Question

Bryan decides to purchase a new car with a price of €14000, but cannot afford the full amount. The car dealership offers two options to finance a loan. Finance option A: A 6 year loan at a nominal annual interest rate of 14% compounded quarterly. No deposit required and repayments are made each quarter

(a) (i) find the repayment made each quarter
(ii) find the total amount paid for the car
(iii) find the interest paid on the loan

Finance option B
A 6 year loan at a nominal annual interest rate of r % compunded monthly. Terms of the loan require a 10% deposit and monthly repayments of $250
(b) (i) find the amount to be borrowed for this option
(ii) find the annual interest rate, r
State which option Bryan should choose. Justify your answer

Answer 1

Bryan's finance options for a \u20ac14,000 car involve calculating quarterly repayments, the total amount paid, and total interest for a compound interest loan (Finance Option A), and determining the loan amount after a deposit and calculating the annual interest rate for a different loan structure (Finance Option B).

Step-by-step explanation:

Given that Bryan needs to finance a \u20ac14,000 car, we analyze the first finance option:

1. Quarterly repayment: To calculate the quarterly payment, we use the formula for annuity payments under compound interest. Since the car loan is for 6 years, with a 14% nominal annual interest rate compounded quarterly, the actual interest rate per quarter is 14%/4 = 3.5%.
2. Total amount paid: The sum of all the quarterly payments made over the 6-year period will provide the total amount paid for the car.
3. Interest paid on the loan: By subtracting the initial loan amount from the total amount paid, we will determine the total interest paid.

For option B:

1. Amount to be borrowed: Since a 10% deposit is required, Bryan will need to finance the remaining 90% of the car's price.
2. Annual interest rate: We need to solve for the annual interest rate r, given the monthly repayments of \u20ac250 and the loan term. This typically involves using the present value of an annuity formula or financial calculators.

To determine which option is better, we will compare the total costs of both options, taking into account the repayment amounts, interest rates, and loan terms.

Answer 2

Bryan should choose Finance option A because it offers a lower total amount paid for the car compared to Finance option B. Finance option A has a total amount paid of €9273.44, while Finance option B requires a larger loan amount and monthly repayments, resulting in a higher total cost for the car.

Step-by-step explanation:

(a) (i) Repayment made each quarter:

First, let's convert the annual interest rate to a quarterly interest rate. Since the interest is compounded quarterly, we divide the annual interest rate by 4:

Quarterly interest rate = Annual interest rate / 4 = 14% / 4 = 3.5%

Next, we calculate the quarterly repayment using the formula for compound interest:

Quarterly repayment = Principal amount * (Quarterly interest rate / (1 - (1 + Quarterly interest rate)^(-n)))

Using the given principal amount of €14000 and a loan term of 6 years (24 quarters), we can substitute these values into the formula to find the quarterly repayment:

Quarterly repayment ≈ €386.06

(ii) Total amount paid for the car:

Total amount paid = Quarterly repayment * Number of quarters

Total amount paid ≈ €386.06 * 24 ≈ €9273.44

(iii) Interest paid on the loan:

Interest paid = Total amount paid - Principal amount = €9273.44 - €14000 ≈ -€4726.56 (negative because this represents the interest paid)

(b) (i) Amount to be borrowed:

First, we need to find the deposit amount. The deposit is 10% of the car's price:

Deposit amount = 10% * €14000 = €1400

The amount to be borrowed is the car's price minus the deposit:

Amount to be borrowed = €14000 - €1400 = €12600

(ii) Annual interest rate, r:

The monthly repayment and loan term don't provide enough information to directly calculate the annual interest rate. Additional information is needed to solve this part of the question.

Which option should Bryan choose and why?

Bryan should choose Finance option A because it offers a lower total amount paid for the car compared to Finance option B. Finance option A has a total amount paid of €9273.44, while Finance option B requires a larger loan amount and monthly repayments, resulting in a higher total cost for the car.