# Solve for Number of Periods - PV & FV

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 $$FV$$ can be calculated using the compounding interest formula as shown below:

$FV = PV \times (1 + r)^n$

with:

• $$PV$$ is the present value
• $$FV$$ is the future value
• r is the nominal annual interest rate
• n is the number of years

Now, suppose you know the given rate $$r$$, the present value $$PV$$ and the future value $$FV$$. How can you calculate $$n$$?

From the above formula we can derive that:

$\frac{FV}{PV} = (1 + r)^n$

Taking the logarithm $$ln$$ on both sides we get:

$ln(\frac{FV}{PV}) = ln(1 + r)^n$

Simplifying the exponent on the right side of the above equation, we have:

$ln(\frac{FV}{PV}) = n \times ln(1 + r)$

Finally, solving the above equation for $$n$$, we get:

$n = \frac{ln(\frac{FV}{PV})}{ln(1 + r)}$

So if we want to know how long it takes for a value to double given a rate of $$10\%$$ a year, we would simply do:

$n = \frac{ln(\frac{2}{1})}{ln(1 + 0.1)} \approx 7$
Written on August 10, 2021