For the past $25$ years, Eric has contributed $\$700$ to his RRSP at the end of every month. The plan earned $7.20\%$ compounded quarterly for the first $12$ years and $7.10\%$ compounded monthly for the subsequent $13$ years. What is the value of his RRSP today? For full marks your answer(s) should be rounded to the nearest cent

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Q. For the past

25

years, Eric has contributed

\$700

to his RRSP at the end of every month. The plan earned

7.20\%

compounded quarterly for the first

12

years and

7.10\%

compounded monthly for the subsequent

13

years. What is the value of his RRSP today? For full marks your answer(s) should be rounded to the nearest cent

Calculate FV for $1$ st $12$ years: First, calculate the future value of the contributions for the first $12$ years with a $7.20\%$ annual interest rate compounded quarterly. $\newline$ Use the future value of an annuity formula: $FV = P \times \left[(1 + \frac{r}{n})^{(nt)} - 1\right] / \left(\frac{r}{n}\right)$ , where $P$ is the payment, $r$ is the annual interest rate, $n$ is the number of times the interest is compounded per year, and $t$ is the time in years. $\newline$ $FV = 700 \times \left[(1 + \frac{0.072}{4})^{(4\times12)} - 1\right] / \left(\frac{0.072}{4}\right)$
Calculate FV for $1$ st $12$ years: Calculate the future value for the first $12$ years. $\newline$ $FV = 700 \times [(1 + 0.018)^{48} - 1] / 0.018$ $\newline$ $FV = 700 \times [(1.018)^{48} - 1] / 0.018$ $\newline$ $FV = 700 \times [2.0398873 - 1] / 0.018$ $\newline$ $FV = 700 \times 1.0398873 / 0.018$ $\newline$ $FV = 700 \times 57.771517$ $\newline$ $FV = 40439.06$
Calculate FV for last $13$ years: Now, calculate the future value of the contributions for the remaining $13$ years with a $7$ . $10$ % annual interest rate compounded monthly. $\newline$ Use the same formula with the new interest rate and compounding period. $\newline$ $FV = 700 \times \left[(1 + 0.071/12)^{(12\times13)} - 1\right] / (0.071/12)$
Calculate FV for last $13$ years: Calculate the future value for the last $13$ years. $\newline$ $FV = 700 \times [(1 + 0.0059167)^{156} - 1] / 0.0059167$ $\newline$ $FV = 700 \times [(1.0059167)^{156} - 1] / 0.0059167$ $\newline$ $FV = 700 \times [2.113775 - 1] / 0.0059167$ $\newline$ $FV = 700 \times 1.113775 / 0.0059167$ $\newline$ $FV = 700 \times 188.3025$ $\newline$ $FV = 131811.75$
Add FV for total: Add the future values of both periods to get the total value of the RRSP today. $\newline$ Total $FV = FV$ from first $12$ years + $FV$ from last $13$ years $\newline$ Total $FV = 40439.06 + 131811.75$
Calculate total FV: Calculate the total future value. $\newline$ Total FV = $$172250$.$81$