Question

A young person with no initial capital invests k dollars per year at an annual rate of return r. Assume that 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 = 6.0%, determine k so that $1 million will be available for retirement in 43 years.

(c) If k=2500/year, determine the return rate r that must be obtained to have $1 million available in 43 years.

Answer 1

(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)

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

Answer: s(t) = (k/r) * ((e^rt) - 1)
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b) r = 0.06
t = 43 years
s = 10^6

10^6 = (k/0.06) * (e^(2.58) - 1)

Answer: k = 4919.1867
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c) k = 200
s = 10^6
t = 43 years

10^6 = (2500/r) * ((e^43r) - 1)

400r = e^43r - 1
r= 0.08182
Answer: r = 8.18 %