Question

Suppose you want to have $400,000 for retirement in 25 years. Your account earns 4% interest.

a) How much would you need to deposit in the account each month?

$


b) How much interest will you earn?

$

Answer 1

`~~~~~~~~~~~~\stackrel{\textit{payments at the beginning of the period}}{\textit{Future Value of an annuity due}} \\\\ A=pmt\left[ \cfrac{\left( 1+(r)/(n) \right)^(nt)-1}{(r)/(n)} \right]\left(1+(r)/(n)\right)`

`\qquad \begin{cases} A=\textit{accumulated amount}\dotfill&\$400000 \\ pmt=\textit{periodic payments}\\ r=rate\to 4\%\to (4)/(100)\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &25 \end{cases}`

`400000=pmt\left[ \cfrac{\left( 1+(0.04)/(12) \right)^(12 \cdot 25)-1}{(0.04)/(12)} \right]\left(1+(0.04)/(12)\right)`

`\cfrac{400000}{\left[ (\left( 1+(0.04)/(12) \right)^(12 \cdot 25)-1)/((0.04)/(12)) \right]\left(1+(0.04)/(12)\right)}=pmt\implies \cfrac{400000}{\left[ (\left( (301)/(300) \right)^(300)-1)/((1)/(300)) \right]\left((301)/(300)\right)}=pmt \\\\\\ \cfrac{400000}{515.84}\approx pmt\implies {\Large \begin{array}{llll} 775.43\approx pmt \end{array}}`

how much will it be in interest alone?

well, for 25 years every month you'd have been putting down that much, so we can just subtract what you put it from the 400,000 and what's left is the interest earned

`400000~~ - ~~(25)(12)(775.43) ~~ \approx ~~ \text{\LARGE 167317}`