Compound Interest
Let P be the principal (initial investment), r be the annual compounded rate, i^((n)) the "nominal rate," n be the number of times interest is compounded per year (i.e., the year is divided into n conversion periods), and t be the number of years (the "term"). The interest rate per conversion period is then
| [画像: r=(i^((n)))/n. ] |
(1)
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If interest is compounded n times at an annual rate of r (where, for example, 10% corresponds to r=0.10), then the effective rate over 1/n the time (what an investor would earn if he did not redeposit his interest after each compounding) is
| (1+r)^(1/n). |
(2)
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The total amount of holdings A after a time t when interest is re-invested is then
Note that even if interest is compounded continuously, the return is still finite since
where e is the base of the natural logarithm.
The time required for a given principal to double (assuming n=1 conversion period) is given by solving
| 2P=P(1+r)^t, |
(5)
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or
| [画像: t=(ln2)/(ln(1+r)), ] |
(6)
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where ln is the natural logarithm. This function can be approximated by the so-called rule of 72:
| [画像: t approx (0.72)/r. ] |
(7)
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See also
e , Interest, Ln, Natural Logarithm, Principal, Rule of 72, Simple InterestExplore with Wolfram|Alpha
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References
Kellison, S. G. The Theory of Interest, 2nd ed. Burr Ridge, IL: Richard D. Irwin, pp. 14-16, 1991.Milanfar, P. "A Persian Folk Method of Figuring Interest." Math. Mag. 69, 376, 1996.Referenced on Wolfram|Alpha
Compound InterestCite this as:
Weisstein, Eric W. "Compound Interest." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CompoundInterest.html