You have just taken out an installment loan totaling $50,000 to purchase a 1969 Pontiac Lemans convertible at 4% interest rate. The loan will be repaid over three years with equal annual payments. What is the amount of the equal annual payment and how much will your total payments throughout the loan exceed the $50,000 principal amount?
You have just taken out an installment loan totaling $50,000 to purchase a 1969 Pontiac Lemans convertible at 4% interest rate. The loan will be repaid over three years with equal annual payments. What is the amount of the equal annual payment and how much will your total payments throughout the loan exceed the $50,000 principal amount?
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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Question
You have just taken out an installment loan totaling $50,000 to purchase a 1969 Pontiac Lemans convertible at 4% interest rate. The loan will be repaid over three years with equal annual payments. What is the amount of the equal annual payment and how much will your total payments throughout the loan exceed the $50,000 principal amount?
![### Installment Loan Calculation Example
You have just taken out an installment loan totaling $50,000 to purchase a 1969 Pontiac Lemans convertible at 4% interest rate. The loan will be repaid over three years with equal annual payments.
**Problem Statement:**
- What is the amount of the equal annual payment?
- How much will your total payments throughout the loan exceed the $50,000 principal amount?
**Solution Steps:**
1. **Determine the Annual Payment Amount:**
To find the annual payment amount for an installment loan, you can use the formula for an installment loan amortization:
\[
P = \frac{r(PV)}{1 - (1 + r)^{-n}}
\]
- \( P \) is the annual payment
- \( PV \) is the present value of the loan (principal = $50,000)
- \( r \) is the annual interest rate (4% or 0.04)
- \( n \) is the number of years (3 years)
2. **Calculate the Total Payment Amount:**
Multiply the annual payment amount by the number of payments (3 years).
3. **Calculate the Total Interest Paid:**
Subtract the principal from the total payments to find out how much the total payments exceed the $50,000 principal amount.
### Step-by-Step Calculation:
1. **Calculate the Annual Payment Amount:**
\[
P = \frac{0.04 \times 50000}{1 - (1 + 0.04)^{-3}}
\]
Plugging in the numbers:
\[
P = \frac{2000}{1 - (1.04)^{-3}}
\]
\[
P = \frac{2000}{1 - 0.889}
\]
\[
P = \frac{2000}{0.111}
\]
\[
P \approx 18,018.02
\]
2. **Calculate the Total Payment Amount:**
\[
\text{Total Payment} = P \times n
\]
\[
\text{Total Payment} \approx 18,018.02 \times 3
\]
\[
\text{Total Payment} \approx 54,054.06
\]
3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d6ce828-7421-4579-a347-7b809a44673d%2F46574bb1-8a07-476a-8019-6762b428b2cb%2Fojo2tyb_processed.png&w=3840&q=75)
Transcribed Image Text:### Installment Loan Calculation Example
You have just taken out an installment loan totaling $50,000 to purchase a 1969 Pontiac Lemans convertible at 4% interest rate. The loan will be repaid over three years with equal annual payments.
**Problem Statement:**
- What is the amount of the equal annual payment?
- How much will your total payments throughout the loan exceed the $50,000 principal amount?
**Solution Steps:**
1. **Determine the Annual Payment Amount:**
To find the annual payment amount for an installment loan, you can use the formula for an installment loan amortization:
\[
P = \frac{r(PV)}{1 - (1 + r)^{-n}}
\]
- \( P \) is the annual payment
- \( PV \) is the present value of the loan (principal = $50,000)
- \( r \) is the annual interest rate (4% or 0.04)
- \( n \) is the number of years (3 years)
2. **Calculate the Total Payment Amount:**
Multiply the annual payment amount by the number of payments (3 years).
3. **Calculate the Total Interest Paid:**
Subtract the principal from the total payments to find out how much the total payments exceed the $50,000 principal amount.
### Step-by-Step Calculation:
1. **Calculate the Annual Payment Amount:**
\[
P = \frac{0.04 \times 50000}{1 - (1 + 0.04)^{-3}}
\]
Plugging in the numbers:
\[
P = \frac{2000}{1 - (1.04)^{-3}}
\]
\[
P = \frac{2000}{1 - 0.889}
\]
\[
P = \frac{2000}{0.111}
\]
\[
P \approx 18,018.02
\]
2. **Calculate the Total Payment Amount:**
\[
\text{Total Payment} = P \times n
\]
\[
\text{Total Payment} \approx 18,018.02 \times 3
\]
\[
\text{Total Payment} \approx 54,054.06
\]
3.
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