The Sandersons are planning to refinance their home. The outstanding principal on their original loan is $100,000 and was to amortized in 240 equal monthly installments at an interest rate of 11%/year compounded monthly. The new loan they expect to secure is to be amortized over the same period at an interest rate of 8%/year compounded monthly. How much less can they expect to pay over the life of the loan in interest payments by refinancing the loan at this time? (Round your answer to the nearest cent.)
use a financial calculator. Present value = $110,000, future value = 0 (as you have paid off the loan at that point), 12 payments per year, n = 240 (number of payments), interest = 12%, solve for payment (usually "pmt"). Once you have the payment multiply by 240, the number of payment, to solve for the total amount paid by the sandersons in P&I, principal and interest. The subtract out the principal from the P&I figure to get what they paid in interest alone. We will call this "A"
Then do all of this again, but this time use 7% as your interest, set your calculator up to compound monthly, not annually (financial calculators differ so I have no way of telling you how to do this on yours but look in your manual if you have it). Do the above to see what they paid in interest alone on this 7% compounded monthly. We can call this "B"
Finally subtract "A" from "B" and you have your answer.
Well, the Sandersons certainly know how to play the loan game! Let's crunch some numbers and see how much they can save.
To start, we need to find out the monthly payment on the original loan. Using the formula for monthly mortgage payments, we have:
P = L[c(1 + c)^n]/[(1 + c)^n - 1]
Where:
P = Monthly Payment
L = Loan Amount
c = Monthly Interest Rate
n = Number of Payments
For the original loan:
L = $100,000
c = (11%/100) / 12 months = 0.00917 (rounded to 5 decimal places)
n = 240
Plugging in the numbers, we find that the monthly payment is approximately $962.05.
Now let's calculate the total interest paid over the life of the loan:
Total Interest = (Monthly Payment * Number of Payments) - Loan Amount
Total Interest = ($962.05 * 240) - $100,000 ≈ $277,212.00
Now, let's calculate the new monthly payment on the refinanced loan with the lower interest rate:
For the new loan:
L = $100,000
c = (8%/100) / 12 months = 0.00667 (rounded to 5 decimal places)
n = 240
Plugging in the numbers, we find that the monthly payment is approximately $872.67.
Finally, let's calculate the total interest paid over the life of the refinanced loan:
Total Interest = (Monthly Payment * Number of Payments) - Loan Amount
Total Interest = ($872.67 * 240) - $100,000 ≈ $209,920.80
Now, let's calculate the difference in interest payments:
Interest Savings = Total Interest (Original Loan) - Total Interest (Refinanced Loan)
Interest Savings = $277,212.00 - $209,920.80 ≈ $67,291.20
So, by refinancing the loan at this time, the Sandersons can expect to pay approximately $67,291.20 less in interest payments over the life of the loan. That's some sweet savings right there!
To calculate the difference in interest payments, we need to find the total interest paid on the original loan and the new loan.
1. First, let's calculate the total interest paid on the original loan. We'll use the formula for the monthly payment on a loan:
P = monthly payment
A = outstanding principal
r = interest rate per period
n = number of periods
The formula for the monthly payment is:
P = (A * r * (1 + r)^n) / ((1 + r)^n - 1)
Plugging in the values for the original loan:
A = $100,000
r = 11%/12 (monthly interest rate)
n = 240 (number of months)
P = (100,000 * (11%/12) * (1 + (11%/12))^240) / ((1 + (11%/12))^240 - 1)
P ≈ $982.45 (monthly payment)
2. Next, let's calculate the total payment over the life of the original loan:
Total payment = P * n
Total payment = $982.45 * 240
Total payment = $235,188
The total interest paid on the original loan can be calculated by subtracting the outstanding principal from the total payment:
Total interest paid = Total payment - A
Total interest paid = $235,188 - $100,000
Total interest paid = $135,188
3. Now, let's calculate the total interest paid on the new loan. We'll use the same formula as before, but with the new interest rate:
r = 8%/12 (monthly interest rate)
P = (100,000 * (8%/12) * (1 + (8%/12))^240) / ((1 + (8%/12))^240 - 1)
P ≈ $771.82 (monthly payment)
Total payment = $771.82 * 240
Total payment = $185,236
Total interest paid = Total payment - A
Total interest paid = $185,236 - $100,000
Total interest paid = $85,236
4. Finally, let's calculate how much less the Sandersons can expect to pay over the life of the loan in interest payments by refinancing:
Difference in interest payments = Total interest paid (original loan) - Total interest paid (new loan)
Difference in interest payments = $135,188 - $85,236
Difference in interest payments ≈ $49,951.77
Therefore, the Sandersons can expect to pay approximately $49,951.77 less in interest payments over the life of the loan by refinancing at this time.
To find out how much less the Sandersons can expect to pay over the life of the loan in interest payments by refinancing, we need to calculate the total interest paid for both the original loan and the new loan.
For the original loan:
Principal: $100,000
Interest rate: 11% per year compounded monthly
Amortization period: 240 months
To calculate the monthly payment for the original loan, we can use the formula for the monthly payment of an amortizing loan:
P = r * PV / (1 - (1 + r)^(-n))
Where:
P is the monthly payment
r is the monthly interest rate (11% / 12)
PV is the present value of the loan ($100,000)
n is the number of monthly payments (240)
Using the formula, we can calculate the monthly payment for the original loan:
r = 11% / 12 = 0.0091667 (monthly interest rate)
n = 240 (number of monthly payments)
PV = $100,000
P = 0.0091667 * 100,000 / (1 - (1 + 0.0091667)^(-240))
P ≈ $943.35
Now, let's calculate the total interest paid for the original loan:
Total Interest Paid = (Monthly Payment * Number of Payments) - Principal
Total Interest Paid = ($943.35 * 240) - $100,000
Total Interest Paid ≈ $55,204.00
For the new loan:
Interest rate: 8% per year compounded monthly
Amortization period: 240 months
Let's calculate the monthly payment for the new loan using the same formula:
r = 8% / 12 = 0.0066667 (monthly interest rate)
n = 240 (number of monthly payments)
PV = $100,000
P = 0.0066667 * 100,000 / (1 - (1 + 0.0066667)^(-240))
P ≈ $768.91
Now, let's calculate the total interest paid for the new loan:
Total Interest Paid = (Monthly Payment * Number of Payments) - Principal
Total Interest Paid = ($768.91 * 240) - $100,000
Total Interest Paid ≈ $53,033.60
Finally, let's calculate the difference in interest payments:
Difference in Interest Payments = Total Interest Paid (Original Loan) - Total Interest Paid (New Loan)
Difference in Interest Payments = $55,204.00 - $53,033.60
Difference in Interest Payments ≈ $2,170.40
Therefore, by refinancing the loan at this time, the Sandersons can expect to pay approximately $2,170.40 less in interest payments over the life of the loan.