Dave takes out a 24-year mortgage of 210,000 dollars for his new house. Dave gets an interest rate of 13.2 present compounded monthly. He agreed to make equal monthly payment, the first coming in one month. After making the 68th payment, Dave wants to buy a boat, so he wants to refinance his house to reduce his monthly payment by 500 dollars, and to get a better interest rate. In particular, he negotiates a new rate of 7.2 percent compounded monthly, and agress to make equal monthly payments (each 500 dollars less than his orginal payments) for as long as necessary, followed by a single smaller payment. How large will Dave's final loan payment be?
original loan:
payment -- p
i = .132/12 = .011
n = 288
PV = 21000
p( 1 - 1.011^-288)/.011 = 210000
I got p = $2413.34
balance after 68th payment
= 210000(1.011^87) - 2413.34(1.011^87 - 1)/.011
= $195,059.83
new payment - 2413.3-500 = 1913.34
new i = .072/12 = .006
PV = 195059.83
n = ??
1913.34( 1 - 1.006^-n)/.006 = 195059.83
1 - 1.006^-n = .061168374
1.006^-n = .388316259
using logs
-n log 1.006 = log .388316259
-n = -158.128
n = 158.128
So 158 full payments plus a partial payment are needed.
Your turn:
find the outstanding balance after 158 payments following the same steps I used above.
Find the interest on the remaining balance by multiplying it by .006, add that to the outstanding balance you just found and you got it.
98
To find out the size of Dave's final loan payment, we need to calculate the new monthly payment and determine how many payments he will make before making the final smaller payment.
First, let's calculate the initial monthly payment using the original mortgage terms:
Step 1: Convert the annual interest rate to a monthly interest rate.
Monthly interest rate = (1 + annual interest rate)^(1/12) - 1
Monthly interest rate = (1 + 0.132)^(1/12) - 1
Monthly interest rate = 0.010648
Step 2: Calculate the total number of payments over the 24-year mortgage.
Number of payments = 24 years * 12 months/year
Number of payments = 288
Step 3: Use the formula to calculate the monthly payment amount.
Monthly payment = (Loan amount * Monthly interest rate) / (1 - (1 + Monthly interest rate)^(-Number of payments))
Monthly payment = (210,000 * 0.010648) / (1 - (1 + 0.010648)^(-288))
Monthly payment ≈ $2,113.13
After making the 68th payment, Dave decides to refinance his house. Now, let's calculate the new monthly payment and the number of payments for the second loan:
Step 4: Calculate the new monthly interest rate for the second loan.
Monthly interest rate = (1 + annual interest rate)^(1/12) -1
Monthly interest rate = (1 + 0.072)^(1/12) -1
Monthly interest rate = 0.005924
Step 5: Calculate the new monthly payment with $500 less than the original monthly payment.
New monthly payment = $2,113.13 - $500
New monthly payment = $1,613.13
Step 6: Calculate the new loan amount using the annuity formula.
Loan amount = (Monthly payment * (1 - (1 + Monthly interest rate)^(-Number of payments))) / Monthly interest rate
Loan amount = ($1613.13 * (1 - (1 + 0.005924)^(-N))) / 0.005924
We have to solve for N, the number of remaining payments.
Step 7: Rearrange the formula to solve for N.
(1 - (1 + Monthly interest rate)^(-N)) = (Monthly interest rate * Loan amount) / Monthly payment
(1 + 0.005924)^(-N) = 1 - ((0.005924 * Loan amount) / Monthly payment)
(-N) * ln(1.005924) = ln(1 - ((0.005924 * Loan amount) / Monthly payment))
N = -ln(1 - ((0.005924 * Loan amount) / Monthly payment)) / ln(1.005924)
N ≈ 241.19
Step 8: Calculate the final loan payment.
Final loan payment = Loan amount * (1 + Monthly interest rate)^(-N)
Final loan payment ≈ Loan amount * (1 + 0.005924)^(-241.19)
Calculating the final loan payment requires knowing the exact loan amount, which can only be determined if Dave provides the remaining loan balance details after the 68th payment.
To find the size of Dave's final loan payment, we need to break down the problem into several steps:
Step 1: Calculate the original monthly payment
Using the mortgage formula, we can calculate the original monthly payment for a mortgage.
P = principal amount (loan amount)
r = monthly interest rate (annual rate divided by 12)
n = total number of payments (number of years multiplied by 12)
M = monthly payment
Using the given values:
P = $210,000
r = 13.2% / 12 = 0.011
n = 24 years * 12 = 288
The formula for the monthly payment is:
M = (P * r * (1 + r)^n) / ((1 + r)^n - 1)
Now let's calculate the original monthly payment (M1):
M1 = ($210,000 * 0.011 * (1 + 0.011)^288) / ((1 + 0.011)^288 - 1)
Step 2: Calculate the new monthly payment
Dave wants to reduce his monthly payment by $500. Therefore, the new monthly payment will be the original monthly payment (M1) minus $500.
Now let's calculate the new monthly payment (M2):
M2 = M1 - $500
Step 3: Calculate the number of remaining payments
Dave has already made 68 payments. So, the number of remaining payments will be the total number of payments (288) minus 68.
Now let's calculate the number of remaining payments (n_remaining):
n_remaining = n - 68
Step 4: Calculate the final loan payment
Since Dave agrees to make equal monthly payments for as long as necessary, followed by a single smaller payment, the final loan payment will be the remaining loan balance.
Using the refinanced interest rate and the new monthly payment, we can calculate the remaining loan balance (B) using the mortgage formula:
B = remaining loan balance
r_new = 7.2% / 12 = 0.006
n_remaining = number of remaining payments
The formula for remaining loan balance is:
B = (M2 * ((1 + r_new)^n_remaining - 1)) / r_new
Now let's calculate the final loan payment (B):
B = (M2 * ((1 + 0.006)^n_remaining - 1)) / 0.006
The resulting value will be the size of Dave's final loan payment.