Dave takes out a 30-year mortgage of 200000 dollars for his new house. Dave gets an interest rate of 16.8 percent compounded monthly. He agrees to make equal monthly payments, 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 8.4 percent compounded monthly, and agrees to make equal monthly payments (each 500 dollars less than his original payments) for as long as necessary, followed by a single smaller payment. How large will Dave's final loan payment be?

Please help to find the answer!

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To find the final loan payment, we need to determine the loan balance after the 68th payment has been made. We can use the formula for the present value of an annuity to calculate this.

First, let's calculate the original monthly payment amount.

The loan amount is $200,000 with a 16.8% annual interest rate compounded monthly for 30 years. Let's convert the interest rate to a monthly rate by dividing it by 12 and converting it to decimal form:

r = (16.8%/12) / 100 = 0.014

Now, let's calculate the number of monthly payments based on a 30-year mortgage:

n = 30 years * 12 months/year = 360 months

Using the present value of an annuity formula, we can find the original monthly payment (PMT):

PMT = PV * r / (1 - (1 + r)^(-n))

Where:
PV is the loan amount = $200,000
r is the monthly interest rate = 0.014
n is the number of monthly payments = 360

Substituting these values into the formula:

PMT = 200,000 * 0.014 / (1 - (1 + 0.014)^(-360))
PMT ≈ $1,911.07 (rounded to the nearest cent)

Dave wants to reduce his monthly payment by $500, so the new monthly payment will be $1,911.07 - $500 = $1,411.07.

Now, let's find out how long he needs to make these reduced payments before making a single smaller payment. Let's assume the final payment will be made after t months.

Using the same formula, we can calculate the remaining loan balance after the 68th payment:

PV = PMT * ((1 - (1 + r)^(-(n - 68)))) / r

Where:
PMT is the reduced monthly payment = $1,411.07
r is the new monthly interest rate = (8.4%/12) / 100 = 0.007
n is the remaining number of monthly payments = t

Substituting these values into the formula and setting it equal to the remaining loan balance:

200,000 = 1,411.07 * ((1 - (1 + 0.007)^(-(t - 68)))) / 0.007

To solve for t, we would need to use numerical methods or a financial calculator. The goal is to find the value of t that satisfies the equation. You can use Excel or financial calculators like the Texas Instruments BA II Plus to find the solution.

Once you have the value of t, you can calculate the final loan payment by substituting it back into the original loan balance formula:

PV = PMT * ((1 - (1 + r)^(-n))) / r

Where:
PMT is the reduced monthly payment = $1,411.07
r is the new monthly interest rate = 0.007
n is the total number of monthly payments = 360

Substituting these values into the formula:

Final Loan Payment = 1,411.07 * ((1 - (1 + 0.007)^(-360))) / 0.007

Calculate this expression to get the final loan payment amount.

Please note that due to rounding and the complexity of the equation, the final loan payment may vary slightly from this calculation.