Simple, compound and continuous interest

An initial capital $C_0$ invested at the annual rate $r$ can yield in three different ways, depending on how and when the interest is reinvested.

Regime	Formula after $t$ years	Notes
Simple	$C(t)=C_0(1+rt)$	interest NOT reinvested
Annual compound	$C(t)=C_0(1+r)^t$	interest reinvested every year
Compounded $n$ times a year	$C(t)=C_0\bigl(1+\frac{r}{n}\bigr)^{nt}$	fractional compounding
Continuous	$C(t)=C_0\,e^{rt}$	limit $n\to\infty$ of the previous

Example — 1000 euros at 5% for 10 years

Numerical comparison:

• Simple: $1000\cdot(1+0{,}5)=1500$.
• Annual compound: $1000\cdot 1{,}05^{10}\approx 1628{,}89$.
• Monthly compound: $1000\cdot (1+0{,}05/12)^{120}\approx 1647{,}01$.
• Continuous: $1000\cdot e^{0{,}5}\approx 1648{,}72$.

Continuous compounding is the limit towards which ever more frequent compounding tends. In Year 5 we shall see that the limit $\lim_{n\to\infty}(1+1/n)^n=e$ is exactly the same calculation, done at the rate of $100\%$.

The number $e\approx 2{,}718$, called Napier’s number, is the “natural” base of the exponentials and emerges precisely from this passage to the limit of continuous compounding.

Links

Topics: Exponential function
Concepts: Continuous compounding · Compound interest · Napier’s number
Functions: Exponential function
Skills: Modelling · Using formulae
People: John Napier (Nepero)