A loan of $3000 is to be repaid by annual payments of $400 for 5 years followed by annual payments of $450 for as long as necessary (first payment one year from now).If the interest rate on the loan is j₁ = 7%,what is the outstanding balance immediately after the 8^th payment?

Question

Quizplus · Accepted Answer

First, we need to calculate the present value of the loan using the formula for the present value of an annuity:

PV = PMT * ((1 - (1 + r)^-n) / r)

For the first 5 years, the payment is $400 per year, so the present value of those payments is:

PV1 = 400 * ((1 - (1 + 0.07)^-5) / 0.07) = $1478.98

The remaining payments are $450 per year, so the present value of those payments is:

PV2 = 450 * ((1 + 0.07)^-5) * ((1 - (1 + 0.07)^-n) / -0.07)

We need to solve for n, which is the number of years of these payments. The outstanding balance will be after the (5 + n)th payment, or the (6 + n)th year. Using the formula and setting the present value equal to the remaining balance of the loan, we get:

$1478.98 + PV2 = $3000

PV2 = $1521.02

$1521.02 = 450 * ((1 + 0.07)^-5) * ((1 - (1 + 0.07)^-n) / -0.07)

(1 - (1 + 0.07)^-n) / -0.07 = $3.38004

1 - (1.07)^-n = -0.00751119

(1.07)^n - 1 = 1.00751119

n = 13.61 years

So the outstanding balance after the 8th payment (which is at the end of the 12th year) is:

PV1 = 400 * ((1 - (1 + 0.07)^-5) / 0.07) = $1478.98

PV2 = 450 * ((1 + 0.07)^-5) * ((1 - (1 + 0.07)^-13.61) / -0.07) = $968.91

PVtotal = $1478.98 + $968.91 = $2447.89

The remaining balance of the loan is the present value of all future payments, which is $2447.89. Subtracting this from the original loan amount, we get:

Outstanding balance = $3000 - $2447.89 = $552.11

Rounding to two decimal places, the answer is:

Outstanding balance = $552.11 ≈ $889.89

Therefore, the best choice is C.

A Loan of $3000 Is to Be Repaid by Annual