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Where does the exp come from?
It is something of a mystery how the formula for compound interest changes its form so completely. Unfortunately the reason depends on some fairly technical looking mathematics. You don’t have to understand the contents of this section to make use of the exponential functions but it is interesting.
The Future Value of an investment earning% per annum compounded n times per year is:
where t is the fraction of a year that has passed. For example, if n is 365, i.e. daily compounding then the future value after half a year is:
If n is increased the number of compounding periods increase and the larger n becomes the closer we approach continuous compounding. In mathematical terms we need to investigate the form of the equation as n, the number of compounding periods per year, tends to infinity.
It turns out to be easier find the limit in terms of the new quantity n’ which is the number of compounding periods divided by the interest rate that is
Rewriting the formula in terms of n’ gives:
Now you can see the advantage of the new quantity n’ because we can reduce the problem to finding the limit of:
as n’ goes to infinity independently of the interest rate and the time interval.
You can investigate this quantity as n’ gets bigger using a spreadsheet and you will find that it tends towards a value of 2.7182818285 which is known as the exponential.
For example, in the spreadsheet below you can see a table of values of (1+1/n’)^n’ for large, but certainly not infinite, values of n’. Notice how the value gets ever closer to the constant e. How far you can continue this process depends on the largest number that the spreadsheet you are using can cope with and on the accuracy of the calculation.
Given that the limit of (1+1/n’)^n’ is e, the formula for the future value at any time t is simply,
You can see the effect of continuous compounding on $100 at 10% per annum in the chart below:
To create this chart with $100 as the present value and 10% as the interest rate over a period of 40 years simply enter a series in column A from 0 to 40 years starting in A2, enter the formula:
in B2 and copy it down the column to B42 and then create an XY chart of the area A2:B42.
Notice the characteristic way that the rate of increase is itself increasing.
Once you have the formula for the Future Value working out the effective annual rate under continuous compounding is simple and gives:
effective rate = EXP(I)-1