If you start making $150 monthly contributions today and continue them for five years, what’s their future value if the compounding rate is 8.75 percent APR? (Do not round intermediate calculations and round your final answer to 2 decimal places.)

What is the present value of this annuity? (Do not round intermediate calculations and round your final answer to 2 decimal places.)

To solve these problems, we need to determine both the future value and the present value of an annuity with regular monthly contributions.

### Future Value of the Annuity
The future value (FV) of an annuity can be calculated using the formula:
\[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \]
where:
- $ P $ is the monthly contribution ($150),
- $ r $ is the monthly interest rate,
- $ n $ is the total number of contributions.

First, we need to convert the annual percentage rate (APR) to a monthly interest rate:
\[ r = \frac{8.75\%}{12} = \frac{0.0875}{12} = 0.00729167 \]

Next, calculate the total number of contributions over five years:
\[ n = 5 \times 12 = 60 \]

Now we can plug these values into the formula:
\[ FV = 150 \times \left( \frac{(1 + 0.00729167)^{60} - 1}{0.00729167} \right) \]

Calculating the term inside the parentheses:
\[ (1 + 0.00729167)^{60} = 1.561676 \]

So:
\[ FV = 150 \times \left( \frac{1.561676 - 1}{0.00729167} \right) \]
\[ FV = 150 \times \left( \frac{0.561676}{0.00729167} \right) \]
\[ FV = 150 \times 77.05102 \]
\[ FV \approx 11557.65 \]

### Present Value of the Annuity
The present value (PV) of an annuity can be calculated using the formula:
\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]

Using the same values for $ P $, $ r $, and $ n $:
\[ PV = 150 \times \left( \frac{1 - (1 + 0.00729167)^{-60}}{0.00729167} \right) \]

Calculating the term inside the parentheses:
\[ (1 + 0.00729167)^{-60} = 0.640208 \]

So:
\[ PV = 150 \times \left( \frac{1 - 0.640208}{0.00729167} \right) \]
\[ PV = 150 \times \left( \frac{0.359792}{0.00729167} \right) \]
\[ PV = 150 \times 49.32757 \]
\[ PV \approx 7399.14 \]

### Summary
- The future value of the annuity is approximately $ \$11,557.65 $.
- The present value of the annuity is approximately $ \$7,399.14 $.

Would you like more details on any part of the calculation or have any further questions?

### Follow-up Questions:
1. How does changing the monthly contribution amount affect the future value?
2. What is the impact of a higher interest rate on the future value?
3. How would extending the contribution period to 10 years affect the future value?
4. What if the interest compounds quarterly instead of monthly?
5. How does inflation impact the present value of future contributions?
6. Can we use a similar formula to calculate the future value of different types of annuities (e.g., annuities due)?
7. How do we calculate the present value of an annuity if payments are made at the beginning of each period?
8. What are some real-life applications of these calculations?

### Tip:
Always double-check your intermediate calculations for accuracy to ensure your final results are correct, especially when dealing with compound interest formulas.

Learn how to calculate the future and present value of an annuity with $150 monthly contributions over five years, compounded at 8.75% APR. Discover step-by-step solutions and formulas used in financial mathematics. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation