Martina has already accumulated $32,000 in her RRSP. If she contributes $2,500 at the end of every six months for the next 12 years and $400 at the end of every month for the subsequent eight years, how much will she have at the end of the 20 years? Assume that her plan will earn 6% compounded semi-annually for the first 12 years and 6% compounded monthly for the last 8 years.
A) $293,054
B) $200,247
C) $204,132
D) $154,398
E) $217,873

Question

Quizplus · Accepted Answer

To solve this, we need to break down the problem into two parts because of the change in contribution frequency and compounding period.1. **First 12 years (semi-annual contributions and compounding):**   - Initial amount: $32,000   - Semi-annual contribution: $2,500   - Interest rate: 6% per year (or 3% per half-year because of semi-annual compounding)   - Number of periods (n) for 12 years, with semi-annual contributions: 12 * 2 = 24The future value (FV) for the first 12 years can be calculated using the future value of an annuity formula for the contributions, plus the future value of a single sum for the initial amount.$$ FV_{contributions} = P \times \left( \frac{(1 + r)^n - 1}{r} \right) $$$$ FV_{initial} = P_0 \times (1 + r)^n $$Where:- $ P $ is the semi-annual contribution ($2,500),- $ r $ is the semi-annual interest rate (3% or 0.03),- $ n $ is the total number of periods (24),- $ P_0 $ is the initial amount ($32,000).2. **Last 8 years (monthly contributions and compounding):**   - We first need to calculate the amount at the end of 12 years from step 1, which will serve as the initial amount for this period.   - Monthly contribution: $400   - Interest rate: 6% per year (or 0.5% per month because of monthly compounding)   - Number of periods (n) for 8 years, with monthly contributions: 8 * 12 = 96The future value for the last 8 years is calculated similarly, using the future value of an annuity formula for the contributions, plus the future value of a single sum for the amount at the end of the first 12 years.By calculating both parts and adding them together, we get the total amount at the end of 20 years. Given the complexity of the calculation and the need for precision, using a financial calculator or spreadsheet is recommended to accurately compute the future values. The correct answer, based on these calculations, is approximately $293,054, which matches option A. This takes into account the compound interest over both periods and the different contribution schedules.

Martina Has Already Accumulated $32,000 in Her RRSP