Continuous Compounding – The Curious Case of e

Let’s look at a shortcut to calculate continuous compound interest using Euler’s number e.

The popular formula for compound interest is:

Where:
  • t=number of years,
  • A = final amount after t years,
  • r=interest rate per annum.

      The formula assumes that the interest rate is compounded once a year. If interest rates arecompounded more frequently, the formula is:
Where n is the yearly compounding frequency.

Say we want to find out the compound rate of return, r, given the following: P=100000, A=8000000, t=25 years, with compounding frequency set to 1 per year.
To calculate r, we can use (2) with n=1:
Rearranging the terms, we get:
To simplify the calculation, we may also use logarithms on the expression for (2):

Therefore, r=0.1915 or 19.15% per year. In other words, if we compounded 100000 at the rate of 19.15%, once a year, for 25 years, we would end up with 8000000.

Let’s increase the compounding frequency n to a large number and see what happens to the return.

n=100: 

r = 0.1754 or 17.54%

If we keep increasing n further and further, the r value decreases, and finally it converges asymptotically as shown below:

As n approaches ∞, as it does in continuous compounding, we can use the following identity and simplify the calculation:

Using the above, let us rewrite (2): let k=n/r, such that:
So (2) becomes:
In other words, when continuously compounding for t years at rate r, the above formula using e gives us the final amount in a single step.

This formula is widely used in financial applications, including derivative and many other exotic financial products. The branch of finance dealing with continuous rate of growth is also known as ‘Continuous Time’ finance, which uses quantitative techniques involving statistics, stochastic calculus, finite differences, numerical methods, etc.

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