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)**

--------------------------------------…

b) r = 0.06

t = 43 years

s = 10^6

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

Answer: **k = 4919.1867**

--------------------------------------…

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 %**

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