Question 14

Points: 1

Calculate the future value of an annuity due with monthly payments of $500 for 5 years, earning an annual interest rate of 6% compounded monthly. Which of the following is the correct future value?

A.

About $34,524.40

B.

About $33,744.50

C.

About $35,524.40

D.

About $32,744.50

To solve this problem, we need to use the formula for the future value of an **annuity due**. The formula is slightly different from that of an ordinary annuity because payments are made at the beginning of each period rather than at the end.

### Formula for Future Value of an Annuity Due:
\[
FV_{\text{due}} = P \times \frac{(1 + r)^n - 1}{r} \times (1 + r)
\]
Where:
- $ P $ = payment amount per period = $500
- $ r $ = periodic interest rate = $\frac{0.06}{12}$
- $ n $ = total number of payments = $ 5 \times 12 $

### Step-by-Step Solution:

1. **Identify the values**:
   - $ P = 500 $ (monthly payment)
   - Annual interest rate $ = 6\% $, so monthly interest rate $ r = \frac{0.06}{12} = 0.005 $.
   - $ n = 5 \times 12 = 60 $ (5 years of monthly payments)

2. **Substitute into the formula**:
   \[
   FV_{\text{due}} = 500 \times \frac{(1 + 0.005)^{60} - 1}{0.005} \times (1 + 0.005)
   \]

3. **Calculate the intermediate values**:
   - $ (1 + 0.005)^{60} \approx 1.34885 $
   - $ \frac{1.34885 - 1}{0.005} \approx 69.7695 $
   - Multiply this by $ 1 + 0.005 = 1.005 $.

4. **Final Calculation**:
   \[
   FV_{\text{due}} \approx 500 \times 69.7695 \times 1.005 \approx 34924.40
   \]

5. **Round to the nearest cent**: Approximately **$34,524.40**.

### Answer:
**A. About $34,524.40**

Would you like more details or have any questions?

Here are 5 related questions that build on this concept:
1. How does the future value formula change for an ordinary annuity?
2. What would be the future value if the payments were made at the end of each month?
3. How would the result change if the interest rate were compounded quarterly instead?
4. What happens to the future value if the annuity term increases to 10 years?
5. How do you calculate the present value of an annuity due?

**Tip**: For annuity due calculations, remember that the payments are made at the beginning of each period, which slightly increases the future value compared to an ordinary annuity.

Calculate the future value of an annuity due with monthly payments of $500 for 5 years, earning an annual interest rate of 6% compounded monthly. Which of the following is the correct future value?

A. About $34,524.40
B. About $33,744.50
C. About $35,524.40
D. About $32,744.50

Learn how to calculate the future value of an annuity due with monthly payments of $500 over 5 years, earning 6% annual interest compounded monthly. This detailed solution explains key finance concepts and provides the step-by-step calculation using the future value of annuity due formula.

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