The table shows the specifications of an adjustable rate mortgage (ARM). Assume no caps apply. Find a) the initial monthly payment; b) the monthly payment for the second adjustment; and c) the change in monthly payment at the first adjustment.
*The principal balance at the time of the first rate adjustment.
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Part 1
Beginning Balance
$85,000
Term
20 years
Initial index rate
5.3%
Margin
2.7%
Adjustment period
1 year
Adjusted index rate
6.8%
*Adjusted balance
$83,203.43
What is the initial monthly payment?
What is the monthly payment for the second adjustment period?
To find the initial monthly payment, we need to use the formula:
Initial Monthly Payment = (Adjusted Balance * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate) ^ (-number of payments))
1. Calculate the Monthly Interest Rate:
Monthly Interest Rate = (Initial Index Rate + Margin) / 12
= (5.3% + 2.7%) / 12
= 0.88% / 12
= 0.00733
2. Calculate the Number of Payments:
Number of Payments = Term * 12
= 20 * 12
= 240
3. Insert the values into the formula:
Initial Monthly Payment = (83,203.43 * 0.00733) / (1 - (1 + 0.00733) ^ (-240))
Using a financial calculator or software, the initial monthly payment is approximately $590.57.
To find the monthly payment for the second adjustment period, we need to use the adjusted index rate.
1. Calculate the Monthly Interest Rate for the second adjustment period:
Monthly Interest Rate (2nd Adjustment Period) = Adjusted Index Rate / 12
= 6.8% / 12
= 0.5667%
2. Insert the values into the formula used in the previous step:
Monthly Payment (2nd Adjustment Period) = (83,203.43 * 0.005667) / (1 - (1 + 0.005667) ^ (-240))
Using a financial calculator or software, the monthly payment for the second adjustment period would be approximately $584.35.
To find the change in monthly payment at the first adjustment, we subtract the initial monthly payment from the monthly payment for the second adjustment.
Change in Monthly Payment = Monthly Payment (2nd Adjustment Period) - Initial Monthly Payment
Change in Monthly Payment = $584.35 - $590.57
The change in monthly payment at the first adjustment would be approximately -$6.22.
To find the initial monthly payment, we need to use the formula for calculating the monthly payment of an adjustable rate mortgage (ARM). The formula is:
Monthly Payment = (Adjusted Balance * (Adjusted rate / 12)) / (1 - (1 + Adjusted rate / 12)^(-number of payments))
a) To calculate the initial monthly payment, we will use the Beginning Balance of $85,000, the Term of 20 years (240 months), the Adjusted index rate of 6.8%, and the Adjusted balance of $83,203.43:
Monthly Payment = ($83,203.43 * (6.8% / 12)) / (1 - (1 + 6.8% / 12)^(-240))
Using this formula, we find that the initial monthly payment is approximately $654.92.
b) To calculate the monthly payment for the second adjustment period, we need to use the Adjusted balance of $83,203.43, the Term of 20 years (240 months), the Adjusted index rate of 6.8%, and the Adjusted balance (which remains the same) of $83,203.43:
Monthly Payment = ($83,203.43 * (6.8% / 12)) / (1 - (1 + 6.8% / 12)^(-240))
Using the same formula, we find that the monthly payment for the second adjustment period is approximately $654.92 (same as the initial monthly payment).
The monthly payment remains the same in this case as there are no caps on the adjustable rate mortgage (ARM).
c) The change in monthly payment at the first adjustment can be found by subtracting the initial monthly payment from the monthly payment for the second adjustment period:
Change in Monthly Payment = Monthly Payment for second adjustment period - Initial Monthly Payment
Change in Monthly Payment = $654.92 - $654.92
The change in monthly payment at the first adjustment is $0.
To calculate the initial monthly payment, you need to know the loan amount, the interest rate, and the loan term.
Since the principal balance at the time of the first rate adjustment is not given in the question, we will assume that it is the same as the beginning balance, which is $85,000.
First, calculate the monthly interest rate by adding the initial index rate (5.3%) to the margin (2.7%):
Monthly interest rate = (5.3% + 2.7%) / 12 = 0.88%
Next, determine the loan term in months. Since the term is 20 years, multiply it by 12:
Loan term = 20 years * 12 months/year = 240 months
Now, you can use the loan amount, interest rate, and loan term to calculate the monthly payment using the formula for a fixed-rate mortgage:
Monthly payment = P * (r * (1 + r)^n) / ((1 + r)^n - 1)
P = Principal balance = $85,000
r = Monthly interest rate = 0.0088
n = Loan term in months = 240
Substituting the values into the formula, you can calculate the initial monthly payment.
For the second adjustment period, the adjustment is from the adjusted index rate of 6.8% (given in the question). Use the same formula with the new interest rate to calculate the monthly payment.
To find the change in the monthly payment at the first adjustment, subtract the initial monthly payment from the monthly payment for the second adjustment period.
Please note that this calculation assumes no caps apply.