The calculator below would allow you to simply determine how many years, quarters, months, weeks or days it would take for a sum invested, to double, treble, quadruple or increase by any other multiple desired.
In these calculations we adjust the discount interest rate to the time period in which we are interested. For example, to determine a monthly rate we don't simply divide a yearly rate by 12 but we recognize that a sum received earlier compounds over time and thus, will be lower than the average.
As such, a conversion of a yearly rate to monthly would use the formula (1+r)1/12 - 1 where r is the yearly rate. For example, a yearly rate of 7.5%, instead of being simply 0.625 % per month (7.5/12) would become, (1.075)1/12 - 1 = 0.6045 % per month.
Another concept involved is cumulative discount factors. A cumulative discount factor of 2, 3, 4 represents a doubling, trebling, quadrupling respectively, of an accumulated sum.
This formula is [1 - (1 + r) -t ] / r where r is the period interest rate expressed as a decimal and t is the number of corresponding periods involved.
For example, 7.5% is expressed as 7.5 / 100 or 0.075; t is the number of periods. The cumulative compound factor for a sum after 5 years at 7.5 % per annum would be worked out as :
[1 - (1.075)-5] / 0.075 = (1 - 0.6966) / 0.075 = 4.0459
On a monthly basis the results would be as follows :
r monthly = 0.006045 (already calculated).
t = 5 x 12 = 60
[1 - (1.006045) -60 ] / 0.006045 = 50.1976.
So the question is how long does it take a sum invested at 7.5% per annum for 5 years to rise to a Cumulative Discount Factor (CDF) of 4.0459 ? Note, normally this should be 5 but the power of compounding has reduced the necessary number to 4.0459.
For a monthly calculation, how many months would it take for a sum invested at 7.5% per annum for 5 years, to rise to 50.1976 ?. Again, the number would have been expected to be 60 (5 x 12) but the compounding effect has allowed it to be reduced to 50.1976 to produce the same final value.
The answers to the above questions can be had from the calculator below. We use these figures and iterate in steps of one-hundredths of a time-unit to arrive at a solution.
There is, in addition, at least one more approach; a formula that uses the same variables in a log format. This formula is : Log (x) / Log (y)
where x is the period to maturity and y is the interest rate for the period.
Input the rate of interest and select yearly, quarterly or monthly.
Input the maturity period, indicating whether expressed in years, quarters, months, weeks or days.