Continuous compounding

Text

A principal of $2000$ € is invested at an annual rate of $3\%$ with continuous compounding, following $M(t)=2000\,e^{0.03\,t}$.

• (a) What is the balance after $5$ years?
• (b) After how many years does the capital double?

Solution

(a) $M(5)=2000\,e^{0.03\cdot 5}=2000\,e^{0.15}\approx 2000\cdot 1.16183\approx 2323.67\ \text{€}.$ (b) Impose $2000\,e^{0.03\,t}=4000$, i.e. $e^{0.03\,t}=2$; taking the natural logarithm, $0.03\,t=\ln 2\;\Rightarrow\; t=\frac{\ln 2}{0.03}\approx 23.10\ \text{years}.$ $\boxed{\,M(5)\approx 2323.67\ \text{€},\quad t\approx 23.10\ \text{years}\,}$