Often times when you read something like “Annual increases at a 10% rate would lead to the doubling of prices every seven years“, you might be wondering how one would go about calculating the length of time required for a single cash flow(present value) to reach a certain amount(future value) based on the given rate.
We all know that a future value can be calculated using the compounding interest formula as shown below:
- is the present value
- is the future value
- r is the nominal annual interest rate
- n is the number of years
Now, suppose you know the given rate , the present value and the future value . How can you calculate ?
From the above formula we can derive that:
Taking the logarithm on both sides we get:
Simplifying the exponent on the right side of the above equation, we have:
Finally, solving the above equation for , we get:
So if we want to know how long it takes for a value to double given a rate of a year, we would simply do: