Question
Derek decides to buy a new car. The dealership offers him a choice of paying $571.00 per month for 5 years (with the first payment due next month) or paying some $28,066.00 today. He can borrow money from his bank to buy the car. What interest rate makes him indifferent between the two options?
Answers
GPT-4o mini
To find the interest rate that makes Derek indifferent between the two payment options, we need to equate the present value of the monthly payments to the upfront payment.
### Step 1: Calculate the Present Value of Monthly Payments
The present value (PV) of an annuity formula is:
\[
PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)
\]
Where:
- \( PMT = 571 \) (monthly payment)
- \( r \) = monthly interest rate (annual interest rate divided by 12)
- \( n = 5 \times 12 = 60 \) (total number of payments)
We need to find \( r \) such that the present value of the annuity equals the upfront payment of $28,066. This gives us the equation:
\[
571 \times \left(\frac{1 - (1 + r)^{-60}}{r}\right) = 28066
\]
### Step 2: Rearranging the Equation
We rearrange the equation to isolate the annuity factor:
\[
\frac{1 - (1 + r)^{-60}}{r} = \frac{28066}{571}
\]
Calculating the right side:
\[
\frac{28066}{571} \approx 49.073
\]
So, we set up the equation:
\[
\frac{1 - (1 + r)^{-60}}{r} = 49.073
\]
### Step 3: Numerical Method to Solve for \( r \)
This is a non-linear equation in \( r \) and does not have a simple algebraic solution, so we will solve it numerically.
Using numerical methods (such as trial and error, goal seek in Excel, or a financial calculator), we can approximate \( r \).
### Step 4: Use Trial and Error to Find \( r \)
Let's try several values for \( r \):
1. **Assume: \( r = 0.005 \) (0.5% monthly)**
\[
\frac{1 - (1 + 0.005)^{-60}}{0.005} \approx 49.773
\]
2. **Assume: \( r = 0.006 \) (0.6% monthly)**
\[
\frac{1 - (1 + 0.006)^{-60}}{0.006} \approx 49.426
\]
Continuing this process (or using computational tools like Excel's Solver):
After continued trials, we narrow it down to:
- For \( r = 0.0055 \):
\[
\frac{1 - (1 + 0.0055)^{-60}}{0.0055} \approx 49.067
\]
Thus, when \( r \approx 0.0055 \) or **0.55% monthly** is very close to our required value.
### Step 5: Convert Monthly Rate to Annual Rate
To find the annual interest rate, multiply the monthly interest rate by 12:
\[
Annual\ Rate = 0.0055 \times 12 \approx 0.066 \text{ or } 6.6\%
\]
### Conclusion
The interest rate that makes Derek indifferent between the two options is approximately **6.6% annual interest rate**.
### Step 1: Calculate the Present Value of Monthly Payments
The present value (PV) of an annuity formula is:
\[
PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)
\]
Where:
- \( PMT = 571 \) (monthly payment)
- \( r \) = monthly interest rate (annual interest rate divided by 12)
- \( n = 5 \times 12 = 60 \) (total number of payments)
We need to find \( r \) such that the present value of the annuity equals the upfront payment of $28,066. This gives us the equation:
\[
571 \times \left(\frac{1 - (1 + r)^{-60}}{r}\right) = 28066
\]
### Step 2: Rearranging the Equation
We rearrange the equation to isolate the annuity factor:
\[
\frac{1 - (1 + r)^{-60}}{r} = \frac{28066}{571}
\]
Calculating the right side:
\[
\frac{28066}{571} \approx 49.073
\]
So, we set up the equation:
\[
\frac{1 - (1 + r)^{-60}}{r} = 49.073
\]
### Step 3: Numerical Method to Solve for \( r \)
This is a non-linear equation in \( r \) and does not have a simple algebraic solution, so we will solve it numerically.
Using numerical methods (such as trial and error, goal seek in Excel, or a financial calculator), we can approximate \( r \).
### Step 4: Use Trial and Error to Find \( r \)
Let's try several values for \( r \):
1. **Assume: \( r = 0.005 \) (0.5% monthly)**
\[
\frac{1 - (1 + 0.005)^{-60}}{0.005} \approx 49.773
\]
2. **Assume: \( r = 0.006 \) (0.6% monthly)**
\[
\frac{1 - (1 + 0.006)^{-60}}{0.006} \approx 49.426
\]
Continuing this process (or using computational tools like Excel's Solver):
After continued trials, we narrow it down to:
- For \( r = 0.0055 \):
\[
\frac{1 - (1 + 0.0055)^{-60}}{0.0055} \approx 49.067
\]
Thus, when \( r \approx 0.0055 \) or **0.55% monthly** is very close to our required value.
### Step 5: Convert Monthly Rate to Annual Rate
To find the annual interest rate, multiply the monthly interest rate by 12:
\[
Annual\ Rate = 0.0055 \times 12 \approx 0.066 \text{ or } 6.6\%
\]
### Conclusion
The interest rate that makes Derek indifferent between the two options is approximately **6.6% annual interest rate**.