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Compound interest

Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest:

An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00 × 1.5 = $2.25 at the end of the year. Compounding quarterly yields $1.00 × 1.25 = $2.44140625, and compounding monthly yields $1.00 × (1 + 1/12) = $2.613035.... If there are n compounding intervals, the interest for each interval will be 100%/ n and the value at the end of the year will be $1.00 × (1 + 1/ n).

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals. Compounding weekly (n = 52) yields $2.692596..., while compounding daily (n = 365) yields $2.714567... (approximately two cents more). The limit as n grows large is the number that came to be known as e. That is, with continuous compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate of R will, after t years, yield e dollars with continuous compounding. Here, R is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, R = 5/100 = 0.05.