Question

You have agreed to pay off an $8,000 loan in 30 monthly payments of $298.79 per month. The annual interest rate is 9% on the unpaid balance.

(a) How much of the first month’s payment will apply towards reducing the principal of $8,000?

(b) What is the unpaid balance (on the principal) after 12 monthly payments have been made?

Answer 1

According to given information the principal amount is $8000 to be repaid in 30 months with monthly payment of $298.79

The annual rate is 9% per annum

So here P = $8000

Number of months = n = 30

Rate of interest = 9% = 9/100 = 0.09

And compounded monthly so r = 0.09/12 = 0.0075

a) For the first month the interest amount on principal amount

Interest amount = 8000 x 0.0075 = $60

And principal amount in monthly payment = payment amount – interest

                                                                                        = 298.79 – 60 = $238.79

So when first month was done then principal reduced to 8000 – 238.79 = 7761.21

So the principal reduced to $7761.21

The next step is to calculate the outstanding loan balance after 12 payments by calculating the present value of the remaining installments, using the present value of an annuity formula

From the given information we need to find the present value of given annuity

Pmt = Periodic monthly payment = 298.79

i = Mortgage interest rate per period = 0.0075

when calculation for after 12 payments then Number of remaining payments = 30-12 = 18

so number of payments = 18

we can use below formula

PV = Pmt x [(1 - 1 / (1 + i)ⁿ)] / i

PV = 298.79 x [(1 - 1 / (1 + 0.0075)¹⁸)] / (0.0075)

PV = 298.79 x [(1 - 1 / (1.0075)¹⁸)] / (0.0075)

PV = 298.79 x [(1 - 1 / (1.14396)] / (0.0075)

PV = 298.79 x [(1 – 0.87415)] / (0.0075)

PV = 298.79 x [0.12585] / (0.0075)

PV = 37.6027 / 0.0075

PV = 5013.6962 ~ 5013.7

So the principal payment due after 12 payments = $5013.7