Refinancing a Mortgage Fifteen years ago, the Budai family...

Refinancing a Mortgage Fifteen years ago, the Budai family bought a home and financed $\$ 150,000$ with a 30 -year mortgage at 8.2$\% .$ a. Find their monthly payment, the total amount of their payments, and the total amount of interest they will pay over the life of this loan. b. The Budais made payments for 15 years. Estimate the unpaid balance using the formula $$y=R\left[\frac{1-(1+i)^{-(n-x)}}{i}\right]$$ and then calculate the total of their remaining payments. c. Suppose interest rates have dropped since the Budai family took out their original loan. One local bank now offers a 30 -year mortgage at 6.5$\% .$ The bank fees for refinancing are $\$ 3400 .$ If the Budais pay this front and refinance the balance of their loan, find their monthly payment. Including the refinancing fee, what is the total amount of their payments? Discuss whether or not the family should refinance with this option. d. A different bank offers the same 6.5$\%$ rate but on a 15 -year mortgage. Their fee for financing is $\$ 4500$ If the Budais pay this fee up front and refinance the balance of their loan, find their monthly payment. Including the refinancing fee, what is the total amount of their payments? Discuss whether or not the family should refinance with this option.

Refinancing a Mortgage Fifteen years ago, the Budai family bought a home and financed $\$ 150,000$ with a 30 -year mortgage at 8.2$\% .$
a. Find their monthly payment, the total amount of their payments, and the total amount of interest they will pay over the life of this loan.
b. The Budais made payments for 15 years. Estimate the unpaid balance using the formula
$$y=R\left[\frac{1-(1+i)^{-(n-x)}}{i}\right]$$
and then calculate the total of their remaining payments.
c. Suppose interest rates have dropped since the Budai family took out their original loan. One local bank now offers a 30 -year mortgage at 6.5$\% .$ The bank fees for refinancing are $\$ 3400 .$ If the Budais pay this front and refinance the balance of their loan, find their monthly payment. Including
the refinancing fee, what is the total amount of their payments? Discuss whether or not the family should refinance with this option.
d. A different bank offers the same 6.5$\%$ rate but on a 15 -year mortgage. Their fee for financing is $\$ 4500$ If the Budais pay this fee up front and refinance the balance of their loan, find
their monthly payment. Including the refinancing fee, what is the total amount of their payments? Discuss whether or not the family should refinance with this option.

Key Concepts

Mortgage Amortization

Mortgage amortization refers to the process by which a loan is repaid over time with regular payments that cover both principal and interest. Each payment reduces the outstanding balance of the loan, and over the term of the mortgage, the interest portion of each payment gradually decreases while the principal portion increases.

Monthly Payment Formula

The monthly payment for a mortgage is determined using a formula that incorporates the loan amount, the periodic (typically monthly) interest rate, and the total number of payments. This formula precisely calculates the fixed payment that will fully amortize the loan over its designated term, ensuring that the borrower pays off both interest and principal.

Outstanding Loan Balance Calculation

Calculating the outstanding loan balance involves determining how much of the original loan remains unpaid after a certain number of payments have been made. This is done using an amortization formula that takes into account the number of payments made, the total number of payments, the interest rate, and the periodic payment amount, providing an estimate of the remaining principal at any point in time.

Total Payment and Total Interest Calculation

The total amount paid over the life of a mortgage is the sum of all the monthly payments made over the loan term. By subtracting the original principal from this total, one can determine the total interest paid. This total interest amount represents the cost of borrowing and is a key factor when comparing loan options or evaluating the benefits of refinancing.

Refinancing

Refinancing involves taking out a new mortgage to pay off an existing one, typically to take advantage of a lower interest rate, change the term of the loan, or access cash from home equity. It requires an analysis of the new loan terms, closing costs or fees associated with refinancing, and the potential financial benefits to determine if it is a worthwhile decision.

Impact of Refinancing Fees and Costs

When considering refinancing, it is important to account for all associated fees and costs, such as application fees, closing costs, and other charges. These costs affect the overall savings a borrower might achieve from a lower interest rate or shorter loan term, making it crucial to include them in any total payment or cost analysis.

Transcript

00:01 This is a lengthy multi -part problem, but we're going to just start and work through it.

00:04 So part a, we're starting with a present value of a loan of $150 ,000.

00:11 It wants us to calculate the r value or the monthly payment on this mortgage.

00:18 It's a 30 -year mortgage at 8 .2%.

00:21 So to calculate the r -value, we're going to do 1 minus 1 plus 0 .082 divided by 12.

00:33 It's 30 years, so it's going to be to the negative 360 power, all divided by 0 .082 divided by 12.

00:47 This is something that just gets put into a calculator normally, so let's just go do that.

00:53 So i have 150 ,000, i have 1 minus 1 plus 0 .082 divided by 12, negative 360 power divided by 0 .082 divided by 12.

01:07 Okay, so the monthly payment is going to be $1 ,121 .63.

01:24 Okay.

01:26 The second part to part a wants the total amount of their payments.

01:30 So we're going to multiply that by $360.

01:42 The total amount of their payments is $403 ,786 .80.

01:53 Cents.

01:57 The last part of part a wants the total amount of interest they'll pay over the life of the loan.

02:04 So the total interest paid is going to be 253 ,000.

02:08 That's just taking away 150 ,000 from the total paid, $786 .80.

02:19 Okay, that's part a.

02:24 Part b says they made payments for 15 years, and to estimate the unpaid balance using the formula and capital of the remaining payments.

02:36 Okay, so we have the amount paid.

02:41 It gives us the formula for that.

02:43 So we're just going to plug stuff in.

02:44 So we're going to have 1121 63 times a one minus our 1 plus the 0 .082 divided about 12 to the negative we have 360 payments and we've only we've already made 15, all divided by that interest rate of 0 .082 divided about 12.

03:18 Okay.

03:19 So again, i'm going to go put that in the calculator.

03:32 Okay, this is just one long formula to put in the calculator.

03:36 So we have 1 ,121 and 63 cents, times that large 1 minus 1 plus 0 .082 over 12 to the negative 345 power divided by 0 .082 over 12.

03:54 So that means so far they have paid 148 ,478 .87s.

04:15 So that's how much they've paid so far and how much they have to go.

04:18 We can just look back to our part a.

04:21 It's subtract this amount from the 403 ,000.

04:27 Okay, so we're going to have, so 403 -786 .8 minus the 148.

04:42 478 .87.

04:46 They have left the pay this market is left $255 ,307 and 93 cents.

05:07 I think that finishes part b.

05:19 Okay, so part c.

05:26 Okay, so i just realized that i only took away 15, but they made payments for 15 years.

05:34 So we actually need to go back and change that 15 to 180 because they paid off for 180 months, not just 15 months.

05:44 So let's go back up a little bit.

05:47 We need to redo some of this.

05:52 I was just getting numbers that were just too high for paying off for 15 years, so it just didn't quite make sense.

06:01 So we're going to do 360 minus 180, so it's going to be to the 180th power.

06:05 Luckily for me, it's still in my calculator, so i can just go back and change that 15 to 180.

06:23 So that formula gives us the unpaid balance of 115 ,962.

06:37 Okay, that makes a whole lot more sense.

06:45 And 66 cents.

06:54 Okay, so, and so because we have 180 payments left, the way we figure out payments, is we still take our original r value.

07:04 So we're going to do 112163 and multiply by 180.

07:13 Okay, so the number of payments we have left, or the total amount of payments we have left is $201 ,893.

07:26 And 40 cents.

07:31 Okay, so that feels better for part b.

07:34 Okay, now part c.

07:38 So it says the bank fees for refinancing are 3 ,400, and they want to refinance at 6 .5%.

07:48 So it says they're going to pay this, they're going to pay the upfront, the $3 ,400, and refinance.

07:58 So if they refinance, that means the r value is going to change, and we're looking back at this $115 ,000 amount that's left.

08:09 So our new r value that we had to pay, we got to remember we have to pay $3 ,400 to get this.

08:18 So let's see if it comes out well...

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