Answer
given,
Sum = S₀
annual rate of return = r
T is the time
Ordinary differential equations is
[tex]\dfrac{dS}{dt}=rs[/tex]
[tex]\dfrac{dS}{S}=r dt[/tex]
integrating both side
[tex]\int\dfrac{dS}{S}=\int r dt[/tex]
[tex] ln (S)= rt + C[/tex]
[tex]S = e^{rt+C}[/tex]
[tex]S=e^C.e^{rt}[/tex]
e^C = S₀
[tex]S=S_0 e^{rt}[/tex]
a) time when sum is doubled
[tex]\dfrac{S}{S_0}=2[/tex]
[tex] 2 = e^{rT}[/tex]
[tex]T= \dfrac{ln(2)}{r}[/tex]
b) Time T if r = 7 %
[tex]T= \dfrac{ln(2)}{0.07}[/tex]
T = 9.9 years.
c) return rate, r = ? T = 8 years
[tex]r= \dfrac{ln(2)}{T}[/tex]
[tex]r= \dfrac{ln(2)}{8}[/tex]
r = 0.0866
r = 8.67%
