Using Geometric Mean to Become a Millionaire

From the hoity-toity pseudoacademic financial community, we hear terms “arithmetic mean” and “geometric mean” more often than we see most relatives. The terms crop up when multi-year stock market returns are referenced, and basically they’re invoked because they sound fancy.

These means can apply to any kind of market’s returns (and any time horizon), but most talking heads prefer to do their talking about stocks (and usually about yearly returns).

So, like unwanted houseguests, that’s where the terms pop up most.

The trouble is some talking heads aren’t simultaneously thinking heads; their usage of the terms can sometimes make “arithmetic” and “geometric” seem interchangeable or as though either can be equivalently useful.

Let’s go ahead right now and throw that sack of pre-algebra stupidity to the mat and lean into a couple of heel stomps while no one’s looking.

Arithmetic mean (or average) and geometric mean are not the same, and only one is really useful when we think about periodic stock market gains and losses. Real investment professionals will only refer to geometric means when considering returns.

The reason is compounding. You’ll also hear geometric mean referred to as “compound annual growth rate,” or CAGR. Whatever the name, it’s how we become millionaires without any real effort.

I'll Take My Millions in Fives, Please
I’ll Take My Millions in Fives, Please

Getting Figured

The arithmetic mean is the straight-line average. Sum all the numbers in a set, divide by the count of observations, and – boom.

A stock market returns sequence of -50%, +60% and +10% yields an arithmetic mean of ((-50 + 60 + 10) / 3) = 6.67%. Yay! We won!

Actually, sorry, sport. We didn’t. You put that game-winning layup into the opponent’s net: we lost (12%).

The trouble with an arithmetic mean in contemplating stock market returns is that the arithmetic average ignores compounding.

Yes, -50 + 60 + 10 = 20 (i.e., more than zero, implying a gain), but the presence of a big loss in the first year reduces our portfolio base for the subsequent years’ gains. The actual result for our portfolio is a bad one. Rather than a 6.67% arithmetic return, we actually end up with a 12% loss, represented by a geometric mean annual loss of 4%.

Which makes us less prone to cartwheels than to laying down in traffic.

And is why we really care about geometric means when thinking about multi-period market returns.

Number Crunches

How do we calculate the geometric mean?

The best way, really, is to plug periodic returns into a spreadsheet and use the “geomean” function. (Note: each periodic return must have a 1 added to it for the function to work properly. For instance, if a year’s return is 8%, use 1.08; if the year’s return is -10%, use 0.90).

Using data from the U.S. stock market for 2013, 2014 and 2015, the following example walks through the mechanics, just in case Excel gives you migraines.

U.S. S&P 500 stock index annual returns in 2013, 2014 and 2015 were 32.15%, 13.52% and 1.36%, respectively. (These figures include dividends, of course. Dividend values for 2015 are estimated using 2014 figures, which means the 2015 return number may vary slightly from that reported by other sources, but, for our purposes here, that’s not too important.)

When calculating the geometric mean, add 1 to each of the observed return values (e.g., 1 + 32.15% = 1.3215). So we’re left with 1.3215; 1.1352; and 1.0136.

Now multiply the three values: 1.3215 x 1.1352 x 1.0136 = 1.5205.

Then, take the nth root of the resulting figure (where n = number of periods observed; in this case 3): 1.5205 ^ (1/3) = 1.1499.

And finally deduct 1, which gives us 0.1499.

So, the geometric mean for our series is 14.99%. (The arithmetic mean is a dastardly 15.68%.)

Which brings us back to an important implication of all this for early retirement.

A critical variable determining portfolio strength and longevity is how well that portfolio performs during the earliest years in which you’re pulling from it for income (or, equivalently, during the last year or two just before retirement – which is why we often see spikes in retirement rates following a strong bull market).

If this seems frightening, and a bit more like gambling than you’d prefer for your doughy nut, consider the following.

It’s All Rigged In Your Favor

From 1928 through 2015, the S&P 500 has given positive annual returns 73% of the time. The last time you had a 73% chance of winning at any gamble?

That would be exactly never.

And even if the index falls during a year, it’s unusual for the following year to be negative as well.

From 1928 through 2015, a negative year for the S&P 500 has been followed by a gain the subsequent year two-thirds of the time (i.e., consecutive negative years have occurred 8 times out of 24 total loss years). That includes 5 instances during the Great Depression and World War II. If those unusually nasty eras of history get pulled out, good follows bad 80+ percent of the time.

So, yeah, stocks go up most of the time. And even when they don’t manage to go up one year, they’ve got better than good odds of ticking up the next.

Which means that, even if you have no idea how stocks will perform during the first year of dependence on your portfolio (and you, of course, don’t), you’re still better off with exposure to the U.S. stock market than not. You’re actually much better off.

And to really take advantage, your exposure should be a felony-level opening of the economic trench coat.

The takeaway from this and from economic studies looking at investment strategy and questions of placing money into the stock market is this: The earlier you dip into the market, the better. Because the market tends to go up, you’re far more likely to miss gains by waiting to invest than you are to incur losses by investing too early.

In fact, in many ways, the market is rigged to always go up over the long term. For those who think stock market investing is like gambling, the analogy is fair as long as it’s recognized that those invested in the stock market are most definitely the “house.”

After all, at their most fundamental level, stock indexes are forecasts of long-term economic growth. If future economic growth is positive, stock values will rise. (And over the long run, economic growth has always been positive. How many of us still live in caves?)

Now you’re all hot and bothered about delving into the particulars of stock valuation. I know, I know; it’s scintillating stuff, those discounted cash flows and risk premia. And we’ll slap all that around to our heart’s content another time.

Making It (to 6 Zeroes)

Investing and holding low-cost index-based U.S. stock issues like ETFs over the long term is one of the surest, awesomest ways of building wealth and guaranteeing a prosperous permanent vacation from cubicles and staff meetings.

Using our newly minted knowledge of geometric means, let’s indulge ourselves with an illustration of just how awesome stocks can be for our doughy nuts.

Assume for a second you’re a bit like me. You plunged into your career ready to conquer the world only to be dealt a ruthless and persistent lesson in how suffocating it can be to slog through the workday routine.

So you decided very quickly to cut your prison term from the average 35+ years to something more like 10.

You saved aggressively and invested smartly. You made regular deposits into broad-based ETFs that emphasized the U.S. market and you watched your doughy nut grow and grow and grow.

Here’s how to set up the calculation in Excel.

You’ll need six columns: Year; Return; Investment; Applicable Geomean; Years; and Result.

Year (singular) is the year in which the lump-sum investment is made. We’ll start our analysis in 2003 and run it through 2014, roughly mirroring my personal investing timeline before saying sayonara to the traditional career.

Return is the annual percentage gain in the applicable index. Here we’re using S&P 500 index returns, including dividends.

Investment is how much you placed into the market. For simplicity, let’s assume you do all your investing on January 1 of each year, and let’s choose a conservative annual savings figure – one that can be attained by most – of $45,000. (If you think saving $45k a year is a pipe dream, you’re right for most people – but if you happen to be smoking FL-branded knowledge grass, we can make it happen.)

Geomean is the geometric mean of annual returns for the years your investment is in the market. This is calculated using the “geomean” function. Each year’s geomean equals the geometric mean of returns for the current and subsequent years in our analysis. For instance, the geomean shown for 2005 equals the geometric mean for 2005 through 2014.

Result is a calculation: Investment Amount * (1 + Applicable Geomean)^Years

Total is a calculation: Result, current year + Total, prior year

Here’s the resulting table below covering 12 years, with a meager $45,000 invested annually. Man, we get to millionaire status pretty quick, don’t we? (Still think you can’t save $45k a year? Your future millionaire self says: Think harder.

Year S&P 500 Return Investment Geomean Years Result
2003 28.36% $45,000 0.0946 12 $133,178
2004 10.74% $45,000 0.0789 11 $103,757
2005 4.83% $45,000 0.0761 10 $93,692
2006 15.61% $45,000 0.0792 9 $89,371
2007 5.48% $45,000 0.0700 8 $77,303
2008 -36.55% $45,000 0.0722 7 $73,283
2009 25.94% $45,000 0.1701 6 $115,502
2010 14.82% $45,000 0.1530 5 $91,715
2011 2.10% $45,000 0.1543 4 $79,877
2012 15.89% $45,000 0.2024 3 $78,235
2013 32.15% $45,000 0.2248 2 $67,508
2014 13.52% $45,000 0.1352 1 $51,086
Source: FinanciaLibre $1,054,507

One final note here that I find particularly interesting. A nifty side-benefit of constructing our table using the geometric mean is we can see just how powerful the stock market is for building wealth, even when the initial year or two of investing is ho-hum (or downright nasty).

Take a look at the geomean values for 2007 and 2008. If you invested on January 1, 2007, your first year’s return was a measly five-and-a-half percent. But you still garnered a compound return of 7% through 2014. And that includes the horrendous year markets had in 2008. In the graph below, that market meltdown barely registers as a speed bump on our road to millionaire status.

That’s another side-benefit of arranging our calculation and graph this way. They illustrate how stock investing builds wealth over the long term. Stock investing isn’t a useful short-term wealth-building strategy because of price fluctuations. This rendering takes out the short-term gyrations and leaves a more stable long-run trendline.



Let’s say, though, that you happened to be one of the poor souls who dumped a bunch of change into the market on January 1, 2008. Your first year of market exposure stripped out more than a third of your portfolio’s value. Your whole portfolio got rocked by price fluctuation. And yet… Look at the geomean for that year in our table: more than 7%. You still gained around $30,000 on that initial $45,000 invested by the end of 2014. (A total gain of 63%.) You just had to leave your money in the market.

It’s so easy. Becoming a millionaire, that is. So get out there and get some.

Luchadores, drop me a note in the comments section quantifying your annual stock contributions – whether they be via IRAs, 401(k)s, SEPs, personal brokerage accounts, etc.


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