Ms. Guo’s monthly insurance premium is ₱500.00, payable at the end of each month. Her policy matures 20 years later, after which she can withdraw all her payments plus the interest earned. If the money is worth 15% compounded monthly, how much does she expect to withdraw on the maturity of her policy?

To determine how much Ms. Guo can expect to withdraw at the maturity of her policy, we need to calculate the future value of an annuity. In this case, Ms. Guo is making regular monthly payments into an account that earns interest compounded monthly.

1. Break Down the Problem

We need to calculate the future value of a series of monthly payments (an annuity) with a given interest rate compounded monthly over a period of 20 years.

2. Relevant Concepts

The formula for the future value of an annuity compounded periodically is:

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

Where:

• $FV$ is the future value of the annuity.
• $P$ is the monthly payment (₱500.00).
• $r$ is the monthly interest rate (annual rate divided by 12).
• $n$ is the total number of payments (months).

3. Analysis and Detail

1. Determine the monthly interest rate:

   • Annual interest rate = 15%
   • Monthly interest rate, $r = \frac{15\%}{12} = \frac{0.15}{12} = 0.0125$.

2. Calculate the total number of payments:

   • 20 years = 20 × 12 months = 240 months.

3. Substitute the values into the formula: $FV = 500 \times \frac{(1 + 0.0125)^{240} - 1}{0.0125}$

4. Calculate the future value:

   • First, calculate $(1 + 0.0125)^{240}$.
   • Then, subtract 1 from the result.
   • Divide by the monthly interest rate (0.0125).
   • Multiply by the monthly payment (₱500.00).

4. Verify and Summarize

Let's compute the values:

1. $(1 + 0.0125)^{240} \approx 148.8822311$
2. $148.8822311 - 1 = 147.8822311$
3. $\frac{147.8822311}{0.0125} \approx 11830.57849$
4. $11830.57849 \times 500 \approx 5,915,289.245$

Final Answer

Ms. Guo can expect to withdraw approximately ₱5,915,289.25 at the maturity of her policy. This amount includes all her payments plus the interest earned over 20 years.