Question

Suppose you want to purchase a home for $425,000 with a 30-year mortgage at 5.44% interest. Suppose also that you can put down 30%. What are the monthly

payments? (Round your answer to the nearest cent.)
$
What is the total amount paid for principal and interest? (Round your answer to the nearest cent.)
$
What is the amount saved if this home is financed for 15 years instead of for 30 years? (Round your answer to the nearest cent.)
$
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Answer 1

a) payment: $1677.99

b) total repaid: $604,076.40

c) saved: $168,231.60

Explanation:

The amortization formula can be used to find the monthly payment on the loan. The total repaid will be the sum of payments made. The amount saved will be the difference between the total repaid amounts for the different loans.

Amortization formula

The payment A on a loan of principal P at annual rate r for t years is given by ...

A = P(r/12)/(1 -(1 +r/12)^(-12t))

Application

The loan of interest is for $425,000 × (1 -30%) = $297,500. Then the payments on a 30 year loan are ...

A = $297,500(0.0544/12)/(1 -(1 +0.0544/12)^-360) ≈ $1677.99

For a 15-year loan, the payments are ...

A = $297,500(0.0544/12)/(1 -(1 +0.0544/12)^-180) ≈ $2421.36

a)

The above calculation shows the monthly payment is $1677.99.

b)

The total repaid is the sum of 360 monthly payments:

360×$1677.99 = $604,076.40

c)

The total repaid is the sum of 180 monthly payments:

180×$2421.36 = $435,844.80

and the savings is ...

$604,076.40 -435,844.80 = $168,231.60

__

Additional comment

We are asked to round to the nearest cent. The payment amounts here are accurately calculated to that precision. The total repaid on an actual loan will be slightly different, because the final payment will be slightly different.

The 30-year mortgage will have a slightly lower final payment because the monthly amount is rounded up. The 15-year mortgage will have a slightly higher final payment because the monthly amount is rounded down.

Overall, this means the totals of payments shown here, and their difference, vary slightly from the reality of loans of this kind. (They're probably not accurate to the nearest cent.)

Even using the "exact" payment value may not give the actual loan values. This is because the actual amortization schedule is rounded to the nearest cent each month. The cumulative effect of this rounding seems to be calculable only by figuring the 360 (or 180) actual monthly balance amounts.

We have calculated the answers in the manner we suppose is expected. Producing an actual amortization spreadsheet is a bit more work.