# Arithmetic Mean vs Geometric Mean – Why are Financial Institutions Allowed to Report Returns the Way they Do?

## Arithmetic Mean

Arithmetic Mean, more commonly referred to as mean or average, is the summation of observation divided by the number of observations.  Or in plain English:

(1+2+3+4)/4

= 2.5

In finance, we are typically concerned with average rates of return over periods of time.  Returns of the stock market indices, mutual funds, ETFs and individual stocks for 1, 3, 5, and 10 year periods typically.  Calculating the average is real simple.  However it may not be the best form of calculating an average due to outlying data.  Basically arithmetic mean understates the effect a large single data point can have on the average.

Using arithmetic mean (which is used by mutual funds, ETFs, etc.) will lead investors to believing they made more money over time than they actually did.

What does that gibberish from the last sentence mean.  Lets use an example with a 5 year sample of returns for the S&P500 from the years 2006-2010.  I am using this 5 year time frame to include a negative year.

2006= +15.79

2007= +5.49

2008= -37.00

2009= +26.46

2010= +15.06

So in calculating the arithmetic mean we simply add up our returns: 15.79+5.49-37.00+26.46+15.06= 25.8

Next we divide the summation of our observation from above, by the number of occurrences (5).  25.8/5= 5.16%

So any reasonable investor would think  \$100 invested for 5 years at an average return of 5.16%  would mean you have \$128.60 at the end of 2010.  But is this necessarily correct?

Lets look a bit deeper and see if this is correct, by using \$100 invested at the start of 2006 and figuring the value we would have at the end of each year.

At the end of 2006, our \$100 investment would be worth \$115.79 (calculated 100*1.1579)

At the end of 2007, our \$115.79 investment would be worth \$122.14 (calculated 115.79*1.0549)

At the end of 2008, our \$122.14 investment would be worth \$76.95 (calculated 122.14*(1-0.37)).  Oh no, we are in the hole now.

At the end of 2009, our \$76.95 investment would be worth \$97.31 (calculated 76.95*1.2646)

At the end of 2010, our \$97.31 investment would be worth \$111.96 (calculated 97.31*1.1506)

Hold the fuck up!  Above we showed an average 5 year return on the S&P500 of 5.16% and applying that average return on \$100 we showed a closing value of \$128.60.  However when we calculate what \$100 actually invested in the S&P500 would be worth, we come up with \$111.96.  \$111.96 vs \$128.60 are not even close with a difference of \$16.64 overstated.  Don’t think it’s a big deal.  Imaging \$100,000 or some other large dollar amount, and the difference is more substantial.  Is there a better means of calculating mean? (See what I did there)

## Geometric Mean

Geometric mean is a way of calculating an average which uses the product (multiplication) of the observations as opposed to arithmetic using the summation (addition).  In other words, geometric mean is the nth root of the product of n observations.

Calculating Geometric Mean for returns by hand is actually quite simple:

{[(1+r)(1+r)…]^(1/n)} -1

where r is the return for each observation point

and n is the total number of observations

Lets use our data points from the S&P 500 again from above.

2006= +15.79

2007= +5.49

2008= -37.00

2009= +26.46

2010= +15.06

Using geometric mean, we arrive at an average of:

{[(1.1579)*(1.0549)*(0.63)*(1.2646)*(1.1506)]^(1/5)}-1

{[1.11969682]^(.20)}-1

(1.02286917)-1

= 0.02286917 or 2.28%

Geometric mean better handles the effect of outliers, but also negative returns.  Here we come up with a geometric mean of 2.28% vs the arithmetic mean of 5.16% for the same sets of data.  In this case, the arithmetic mean is more than twice as high as the geometric mean.

So, now lets apply our geometric mean of 2.28% to the \$100 initial investment and see where we are after 5 years and compare to arithmetic.

Using the 2.28% geometric mean on an initial \$100 investment, after 5 years we have a closing value of \$111.93.  Compared to the arithmetic mean ending 5 year value of \$128.60.  A difference between the 2 averages of \$16.67 or  what I’d say is statistically significant when it comes to money.

More interesting is comparing the 5 year ending value of the geometric mean of \$111.93 to the actual ending value of investing in the S&P500 from above of \$111.96.  A difference of only \$0.03 understated. Compare that to the difference of using arithmetic mean at \$16.64 overstated (\$128.60-\$111.96).

## Why Does This Matter?

Think about the differences in numbers above using arithmetic vs geometric mean.  When you pull information for returns on mutual funds and ETFs, they are reporting arithmetic mean.  If a return was constant year after year, no big deal.  However, because of variability/volatility of returns and compounding, arithmetic mean will overstate the performance of a mutual fund, which in turn translates to overstating your potential account balance.  This sucks, because in reality your balance will be lower.  Geometric mean offers a better may of stating the returns of mutual funds over time and better reflect the actual dollar balance an investor has.  That’s why this matters.  It affects your money.

So what do the SEC and FINRA actually do to protect investors?  Returns are being misquoted to investors.  Add on top of this the ridiculous fees that many mutual funds have, and investors are fighting even more of an uphill battle with no actual regulators doing a thing to protect them.

Coming soon, we will take another look at average returns and how the timing of returns also makes a big difference on portfolio values.

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