# Math Behind Doubling Your Investment with Compounding

Let’s attack, mathematically, the common question of how long it takes to double my investment when the investment is one that earns compound interest.

There are simple guidelines (Rule of 72, for example) to estimate it.

There is also a fairly simple way to derive a formula to calculate this mathematically so that we can be precise about this duration.

As you will see here, the exact formula requires the use of algebra and logarithms and so an approximation using simple arithmetic is the popular alternative. However, it is good to understand the exact calculation so that we can appreciate what the approximation offers us.

### Let’s dive in

Say you start with a capital of C and invest it in an instrument that gives you an average percentage annual return of r.

At the end of the first year, you’ll add C*r/100 to your capital by way of these returns.

So, the total account balance at the end of the first year is C + C*r/100.

Or, using simple algebra,

balance at the end of the first year = C*(1+r/100)

Assuming compounding, this is also the starting capital for the second year.

Following a similar calculation,

balance at the end of year 2 = (C*(1+r/100))*(1+r/100)

Or, with more algebra,

balance at the end of year 2 = C*(1+r/100)2

Repeating this process, we can establish the compounded account balance after n years:

balance after n years = C*(1+r/100)n

We want to find out n when it results in the balance being twice the starting capital, C. For such an n, this can be expressed mathematically as

2*C = C*(1+r/100)n

Removing the common factor C from the equation, we get

2 = (1+r/100)n

as the relationship that holds regardless of the starting capital. Taking logarithm of both sides, we get

log 2 = n log (1+r/100)

Solving for n, we get

n = (log 2)/log (1+r/100)

Choosing 2 as the base of the logarithm simplifies it further:

n = 1/log2 (1+r/100)

When n is a whole number, the only assumption made here is that the annual return for the investment is a steady r percent.

When n is not a whole number, this formula assumes that the steady annual return is the result of steady returns at all the sub-intervals of the year: month, week, day, and even smaller units.

While these assumptions are not realistic, the derived formula provides a reasonable basis for devising simpler approximations.

### The flip side

While we are at it, let’s ask a different question.

What interest rate r should we be posting annually to accomplish doubling of our investment in n years?

This question arises when you are trying to find an investment that satisfies your growth objective.

Starting with the same equation

log 2 = n log (1+r/100)

from above, or, modified for logarithm using base 2,

1 = n log2 (1+r/100)

we can solve for r instead of n. The following math jugglery leads to our desired result:

Restating the equation,

n log2 (1+r/100) = 1

Using simple algebra,

log2 (1+r/100) = 1/n

Taking inverse of the logarithm,

(1+r/100) = 21/n

resulting finally in

r = 100*(21/n-1)

with a little algebra again.

Phew! Yes, this will do the job, but it begs the use of a calculator! Taking the nth root of 2 is a non-starter in most situations.

But, since we can work wonders with just close enough estimations, we can leverage the Rule of 72 to make it quick and painless!