Here are some calculations for an investment which yields a final value of $ e million ($2,718,282) from an initial value of $ 1 m.
The logarithmic or continuously compounded return is given as:-
Vf = 2,718,282
Vi = 1,000,000
rlog = ln(Vf/Vi) = 1.0 = 100 %
This is a logarithmic return, or nominal return (continuously compounded) of 100%.
The effective annual rate can be calculated by:-
i is the logarithmic or continuously compounded nominal rate.
i = rlog = 1.0 = 100%
r = e^i - 1 = 1.718282 = 171.8282 %
An effective annual return of 171.8282% produces the final value of $ e million.
Of course, the effective return can also be calculated as:
r = Vf/Vi - 1 = 1.718282 = 171.8282 %
Now considering monthly periodic returns
The effective annual rate calculated from a periodically compounded nominal return is:
n is the number of compounding periods.
Rearranging this formula, and using the previously calculated effective annual rate (which produced $ e million), the annual nominal rate compounded monthly is calculated:
r = 1.718282 = 171.8282 %
n = 12
i = n*((r + 1)^(1/n) - 1) = 1.0428486 = 104.28486 %
Note how this differs from the 100% calculated for the continuously compounded nominal rate. As
n increases the periodic nominal rate approaches the continuously compounded nominal rate, as demonstrated by the limit formula:
For example, the nominal rate compounded daily (with
n = 365) is 100.137%, which is somewhat closer to 100% than the nominal rate compounded monthly.
From the annual nominal rate compounded monthly, the monthly compounding rate can be found:
m = i/n = 1.0428486/12 = 0.08690405 = 8.690405 %
Checking by compounding for 12 months:
(m + 1)^n - 1 = 1.718282
The monthly compounding rate can also be calculated directly from the logarithmic rate, or annual continuously compounded nominal rate:
i = rlog = 1.0 = 100 %
n = 12
m = e^(i/n) - 1 = 8.690405 %