Comparison of compounding frequencies

Statement

We invest $10\,000\ \text{€}$ for $5$ years at a nominal rate of $8\%$. Find the accumulated value with (a) annual, (b) quarterly, (c) continuous compounding.

Solution

(a) $M = 10\,000\cdot 1{.}08^{5} = 10\,000\cdot 1{.}469328 \approx \boxed{14\,693{.}28\ \text{€}}$. (b) $M = 10\,000\left(1 + \dfrac{0{.}08}{4}\right)^{20} = 10\,000\cdot 1{.}02^{20} \approx \boxed{14\,859{.}47\ \text{€}}$. (c) $M = 10\,000\,e^{0{.}08\cdot 5} = 10\,000\,e^{0{.}4} \approx \boxed{14\,918{.}25\ \text{€}}$. The accumulated value grows as the compounding frequency increases.