Monthly IRR

[Cypher2841]

I understand that if you wish to convert an annual IRR to monthly you need to employ the formula (1+IRR)^(1/12)-1. The reasoning (as I understand it) is that this formula will account for the reinvestment effect which is why you don't simply divide the annual IRR by 12.

What my simple mind can't quite get from a pure math perspective is why do we add 1 to the IRR and then subtract 1 at the end of the formula?

Any help is appreciated.

 

[JoeMo]

Welcome to the board.
Simply dividing the annual return by 12 would yield the arithmetic mean, which is not what you want. If the value at the start of the year is V1 and the value at the end of that year is V2, then if the average monthly return is R:
V2 = V1(1+R)^12 or
R= (V2/V1)^(1/12) -1
R is the geometric mean not the arithmetic mean.
V2/V1 = 1 + IRR

 

[shg]

Think of it backwards. If the opening balance is b and the monthly rate is m, then after one month the balance is b*(1+m), after two months, b*(1+m)^2, ..., and after 12, b*(1+m)^12. The change in value over a year is the final balance minus the opening balance:

b*(1+m)^12 - b == b * ((1+m)^12 - 1)

That last term, (1+m)^12 -1, is, by definition, the annual rate a:

a = (1+m)^12 -1

So

a + 1 = (1+m)^12

Rooting both sides,

(a + 1)^(1/12) = 1+m

m = (a + 1)^(1/12) - 1