One of the more confusing elements of the mathematics of finance is the issue of "average annualized return." It's not that we don't understand what an average is, it's just that most of us were introduced early and often to only one of two important means: namely, the arithmetic mean. Perhaps lesser known-but critically important-is the geometric mean. Other means (that won't be discussed here) include harmonic mean, quadratic mean, trimmed mean and winsorized mean.

The word "average" in "average annualized return" may be part of the problem because it triggers that arithmetic averaging method we've all been overtaught. But even though "average" is synonymous with "mean," one mean or average is appropriate for investments while the other is not.

A simple example might help at this point.

Year 1       50% gain

Year 2       50% loss

The too-casual observer using an arithmetic mean method might conclude that the average annualized return over this two-year period equals zero. You would add the numbers and then divide by the number of numbers.

[.50 + (-50)] / 2 = 0% return

Now, let's put dollar figures into the example and see if the arithmetic mean holds up. Assume a $1,000 initial investment.

        Starting Account    Account           Annual $      Ending Account 
              Value               Return         Gain (or Loss)          Value

Yr 1        $1,000               50% gain            $500                $1,500

Yr 2        $1,500               50% loss           ($750)                  $750

Clearly, the two-year average annualized return cannot be zero if the starting account value was $1,000 and the ending account value was $750 two years later. What then is the average return in this scenario? On a financial calculator, we would enter the following:

PV    = <1000>

FV    = 750

n      = 2

i       = -13.397%

The two-year average annualized return is -13.397%, which can be proven by the following:

Year 1   $1,000.00   -13.397% = $866.03

Year 2      $866.03   -13.397% = $750.00

The financial calculator solved for the geometric mean, not the arithmetic mean. All data providers, such as Morningstar, Lipper and Ibbotson, compute the geometric mean when reporting "average annualized return."

Let's look at another example using a hypothetical mutual fund.

Manhattan
Transfer Fund     Year 1        Year 2      Year 3

Total Return        31.65%     -1.77%      17.13%

A client asks for your help in calculating the "average return" for the Manhattan Transfer Fund over a three-year period. Summing the three returns and dividing by three yields the following average:

(.3165 - .0177 + .1713) / 3 = .1567 or 15.67%.

MULTIPLY, DON'T ADD

The 15.67% above is an arithmetic mean. The math (the sum of the numbers divided by the number of numbers) is correct, but the application is wrong. When numbers, such as test scores, are simply added, calculating the arithmetic mean is appropriate.

Investments grow in a multiplicative, rather than an additive, manner, however. To gauge the growth of an investment, numbers should be multiplied in sequence to get the geometric mean.

To calculate the three-year average annualized return (or geometric mean) of the Manhattan Transfer Fund for the client you first need to compute the three-year cumulative return:

Cumulative
Return       = [(1 + Yr 1 Return) * (1 + Yr 2 Return) * (1 + Yr 3 Return)] - 1

                 = [(1 + .3165) * (1 - .0177) * (1 + .1713)] -1

                 = [1.3165 * .9823 * 1.1713] - 1

                 = 1.5147 1

                 = .5147 or 51.47%

(Some fund companies report or advertise cumulative returns, which can be misleading because they are always higher than the average annualized return. However, the average annualized return is much more useful for comparing performance with other assets, and hence is the industry standard.)

The Manhattan Transfer Fund had a cumulative return of 51.47% over these three years. With that information, we can compute the three-year average annualized return or the geometric mean.

Using a financial calculator you would enter the following:

PV = -1 (representing a Present Value investment of $1, which should be entered as a negative number)

FV = 1.5147 (representing the Future Value, i.e. the cumulative return + 1).

n = 3 (a three-year investment period)

Solving for i -- interest per year or annualized rate of return -- we obtain:

i = 14.84% average annualized return or geometric mean

Thus, the average annualized return of the Manhattan Transfer Fund is actually 14.84%, not 15.67%.

When solving for "i" (or "i/yr") on a financial calculator (for example, Hewlett Packard 12C, Hewlett Packard 10B or 10BII, Texas Instruments BA II+, etc.) we are solving for the geometric mean.

The geometric mean is the return that, if held constant, will generate the stipulated future value over the stated period. In this example the stipulated FV was $1.5147 and the time period was three years.

If a financial calculator isn't handy, you can use the following algebraic equation:

i = (FV / PV)^1/n - 1

or

(1.5147 / 1)^1/3 - 1

which reduces to:

(1.5147)^.33333 - 1 = .1484 or 14.84%

The problem with calculating and reporting an arithmetic mean is that it always overstates the correct (geometric) annualized return. Interestingly, how much it overstates the actual geometric mean is highly related to the volatility of the asset's returns.

BIBLICAL PROPORTIONS

 

Consider the funds in the table "A Meaningful Difference" (above). These are ranked from lowest standard deviation of annual return over a 10-year period to highest standard deviation of return. For example, Fidelity Asset Manager: Income had a 10-year geometric mean return (the correct average) of 7.69% from January 1, 1995 to December 31, 2004. By contrast, its arithmetic mean return over the same time period was 7.82%. Only 13 basis points of difference separated the two different returns because the standard deviation of the annual returns was a very small 5.6%.

On the other end is AIM Technology with a 10-year standard deviation of annual returns of 55.7%. Its arithmetic mean is 18.25%, but its geometric mean is 7.31%-a difference of over 1,000 basis points. That sort of error is a compliance problem of biblical proportions.

The relationship between standard deviation of return and the gap between geometric and arithmetic mean is extremely predictable, as shown in the chart "Growing Gaposis" (below). It shows a total of 275 U.S. no-load equity funds with at least 11 years of performance history as of July 31, 2005 and net assets exceeding $750 million. Raw annual return data were extracted from the August 2005 release of Morningstar Principia.

 

You see that as the standard deviation (or volatility) of annual returns increases, the arithmetic mean grows larger than (and therefore, further away from) the correct geometric mean.

The good news is that no significant data provider reports multi-year annualized returns as arithmetic means. The bad news is that careless individuals may calculate an arithmetic mean and portray it as a geometric mean-without realizing the important difference.

The moral of the story is you shouldn't ever use the arithmetic mean method to calculate average annual returns of an investment, but if you're ever tempted to do so, pick a fund with low annual volatility.

Craig L. Israelsen is an associate professor in the department of home and family living at Brigham Young University, where he teaches personal and family finance. His e-mail is craig_israelsen@byu.edu.