Annualized Returns

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 + N-month-Gain)^{12 / N}
or, equivalently
12-month-Gain Factor = (1 + N-month-Gain)^{12 / N}

Of course, all months aren't the same length!

It's better to use:

Annualized Gain Factor = (1 + M-day-Gain)^{365 / M}
or, equivalently
365-day-Gain Factor = (1 + M-day-Gain)^{365 / M}

Of course, this assumes a 365-day 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):