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Lesson 45 · Compounding frequency
Reading the fine print
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Reading the fine print

Banks rarely compound once a year. The fine print says compounded monthly or semi-annually, and the formula grows two small sockets to cope: A=P(1+rn)ntA = P\left(1 + \dfrac{r}{n}\right)^{n t} (read aloud: A equals P times one-plus-r-over-n, to the power n t). The yearly rate gets SPLIT into nn slices, and the clock ticks nn times as often. Split the rate, multiply the ticks — both, always. The classic slip is forgetting the split and charging the full yearly rate every period.

Does faster compounding help? A little — never as much as the ads imply. The honest measure of any quoted rate is the effective annual rate, EAR=(1+rm)m−1\mathit{EAR} = \left(1 + \dfrac{r}{m}\right)^{m} - 1 — EAR equals one-plus-r-over-m, to the m, minus one. Here rr is still the quoted yearly rate as a decimal, and mm is the same counter nn was: how many times a year the bank compounds (semi-annually means m=2m = 2). What comes out is what the year actually pays once the compounding is counted. It is how you compare two banks telling two different stories.

PArnt

A=P(1+rn)ntA = P \left( 1 + \frac{r}{n} \right)^{n t}

  • AA= Final amount
  • PP= Principal
  • rr= Interest rate per period (decimal)
  • nn= Compounds per period
  • tt= Number of periods
Compound Interest (Periodic) solver →
rmEAR

EAR=(1+rm)m−1\mathit{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1

  • EAR\mathit{EAR}= Effective annual rate
  • rr= Nominal annual rate
  • mm= Compounds per year
Effective Annual Rate from a Nominal Rate solver →