A family has a $121,857, 15-year mortgage at 7.8% compounded monthly. (A) Find the monthly payment and the total interest paid. (B) Suppose the family decides to add an extra $100 to its mortgage payment each month starting with the very first payment. How long will it take the family to pay off the mortgage? How much interest will the family save?
A family has a $121,857, 15-year mortgage at 7.8% compounded monthly. (A) Find the monthly payment and the total interest paid. (B) Suppose the family decides to add an extra $100 to its mortgage payment each month starting with the very first payment. How long will it take the family to pay off the mortgage? How much interest will the family save?
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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![### Mortgage Calculation Problem
#### Problem Statement:
A family has a $121,857, 15-year mortgage at 7.8% compounded monthly.
**(A)** Find the monthly payment and the total interest paid.
**(B)** Suppose the family decides to add an extra $100 to its mortgage payment each month starting with the very first payment.
- How long will it take the family to pay off the mortgage?
- How much interest will the family save?
#### Solution:
**(A) Monthly payment:**
\[
\text{Monthly payment:} \, \$ \_\_\_\_\_\_ (\text{Round to two decimal places}.)
\]
To solve part (A), you need to use the formula for the monthly payment of a fixed-rate mortgage:
\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( M \) is the monthly payment
- \( P \) is the loan principal (amount borrowed)
- \( r \) is the monthly interest rate (annual rate divided by 12)
- \( n \) is the number of payments (loan term in months)
For a 15-year mortgage:
- Annual interest rate \( = 7.8\% \), so the monthly interest rate \( r = \frac{7.8}{12 \cdot 100} = 0.0065 \)
- Loan term \( = 15 \text{ years} \cdot 12 = 180 \text{ months} \)
- Principal \( P = \$121,857 \)
Substitute these values into the formula to find \( M \).
**(B)**
1. To find out how long it will take to pay off the mortgage with an additional $100 each month, you will need to adjust the payment amount in the amortization formula.
2. The interest saved can be calculated by comparing the total interest paid with and without the extra $100 payment.
Note: A detailed solution would involve iterative calculations or the use of an amortization schedule to find the precise time and interest savings.
---
#### Explanation
Graphs and diagrams (if applicable):
* **Amortization Schedule:** This graph typically shows the breakdown of each mortgage payment into principal and interest over time. The graph will illustrate how the principal diminishes and how the interest component decreases more](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb3f5949-3440-4952-a068-9f8891699419%2F8dd93fcb-bb96-40bb-8105-597ffda09f13%2F54u1j1b.jpeg&w=3840&q=75)
Transcribed Image Text:### Mortgage Calculation Problem
#### Problem Statement:
A family has a $121,857, 15-year mortgage at 7.8% compounded monthly.
**(A)** Find the monthly payment and the total interest paid.
**(B)** Suppose the family decides to add an extra $100 to its mortgage payment each month starting with the very first payment.
- How long will it take the family to pay off the mortgage?
- How much interest will the family save?
#### Solution:
**(A) Monthly payment:**
\[
\text{Monthly payment:} \, \$ \_\_\_\_\_\_ (\text{Round to two decimal places}.)
\]
To solve part (A), you need to use the formula for the monthly payment of a fixed-rate mortgage:
\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( M \) is the monthly payment
- \( P \) is the loan principal (amount borrowed)
- \( r \) is the monthly interest rate (annual rate divided by 12)
- \( n \) is the number of payments (loan term in months)
For a 15-year mortgage:
- Annual interest rate \( = 7.8\% \), so the monthly interest rate \( r = \frac{7.8}{12 \cdot 100} = 0.0065 \)
- Loan term \( = 15 \text{ years} \cdot 12 = 180 \text{ months} \)
- Principal \( P = \$121,857 \)
Substitute these values into the formula to find \( M \).
**(B)**
1. To find out how long it will take to pay off the mortgage with an additional $100 each month, you will need to adjust the payment amount in the amortization formula.
2. The interest saved can be calculated by comparing the total interest paid with and without the extra $100 payment.
Note: A detailed solution would involve iterative calculations or the use of an amortization schedule to find the precise time and interest savings.
---
#### Explanation
Graphs and diagrams (if applicable):
* **Amortization Schedule:** This graph typically shows the breakdown of each mortgage payment into principal and interest over time. The graph will illustrate how the principal diminishes and how the interest component decreases more
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