Business Mathematics and Statistics · Ch 4 — Simple and Compound Interest

Different Compounding Periods

So far interest has been compounded once a year. In practice, banks and financial institutions often compound interest more frequently — half-yearly, quarterly or monthly. When this happens, two adjustments must be made to the annual figures before using the formula, and they always go together:

The rate per conversion period is the annual rate divided by the number of periods in a year.
The number of conversion periods is the number of years multiplied by the number of periods in a year.

The general form of the formula stays exactly the same — only $R$ and $n$ are replaced by the per-period rate and the total number of periods:

A = P\left(1 + \dfrac{r}{100}\right)^{m}, \quad \text{where } r = \dfrac{R}{k}, \; m = k \times T.

Here $R$ is the nominal annual rate, $T$ the time in years, and $k$ the number of conversion periods per year. The specific cases:

Compounding	Periods per year, $k$	Rate per period, $r$	No. of periods, $m$
Annually	1	$R$	$T$
Half-yearly	2	$R/2$	$2T$
Quarterly	4	$R/4$	$4T$
Monthly	12	$R/12$	$12T$

Definition 1Rate per conversion period

The nominal annual rate divided by the number of times interest is compounded per year (R/2 for half-yearly, R/4 for quarter …

Definition 2Number of conversion periods

The time in years multiplied by the number of compoundings per year (2T half-yearly, 4T quarter …

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