Continuous Compounding
IntermediateContinuous compounding is the theoretical limit of compounding interest more and more frequently, so that interest is added at every instant, producing the maximum possible growth for a given annual rate.
Interest can be compounded yearly, monthly, daily or even more often. The more frequently it is compounded, the faster money grows, because interest starts earning interest sooner. Continuous compounding takes this to the limit.
For example, 1,000 invested at 5% a year grows to 1,050 with annual compounding and to about 1,051.16 with monthly compounding. With continuous compounding, it grows to about 1,051.27. The difference from daily compounding is tiny, which shows that the benefit of more frequent compounding levels off.
Continuous compounding uses the mathematical constant e, approximately 2.718. The value after a period is the starting amount multiplied by e raised to the power of the rate multiplied by the time.
In practice, banks rarely compound continuously, but the concept is widely used in finance because it simplifies calculations. It is used in option pricing models, such as Black–Scholes, in calculating log returns and in valuing bonds and derivatives.
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