A couple received a $129,000 inheritance the year they turned 50 and invested it in a fund that earns 6.1% compounded semiannually. If this amount is deferred for 12 years (until they retire), how much will it provide at the end of each half year (in dollars) for the next 20 years after they retire? (Round your answer to two decimal places.)
$  

This problem involves a two-step process: first, calculating the future value (FV) of the $129,000 investment over 12 years with compounded interest, and then determining the annuity payments that will be made over 20 years based on the future value.

### Step 1: Calculate the Future Value (FV)
The inheritance of $129,000 is invested in a fund that earns 6.1% interest, compounded semiannually, for 12 years.

The formula for compound interest is:

\[
FV = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $P = 129,000$ (the initial investment)
- $r = 0.061$ (the annual interest rate as a decimal)
- $n = 2$ (because the interest is compounded semiannually)
- $t = 12$ years

Now, plug the values into the formula:

\[
FV = 129,000 \left(1 + \frac{0.061}{2}\right)^{2 \times 12}
\]

\[
FV = 129,000 \left(1 + 0.0305\right)^{24}
\]

\[
FV = 129,000 \times (1.0305)^{24}
\]

Calculating:

\[
FV \approx 129,000 \times 2.0387 \approx 262,000.23
\]

So, the future value after 12 years is approximately $262,000.23.

### Step 2: Calculate the Semiannual Payments
Now, the couple wants to withdraw an equal amount at the end of each half-year for the next 20 years (40 periods). To calculate the payment amount, we use the annuity payment formula:

\[
PMT = \frac{FV \times \frac{r}{n}}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}
\]

Where:
- $FV = 262,000.23$ (the future value calculated in step 1)
- $r = 0.061$ (the annual interest rate)
- $n = 2$ (semiannual payments)
- $t = 20$ years

Now, plug the values into the formula:

\[
PMT = \frac{262,000.23 \times \frac{0.061}{2}}{1 - \left(1 + \frac{0.061}{2}\right)^{-2 \times 20}}
\]

\[
PMT = \frac{262,000.23 \times 0.0305}{1 - (1.0305)^{-40}}
\]

\[
PMT = \frac{7,991.01}{1 - 0.3086}
\]

\[
PMT = \frac{7,991.01}{0.6914}
\]

\[
PMT \approx 11,557.55
\]

### Final Answer:
The amount they can withdraw at the end of each half year is approximately **$11,557.55**.

---

Would you like a deeper explanation of any part of this solution?

Here are five related questions to think about:
1. What would the final payment be if the interest rate changed to 5%?
2. How would the payment schedule change if the funds were withdrawn quarterly?
3. What would be the effect if the inheritance were invested for 15 years instead of 12?
4. How would monthly compounding affect the final withdrawal amount?
5. What if the couple deferred the withdrawal for only 10 years?

**Tip:** When dealing with interest problems, always be mindful of the compounding period as it can significantly change the final outcome.

A couple received a $129,000 inheritance the year they turned 50 and invested it in a fund that earns 6.1% compounded semiannually. If this amount is deferred for 12 years (until they retire), how much will it provide at the end of each half year (in dollars) for the next 20 years after they retire? (Round your answer to two decimal places.)

Learn how to calculate the future value of a $129,000 inheritance invested at 6.1% compounded semiannually for 12 years, and find the semiannual annuity payments over 20 years. This problem covers compound interest and annuity payment formulas. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation