You just borrowed 90,142. You and your heirs plan to repay this loan by making regular monthly payments of X forever and a special payment of $21,800 in 4 months. The interest rate on the loan is 4.30 percent per month and the first regular payment will be made in 1 month. What is X?

We are tasked with determining the value of $ X $, the regular monthly payment, given the following conditions:

- The loan amount borrowed is \$90,142.
- The repayment includes a special payment of \$21,800 in 4 months.
- The regular monthly payments $ X $ continue indefinitely.
- The interest rate is 4.30% per month.
- The first regular payment is made in 1 month.

### Step-by-Step Solution:

#### 1. **Present Value of Regular Payments (Perpetuity):**

For an infinite series of regular monthly payments starting in one month, the present value $ PV $ of the perpetuity can be computed using the formula:

\[
PV = \frac{X}{r}
\]

where:
- $ X $ is the monthly payment,
- $ r $ is the monthly interest rate, 4.30% or 0.043 in decimal.

\[
PV_{\text{perpetuity}} = \frac{X}{0.043}
\]

#### 2. **Present Value of the Special Payment:**

The special payment of \$21,800 is made in 4 months. We need to discount this amount back to the present value using the formula:

\[
PV_{\text{special}} = \frac{21,800}{(1 + r)^n}
\]

where:
- $ r = 0.043 $,
- $ n = 4 $ (the payment is made in 4 months).

\[
PV_{\text{special}} = \frac{21,800}{(1.043)^4}
\]

#### 3. **Total Present Value:**

The total loan amount $ 90,142 $ is equal to the sum of the present value of the perpetuity and the present value of the special payment:

\[
90,142 = \frac{X}{0.043} + \frac{21,800}{(1.043)^4}
\]

We can solve this equation for $ X $.

Let's compute $ PV_{\text{special}} $ and then solve for $ X $.The regular monthly payment $ X $ is approximately **\$3,084**.

Would you like a detailed explanation of any part of the solution or have further questions? Here are five related questions you might explore:

1. How would the payment change if the interest rate was lower?
2. What if the loan had to be repaid over a fixed number of months instead of indefinitely?
3. How would an increase in the loan amount affect the monthly payments?
4. What would the monthly payment be if the special payment was delayed to 6 months instead of 4?
5. How does a perpetuity differ from other types of loan repayment methods?

**Tip:** The present value of future payments decreases as the payment is further delayed, since interest rates discount the value over time.

Determine the regular monthly payment for a $90,142 loan with perpetual repayments and a special payment of $21,800 after 4 months, using a 4.30% monthly interest rate. This problem involves key finance concepts like perpetuity and present value.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation