Question

A. Sebastian checks Daniel’s credit rating and determines that he will qualify for a 6% auto loan, and they agree that his trade-in is worth $6,000. The cost of the car is $22,000. If he is planning to finance the loan for 5 years, how much does he have to pay every month?

B. Repeat for if he is planning to finance the loan for 3 years.

Answer 1

Loan taken = $22000 - $6000 = $16000

(a) Here, the payments will be same every month, so it is an annuity. We will use the present value of annuity formula to calculate the monthly payments:

PVA = P * (1 - (1 + r)-n / r)

where, PVA = Present value of annuity = $16000, P is the periodical amount, r is the rate of interest = 6%. Monthly rate = 6% / 12 = 0.5% and n is the time period = 5 * 12 = 60 months

Now, putting these values in the above formula, we get,

$16000 = P * (1 - (1 + 0.5%)-60 / 0.5%)

$16000 = P * (1 - ( 1+ 0.005)-60 / 0.005)

$16000 = P * (1 - ( 1.005)-60 / 0.005)

$16000 = P * (1 - 0.74137219624) / 0.005)

$16000 = P * (0.25862780375 / 0.005)

$16000 = P * 51.7255607511

P = $16000 / 51.7255607511

P = $309.32

So, monthly payments are for $309.32

(b) Here, the payments will be same every month, so it is an annuity. We will use the present value of annuity formula to calculate the monthly payments:

PVA = P * (1 - (1 + r)-n / r)

where, PVA = Present value of annuity = $16000, P is the periodical amount, r is the rate of interest = 6%. Monthly rate = 6% / 12 = 0.5% and n is the time period = 3 * 12 = 36 months

Now, putting these values in the above formula, we get,

$16000 = P * (1 - (1 + 0.5%)-36 / 0.5%)

$16000 = P * (1 - ( 1+ 0.005)-36 / 0.005)

$16000 = P * (1 - ( 1.005)-36 / 0.005)

$16000 = P * (1 - 0.8356449188) / 0.005)

$16000 = P * (0.16435508119 / 0.005)

$16000 = P * 32.8710162393

P = $16000 / 32.8710162393

P = $486.75

So, monthly payments are for $486.75