A family takes out a mortgage for $251,200.00 from the local bank. The loan is for 30 years of monthly payments at a 5.76% APR (monthly compounding). What will the family’s balance be on the mortgage after 8.00 years?

Question

GPT-4o mini · Answer

To calculate the remaining balance on a mortgage after a certain period, we can use the formula for the remaining balance on an amortizing loan. The formula for the remaining balance $ B $ after $ n $ payments is given by:

\[ B = P \times \frac{(1 + r)^N - (1 + r)^n}{(1 + r)^N - 1} \]

where:

$ P $ is the principal of the loan (initial mortgage amount),
$ r $ is the monthly interest rate,
$ N $ is the total number of payments,
$ n $ is the number of payments made.

Given the details of the mortgage:

Principal ($ P $) = $251,200.00
Annual interest rate (APR) = 5.76% = 0.0576
Monthly interest rate ($ r $) = $ \frac{0.0576}{12} = 0.0048 $
Total payments ($ N $) = 30 years × 12 months/year = 360 months
Payments made ($ n $) = 8 years × 12 months/year = 96 months

Now, we can substitute these values into the formula:

\[ B = 251200 \times \frac{(1 + 0.0048)^{360} - (1 + 0.0048)^{96}}{(1 + 0.0048)^{360} - 1} \]

First, we compute $ (1 + r)^{N} $ and $ (1 + r)^{n} $:

$ (1 + 0.0048)^{360} $
$ (1 + 0.0048)^{96} $

Using a calculator:

$ (1 + 0.0048)^{360} \approx 5.061295 $
$ (1 + 0.0048)^{96} \approx 1.487708 $

Substituting these values back into the balance formula gives:

\[ B = 251200 \times \frac{5.061295 - 1.487708}{5.061295 - 1} \]

Calculating the numerator:

\[ 5.061295 - 1.487708 \approx 3.573587 \]

And the denominator:

\[ 5.061295 - 1 \approx 4.061295 \]

Now substituting these results into the equation:

\[ B = 251200 \times \frac{3.573587}{4.061295} \approx 251200 \times 0.879128 \approx 220,925.04 \]

Thus, the family’s remaining balance on the mortgage after 8 years will be approximately $220,925.04.

123 · Answer

What values do I enter for N, i/y, PV, PMT, and FV to solve on a finance calculator?

GPT-4o mini · Answer

To solve for the remaining balance on the mortgage using a financial calculator, you need to enter values for the following variables:

N: The total number of payments for the loan.
i/y: The interest rate per period (monthly interest rate).
PV: The present value or initial loan amount (mortgage amount).
PMT: The payment amount per period (monthly payment).
FV: The future value of the loan after the specified number of periods (remaining balance you want to find).

Here's how to set it up for your specific mortgage scenario:

N: Total payments = $ 30 \text{ years} \times 12 \text{ months/year} = 360 $.
i/y: Monthly interest rate = $ \frac{5.76%}{12} = 0.48% $ per month (or simply enter 0.48).
PV: Present value (initial mortgage) = $-251200$ (the negative sign indicates cash outflow).
PMT: You need to first calculate this value because it is not directly given. The formula to calculate the monthly payment is:

\[ PMT = P \times \frac{r(1 + r)^N}{(1 + r)^N - 1} \]

Substituting in the values:
- $ P = 251200 $
- $ r = 0.0048 $
- $ N = 360 $
Calculating $ PMT $:

\[ PMT = 251200 \times \frac{0.0048(1 + 0.0048)^{360}}{(1 + 0.0048)^{360} - 1} \]

From previous calculations, we found $ (1 + 0.0048)^{360} \approx 5.061295 $:

\[ PMT = 251200 \times \frac{0.0048 \times 5.061295}{5.061295 - 1} \approx 1,464.36 \]

So, enter $ PMT = -1464.36 $ (negative, because it's a cash outflow).
FV: This is what you want to find, which is the remaining balance after $ n = 96 $ payments (8 years).

Summary of what to enter into the calculator:

$ N = 360 $ (Total payments for the loan)
$ i/y = 0.48 $ (Monthly interest rate)
$ PV = -251200 $ (Present Value or mortgage amount)
$ PMT = -1464.36 $ (Monthly payment)
$ FV = ? $ (This is what you are solving for)

After entering these values, set your calculator to compute the Future Value (FV), which will give you the remaining balance on the mortgage after the specified period (8 years or 96 payments).