Suppose your client wishes to purchase an annuity that pays $100,000 each year for 6 years, with the first payment 7 years from now. At an interest rate of 7%, how much would the client need to invest now?

Please round your answer to the nearest hundredth.

To determine how much the client needs to invest now to purchase an annuity that pays $100,000 each year for 6 years, starting 7 years from now, we need to follow these steps:

### Step 1: Calculate the Present Value of the Annuity
First, calculate the present value (PV) of the annuity payments at the time when the first payment is made, which is 7 years from now.

The formula for the present value of an annuity is:

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

Where:
- $ P $ is the annuity payment ($100,000).
- $ r $ is the interest rate (7% or 0.07).
- $ n $ is the number of payments (6 years).

\[
PV = 100,000 \times \frac{1 - (1 + 0.07)^{-6}}{0.07}
\]

### Step 2: Discount the Present Value Back to Today
Next, discount this PV back to the present time (0 years). This step accounts for the 7-year period before the first payment is made.

The formula to discount the present value back to today is:

\[
PV_{\text{today}} = PV_{\text{annuity}} \times \frac{1}{(1 + r)^t}
\]

Where:
- $ t $ is the number of years until the first payment (7 years).

Let's calculate this.The client would need to invest **$296,836.13** today to purchase an annuity that pays $100,000 each year for 6 years, with the first payment 7 years from now, assuming a 7% interest rate.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How does the interest rate affect the present value of the annuity?
2. What would happen if the first payment were to be made immediately instead of after 7 years?
3. How would a change in the number of years of payment (e.g., from 6 to 8 years) impact the investment amount?
4. What if the payments were made semi-annually instead of annually?
5. How would an increase in the interest rate to 8% affect the initial investment?

**Tip:** Understanding how to discount future cash flows is essential for evaluating long-term investments like annuities.

Learn how to calculate the present value of an annuity that pays $100,000 annually for 6 years, starting 7 years from now, at a 7% interest rate. Understand the formula for annuities and how to discount future cash flows. Suitable for advanced high school students studying finance and time value of money.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation