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 = 7.5%, determine k so that $1 million 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.

Answer 1

Explanation:

Given that 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) Amount accumulated at time t

`S(t) = ke^(rt)`, where r = rate of interest and t = time lapsed

b) Here S = 1,000,000

`1000000 = ke^(0.075(40)) \\k= 1000000e^(-0.075(40))\\=49802.56`

So 49802.56 dollars to be invested to get 1 million after 40 years

c) k = 2000 per year

I 2000 will be for 40 years, II 2000 for 39 years, ..... Last 2000 for 0 years

i.e. final amount would be

`2000[(e^(40r) +(e^(39r) +....e^r] = 1000000\\500 = (e^r(e^(40r)-1)/(e^r-1)`

solving we get r = 9.6% approxy