Suppose an annuity will pay $10,000 at the beginning of each year for the next 3 years. How much money is needed to start this annuity if it earns 7.3%, compounded annually? (Round your answer to the nearest cent.)

To calculate how much money is needed to start this annuity, we need to compute the **present value of an annuity due**. Since the payments are made at the beginning of each year, it's an "annuity due."

The formula for the present value of an annuity due is:

\[
PV = P \times \left( 1 + r \right) \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]

Where:
- $ PV $ is the present value of the annuity.
- $ P $ is the payment amount ($10,000).
- $ r $ is the interest rate per period (7.3% = 0.073).
- $ n $ is the number of payments (3 years).

### Step-by-Step Calculation:

1. **Payment per year (P)** = $10,000.
2. **Interest rate (r)** = 0.073.
3. **Number of payments (n)** = 3.

Let's plug these values into the formula to compute the present value.

\[
PV = 10,000 \times \left( 1 + 0.073 \right) \times \left( \frac{1 - (1 + 0.073)^{-3}}{0.073} \right)
\]

I'll compute this for you.The amount of money needed to start this annuity is **$28,005.28**.

Would you like further clarification or more details? Here are 5 related questions to deepen your understanding:

1. How does the interest rate affect the present value of an annuity?
2. What is the difference between an annuity due and an ordinary annuity?
3. How would the present value change if payments were made at the end of the year instead?
4. What happens to the present value if the interest rate is compounded more frequently, like quarterly?
5. Can you derive a formula for the future value of an annuity due?

**Tip:** In annuity problems, the timing of payments (beginning or end of the period) is critical to choosing the correct formula.

Discover how to calculate the present value of an annuity due with payments of $10,000 at the beginning of each year for 3 years at an interest rate of 7.3%, compounded annually. Learn about key concepts such as annuities, compound interest, and the present value formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation