Question

Phil Pittman is interested in a fwed-rate mortgage for $300,000. He is undecided whether to choose a 15- or 30-year mortgage. The current mortgage rate is 4.5% for the 15-year mortgage and 5% for the 30-year mortgage (Round your answers to the nearest dolor. Use this tablet, if necessary (a) What are the monthly principal and interest payments (in s) for each loan? 15-year mortgage $ 30-year mortgage $ Yb) What is the total amount of interest (in $) paid on each loan? 15 year mortgage 5 30-year mortgage $ (c) Overall, how much more interest in ) is paid by choosing the 30-year mortgage?

Answer 1

(a) The monthly principal and interest payments for each loan are as follows:

- 15-year mortgage: $2,271

- 30-year mortgage: $1,610

(b) The total amount of interest paid on each loan is as follows:

- 15-year mortgage: $88,017

- 30-year mortgage: $256,017

(c) Overall, by choosing the 30-year mortgage, you would pay $168,000 more in interest compared to the 15-year mortgage.

Now, let's break down the calculations step by step:

(a) Monthly Principal and Interest Payments:

To calculate the monthly principal and interest payments, you can use the formula for the monthly payment on a fixed-rate mortgage:

Monthly Payment (M) = P [r(1+r)^n] / [(1+r)^n - 1]

Where:

- P = Loan amount ($300,000)

- r = Monthly interest rate (annual rate / 12 / 100)

- n = Number of monthly payments (15 years = 180 months for the 15-year mortgage and 30 years = 360 months for the 30-year mortgage)

For the 15-year mortgage:

r = 4.5% / 12 / 100 = 0.00375

n = 15 years * 12 months/year = 180 months

M = 300,000 [0.00375(1+0.00375)^180] / [(1+0.00375)^180 - 1]

M ≈ $2,271

For the 30-year mortgage:

r = 5% / 12 / 100 = 0.004167

n = 30 years * 12 months/year = 360 months

M = 300,000 [0.004167(1+0.004167)^360] / [(1+0.004167)^360 - 1]

M ≈ $1,610

(b) Total Amount of Interest Paid:

To find the total amount of interest paid on each loan, you can use the following formula:

Total Interest = (Monthly Payment * Number of Payments) - Loan Amount

For the 15-year mortgage:

Total Interest = ($2,271 * 180) - $300,000

Total Interest ≈ $88,017

For the 30-year mortgage:

Total Interest = ($1,610 * 360) - $300,000

Total Interest ≈ $256,017

(c) To find the overall difference in interest paid, subtract the total interest paid on the 15-year mortgage from the total interest paid on the 30-year mortgage:

Overall Difference = $256,017 (30-year mortgage) - $88,017 (15-year mortgage)

Overall Difference ≈ $168,000

So, choosing the 30-year mortgage would result in paying approximately $168,000 more in interest compared to the 15-year mortgage over the life of the loan.

Answer 2

15-year mortgage: Monthly payment: $2,793.41, Total interest: $119,012.14

30-year mortgage: Monthly payment: $1,951.54, Total interest: $285,462.92

Difference in interest: $166,450.78

a) Monthly Payments:

15-year mortgage: Using the formula for a fixed-rate mortgage payment, we get:

Monthly payment = $300,000 * (4.5% * (1 + 4.5%)^15) / ((1 + 4.5%)^15 - 1)

Monthly payment ≈ $2,793.41 (rounded to nearest dollar)

30-year mortgage: Similarly, using the 5% interest rate and 30-year term:

Monthly payment = $300,000 * (5% * (1 + 5%)^30) / ((1 + 5%)^30 - 1)

Monthly payment ≈ $1,951.54 (rounded to nearest dollar)

b) Total Interest Paid:

15-year mortgage: Total interest = (15 years * $2,793.41) - $300,000

Total interest ≈ $119,012.14 (rounded to nearest dollar)

30-year mortgage: Similarly, using the 30-year term:

Total interest ≈ $285,462.92 (rounded to nearest dollar)

c) Difference in Interest:

Choosing the 30-year mortgage results in paying $285,462.92 - $119,012.14 ≈ $166,450.78 more in interest compared to the 15-year option.