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:

$$n = \frac{ln(\frac{2}{1})}{ln(1 + 0.1)} \approx 7$$