Capital invested at 6 percent

Statement

A banker invests two million euros at an annual interest rate of $6\%$.

• (a) How much is his capital worth after $4$ years?
• (b) How many years does it take to double the capital?
• (c) At what rate must he invest it so that it doubles in $4$ years?

Solution

Model $C(t)=2\cdot(1+i)^{t}$ (in millions of euros).

(a) $C_4=2\cdot 1{.}06^{4}=2\cdot 1{.}26248\approx 2{.}525$ million euros.

(b) $1{.}06^{\,n}=2 \Rightarrow n=\dfrac{\ln 2}{\ln 1{.}06}\approx 11{.}90$ years.

(c) $(1+r)^4=2 \Rightarrow 1+r=2^{1/4}\approx 1{.}18921$, so $r\approx 18{.}92\%$.

$\boxed{C_4\approx 2{.}525\ \text{M};\quad n\approx 11{.}90\ \text{yr};\quad r\approx 18{.}92\%}$