Compound Interest

Summary: The calculation of interest in which each year’s interest is added to the principal before computing the next year’s interest, leading to exponential growth described by a geometrical progression.

Sources: chapter-1.3.13

Last updated: 2026-05-02

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Core formula

At rate $p$% per year, a principal $a$ grows to

$A_n={\left(\frac{100+p}{100}\right)}^{n}a$

after $n$ years. At 5% this is $(21/20{)}^{n}a$; at 6%, $(53/50{)}^{n}a$.

The sequence $a,(21/20)a,(21/20{)}^{2}a,\dots$ is a geometrical progression with ratio $21/20$ (§544–545).

Logarithmic computation

Because $A_n$ involves a power, large exponents are handled by logarithms:

$\log A_n=n\log \left(\frac{21}{20}\right)+\log a,\log \left(\frac{21}{20}\right)=0.0211893$

Finding the doubling time

To find when $A_n=k\cdot a$, take logarithms: $n=\log k/\log (21/20)$.

Annual additions

Adding fixed sum $b$ each year (§550–552):

$A_n={\left(\frac{21}{20}\right)}^n(a+20b)-20b$

Annual withdrawals

Withdrawing fixed sum $b$ each year (§554–555):

$A_n={\left(\frac{21}{20}\right)}^n(a-20b)+20b$

If $a<20b$ the principal eventually reaches zero.

Fractional-year periods

The exponent $n$ may be a fraction $d/365$ to compute interest for $d$ days.

Present value

A sum $a$ due $n$ years hence is worth $(20/21{)}^{n}a$ today. An annuity of $a$ per year for $n$ years starting now has present value $21a-21(20/21{)}^{n+1}a$.

Related pages

• geometrical-progressions
• logarithms
• ch1.3.13-calculation-of-interest
• geometrical-proportion