Capital growth and doubling rate

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Answer:

• (a) a banker invests $3$ million euros in shares growing $3\%$ per year. After how many years does he reach $10$ million?
• (b) which annual interest rate doubles a capital in ten years?

Solution

(a) $3\cdot 1{,}03^{\,n}=10 \Rightarrow 1{,}03^{\,n}=\tfrac{10}{3}$, so $n=\frac{\ln(10/3)}{\ln 1{,}03}\approx 40{,}73\ \text{years}\quad(\text{i.e. in the 41st year it exceeds 10 million}).$

(b) $(1+r)^{10}=2\Rightarrow 1+r=2^{1/10}\approx 1{,}0718$, so $r\approx 7{,}18\%$.

$\boxed{n\approx 40{,}73\ \text{years};\qquad r\approx 7{,}18\%}$