Use PMT equals StartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis Over left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt right bracket EndFraction

Question

to determine the regular payment​ amount, rounded to the nearest dollar. In terms of paying less in​ interest, which is more economical for a ​$170 comma 000

​mortgage: a​ 30-year fixed-rate at 7.5
​%
or a​ 20-year fixed-rate at 7
​%?
How much is saved in​ interest?

Answer 1

To find out which option is more economical, we need to compute the regular payment amount for both mortgages using the provided formula and then find out how much interest is paid over the term of the loan.

1. For a 30-year fixed-rate mortgage at 7.5 %:

In this case:
P = $170,000
r = 7.5 / 100 = 0.075 (annual interest rate in decimal)
n = 12 (payments per year)
t = 30 (number of years)

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

PMT = 170,000 * ((0.075/12) / (1 - (1 + (0.075/12))^(-12*30)))

PMT = 1,187.86

Round to the nearest dollar: PMT = $1,188

Total amount paid over 30 years = PMT * n*t = 1,188 * 12 * 30 = $427,680
Total interest paid = Total amount paid - Initial loan amount = $427,680 - $170,000 = $257,680

2. For a 20-year fixed-rate mortgage at 7 %:

In this case:
P = $170,000
r = 7 / 100 = 0.07 (annual interest rate in decimal)
n = 12 (payments per year)
t = 20 (number of years)

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

PMT = 170,000 * ((0.07/12) / (1 - (1 + (0.07/12))^(-12*20)))

PMT = 1,316.36

Round to the nearest dollar: PMT = $1,316

Total amount paid over 20 years = PMT * n*t = 1,316 * 12 * 20 = $315,840
Total interest paid = Total amount paid - Initial loan amount = $315,840 - $170,000 = $145,840

Comparing the interest paid for both loans, the 20-year fixed-rate mortgage at 7% is more economical as it results in less interest being paid ($145,840) compared to the 30-year fixed-rate mortgage at 7.5% where the total interest paid is $257,680.

The amount saved in interest with the 20-year fixed-rate mortgage, compared to the 30-year fixed-rate mortgage, is $257,680 - $145,840 = $111,840.