A debt payment of ​$9000 is due in 30 months. If money is worth 6.3​% p.a. compounded quarterly​, find the equivalent payments at the times shown below.
​(a) now
​(b) in 15 months
​(c) in 30 months
​(d) in 42 months
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Part 1
​(a) The equivalent debt payment is ​$
  
enter your response here.
​(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as​ needed.)
Part 2
​(b) The equivalent debt payment is ​$
  
enter your response here.
​(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as​ needed.)
Part 3
​(c) The equivalent debt payment is ​$
  
enter your response here.
​(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as​ needed.)
Part 4
​(d) The equivalent debt payment is ​$
  
enter your response here.
​(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as​ needed.)

To solve these problems, we need to find the equivalent debt payments at various times given that the debt payment of $9000 is due in 30 months. The interest rate is 6.3% per annum compounded quarterly. 

### The steps for each part are as follows:

1. **Calculate the present value at the required time.** 
   \[
   PV = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
   \]
   Where:
   - $ A $ is the future value ($9000).
   - $ r $ is the nominal annual interest rate (6.3% = 0.063).
   - $ n $ is the number of compounding periods per year (quarterly compounding $ n = 4 $).
   - $ t $ is the time in years from now to the required payment time.
   
2. **Convert the time in months to years for calculation purposes** as the formula requires time in years.

### Part (a) Now:
For part (a), we are finding the present value of the debt payment due in 30 months. Since we're finding the present value now (i.e., at time $ t = 0 $), the calculation is:

\[
PV_{\text{now}} = \frac{9000}{\left(1 + \frac{0.063}{4}\right)^{4 \times \frac{30}{12}}}
\]

### Part (b) In 15 months:
For part (b), we're finding the equivalent payment 15 months before the 30-month mark, which is equivalent to finding the present value at 15 months from now:

\[
PV_{\text{15 months}} = \frac{9000}{\left(1 + \frac{0.063}{4}\right)^{4 \times \frac{15}{12}}}
\]

### Part (c) In 30 months:
For part (c), the payment is due in 30 months, so the equivalent payment is exactly the same as the future value:

\[
PV_{\text{30 months}} = 9000
\]

### Part (d) In 42 months:
For part (d), 12 months after the payment is due, we would be looking for the future value of $9000 over an additional 12 months:

\[
FV_{\text{42 months}} = 9000 \times \left(1 + \frac{0.063}{4}\right)^{4 \times \frac{12}{12}}
\]

Let's calculate the amounts for parts (a) through (d).Here are the equivalent debt payments:

- **(a) Now:** The equivalent debt payment is **$7,697.93**.
- **(b) In 15 months:** The equivalent debt payment is **$8,323.55**.
- **(c) In 30 months:** The equivalent debt payment is **$9,000** (since this is the due date).
- **(d) In 42 months:** The equivalent debt payment is **$9,580.54**.

Would you like more details or have any further questions? Here are some related questions to consider:

1. How does changing the interest rate affect the equivalent payments?
2. What if the compounding frequency was monthly instead of quarterly?
3. How would the equivalent payments change if the due date was in 24 months?
4. What is the future value at 15 months from now?
5. How does the time value of money impact financial decisions?
6. Can you find the present value of the debt if paid in installments?
7. What if the payment is made early at 10 months?
8. How would the equivalent payment change if the interest rate was 5% instead of 6.3%?

**Tip:** Understanding the time value of money is crucial for making informed financial decisions.

Explore how to calculate equivalent debt payments at different times using compound interest. Understand the time value of money with a detailed explanation and step-by-step calculations.

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