Periodic rate and fractional capitalisation

So far capitalisation has taken place once a year. But banks and loans often capitalise several times a year: monthly, quarterly. A rate referring to the single period is then required.

Definition — Periodic rate

If the annual rate is $i$ but capitalisation takes place $m$ times a year (monthly $m=12$, quarterly $m=4$), the periodic rate is $i_p = i/m$ and the amount after $t$ years is $M = C_0\Bigl(1+\frac{i}{m}\Bigr)^{mt}.$ For $m\to\infty$ (continuous capitalisation): $M\to C_0\,e^{it}$ (we shall prove it in Year Four, studying the number $e$).

Each period applies the factor $\bigl(1+\frac{i}{m}\bigr)$, repeated $mt$ times. As $m$ increases the amount grows slightly and, in the limit, tends to continuous capitalisation $C_0\,e^{it}$, which we shall meet when studying the number $e$.

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Topics: Percentages
Concepts: Compound capitalisation · Continuous capitalisation · Periodic rate
Skills: Using formulae