A loan is taken out at j₁₂ = 18% and is paid back with monthly payments of $279.00,plus a smaller concluding payment.With one month left on the loan,the outstanding balance is $268.66.What is the size of the final loan payment?

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First, we need to calculate the original loan amount. Using the formula for present value of an ordinary annuity, we get:

PV = PMT x (1 - (1 + r/n)^(-nt)) / (r/n)
PV = $279 x (1 - (1 + 0.18/12)^(-12*5)) / (0.18/12)
PV = $279 x 3.4337
PV = $958.69

So the original loan amount was $958.69.

Next, we need to calculate the remaining balance after 59 payments. Using the formula for remaining balance of an ordinary annuity, we get:

B = PV x (1 + r/n)^(n*t) - PMT x ((1 + r/n)^(n*t) - 1) / (r/n)
B = $958.69 x (1 + 0.18/12)^(12*5) - $279 x ((1 + 0.18/12)^(12*5) - 1) / (0.18/12)
B = $268.66

So the remaining balance is indeed $268.66.

To calculate the final payment, we can use the formula for a single payment present value:

PV = FV / (1 + r/n)^nt

Where PV is the present value of the final payment, FV is the future value (which we want to find), r is the interest rate, n is the number of compounding periods per year, and t is the number of years. We know that there is one month left on the loan, so t = 1/12. We also know that the outstanding balance is $268.66, so that is the future value we input into the formula.

$268.66 = FV / (1 + 0.18/12)^(1/12)
$268.66 x 1.015 = FV
FV = $272.69

Therefore, the final loan payment is $272.69.

A Loan Is Taken Out at J12 = 18% and Is

A Loan Is Taken Out at J₁₂ = 18% and Is