Full solution

Q. You would like to have

\$ 900,000

when you retire in

40

years. How much should you invest each quarter if you can earn a rate of

3.2 \%

compounded quarterly?

\newline

a) How much should you deposit each quarter?

Identify Variables: Identify the variables for the future value annuity formula. $\newline$ Future Value (FV) = $\$900,000$ $\newline$ Annual interest rate (r) = $3.2\%$ or $0.032$ $\newline$ Number of years (t) = $40$ $\newline$ Compounding frequency per year (n) = $4$ (quarterly) $\newline$ We need to find the regular deposit amount per quarter (PMT).
Convert Interest Rate: Convert the annual interest rate to the quarterly interest rate. $\newline$ Quarterly interest rate = Annual interest rate / Number of quarters in a year $\newline$ Quarterly interest rate = $\frac{0.032}{4}$ $\newline$ Quarterly interest rate = $0.008$
Calculate Compounding Periods: Calculate the total number of compounding periods. $\newline$ Total compounding periods $N$ = Number of years $\times$ Compounding frequency per year $\newline$ Total compounding periods = $40 \times 4$ $\newline$ Total compounding periods = $160$
Use Annuity Formula: Use the future value of an annuity formula to find the regular deposit amount (PMT). The formula for the future value of an annuity is: $FV = PMT \times \left[\left(\frac{1 + r/n}{n}\right)^{nt} - 1\right] / \left(\frac{r}{n}\right)$ We need to rearrange the formula to solve for PMT: $PMT = \frac{FV}{\left[\left(\frac{1 + r/n}{n}\right)^{nt} - 1\right] / \left(\frac{r}{n}\right)}$
Calculate PMT: Plug the values into the rearranged formula to calculate PMT. $\newline$ PMT = $\$900,000 / [((1 + 0.008)^{160} - 1) / 0.008]$ $\newline$ PMT = $\$900,000 / [((1.008)^{160} - 1) / 0.008]$ $\newline$ First, calculate the term $(1.008)^{160}$ : $\newline$ $(1.008)^{160} \approx 4.8012$ $\newline$ Now, subtract $1$ from this term: $\newline$ $4.8012 - 1 \approx 3.8012$ $\newline$ Now, divide by the quarterly interest rate: $\newline$ $3.8012 / 0.008 \approx 475.15$ $\newline$ Finally, divide the future value by this result to find PMT: $\newline$ PMT = $\$900,000 / 475.15$ $\newline$ PMT \approx $\$1894.63$