Average vs Annualized Gains 
The year 2000 is over and I look back on my portfolio, for 1999 and 2000.
In 1999 it increased by 60%.
>Really? That's good, eh?
Yeah, but the NASDAQ increased by over 85%. Anyway, in 2000, it fell 40%. That means ...
>That makes an average gain of ... let's see ... (60  40)/2 = 10%.
Not too shabby. You're happy, right?
Wrong! I actually lost money over those two years. A gain of 60% followed by a loss of 40%
gives an overall loss of 4%. It's not like 10% per year. The average gain of 10%
is meaningless when ...
>Hold on! A gain of 60% then a loss of 40% means that every dollar
becomes (1+.6)(1.4) = (1.6)(.6) = .96 and ... uh, that's 96 cents. So what's your real
annual gain if it isn't that average of 10%?
An annual gain of R (for a 23.4% gain, we put R = 0.234),
turns $1.00 into (1+R)^{2} after two years. If that's $0.96, then
(1+R)^{2} = .96 and that makes R = .0202 which is equivalent to a loss
of 2.02% per year.
See also miscellany.
>So, what good are average gains?
Good question. I recently heard some guy on TV say, "The average market gain for the past
seventyfive years has been 11%, so run out and invest" ... or something to that effect.
In fact, on Jan 1, 1970, the S&P500 had a value of 85. The average annual gain, until
Jan 1, 2000, was 10.48% and if the S&P had actually increased by 10.48% per year, for thirty
years, it'd be significantly larger than it was on Jan 1, 2000.
>How much larger?
More than a third larger. The moral here is this: If the annual gains are volatile they vary
significantly from year to year  then the average return won't give a sensible value
for your overall return.

Fig. 1: 30 years of S&P500 with two "Averages" 
Use the Annualized Return
... which is always smaller than the Average
return, so
If you get excited about Average returns ... don't

Fig. 2: The Annualized Gain is bang on! 
>How about a picture?
Okay, here's a wee chart of the actual S&P500 (in black) together with the S&P if, each year, the
30year Average gain was used (the green curve)
... or if the 30year Annualized gain was used.
(The red guy).
Note that the AnnualizedGain curve winds up at the same final value, but the AverageGain
curve overshoots.

>Average or annualized, they're pretty good gains, eh?
For that period, yes. But if we go back earlier, to 1950, and consider, say, all 20year gains
(annualized) starting Jan 1, 1950 then starting Feb 1, 1950 etc., ending with Jan 1, 1980,
we'd get a different picture. Some annualized gains were pretty lousy ... some less than 2.5%.


Let's do a wee bit o' math.
>Do we have to?
Pay attention. We suppose the annual gain factors in two consecutive years are
x and y. By "gain factor", I mean ... uh, if the gain was 34.5%, the "gain factor"
would be 1.345  it's what $1.00 becomes after the year's gain. Just add "1". It's like ...
>Okay! Okay! Keep goin'.
The gain factor after two years is just xy. That's what our $1.00 would be worth.
If, however, we took two year's worth of the Average gain factor ...
that's (x + y)/2 ... our $1.00 would be worth
{(x + y)/2}^{2}
They aren't the same.
In fact:
{(x + y)/2}^{2}
is always larger than xy
which is like saying that
(x + y)/2 is always larger than (xy)^{1/2}
which is like saying that
The Arithmetic Mean (that's our Average) is always larger than the Geometric Mean
>Always?
Well, so long as the numbers, like x and y, aren't negative. In our case, they're
gain factors and are certainly not negative since your $1.00 won't be worth less than $0,
unless, of course, you play with options or sell short or ...
>Okay, I get it.
If we use the Geometric Mean to calculate our annual return (instead of the "Average" or
Arithmetic Mean), it's called the Annualized Return and it's bang on, meaning that
the Annualized Return will turn your $1.00 into exactly the right value after umpteen years of
investing. It won't overestimate your portfolio worth. To get the Geometric Mean (or
Annualized Return) after N years of Gain Factors G_{1}, G_{2},
etc. (like our x and y, above) just use:
{G_{1}G_{2}G_{3} ... G_{N}}^{1/N}

If N = 2, then we'd get (xy)^{1/2}. Further, the Average Gain
Factor (necessarily LARGER) is (1/N){G_{1}+G_{2}+G_{3}+...+G_{N}}
To get the gain, from the gain factor, just subtract "1" ... and multiply by 100 if y'all
want it as a percentage.
In an earlier chart (Fig. 1) we've shown both the Average and Annualized gains
as horizontal lines (after subtracting "1" and changing to a percentage by multplying by 100).
Although the actual gains oscillate about both the Average and the Annualized gains, the
Annualized gain is a wee bit smaller ... and there's quite a difference after thirty years!
After thirty years, with a 1% increase in annual gain (using Average
instead of Annualized gain), your portfolio value would be higher by almost
35%. It'd give you a warm and fuzzy feeling, but it'd be misleading ...
so don't use the Average!
Indeed, the easy way to compute the Annualized Gain Factor over N years is:
{Portfolio(at the end)/Portfolio(at the beginning)
}^{1/N}
Every $1.00 invested N years ago, if increased (or decreased?) by that factor each year,
will give an approximate endvalue for that dollar (after a year)
 just multiply by the Annualized Gain Factor. (See Fig. 2)
Repeat the multiplication N times and you'd get
Portfolio(at the end)/Portfolio(at the beginning)
for every dollar invested
N years ago. If you invested Portfolio(at the beginning) way back then, then
(multiplying) you'd have exactly your Portfolio(at the end).
>That's obvious!
You said it.
Fig. 3: Average and Annualized gains for a mutual fund 
To the left is a chart of the annual returns for a well known Canadian equity mutual fund
(which shall remain nameless),
over the last 1 year, 2 years, 3 years, ... 10 years (ending Dec 31, 2000). The fund company
calls the returns "Average Annual Returns". In fact, the "Average" returns published by the
company are ... uh ... GUESS! Are they the (larger) Arithmetic Means or the (lower)
Geometric Means.
Surprise! Although the company refers to them as "Average Annual Returns", the published returns are the
"Annualized" (or Geometric Mean) returns, identified by the
red dots. Three cheers for Altamira
the fund company!

I put in the green dots myself,
to confuse y'all into thinking
they'd published the misleading (and higher) Arithmetic Mean returns
>In case you're interested, Canadian Mutual Fund companies are
required to state annual returns as "Annualized" (National Policy 39).
Are you sure?
>I have it on good authority (thebox/mikale/KANGAS): see typical
Rules (page 70) where you'll see magic formulas like:
[(redeemable value/initial value)^{1/N}  1]x100
Hmmm ... looks familiar. Thanks.
>You're welcome.
for Part II
