Question

The Smith family just took out a variable-rate mortgage on their new home. The mortgage value is $100,000, the term is 30 years, and initially the interest rate is 8%. The interest rate is guaranteed for 5 years, after which time the rate will be adjusted according to prevailing rates. The new rate can be applied to their loan either by changing the payment amount or by changing the length of the mortgage.

(a) What is the original yearly mortgage payment? (Assume payments are yearly.)
(b) What will be the mortgage balance after 5 years?
(c) If the interest rate on the mortgage changes to 9% after 5 years, what will be the new yearly payment that keeps the termination time the same?
(d) Under the interest change in (c), what will be the new term if the payments remain the same?

Answer 1

(a) $8,882.74

(b) $94,821.32

(c) $9,653.40

(d) 43 years

Explanation:

These calculations are best done using a financial calculator or app, or a spreadsheet. The attachment shows the results using a spreadsheet.

(a)

The annual payment amount is given by the spreadsheet formula ...

=PMT(8%,30,100000)

The yearly payment will be $8,882.74.

__

(b)

The remaining balance on the loan is its future value after 5 payments. That is given by the spreadsheet formula ...

=-FV(8%,5,-8882.74,100000)

The balance after 5 years will be $94,821.32.

__

(c)

The payment on $94,821.32 at 9% amortized over the remaining 25 years of the original loan is given by ...

=PMT(9%,25,94,821.32)

The new yearly payment will be $9,653.40.

__

(d)

The time required to pay off the remaining $94,821.32 with the original payment amount of $8,882.74 and an interest rate of 9% is given by the spreadsheet formula ...

=NPER(9%,-8882.74,94,821.32)

It will take about 37.6 more payments if the payments remain the same. The final payment will be made at the end of the 43rd year.

_____

Additional comment

When using spreadsheet formulas, you need to pay attention to signs. Generally, payments will have a negative sign, and amounts received will have a positive sign. Financial calculators will often have a similar convention. Loan apps may be different.