To find the payment for an annuity due, we use the following formula for the future value of an annuity due:

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

Where:

• $FV$ is the future value of the annuity.
• $P$ is the payment amount per period.
• $i$ is the interest rate per period.
• $n$ is the total number of payments.

Given:

• Future value $FV = 15,500$
• Interest rate $r = 6.2\%$ per year.
• Compounding and payment frequency is quarterly.
• Number of years $t = 20$.

Step 1: Calculate the interest rate per period

Since the interest is compounded quarterly, divide the annual interest rate by 4: $i = \frac{6.2\%}{4} = \frac{0.062}{4} = 0.0155$

Step 2: Calculate the number of periods

Since the payments are quarterly for 20 years: $n = 20 \times 4 = 80$

Step 3: Plug the values into the formula

$15,500 = P \times \left( \frac{(1 + 0.0155)^{80} - 1}{0.0155} \right) \times (1 + 0.0155)$

First, calculate $(1 + 0.0155)^{80}$:

$(1 + 0.0155)^{80} = 1.0155^{80} \approx 3.5119$

Now, calculate the factor inside the brackets:

$\frac{(3.5119 - 1)}{0.0155} \approx \frac{2.5119}{0.0155} \approx 162.7019$

Multiply this by $(1 + 0.0155)$:

$162.7019 \times 1.0155 \approx 165.222$

Step 4: Solve for $P$

Now, substitute this back into the equation:

$15,500 = P \times 165.222$

$P = \frac{15,500}{165.222} \approx 93.81$

Final Answer:

The quarterly payment for the annuity due is approximately $93.81.

Would you like further details or clarifications?

Here are five related questions for you:

1. How would the payment change if the interest rate increased to 7%?
2. What is the difference between an ordinary annuity and an annuity due?
3. How do payments change if the compounding period changes to monthly?
4. Can the future value of an annuity due be calculated with continuous compounding?
5. How does the time period affect the size of each payment?

Tip: Always ensure that the interest rate and the payment period align (e.g., quarterly payments require a quarterly interest rate).