What Do You Mean, Average?

If you get the pun in the title of the post, consider yourself a step ahead of the game for today’s posts.  By nature, I’m a bit of a numbers girl – so today I’m going to let that nerd-flag fly and talk about some of the different numbers that we look at when we’re evaluating investment returns.

Screenshot of MUTF:VTIVX taken September 25 2012 from Google Finance.

First, here in this post, we’ll have an intro on the different types of functions that someone might call an “average” and which ones you should really use when looking at stock returns, and after you master those basics… Head over to post 2, kindly hosted by 6400 Personal Finance (but still written by me, Mrs. PoP) to see why what you really need to be worried about is the variability behind those averages, often measured by standard deviation – Variability Can Eat Your Returns.

So here goes…


Where Do We Care Most About Averages?

Remember when the most important average in your life was your GPA?  Now that most of us have put our schooling years behind us, that’s no longer the case.  Instead what we care about is average performance, and we’re usually talking about the stock market.

Say you’re evaluating mutual funds for your Roth IRA or your 401K, so you’re looking them up on google finance.  This screenshot is taken from the google finance page of a mutual fund that we actually own shares of, the Vanguard 2045 Target Date Fund (MUTF:VTIVX).

Google shows the trailing returns for varying periods in the nice little red and green bar chart, as well as Morningstar’s “star” ratings.  You’ll see this fund is pretty good as it’s had above average returns with below average risk across its entire history if you clicked through to the rest of the Morningstar report.  Also, a push for Vanguard – the fees are really reasonable.  This one has a management fee of 0.19%/year.

But back to the topic at hand…

See the * that Google puts next to the 3-year and 5-year trailing returns?  Below it says that represents annualized returns… so let’s talk about a few ways that we can get there and understand what’s going on behind it.


The Very Basics

I’m going to skim through some of these since I assume you’ve seen them before – but if you have questions, feel free to email and I’ll be happy to go into more detail.

Mean – this is your traditional average.  Add up all the parts, divide by the number of components.  For a stock this would equate to: Add up all the yearly returns, then divide that sum by the number of years.

Median: median means the middle number if you lined up all the components in order.  Yearly returns of 5%, 25%, 10% would have a median of 10% since that is the middle if you put them in order.  The good thing about medians is that if you have numbers that are really unusual or seem “out of whack” with the rest of them, they get ignored since they are on the ends when you line them up in order.

But the mean and median aren’t nuanced enough to get a clear picture about what’s going on.  For that, we need to do a little…



Annualizing a rate basically means that if you have a given stream of returns (usually monthly), we want to figure out what the equivalent yearly return would be.  Annualizing makes it easier to compare to other stocks and investment intstruments.

There are many ways to annualize a calculation, so when you can, you want to know how a calculation is being annualized because it can make a big difference.


Annualizing the Mean With Reinvestment (AM)

This is one of the simplest ways to annualize.  Basically what it means is that you let all gains accumulate and grow gains on top of gains, reinvesting the gains each month.  It also means that any dividends stay in the account and are reinvested into the same fund.  From an equation point of view, where mean = monthly mean, and n = number of months:

AM = (1+mean)^(n/12) – 1

In case it’s not clear, we’re dividing n by 12 since there are 12 months in a year, and we’re trying to calculate an equivalent annual (yearly) rate.

Although the AM is a pretty easy calculation to do, it can be a pretty weak way to calculate annualized returns because it obscures the fact that there’s variability behind the stock.  Because of that, the AM tends overestimate returns when they are both positive and negative.  Kindof the worst of both worlds when it comes to rounding errors.

Compound Annual Growth Rate (CAGR)

A much better way to calculate annualized returns is using the CAGR.  The CAGR (pronounced kager, like kegger but with an “A” sound) is the compound annual growth rate.  Basically it requires you to know the total growth for a given period, and then backs into the rate, that when compounded on a daily basis, would produce that growth.

Finding the CAGR taks a little more time, but if you’re got your data in something like Excel you can do it in a snap.

1 – Calculate the total return by multiplying together the quantities 1+monthly return

(1 + return month 1)*(1+return month 2)*… *(1+return month n) = 1+Total Return

2 – To annualize it, we take the (n/12)th root of the Total Return.  Don’t remember what that is?  Here you go!

CAGR = (1+Total Return)^(12/n) – 1

(Obviously if someone tells you the total return and the number of months, just skip to step 2!)



Why does this matter?  

Well, let’s think about the last six months of SP500 returns.

Month Yr % Gain SP500
Mar 2012 -0.75%
Apr 2012 -6.27%
May 2012 3.96%
Jun 2012 1.26%
Jul 2012 1.98%
Aug 2012 4.21%

The mean of these monthly returns is 0.73%, so if we calculated the AM, we get (1+0.73%)^12 -1 = 9.14%

But is that actually what the SP500 returned over this period?  Nope.  Multiplying the 1+monthly returns together, we see that the SP500 actually was only up 4.1% in that period, so we get a CAGR of (1.0041)^(12/6)-1 = 8.31% for the same period.

Sure, it doesn’t look like a huge difference (9.14% vs 8.31%), but it’s a 10% difference in returns between the AM and the CAGR calculations on the same data.  The AM is overestimating those returns by 10%!  And if you start using the AM to project into the future, it leaves open the distinct possibility that you’re counting on your returns to be 10% better than they are likely to be.  And that could be the difference between running out of money before you reach your expiration date.


Is This The Whole Story?

Not in the least – while I think we’ve pretty firmly established that you should be using a CAGR to calculate annualized “average” returns, thats only part of the story.  Variability matters – and as it turns out, it can matter quite a bit.

For more on how variability can impact your long term expected returns, pop on over to 6400 Personal Finance where Dave is kindly hosting the second post in this “mini-series”, Variability Can Eat Your Returns.  Be sure to check out the beautiful graph I put together for that post =)


When someone quotes you an “average” investment return, do you actually know what method they’re using to calculate it?  Now that you know how big the differences can be, does it make you want to go back and check?


7 comments to What Do You Mean, Average?