## Understanding the Annual Rate of Return and What it Means for You

Quite often, when you see an advertisement trumpeting the amazing annual rate of return of an investment, you’re not seeing the full picture of how good that investment really is. The annual rate of return, while an interesting metric, doesn’t really tell you how much money you can expect to earn in an investment over a period of several years – in fact, you’ll almost always earn substantially less than you think if you expect to get that advertised rate.

How does this work? Let’s take a look at a real world example – the S&P 500. Here are the annual rates of return of the S&P 500 over eight recent years.

In 2000, the S&P 500 returned -10.14%.

In 2001, the S&P 500 returned -13.04%.

In 2002, the S&P 500 returned -23.37%.

In 2003, the S&P 500 returned 26.39%.

In 2004, the S&P 500 returned 9.00%.

In 2005, the S&P 500 returned 3.01%.

In 2006, the S&P 500 returned 12.80%.

In 2007, the S&P 500 returned 3.81%.

### Annual rate of return

Almost all advertising for mutual funds uses the average annual rate of return to talk about how “good” that investment is. It’s pretty easy to calculate the average annual rate of return – just add up all the numbers and divide by the number of years.

In the above case, the average annual rate of return is 1.06% (it’s actually -3.31% if you want to include the abysmal 2008 in your numbers). Considering that this period includes two severe recessionary markets and only one bull market, that’s actually fairly reasonable.

One would expect, then, that an investment of $100 in that fund from 2000 to 2007 would earn 1.06% each year, leaving us with a total of $108.80 after the eight years, right?

Actually, that’s wrong – but it’s what the investment advertisers would like you to think.

### How it actually works:

If you walk through the numbers, year by year, you’ll see that in fact you would wind up with less than $108.08 in your investment.

In 2000, the S&P 500 returned -10.14%, meaning your $100 investment became worth $89.86.

In 2001, the S&P 500 returned -13.04%, meaning your $89.86 investment became worth $78.14.

In 2002, the S&P 500 returned -23.37%, meaning your $78.14 investment became worth $59.88.

In 2003, the S&P 500 returned 26.39%, meaning your $59.88 investment became worth $75.68.

In 2004, the S&P 500 returned 9.00%, meaning your $75.68 investment became worth $82.49.

In 2005, the S&P 500 returned 3.01%, meaning your $82.49 investment became worth $84.97.

In 2006, the S&P 500 returned 12.80%, meaning your $84.97 investment became worth $93.05.

In 2007, the S&P 500 returned 3.81%, meaning your $93.05 investment became $96.60.

So, while the ad might brag about a 1.06% annual rate of return, the truth is that you actually *lost* a bit of money in that investment.

### Why?

The average annual growth rate – which grew 1.06% over the period, remember – is only accurate if you *reset* the investment to $100 each and every year at the end of the year. That means that at the end of each losing year, you contribute enough extra money to bring the balance back to $100, and at the end of each winning year, you take out all of your gains.

**That’s not how most people invest – we tend to buy and hold, not take out our gains every year.** Thus, the “average annual rate of return” really doesn’t mean too much to us. Investment houses use it because it’s a simple and accurate number that is almost always higher than the real returns we would see if we were investing.

### What can you do?

If you’re considering an investment, don’t pay any attention to the average annual rate of return. Instead, focus entirely on the **compound annual growth rate** – that’s the interest rate that truly reflects how much you’ll earn over a longer-term investment if you just buy and hold. Look for that number in the investment literature – and if you can’t find it, ask for it.

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I haven’t really run the numbers, but it seems to me that you could earn a lot of money if you’re dollar-cost-averaging. If the recession years come at the beginning of your accounting period especially, you’re getting more bang for your buck, so the bull years will be disproportionately good. True, a bear market at the end of the accounting period might still have a very negative effect, but not as bad as a simple “buy and hold” strategy, invested in a bear market, would have had.

Am I missing something? (Full disclosure: I’ve just recently started working full-time, and consequently don’t have any money invested in stocks.)

You left out dividends. If you were truly invested in an S&P index fund, your actual returns would be different when you factor in dividends, whether reinvested or not. If you look at a fund like Vanguard 500 (VFINX) you can see the real rate of return (factors in dividends and subtracts expenses), all of the annual numbers are a little bit better. Smaller losses, and slightly higher gains. Total returns since 2000

So it isn’t good enough to just take the start value and end value of the S&P each year and say that’s what your return would be, because it wouldn’t in a real world investment.

I’m not going to do the math to determine whether or not this managed a positive return over that time or not, but it’s important to consider how a real investment would perform, not just index numbers.

This is a really interesting post. For an investment such as an index fund covering the S&P500, where would you go about locating the compounded annual growth rate?

Trent, does this apply to 20 year rolling averages (of, say, the S&P 500) as well, then? Or are those the CAGR for the S&P? I’m think of data like that found here.

Trent, does this apply to 20 year rolling averages (of, say, the S&P 500) as well, then? Or are those the CAGR for the S&P? I’m think of data like that found here.

(Sorry for the duplicate posts… the first had an error in the link code.)

Yep, arithmetic versus geometric means. Many an investor has probably used the arithmetic one to their detriment. This is a great, timely post.

For Excel users, you can use the AVERAGE and GEOMEAN functions to compute these stats. If your dataset includes negative numbers (like above), you’ll have problems with Excel’s GEOMEAN function. To get around that, add one to each value, compute the geometric mean, then subtract the one at the end. In the example above, I got -2.20% for the geometric mean.

A tidbit from my investments textbook: Conventionally, historical returns are represented by geometric means and forecasted returns are represented as arithmetic means. Anyone using the arithmetic mean for historical returns are inflating their numbers (as Geometric <= Arithmetic).

To further elaborate on this:

The geometric mean is calculated like this

(1-0.1014)*(1-0.1304)*…..

Then take the n-th root of that for however many years you compute and convert it into a percentage (subtract 1, multiply by 100).

The geometric mean is always less that arithmetic mean. The difference becomes particularly large if the volatility is high. Just adding up numbers historically would be naive. For future numbers, it matters less. If they are all the same, geometric=arithmetic.

Another thing that should have been learned in school is that a 50% drop in the stock market can not be regained by a 50% gain. You need a 100% gain!

Finally dollar cost averaging buys the harmonic mean of the stock prices. We also have harmonic<=geometric<=arithmetic. Hence DCA is the cheapest way to buy a stock if you make regular investments without considering the value of what you are buying.

One interesting thing to note though is that for the years you mentioned, dollar cost averaging would have served you rather well.

Let’s start with a value of 1.0 for the S&P at the beginning of 2000, so a $100 investment will be buy you 100 shares. If you continue to invest $100 at the beginning of each year (up to beginning of 2007), you will invest $800 and get a total of of 981.61 shares. These are worth $976.77 at the end of 2007. Now applying the internal rate of return calculation for 8 investments of $100 and a final value of $976.77 gives you an annualized return of 4.42%.

Also one nitpick: In your calculation for 2006 you applied the 12.8% gain to the $82.49, but you should have applied it to the $84.97. After taking this into account your final value would be $99.51 instead of $96.60.

This is likely one of your most informative posts.

Kudos for the detailed explanation.

Another reaon why you better know what you are doing if you invest in the market.

I really liked this post for a lot of reasons. The biggest one is that I’ve seen so many people recently tout the 20-year average of the S&P 500 and give recent averages for this decade.

I like stock investing, but most people have not ridden the S&P 500 exactly for the past 20 years with their investments. And in another 5-10 years, how will the 20-year average be? It might actually be a losing 20yrs overall and there will be experts saying “No, don’t invest! Look at the 20-yr avg!!”

It’s silly! Invest if you want to, and if you are aware of the risks. Also invest if you are investing pre-tax dollars and/or get a match from your employer. If you get 100% match from your employer, even if it’s a down year then you might still be in the positive.

Interesting stuff. Always nice to be reminded that simple averages will give different results than the annualized data. Thanks for the good example.

(Also, it’s amazing just how bad the markets have been the past eight years or so (and this is before adding the depressing results from last year).)

GREAT post with clear explanations. I am going to post about this at my site.

Good point. :)

I hate how everyone excludes the ‘abysmal 2008 returns’ when doing posts like this. The year existed for a reason and could happen again in the future (past performance does not tell about future gains). I think it is intellectually dishonest to exclude it normally.

That being said, in this post I see why it was excluded. You wanted a positive initial result leading to a negative end result to make your point.

Nice post.

I bet this post is very eye opening for some. I love ERE’s point as well!

There’s a “simple” way to calculate the compounded annual growth rate over long periods without knowing each year. CAGR = [(Price_final/Price_start)^(1/Num_Years)] – 1.

If we take the starting price at the beginning of 2000 (1455.22) and the ending price at the end of 2007 (1468.36), we get a CAGR of: 0.128%. If we use the ending price of 2008 (903.25), then we get a CAGR of: -5.79%. If we use 114.16 for Jan 1980 all the way through 2008, then the CAGR is a respectable 7.67%.

Plug this formula into excel and it makes this really easy to calculate.

What about the dividend yield you would have received if you invested in an S&P 500 index? For example Vanguard’s 500 Index fund has a dividend yield of 2.84% with a .15% expense ratio.

https://personal.vanguard.com/us/FundsSnapshot?FundId=0040&FundIntExt=INT#hist::tab=0

Coming from the de-lurking post, I’d like to point out that this article is a good example of why I keep coming back to your blog.

Great post, Trent. A lot of people are fooled by strong average yearly returns, and invest accordingly, without knowing anything about the compound annual growth rate. If they looked at the literature, it would tell them some pretty surprising things about their investments!

To Brandon above, it’s quite possible that the 2008 data has not been officially released yet. It’s only the 19th of January. :)

You can calculate your compound annual growth rate (CAGR) “easily” — http://www.investopedia.com/terms/c/cagr.asp

Using the formula in the link, I get 0.13% CAGR for 2000-2007, -5.8% CAGR for 2000-2008, and 7.7% CAGR for 1980 – 2008.

I’m not getting a geometric mean of -2.20% for the above numbers. I’m getting -0.062%. I’ve calculated this same value by hand, with Excel, and with an online geometric mean calculator. Could someone steer me in the right direction, please?

I work for a mutual fund company and (at least for my company) this is not how average annual returns are calculated. If it were, I agree that it would be deceptive.

Average annual total returns (AATR) are calculated using the CAGR formula Trent mentions. The calculation and display of these returns is mandated by FINRA (regulatory body for the industry). I would be surprised to see firms promoting “simple average” returns that covered multiple years as that would be just begging for a costly audit, if not a lawsuit.

So what was the S&P average for 2008?