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

$$r\equiv {i^{(n)}\over n}.$$ (1)

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)

The total amount of holdings $A$ after a time $t$ when interest is re-invested is then

$$A = P\left({1+{i^{(n)}\over n}}\right)^{nt} = P(1+r)^{nt}.$$ (3)

Note that even if interest is compounded continuously, the return is still finite since

$$\lim_{n\to\infty} \left({1+{1\over n}}\right)^n=e,$$ (4)

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)

or

$$t={\ln 2\over\ln(1+r)},$$ (6)

where Ln is the Natural Logarithm. This function can be approximated by the so-called Rule of 72:

$$t\approx {0.72\over r}.$$ (7)

See also e, Interest, Ln, Natural Logarithm, Principal, Rule of 72, Simple Interest

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.