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You see a used sporty car that you would like to own. It costs $9,000 and you would pay 7.2% interest, compounded monthly and finance for 3 years. How much will you loan balance be after the first payment?

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sebastiansierraocm

Answer:

$24,705.8

Explanation:

To find the answer, we will use the present value of an annuity formula:

PV = A (1 - (1 + I)^-n / i

Where:

  • PV = Present value of the investment (in thise case, the cost of the car)
  • A = Value of the annuity (the monthly payments)
  • i = Interest Rate
  • n = number of compounding periods

The monthly payments are an annuity: they are periodic, fall under the same interest rate, and have the same value, therefore, if we find the value of the annuity, we will find the value of the first monthly payment at the same time (both things are the same):

Plugging the amounts into the formula we obtain:

9,000 = A ( 1 - (1 + 0.072)^-36 / 0.072

9,000 = A (12.75)

9,000 / 12.75 = A

705.88 = A

Now, to find the full value of the loan, we multiply the annuity value for 36, because that value will be paid 36 times until the loan is completed:

Full value of the loan = 705.88 x 36

                                   = 25,411.68

Finally, to find the loan balance after the first payment, we take the full value of the loan, and substract the value of the annuity from it:

Loan balance after first payment = 25,411.68 - 705.88

                                                      = 24,705.8

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