Payments of $300$300 are made at the beginning of each month to an RRSP that earns 9.6%9.6% compounded quarterly for the first 2020 years that is followed by a period of 1818 years with no contributions. The RRSP is then converted into a RIF with quarterly payments that will deplete the RIF in 1919 years. Determine the payment received at the end of each quarter if the RIF earns 3.9%3.9% compounded annually.

To solve this problem, we need to break it down into three main parts:

1. **Calculate the future value of the RRSP after 20 years of contributions.**
2. **Determine the value of the RRSP after 18 additional years with no contributions.**
3. **Calculate the quarterly payment from the RIF that will deplete the account over 19 years.**

### 1. Future Value of the RRSP after 20 Years of Contributions

- **Monthly Payment (PMT):** $300
- **Interest Rate:** 9.6% compounded quarterly (which needs to be converted to an equivalent monthly interest rate)
- **Compounding Periods:** 20 years × 12 months/year = 240 months

The quarterly rate $ r_q = 9.6\% / 4 = 2.4\% $ per quarter.

Since contributions are made monthly, we need to convert the quarterly rate to a monthly rate:
\[
r_m = \left(1 + \frac{r_q}{100}\right)^{1/3} - 1
\]
Where:
- $ r_q $ is the quarterly rate in decimal form.
  
Then, we calculate the future value (FV) using the formula for the future value of an annuity due:
\[
FV = PMT \times \frac{(1 + r_m)^n - 1}{r_m} \times (1 + r_m)
\]
Where:
- $ PMT $ is the payment made at the beginning of each month.
- $ n $ is the total number of payments.

### 2. Value of the RRSP after 18 Years with No Contributions

After the 20-year period, no contributions are made for 18 years, but the account still earns interest. The value at the end of 18 years is calculated as:
\[
FV' = FV \times (1 + r_q)^{72}
\]
Where:
- $ r_q $ is the quarterly rate.
- $ 72 $ is the number of quarters in 18 years.

### 3. Quarterly Payment from the RIF

Now, the RRSP is converted into a RIF, and quarterly withdrawals are made over 19 years, with the RIF earning 3.9% compounded annually. The quarterly interest rate is:
\[
r_q' = \frac{3.9\%}{4}
\]
The quarterly payment is then calculated using the formula for the present value of an annuity:
\[
PMT_q = \frac{FV'}{\frac{1 - (1 + r_q')^{-n'}}{r_q'}}
\]
Where:
- $ FV' $ is the future value after 38 years.
- $ n' $ is the total number of quarters in 19 years.

Let's calculate these values step by step.### Results:

1. **Future Value of the RRSP after 20 Years of Contributions:**
   The future value of the RRSP after 20 years of making $300 payments at the beginning of each month is approximately **$215,942.39**.

2. **Value of the RRSP after 18 More Years with No Contributions:**
   After 18 years with no additional contributions, the RRSP grows to approximately **$1,191,063.12**.

3. **Quarterly Payment from the RIF:**
   The payment received at the end of each quarter from the RIF, which will deplete the account over 19 years, is approximately **$22,261.83**.

Would you like more details or have any questions?

### Related Questions:
1. How would the future value change if contributions were made at the end of each month instead?
2. What would be the impact of a higher interest rate during the RIF withdrawal phase?
3. How much more would the RRSP be worth if contributions continued during the 18-year period?
4. How does compounding frequency affect the final value in a savings plan?
5. What is the significance of converting an RRSP to a RIF in terms of tax implications?

### Tip:
Contributing regularly and letting your investments grow over time without withdrawals maximizes the benefits of compounding interest.

Explore how to calculate payments and values for an RRSP and RIF with compounded interest rates. Solve for future values, quarterly payments, and more using specific financial formulas. Get step-by-step solutions and explanations.

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