On January 1, you will purchase a new car. The automobile dealer will allow you to make increasing annual December 31 payments over the following four years. The amounts of these payments are $4,000: $4,500: $5,000: $6,000. On the same January 1, your mother will lend you just enough money to enable you to meet these payments. Interest rates are expected to be 8 percent for the next five years. Assuming that you can earn annual compounding interest by depositing the loan from your mother in a bank, what is the minimum amount your mother must loan you to enable you to meet the car payments?

Question

On January 1, you will purchase a new car. The automobile dealer will allow you to make increasing annual December 31 payments over the following four years. The amounts of these payments are $4,000: $4,500: $5,000: $6,000. On the same January 1, your mother will lend you just enough money to enable you to meet these payments. Interest rates are expected to be 8 percent for the next five years. Assuming that you can earn annual compounding interest by depositing the loan from your mother in a bank, what is the minimum amount your mother must loan you to enable you to meet the car payments?

Answer 1

To calculate the minimum amount your mother must loan you, we need to find the present value of the car payments and account for interest earned on the loan from your mother.

Here's how you can calculate it step by step:

1. Calculate the present value of the car payments:
- The car payments are $4,000, $4,500, $5,000, and $6,000 over four years.
- Assuming an 8% interest rate, you need to discount each payment to its present value.
- Use the present value formula: PV = FV / (1 + r)^n, where PV is the present value, FV is the future value or payment, r is the interest rate, and n is the number of periods.
- Calculate the present value for each payment:
- PV1 = $4,000 / (1 + 0.08)^1
- PV2 = $4,500 / (1 + 0.08)^2
- PV3 = $5,000 / (1 + 0.08)^3
- PV4 = $6,000 / (1 + 0.08)^4

2. Calculate the total present value of the car payments:
- Total PV = PV1 + PV2 + PV3 + PV4

3. Account for interest earned on the loan from your mother:
- To meet the car payments, you will need to earn enough interest on the loan from your mother.
- The interest rate is 8%, and the loan period is five years (including the four years for the car payments).
- Use the compound interest formula: A = P(1 + r/n)^(nt), where A is the future value (amount needed to meet car payments), P is the principal (loan amount), r is the interest rate, n is the compounding frequency per year, and t is the number of years.
- Rearrange the formula to solve for the loan amount (P):
- P = A / (1 + r/n)^(nt)

4. Substitute the future value (total PV from step 2) and solve for the loan amount (P):
- Loan amount (P) = Total PV / (1 + 0.08/1)^(1 * 5)

5. The loan amount that your mother must provide is the solution from step 4.

By following these steps, you will find the minimum amount your mother must loan you to enable you to meet the car payments considering the interest earned on the loan from your mother.