//
archives

# Financial

This category contains 1 post

### Compound interest

Jacob Bernoulli discovered this constant in 1683 by studying a question about compound interest:[5]

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 \$1.00×1.52 = \$2.25 at the end of the year. Compounding quarterly yields \$1.00×1.254 = \$2.4414…, and compounding monthly yields \$1.00×(1+1/12)12 = \$2.613035… If there are ncompounding 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)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 \$2.692597…, while compounding daily (n = 365) yields \$2.714567…, just two cents more. The limit as n grows large is the number that came to be known as e; with continuous compounding, the account value will reach \$2.7182818…. More generally, an account that starts at \$1 and offers an annual interest rate of R will, after t years, yield eRt 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)

Sitworld Analytics Team

Advertisements
Advertisements