Calculating Annualized Returns

We have an investment which begins with $123,456 and, after 78 months, has become $200,000.
The Gain Factor
is 200000/123456 = 1.620 meaning a gain of 62%.
Ah, but that's over 78 months and we'd like the Annualized Gain.
Suppose the Annualized Gain was R
(where R=0.123 means a 12.3% annualized gain), then:
 (1+R) is the
Gain Factor over one year
 (1+R)^{2} would be the
Gain Factor over two years
 (1+R)^{0.75} would be the
Gain Factor over 0.75 years (that's 9 months)
 (1+R)^{78/12} would be the
Gain Factor over 78/12 = 6.5 years (namely 78 months!)
If we know that Gain Factor =
1.620 over 78 months then:
 (1+R)^{78/12} = 1.620
so
1+R = 1.620^{12/78}
so
R = 1.620^{12/78}  1
= 1.077  1 = 0.077 or 7.7%
That gives us our Magic Formula:
Annualized Gain Factor = (1 + NmonthGain)^{12 / N}
or, equivalently
12monthGain Factor =
(1 + NmonthGain)^{12 / N }

Of course, all months aren't the same length!
It's better to use:
Annualized Gain Factor = (1 + MdayGain)^{365 / M}
or, equivalently
365dayGain Factor =
(1 + MdayGain)^{365 / M }

Of course, this assumes a 365day year. You might like to try 366 or maybe 365.25
If you already know the Gain over N days, you can use this:
Now here's a popular (uncanny?) way to calculate an Annualized Return:
 The return over N = 40 days is 2% so R = 0.02
 The return over one day is (2/40)% = (1/20)% so r = R/40 = 0.0005, pretending that the daily returns are not compounded
 The return over 365 days is (1/20)% compounded (!?), namely (1.0005)^{365}  1 = 0.2002 or 20.02%
The prescription would be: (1+R/N)^{365}  1 ... but it's not my favourite formula
Assume, for example, that N = 365 so R is already the annualized return
The above calculation assumes an initial investment without subsequent investments or withdrawals.
If there are further investments and/or withdrawals the problem is more complicated and one
can use (for example) a spreadsheet command:
XIRR, like so (where withdrawals and the
current portfolio value are entered as negative numbers):
See also Average & Annualized Gains and even
Misc. Stuff
