Brooke deposited $1,100 at the end of every month into an RRSP for 8 years. The interest rate earned was 3.50% compounded semi-annually for the first 3 years and changed to 3.75% compounded monthly for the next 5 years. What was the accumulated value of the RRSP at the end of 8 years? Round to the nearest cent

To find the accumulated value of Brooke's RRSP at the end of 8 years, we need to break the problem into two parts: the first 3 years and the next 5 years, due to the change in the interest rate and compounding frequency.

Part 1: First 3 Years

1. Monthly Deposit: $1,100
2. Interest Rate: 3.50% compounded semi-annually
3. Duration: 3 years

First, we need to convert the semi-annual interest rate to an effective monthly interest rate.

• Semi-annual interest rate: 3.50% / 2 = 1.75% per 6 months
• Effective semi-annual rate: $(1 + 0.0175)$
• Effective annual rate: $(1 + 0.0175)^2 - 1$
• Effective annual rate: $1.03530625 - 1 = 0.03530625$ or 3.530625%
• Effective monthly rate: $(1 + 0.03530625)^{1/12} - 1$
• Effective monthly rate: $(1.03530625)^{1/12} - 1 \approx 0.002896$ or 0.2896%

Now, we use the future value of an ordinary annuity formula to find the accumulated value after 3 years:

$FV = P \times \frac{(1 + r)^n - 1}{r}$

Where:

• $P$ = $1,100 (monthly deposit)
• $r$ = 0.002896 (monthly interest rate)
• $n$ = 3 years × 12 months/year = 36 months

$FV_{3 years} = 1100 \times \frac{(1 + 0.002896)^{36} - 1}{0.002896}$

$FV_{3 years} = 1100 \times \frac{(1.002896)^{36} - 1}{0.002896}$

$FV_{3 years} = 1100 \times \frac{1.1094 - 1}{0.002896}$

$FV_{3 years} = 1100 \times \frac{0.1094}{0.002896}$

$FV_{3 years} = 1100 \times 37.78$

$FV_{3 years} \approx 41,558.00$

Part 2: Next 5 Years

1. Initial Amount: $41,558.00 (from Part 1)
2. Monthly Deposit: $1,100
3. Interest Rate: 3.75% compounded monthly
4. Duration: 5 years

The monthly interest rate for the next 5 years is:

• Annual interest rate: 3.75%
• Monthly interest rate: 3.75% / 12 = 0.3125%

We need to calculate the future value of the initial amount after 5 years and the future value of the monthly deposits over 5 years.

Future Value of Initial Amount:

$FV_{initial} = PV \times (1 + r)^n$

Where:

• $PV$ = $41,558.00
• $r$ = 0.003125 (monthly interest rate)
• $n$ = 5 years × 12 months/year = 60 months

$FV_{initial} = 41558 \times (1 + 0.003125)^{60}$

$FV_{initial} = 41558 \times (1.003125)^{60}$

$FV_{initial} = 41558 \times 1.197$

$FV_{initial} \approx 49,748.63$

Future Value of Monthly Deposits:

$FV_{deposits} = P \times \frac{(1 + r)^n - 1}{r}$

Where:

• $P$ = $1,100 (monthly deposit)
• $r$ = 0.003125 (monthly interest rate)
• $n$ = 60 months

$FV_{deposits} = 1100 \times \frac{(1 + 0.003125)^{60} - 1}{0.003125}$

$FV_{deposits} = 1100 \times \frac{(1.003125)^{60} - 1}{0.003125}$

$FV_{deposits} = 1100 \times \frac{1.197 - 1}{0.003125}$

$FV_{deposits} = 1100 \times \frac{0.197}{0.003125}$

$FV_{deposits} = 1100 \times 63.04$

$FV_{deposits} \approx 69,344.00$

Total Accumulated Value:

$FV_{total} = FV_{initial} + FV_{deposits}$

$FV_{total} = 49,748.63 + 69,344.00$

$FV_{total} \approx 119,092.63$

Therefore, the accumulated value of the RRSP at the end of 8 years is approximately $119,092.63.