

Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period.
​$15,500​; quarterly payments for 20 ​years; interest rate 6.2​%

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).

Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period. $15,500; quarterly payments for 20 years; interest rate 6.2%.

Learn how to calculate the quarterly payment for an annuity due with a future value of $15,500, an interest rate of 6.2%, and payments made over 20 years. This detailed solution covers key concepts like compound interest and annuities, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation