In this equation (simple interest), the earned interest on the initial principle is not re-invested to earn additional interest, so the future value grows linearly with time. For instance, a principle of $100 at an interest rate of 10% would earn $10/year. Another possibility, however, is for these returns to be re-invested to increase the rate of accumulation. In the previous scenario, suppose the $10 earned each year is also invested at 10% interest annually. This scenario is called annual compounding:
To explore the consequences of compounding, I compare below the growth of $100 invested at 10% interest under simple interest versus annual, quarterly, and monthly compounding.
Thus, to estimate EAR under continuous compounding:
Real vs nominal interest and the Fischer equation
The distinction between nominal and real interest has little effect on the earlier treatment of compound interest, except that the “real” interest rate is lower than the nominal one. The earlier equations could use either rate to consider the inflation-adjusted or nominal value of the investment, respectively.
Rule of 72
Though the above treatment focused on annually compounded interest, continuous compounding presents an interesting special case. The result of the calculation is simpler, the 69.3 form of the equation comes out as the exact result:
Periodic contributions under compound interest
Now I extend the model, allowing contributions to be made at a frequency of every F months instead of yearly. We can utilize the above equations, grouping the time into increments of F months and expressing interest rate (i) as a monthly rate: