Dollar Young

A young person with no initial capital invests k dollars per year?

A young person with no initial capital invests k dollars per year at an annual rate of return r. Assume that
the investments are made continuously and that the return is compounded continuously.
(a) Determine the sum S(t) accumulated at any time t.
(b) If r = 7.5%, determine k so that 1 million dollars will be available for retirement in 40 years.
(c) If k = $2000/year determine the return rate r that must be obtained to have 1 million available in 40
years.

(a) The differential equation for continuously compounding investment is

dS/dt = rS + k

dS/dt – rS = k

u(t)*dS/dt – u(t)*rS = u(t)*k

du(t)/dt = -u(t) * r

(du(t)/dt) / u(t) = -r

d(ln|u(t)|)/dt = -r

ln|u(t)| = -rt + c

u(t) = ce^(-rt)

d[s * e^(-rt)]/dt = k * e^(-rt)

s * e^(-rt) = k * (int(e^(-rt))

s * e^(-rt) = (-k/r) * e^(-rt) + C

s = (-k/r) + C * (e^rt)

(s,t) = (0,0)

C = -k/r

s = (-k/r) + (k/r) * e ^ (rt)

Answer: s = (k/r) * ((e^rt) – 1)
———————————————
b) r = 0.075
t = 40 years
s = 10^6

10^6 = (k/0.075) * (e^3 – 1)

Answer: k = 3929.68
———————————————-
c) k = 2000
s = 10^6
t = 40 years

10^6 = (2000/r) * ((e^40r) – 1)

500r = e^40r – 1

Answer: r = 9.77 %

This percent was found by trial and error to make the equation work out right.

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