A $50,000 interest only mortgage loan is made for 30 years at a nominal interest rate of 6%....

Question:

A $50,000 interest only mortgage loan is made for 30 years at a nominal interest rate of 6%. Interest is to be accrued daily, but payments are to be made monthly. Assume 30 days each month.

a. What will the monthly payments be on such a loan?

b. What will the loan balance be at the end of 30 years?

c. What is the effective annual rate on this loan?

Effective Rate and Payment Frequency:

The frequency of payments does not affect the effective annual rate. Only the frequency of interest compounding does. This is so because the effective rate formula assumes that the calculated interest is reinvested at the same interest rate. So it does not matter if the interest is paid out immediately or at the end of the year.

Answer and Explanation:

a.

Let,

  • r = daily interest rate = 6% / 360 = 0.016667%
  • PMT = daily interest calculated on loan = Loan * r = 50,000 * 0.0001667 = $8.335
  • k = days in a month = 30

Each day the calculated interest begins earning the same daily interest rate. So the payment at the end of the month is the future value of the annuity of interest payments during the month:

{eq}Monthly \ payment=PMT*\frac{(1+r)^{k}-1}{r}\\ Monthly \ payment=8.335*\frac{(1+0.00016667)^{30}-1}{0.00016667}\\ Monthly \ payment=\$250.66 {/eq}

The monthly payment is $250.66

b.

Since this is an interest only loan, at the end of the loan the balance is equal to the initial loan amount. So the loan balance will be $50,000

c.

The effective annual interest rate (EAR) formula is:

{eq}EAR = (1+r)^{Days}-1\\ EAR = (1+0.00016667)^{360}-1\\ EAR = 6.18\%\\ {/eq}

The effective annual rate is 6.18%


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Effective Annual Rate: Formula & Calculations

from Business 110: Business Math

Chapter 7 / Lesson 6
16K

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