You plan to buy a $400,000 home with a 25% down payment. The bank you want to finance the loan...

Question:

You plan to buy a $400,000 home with a 25% down payment. The bank you want to finance the loan with suggests two options: a 20-year mortgage at 5% APR and a 25-year mortgage at 6% APR. What is the difference in monthly payments (for the first 20 years) between these two options? Use semiannual compounding.

Mortgage:

A mortgage is a loan issued to an investor who wishes to purchase a home or a building. The terms of a mortgage are set at the time of issuance and the borrower will be expected to make periodic payments to settle the mortgage. The payments are calculated based on time value of money factor.

Answer and Explanation:

Price = $400,000

Down payment = (0.25 * 400,000) = 100,000

Loan = 400,000 - 100,000 = $300,000

  • {eq}Loan = Payment * \dfrac{(1 - (1 + r )^{-n}) }{ r} {/eq}

The payments are made monthly but the interest is compounded semi-annually, determine the effective interest rate

Option 1

n = 20 years adjusted for monthly payments = 20 *12 = 240

r =0.05

The payments are made monthly but the interest is compounded semi-annually, determine the effective interest rate

  • {eq}i = q * [(1 + \dfrac{r}{m} )^{m/q)}-1 ] {/eq}

i is the effective annual rate

q is the number of payments in a year = 12

r is the nominal annual rate = 5%

m is the number of compounding periods in a year = 2

  • {eq}i = 12 * [(1 + \dfrac{0.05}{2} )^{2/12)}-1 ] {/eq}
  • {eq}i = 0.0495 {/eq}
  • {eq}i =4.95\% {/eq}

The monthly rate will be =0.0495/12 =0.004125

  • {eq}300,000 = Payment * \dfrac{(1 - (1 + 0.004125)^{-240}) }{ 0.004125} {/eq}
  • {eq}300,000 = Payment * 152.1614315 {/eq}
  • {eq}Payment = \dfrac{300,000}{152.1614315} {/eq}
  • {eq}Payment = $1,971.59 {/eq}

Option 2

n = 25 years adjusted for monthly payments = 25 *12 = 300

r =0.06

Determine the effective interest rate

  • {eq}i = 12 * [(1 + \dfrac{0.06}{2} )^{2/12)}-1 ] {/eq}
  • {eq}i = 0.05926 {/eq}
  • {eq}i =5.926\% {/eq}

The monthly rate will be =0.05926/12

  • {eq}300,000 = Payment * \dfrac{(1 - (1 + (0.05926/12))^{-300}) }{ (0.05926/12)} {/eq}
  • {eq}300,000 = Payment * 156.3023814 {/eq}
  • {eq}Payment = \dfrac{300,000}{156.3023814} {/eq}
  • {eq}Payment = $1,919.36 {/eq}

What is the difference in monthly payments (for the first 20 years) between these two options?

  • Difference = (1,971.59 *12 *20) - (1,919.36 *12 *20)
  • Difference = 473,181.60 - 460,646.40
  • Difference = $ 12,535.20

Learn more about this topic:

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Buying a House: Mortgage Types & Loan Length

from Finance 102: Personal Finance

Chapter 7 / Lesson 4
8.2K

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