Question

Ricky is 24 years old and starting an IRA. He is going to invest $200 at the beginning of each month. The account is expected to earn 2.95% interest, compounded monthly. How much money will Ricky have in his IRA when he retires, at age 65?

Answer 1

he's 24 right now, and retiring at 65 means 41 years later.


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


`\bf \qquad \begin{cases} A= \begin{array}{llll} \textit{accumulated amount}\\ \end{array} \\ pmt=\textit{periodic payments}\to &200\\ r=rate\to 2.95\%\to (2.95)/(100)\to &0.0295\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{they're done monthly, thus} \end{array}\to &12\\ t=years\to &41 \end{cases}`


`\bf A=200\left[ \cfrac{\left( 1+(0.0295)/(12) \right)^(12\cdot 41)-1}{(0.0295)/(12)} \right]\left(1+(0.0295)/(12)\right) \\\\\\ A=200\left[ \cfrac{\left( 1 +(59)/(24000)\right)^(492)-1}{(59)/(24000)}\right]\left(1+(59)/(24000)\right) \\\\\\ A=200\left[ \cfrac{\left( (24059)/(24000)\right)^(492)-1}{(59)/(24000)}\right]\left((24059)/(24000)\right)\\\\\\A\approx 191398.48860411668565454023`

Answer 2

Fv=200×(((1+0.0295÷12)^(41 ×12)−1)÷(0.0295÷12))×(1+0.0295÷12) Fv=191,398.48