Question

PLEASE HELPPatti is just about to take out a 20-year house loan for $160,000 at an APR of 8.4%, compounded monthly. She asked the loan officer, "What will the remaining balance be on my loan after 10 years?" Help the loan officer respond to Patti's question. Part I: What will the periodic interest rate of Patti's house loan be?Part II: How many monthly payments will Patti have made once her house loan is paid off?Part III: What will Patti's monthly payment be?Part IV: How many monthly payments will Patti have made after 10 years?Part V: What will the remaining balance be on Patti's loan after 10 years?

Answer 1

1. 0.7% monthly
2. 240 payments for payoff
3. monthly payment $1378.41
4. 120 payments in 10 years
5. balance $111,655.24 after 10 years

Explanation:

For a 20-year loan of $160,000 at 8.4%, you want to know the number of payments after 10 years and for payoff, the monthly interest rate, the monthly payment, and the remaining balance after 10 years.

Part I

The periodic interest rate applied to monthly payments is the annual rate divided by the number of months in a year.

periodic rate = 8.4%/12

periodic rate = 0.7%

Part II

The number of monthly payments made on a 20-year loan is the number of months in 20 years:

months = (20 yr)×(12 mo/yr) = 240 mo

Patti will have made 240 payments to pay off the loan.

Part III

Patti's monthly payment can be found using the amortization formula.

A = Pr/(1 -(1+r)^-n) . . . . Principal P, periodic rate r, n periods

A = 160000(0.007)/(1 -1.007^-240) ≈ 1378.41

Patti's monthly payment will be $1378.41.

Part IV

The number of monthly payments in 10 years is ...

months = (10 yr)×(12 mo/yr) = 120 mo

Patti will have made 120 payments after 10 years.

Part V

The remaining balance can be found using a future value formula.

FV = P((1+r)^n -((1+r)^n -1)/(1 -(1+r)^-N))

where P is the principal value, r is the periodic rate, n is the number of payments made, and N is the number of payments used to compute the payment amount.

Here, we have P=160,000, n=120, N=240, r=0.007, so the remaining balance is computed to be ...

FV = 160000(1.007^120 -(1.007^120 -1)/(1 -1.007^-240)) ≈ 111,655.77

The amount calculated in the attachment is slightly lower, because we used the actual payment amount. That amount is rounded up from the value computed by the amortization formula, so the loan is paid off very slightly faster than the FV formula shows.

The remaining balance is about $111,655.24.

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Additional comment

These calculations are most easily done using a financial calculator or spreadsheet. Results are most accurate when you compute the actual amortization schedule, with interest charges and payment values rounded to cents each month.

The final payment is almost always different from that calculated using a formula. The rounding of intermediate values affects the actual payoff rate, which may be more or less than the rate using high-precision values.

Here, the payment is rounded up, so the loan is paid faster, and the final payment will be less than the usual monthly payment.