You have decided to purchase a new home. The cost of the home is $$200,000$ and the loan requires a $20\%$ down payment. The mortgage company offers you a variable rate mortgage at $4\%$ nominal for the first five years of the loan and a $6\%$ nominal rate per year for the remaining ten years. Assume interest is compounded monthly. Show all of your calculations.

a$.$ Compute the monthly payment that you will have to pay for the first five years of the loan.

b$.$ Compute the monthly payment that you will have to pay for the last ten years of the loan.

c$.$ Develop a PW equation to solve for the effective monthly interest rate being charged over the fifteen year period. Develop an equation to solve for the nominal rate being charged over the fifteen year period. Develop an equation to compute the effective annual interest rate per year for the variable rate mortgage. You do not have to solve for the interest rates.

$(50$ points$)$