Ways and Means

ne of the potentially 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.

A simple example might help at this point.

Year 1 50% gain Year 2 50% loss

The too-casual observer might assume that the average annualized return over this two year period equals zero if using an arithmetic mean approach. That is, add the numbers and then divide by the number of numbers.

50 + (-50) = 0 / 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 Value	Annual Return	Annual $ Gain or (Loss)	Ending Account Value
Year 1	$1,000	50% gain	$500	$1,500
Year 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 2-year average annualized return was -13.397%, which can be proven by the following:

Year 1 $1,000 - 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 significant data providers (Morningstar, Lipper, Ibbotson, etc.) 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" over the three year period. Summing the three returns and dividing by 3 yields the following "average":

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

The calculation shown above is an arithmetic mean (the sum of the numbers divided by the number of numbers). The math is correct, but the application is wrong.

Investments grow in a multiplicative manner, not in an additive manner. When numbers are simply added together (such as test scores), calculating the arithmetic mean is appropriate. When numbers are multiplied in sequence (such as the growth of an investment) the geometric mean is the appropriate measure.

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, as demonstrated below.

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%

The Manhattan Transfer Fund had a cumulative return of 51.47% over a three year period. With that calculated, we can compute the three year average annualized return (or the geometric mean).

Using a financial calculator you will 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 3 year investment period)

Solving for i 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 (e.g. 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 which, 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, 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 returns of the asset.

Consider the funds in the table "Right vs. Wrong", which are ranked from lowest standard deviation of annual return (over a 10-year period) to highest standard deviation of return. Fidelity Asset Manager: Income had a ten-year geometric mean return (the correct mean) of 7.69% covering the period 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 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 figure "Tight Fit". 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 are represented. Raw annual return data were extracted from the August, 2005 release of Morningstar Principia.

As can be seen, as the standard deviation of annual returns increases (or, said differently, as volatility of return increases) the arithmetic mean is increasingly larger than (and therefore, further away from) the correct geometric mean.

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

In conclusion, if you're tempted to calculate an average annualized return using an arithmetic mean formula, be sure to pick a fund with low annual volatility of return.

Fund Name	Annual  Return  1995	Annual  Return  1996	Annual  Return  1997	Annual  Return  1998	Annual  Return  1999	Annual  Return  2000	Annual  Return  2001	Annual  Return  2002	Annual  Return  2003	Annual  Return  2004	10 Yr Std Deviation of Annual Returns (%)	10 Year Arithmetic Mean (%)   WRONG	10 Yr Geometric Mean (%)   CORRECT	Difference b/w Arithmetic & Geometric Mean (bps)
Fidelity Asset Mgr: Income	16.7	7.8	12.4	10.3	5.7	3.6	1.3	-0.5	14.4	6.4	5.6	7.82	7.69	13
Dodge & Cox Stock	33.4	22.3	28.4	5.4	20.2	16.3	9.3	-10.5	32.3	19.2	13.4	17.63	16.89	74
T. Rowe Price Small Value	29.3	24.6	27.9	-12.5	1.2	19.8	21.9	-1.8	36.4	25.7	15.9	17.26	16.21	105
Wasatch Small Cap Growth	28.1	5.2	19.2	11.2	40.9	16.8	24.2	-23.4	37.4	13.4	18.2	17.30	15.85	144
Vanguard Total Stock	35.8	21.0	31.0	23.3	23.8	-10.6	-11.0	-21.0	31.4	12.5	20.4	13.62	11.81	181
T. Rowe Price New Horizons	55.4	17.0	9.8	6.3	32.5	-1.9	-2.8	-26.6	49.3	17.9	24.9	15.69	13.23	246
AIM Technology Inv	45.8	21.8	8.9	30.1	144.9	-22.8	-45.5	-47.2	43.2	3.4	55.7	18.25	7.31	1,094

Tight Fit

____________________________________________________________________________________
Craig L. Israelsen, Ph.D. is an associate professor at Brigham Young University. He teaches family finance in the Department of Home and Family Living. His research interests include mutual fund analysis. He writes monthly for Financial Planning magazine.