Can anyone help me to answer these three questions? Thank you so much. An idivid

Question

Can anyone help me to answer these three questions? Thank you so much. An idividual with no initial capital invests k dollars per year at an annual interest rate of return r. Assume investments are made continuously and interest is compounded continuously. (a) Determine the sum accumulated S(t) at any time t. (b) If r = 3% determine k so that $1,000,000 will be available for retirement in 40 years. (c) If k $10,000 per year approximate r such that $1,000,000 will be available for retirement in 40 years. This will require solving a nonlinear equation. You can use Newton's method or a software package you have access to.

Explanation / Answer

Each annum a sum of k $ is invested at an annual interest rate of return 'r'
at any time 't' years, sum accumulated will be the individual sum of the amounts invested in each year that have compounded to until that point of time;
S(t) (after 't' years) = k ert + k er(t-1) + k er(t-2) + .... k ert  + k e0
here the factor erm  represents the comupounding that has taken place for that year's sum 'k' until the year 'm' i.e. m times compounded then multiplied with k;

if r=3% = 0.03; such taht S(t) = 1,000,000$ after 40 years
S(t) = k ert + k er(t-1) + k er(t-2) + .... k ert  + k e0 = k ( ert + er(t-1) + er(t-2) + .... + e2r e1r + e0 )
= k  1 (ern-1) / (er-1) = 106 and k is to be found out, n= 40 years and r=0.03
k (e0.03*40 -1) / (e0.03-1) = 106
k (e1.2-1) / 0.03045 = 106
k = 3.045 * 104 / 2.32 = 1.3125*10000 = 13,125$
Thus, k= 13,125$ which means each year total 13,125$ has to be continuously invested as per the above mentioned terms.

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