useful statistic in personal finance is the concept that stocks will return an average return of 8% year after year. This value is based upon a trend of stock market returns from over almost a whole century.

Although historical returns never guarantee future returns, many investors believe that if they put their money into a stock index fund that they will be able to achieve this 8% return over the long haul.

To you and me, that means that an investment of $10,000 would hypothetically look like this:

###### What Does This Mean for My Money?

Unfortunately, as real life goes, your money hardly ever actually looks like this chart. Wouldn’t it be nice if your money just always went up and up?

The true benefit of this statistic is this:

- It helps you plan your finances for the long-term.

For example, if you have 30 years until retirement, how can you estimate how much you should be saving each month to hit your goals? Knowing that an index fund may return you 8% helps to provide some basis for your planning.

###### Arithmetic vs. Geometric Average Return – Don’t Use the Wrong One!

Unfortunately when someone tells you the average return of an investment, they may not be telling you the right (or useful) number.

Consider the following two ways to calculate the average of a data set – the arithmetic average and the geometric average:

What’s the difference? Quite a bit! Consider what $10,000 would look like in 5 years using each of these average percentages:

Which one is the right one?

To answer that, how about we plug some numbers in and actually figure this out the long way:

As you can see, the geometric average was the one that matched! Why is that?

Without getting too technical, the difference is that the geometric average takes into account compounding affects while the arithmetic average does not. If you go back to our 8% index fund example, notice how 8% grows on top of the “total” amount of money from the year before. That’s the power of a compounding rate of return.

Coincidently, the geometric average is also called an annualized rate of return, and this is the figure that most professional documents use.

Remember that if someone presents a set of data to you, they may NOT be using the geometric average or annualized rate of return, and their return rates may lead you astray. Knowing the method behind the numbers will prove useful in validating the data for yourself.

###### Nerd Alert! Here’s the Math:

If you have some annual return data and you want to figure this out yourself, it is pretty easy to do (especially if you’re handy with Microsoft Excel). The equation looks like this:

G=[ R1×R2×…×Rn ]^(1/n)

Where:

R=(1+r)

To put it in words, you start by adding 1 to each of your yearly returns, and then multiply them all together. Next apply an exponent of 1 divided by the number of values you have.

What does that look like in Excel? Here is the above example with the math shown to the right.

Try it for yourself and see if you get the same values I did.

Knowing how to do this yourself will allow you to run dozens of useful “what-if” scenarios. For example, what if I had invested in this type of stock? What if I had invested in that type of bond? All you need is the annual history of any investment to calculate your own rates of return and arrive at your own conclusions. The combinations of comparisons could be endless!

**Readers: Did you know that there was a difference between these two statistics? Have you ever tried to calculate these averages yourself for any potential investments? Did running the numbers help you learn anything new? Did I put you to sleep with the math lesson?**

###### Related Posts:

2) Why Compound Interest Makes You Rich!

3) Six Easy Steps to Figuring Out Your Retirement

John @ Married (with Debt) says

I’m definitely not going to pretend like I knew these formulas. Luckily I have access to blogs like this, where I can get info in areas that my knowledge is weak. Thanks!

MMD says

I’m glad to be a resource! That’s what we’re here for!

Lance@MoneyLife&More says

This is some great information. I knew that is how it ended up working but never took the time to do the math behind it. I know that when the market goes down 50% going up 50% won’t get me to the same amount I had before but this shows me why without having to do all of the math to get to the actual dollar amounts myself! Thanks.

MMD says

Thanks! The percentages can be a little tricky (and sometimes misleading). But by digging a little deeper, we can see through the stats and really know what we’re getting.

AverageJoeMoney says

Dude, you brought it!

This post opens up a whole new can of worms, which is: pros are numbers experts and can manipulate data. The only way to counteract manipulated data? I think it’s posts like this. Thanks.

MMD says

Thanks AJ! That premise was emphasized in every stats class I ever took: Statistics can be manipulated to say whatever you want them to say! That’s why I like to be able to work through these kinds of things myself in case I need to spot check. Hopefully I can encourage others to do the same!

From Shopping to Saving says

I knew there was a difference between those 2 statistics but I’m definitely not competent enough to figure out the difference like you with all of these formulas! I like doing what-if scenarios though and I think this will be beneficial.

MMD says

I have a feeling you are much better equipped to handle figuring these numbers out than you think. All you really need is some solid data and Excel. Doing your own what-if’s can be very liberating for your investing confidence.

Liquid says

I had a feeling it was a trick question haha. It’s always cool to learn about different ways to calculate averages. I never learned this stuff in math class so thanks for sharing :).

MMD says

I had to have a good angle or else it wouldn’t make a good post! 🙂 Glad to have taught a good lesson.

Shilpan says

This is awesome. I didn’t know the difference, but I do know now. Thanks for sharing. I wonder if Einstein’s rule of 72 is using geometric average. Good stuff!

MMD says

Thanks! Glad to be of service! And I do think the Rule of 72 is based on geometric average. Einstein was a big fan of compound interest, so I would infer that the rule is based on the same principle.

Jessica says

I’m sure I learned about these at some point in my high school math class. Must have expelled them from my memory because at first I had no idea what you were talking about. Glad to get a refresher course on arithmatic and geometric averages

MMD says

Well I hope I didn’t put you to sleep! 🙂 Thanks for reading!

Justin @ The Family Finances says

I enjoyed this post. I’m a numbers guy myself, so I knew the answer. Statistics are fun like that; you can make them say almost anything you want, lol.

MMD says

I’m glad you liked it! As you can guess, I’m a numbers guy too.

Stats can be pretty misleading because you never know who prepared them or where they’ve come from. I just like having the know-how to do my own what-if’s if I want to.

Joe Morgan says

Great post, even though the formulas probably scare a good number of people off…

I think the open charts are very powerful and really add enough to the post to be immediately helpful to any investor regardless of their mathematical acumen.