Q$(1)$ You have just taken out a $30-$year fixed$-$rate mortgage for $$500,000$ at an annual interest rate of $6\mathrm{.}5\%.$ Calculate the monthly mortgage payment and create an amortization schedule for the mortgage. Include columns for the month, beginning balance, monthly payment, interest paid, principal paid, and ending balance.

a$.$ What is the interest payment in month $1?(2$pt$)$

b$.$ What is the total mortgage interest paid over $30$ years? $(2$pt$)$

c$.$ What is the total mortgage payment paid over $30$ years? $(2$pt$)$

d$.$ What is the remaining balance of the loan at the end of year $5?(2$pt$)$

e$.$ How much do you pay for your mortgage from year $1$ to year $5$ in total?? $(2$pt$)$

After $5$ years, the $30-$year mortgage rate dropped to $3\mathrm{.}5\%.$ You decide to

refinance your mortgage at $3\mathrm{.}5\%$ for $30$ years. The cost of refinancing is $4\%$ of the loan amount.

f$.$ What is the interest payment in month $1$ on a new loan? $(2$pt$)$

g$.$ What is the total interest paid over $30$ years on a new loan? $(2$pt$)$

h$.$ What is the total mortgage payment paid over $30$ years on a new loan? $(2$pt$)$

i$.$ What is the total mortgage payment of the new mortgage $($refinance$)$ and the previous $5$ years' mortgage payments $($original loan$)?(3$pt$)$

j$.$ Determine the overall financial benefit achieved through refinancing. Specifically, calculate the savings$($$$)$ between the total mortgage payments made under the initial loan and the combined total mortgage payments under the refinance, including the payments made during the previous $5$ years. $(3$pt$)$

Use the original loan mortgage amortization table to answer the next question.

k$.$ If you make $$500/$month extra payments toward the principal at the end of each month for the next $10$ years, how long will it take to pay off the loan? Please answer in the number of months. $(3$pt$)$