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Computing the average annual rate of return (ROR) for a bond, stock, fund, or trading strategy should be straightforward, shouldn’t it? For example, to get the average annual ROR for the last five years, don’t you just sum up the 5 individual annual returns, and divide by 5?

Note: Please read the disclaimer. The author is not providing professional investing advice or recommendations.

Well yes, you can do it that way. But what you’ll end up with is the arithmetic average, which is probably not really what you’re after. Why not? Let’s look at an example.

Say we’re examining a stock that had the following closing prices on December 31 of the following years:

Table 1

Year	2000	2001	2002	2003	2004	2005
Price per Share	$111.19	$97.80	$76.12	$97.82	$108.32	$113.49

This translates into the following annual returns:

Table 2

Year	2001	2002	2003	2004	2005
Annual ROR	-12.0%	-22.2%	+28.5%	+10.7%	+4.8%

Now we’ll follow our original assumption and compute the arithmetic average:

[ (-12.0) + (-22.2) + (28.5) + (10.7) + (4.8) ] / 5 = +2.0%

What’s wrong with our computation? Well nothing technically, but the arithmetic mean can mistakenly lead you to assume that if you had invested money 5 years ago, your investment today would be worth the same as if the fund returned that average +2.0% for each of those 5 years. But didn’t it?

If we had initially invested $10,000:
At the end of year 1 it would be worth: $10,000 x (1-.12) = $8,800
At the end of year 2 it would be worth: $8,800 x (1-.222) = $6,846
At the end of year 3 it would be worth: $6,846 x (1+.285) = $8,798
At the end of year 4 it would be worth: $8,798 x (1+.107) = $9,739
At the end of year 5 it would be worth: $9,739 x (1+.048) = $10,206

Note that this is quite a different return than if we’d made that arithmetic average 2% each year:

At the end of year 1 it would be worth: $10,000 x (1+.02) = $10,200
At the end of year 2 it would be worth: $10,200 x (1+.02) = $10,404
At the end of year 3 it would be worth: $10,404 x (1+.02) = $10,612
At the end of year 4 it would be worth: $10,612 x (1+.02) = $10,824
At the end of year 5 it would be worth: $10,824 x (1+.02) = $11,041

Of course the problem with the arithmetic mean is that it doesn’t take into account the compounding effect of each yearly ROR. This would be fine if we invested say $10,000 each year and whether we made money or lost money, we still set our principal back to $10,000 for each subsequent year.

But most of us don’t invest that way. If we put money in a fund for a number of years, the amount invested in any particular year is what was leftover from the previous year.

Assuming this is the way we’re investing, if we wanted to figure out what equivalent yearly average return would give us what our investment is worth today, we have to use the geometric mean.

To compute the geometric mean, multiply each of the yearly returns (rather than summing them) and then take the n^th root of the product (rather than dividing by n), where n is the number of years we’re averaging over.

For our n=5 years this would work out to be:

[(1-.12) x (1-.222) x (1+.285) x (1+.107) x (1+.048)]^(1/5) = 1.0041 or 0.41%

And in fact if we compound our initial principal at this geometric average, we do get what our investment is worth today:

At the end of year 1: $10,000 x (1+.0041) = $10,041
At the end of year 2: $10,041 x (1+.0041) = $10,082
At the end of year 3: $10,082 x (1+.0041) = $10,124
At the end of year 4: $10,124 x (1+.0041) = $10,165
At the end of year 5: $10,165 x (1+.0041) = $10,207

Actually you’ll see we’re off by one dollar, due to round-off error.

There is a simpler way to compute the geometric mean, and that is to use the beginning and ending prices in Table 1, but annualize the gain, again by taking the n^th root where n is the number of years between beginning and end prices.

If we do this, we get the same as the geometric mean:

(113.49 / 111.19)^(1/5) = 1.0041 or 0.41%

Apart from the calculation being simpler, the nice thing about annualizing is that you can get a yearly geometric mean, even when you have a non-integer number of years.

For example, let’s say that instead of annual prices we’re examining quarterly prices:

Table 3

Quarter	Q1 2005	Q2 2005	Q3 2005	Q4 2004	Q1 2006
Price per Share	$64.08	$67.77	$81.02	$80.85	$83.52

We have 5 quarters, or 1.25 years worth of data, so we annualize to compute the following effective annual ROR:

(83.52 / 64.08) ^(1/1.25) = 1.2361 or 23.6%

So by annualizing, when computing average ROR we don’t have to restrict ourselves to periods where we have a complete year’s worth of data. This is helpful because chances are the stock or fund was not created on January 1 of its inception year and usually we’re part of the way into a new year.

By annualizing we can therefore include all price data available to us to get an average ROR - including fractional years.

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