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:

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