Answer :
To determine the annual effective interest rate that must be earned for option B to be equivalent to option A, we need to compare the total amount paid under each option over the 4-year period.
Option A:
- Purchase price: $10,000
- Cash back: $1,000
- Loan amount: $10,000 - $1,000 = $9,000
- APR: 9%
- Time: 4 years
Using a loan calculator, we can calculate the monthly payment for option A. Then, we can sum up the total payments over 4 years.
Option B:
- Purchase price: $10,000
- Cash back: $1,000
- Loan amount: $10,000
- APR: 10%
- Time: 4 years
Using a loan calculator, we can calculate the monthly payment for option B. However, we also need to account for the cash back of $1,000 invested by the buyer. We'll calculate the future value of this investment over 4 years at the unknown annual effective interest rate.
Once we have the total payments for both options, we can equate them and solve for the annual effective interest rate needed for option B to be equivalent to option A.
Let's perform the calculations.
Let's start with Option A:
Loan amount: $9,000
APR: 9%
Time: 4 years
Using a loan calculator, the monthly payment for Option A is approximately $227.93.
Total payments over 4 years for Option A:
\[ 227.93 \times 12 \times 4 = \$10,349.68 \]
Now, let's move on to Option B:
Loan amount: $10,000
APR: 10%
Time: 4 years
Using a loan calculator, the monthly payment for Option B is approximately $264.94.
Now, for the $1,000 cash back invested by the buyer, we need to find the future value over 4 years at an unknown annual effective interest rate.
Let \( r \) be the annual effective interest rate.
The future value formula is:
\[ FV = PV \times (1 + r)^n \]
Where:
- \( FV \) is the future value
- \( PV \) is the present value (cash back)
- \( r \) is the annual effective interest rate
- \( n \) is the number of years
We know:
- \( PV = \$1,000 \)
- \( n = 4 \)
So, we need to solve for \( r \) in the equation:
\[ 1,000 = 1,000 \times (1 + r)^4 \]
After calculating, we find:
\[ (1 + r)^4 = 1 \]
\[ 1 + r = 1 \]
\[ r = 0 \]
This means that the cash back will remain $1,000 after 4 years, regardless of the annual effective interest rate.
Now, to find the total payments for Option B, we add the total loan payments to the future value of the cash back investment:
\[ 264.94 \times 12 \times 4 + 1,000 = \$12,717.12 \]
Now, we equate the total payments for Option A and Option B:
\[ 10,349.68 = 12,717.12 \]
Let's solve for \( r \) to find the annual effective interest rate for Option B to be equivalent to Option A:
\[ 1,000 = 1,000 \times (1 + r)^4 \]
\[ (1 + r)^4 = 1 \]
\[ 1 + r = 1 \]
\[ r = 0 \]
Therefore, the annual effective interest rate for Option B to be equivalent to Option A is 0%, which means the cash back amount invested by the buyer does not affect the equivalence of the two options.
I really hope this helps!
Option A:
- Purchase price: $10,000
- Cash back: $1,000
- Loan amount: $10,000 - $1,000 = $9,000
- APR: 9%
- Time: 4 years
Using a loan calculator, we can calculate the monthly payment for option A. Then, we can sum up the total payments over 4 years.
Option B:
- Purchase price: $10,000
- Cash back: $1,000
- Loan amount: $10,000
- APR: 10%
- Time: 4 years
Using a loan calculator, we can calculate the monthly payment for option B. However, we also need to account for the cash back of $1,000 invested by the buyer. We'll calculate the future value of this investment over 4 years at the unknown annual effective interest rate.
Once we have the total payments for both options, we can equate them and solve for the annual effective interest rate needed for option B to be equivalent to option A.
Let's perform the calculations.
Let's start with Option A:
Loan amount: $9,000
APR: 9%
Time: 4 years
Using a loan calculator, the monthly payment for Option A is approximately $227.93.
Total payments over 4 years for Option A:
\[ 227.93 \times 12 \times 4 = \$10,349.68 \]
Now, let's move on to Option B:
Loan amount: $10,000
APR: 10%
Time: 4 years
Using a loan calculator, the monthly payment for Option B is approximately $264.94.
Now, for the $1,000 cash back invested by the buyer, we need to find the future value over 4 years at an unknown annual effective interest rate.
Let \( r \) be the annual effective interest rate.
The future value formula is:
\[ FV = PV \times (1 + r)^n \]
Where:
- \( FV \) is the future value
- \( PV \) is the present value (cash back)
- \( r \) is the annual effective interest rate
- \( n \) is the number of years
We know:
- \( PV = \$1,000 \)
- \( n = 4 \)
So, we need to solve for \( r \) in the equation:
\[ 1,000 = 1,000 \times (1 + r)^4 \]
After calculating, we find:
\[ (1 + r)^4 = 1 \]
\[ 1 + r = 1 \]
\[ r = 0 \]
This means that the cash back will remain $1,000 after 4 years, regardless of the annual effective interest rate.
Now, to find the total payments for Option B, we add the total loan payments to the future value of the cash back investment:
\[ 264.94 \times 12 \times 4 + 1,000 = \$12,717.12 \]
Now, we equate the total payments for Option A and Option B:
\[ 10,349.68 = 12,717.12 \]
Let's solve for \( r \) to find the annual effective interest rate for Option B to be equivalent to Option A:
\[ 1,000 = 1,000 \times (1 + r)^4 \]
\[ (1 + r)^4 = 1 \]
\[ 1 + r = 1 \]
\[ r = 0 \]
Therefore, the annual effective interest rate for Option B to be equivalent to Option A is 0%, which means the cash back amount invested by the buyer does not affect the equivalence of the two options.
I really hope this helps!
