You have a loan outstanding. It requires making 

ninenine 

annual payments of 

​$2 comma 0002,000 

each at the end of the next 

ninenine 

years. Your bank has offered to allow you to skip making the next 

eighteight 

payments in lieu of making one large payment at the end of the​ loan's term in 

ninenine 

years. If the interest rate on the loan is 

8 %8%​, 

what final payment will the bank require you to make so that it is indifferent to the two forms of​ payment?

Question content area bottom

Part 1

The final payment the bank will require you to make is 

​$enter your response here. 

 ​(Round to the nearest​ cent.)

To calculate the final payment the bank will require at the end of 9 years, we need to understand the time value of money. Specifically, we are comparing two payment plans:

1. **Original plan**: The loan requires 9 annual payments of $2,000 at the end of each year.
2. **New plan**: The bank allows you to skip the first 8 payments and instead make a larger lump sum payment at the end of 9 years.

The bank will be indifferent between the two payment plans if the present value (PV) of both payment plans is the same. The present value of a cash flow is calculated by discounting future payments at the given interest rate (8%).

### Step 1: Calculate the present value of the original payment plan

The present value of the original 9 payments of $2,000 at the end of each year can be calculated using the formula for the present value of an annuity:

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

Where:
- $ P $ = $2,000 (the annual payment)
- $ r $ = 0.08 (the interest rate)
- $ n $ = 9 (the number of years)

Let's calculate the present value of the original plan:

\[
PV_{\text{original}} = 2000 \times \frac{1 - (1 + 0.08)^{-9}}{0.08}
\]

### Step 2: Calculate the final lump sum payment

For the second plan, we will need to calculate what final payment $ F $ at the end of year 9 would be equivalent to the present value of the original plan.

The formula to calculate the future value $ F $ from a present value $ PV $ is:

\[
F = PV \times (1 + r)^n
\]

Once we have the present value of the original plan, we will use this formula to determine the future value (final payment) after 9 years.

Let me calculate these values.The present value of the original 9 annual payments is approximately \$12,493.78. To make the bank indifferent between the two options, you would need to make a final lump sum payment of approximately **\$24,975.12** at the end of 9 years.

Would you like further details on this calculation or have any questions?

Here are 5 related questions to expand on this topic:
1. How does the interest rate affect the final lump sum payment?
2. What is the formula for calculating the present value of an annuity?
3. Can the future lump sum payment be smaller if the interest rate decreases?
4. How would the final payment change if the loan term was extended?
5. What are some other ways to calculate loan amortization?

**Tip**: Understanding the time value of money is crucial in financial decision-making, especially for loans and investments.

You have a loan outstanding. It requires making nine annual payments of $2,000 each at the end of the next nine years. Your bank has offered to allow you to skip making the next eight payments in lieu of making one large payment at the end of the loan's term in nine years. If the interest rate on the loan is 8%, what final payment will the bank require you to make so that it is indifferent to the two forms of payment? (Round to the nearest cent.)

Solve a loan problem where nine annual payments of $2,000 are replaced by one final payment after skipping eight payments. Learn how to use the Present Value of an Annuity and Future Value formulas to find the final payment required at 8% interest. Suitable for finance 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