Question

Suppose that a sum S0 is invested at an annual rate of return r compounded continuously. a. Find the time T required for the original sum to double in value as a function of r. b. Determine T if r = 7%. c. Find the return rate that must be achieved if the initial investment is to double in 8 years.

Answer 1

given,

Sum = S₀

annual rate of return = r

T is the time

Ordinary differential equations is

`(dS)/(dt)=rs`

`(dS)/(S)=r dt`

integrating both side

`\int(dS)/(S)=\int r dt`

`ln (S)= rt + C`

`S = e^(rt+C)`

`S=e^C.e^(rt)`

e^C = S₀

`S=S_0 e^(rt)`

a) time when sum is doubled

`(S)/(S_0)=2`

`2 = e^(rT)`

`T= (ln(2))/(r)`

b) Time T if r = 7 %

`T= (ln(2))/(0.07)`

T = 9.9 years.

c) return rate, r = ? T = 8 years

`r= (ln(2))/(T)`

`r= (ln(2))/(8)`

r = 0.0866

r = 8.67%