### What Is Continuous Compound Interest?

To understand what Continuous Compound Interest is, first, we need to understand what Compound Interest is.

Compound Interest is interest on interest. In other words, it’s the amount of interest earned on the investment where the amount earned is reinvested rather than being paid out. By using a compound interest an investor gets a faster increase in the value of capital than by using simple interest, where interest is not added to the principal, so there is no compounding.

The more frequent compounding is, the faster the capital increases. An extreme case in which the time gap between the compoundings drops to zero is called continuous compounding.

### How to Calculate Continuous Compound Interest?

$$A = Pe^{rt}$$

Where A = the final amount in the account, P is the initial (principal) sum, r = the rate of interest and t = the time in years. This formula also uses the mathematical constant e (Euler’s number), which is the base of the natural logarithms (e=2,718).

This formula is derived from the Compound Interest formula, which computes the future value of an investment:

$$A = P \times ( 1 + \frac{r}{n})^{nt}$$

Where A = the future value of the investment, P is the initial/principal investment amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the number of years the money is invested/borrowed for.

If compounding happens annually, the formula can be simplified to:

$$A = P \times ( 1 + r )^{n}$$

### Example of Continuous Compound Interest Calculation

• Let’s assume an investment of $100,000 earns 12% interest annually. Future Value of the investment when the interest is compounding continuously is: $$A = \100,000 \times 2,718^{12\%} = \112,749.69$$ To compare this value, lets calculate Future Value of the investment when the interest is compounded annually, quarterly, monthly and daily. When the investment is compounded annually, Future Value of the investment is: $$A = \100,000 \times ( 1 + \frac{ 12\% }{ 1 } )^{1} = \112,000$$ When the investment is compounded quarterly, Future Value of the investment is: $$A = \100,000 \times ( 1 + \frac{ 12\% }{ 4 } )^{1\times 4} = \112,550.88$$ When the investment is compounded monthly, Future Value of the investment is: $$A = \100,000 \times ( 1 + \frac{ 12\% }{ 12 } )^{1\times 12} = \112,682.50$$ When the investment is compounded daily, Future Value of the investment is: $$A = \100,000 \times ( 1 + \frac{ 12\% }{365 } )^{1\times 365} = \112,747.46$$ As we can see, with continuous compounding total earnings of the investment are$12,749.69, while with daily compounding total earrings are $12,747.46. The difference between these numbers is only$2.23. Considering the fact, that the investment was \$100,000, not much more is earned by using a Continuous Compound.