Compound-interest share investment

Statement

An entrepreneur invests his wealth of $87$ million euros in BorghiMotors shares, which increase in value by $4\%$ each year.

• (a) After how many years has he doubled his wealth?
• (b) How much money is in his account after $30$ years?
• (c) A friend invested $12$ million in FantiEnterprises shares and after three years found $17$ million. What is the annual growth rate of these shares?
• (d) Another friend now has $400$ million euros; five years ago he had invested a certain sum in bonds yielding $12\%$ per year. How much did he invest five years ago?

Solution

The model is $C(t)=C_0\,(1+i)^{t}$.

(a) $1{.}04^{\,n}=2 \Rightarrow n=\dfrac{\ln 2}{\ln 1{.}04}\approx 17{.}67$ years (it doubles during the $18^\text{th}$ year).

(b) $C_{30}=87\cdot 1{.}04^{30}=87\cdot 3{.}2434\approx 282{.}18$ million euros.

(c) $(1+r)^3=\dfrac{17}{12}=1{.}41\overline{6} \Rightarrow 1+r=1{.}41\overline{6}^{\,1/3}\approx 1{.}1231$, so $r\approx 12{.}31\%$.

(d) $P\cdot 1{.}12^{5}=400 \Rightarrow P=\dfrac{400}{1{.}76234}\approx 226{.}97$ million euros.

$\boxed{n\approx 17{.}67\ \text{yr};\quad C_{30}\approx 282{.}2\ \text{M};\quad r\approx 12{.}31\%;\quad P\approx 226{.}97\ \text{M}}$