Armando is buying a house worth $500,000,$ and must decide whether to take on a $$xed or adjustable rate mortgage $($ARM$).$ The term of the loan is T $=15$ years, with n $=12$ monthly payments per year.

$($A$)$ Let the $$xed interest rate be r $=0\mathrm{.}04.$ Calculate Armando$$s monthly payment.

$($B$)$ Suppose that the ARM has a starter rate of r$\_0=0\mathrm{.}03,$ guaranteed for at least the $$rst $5$ years. Calculate Armando$$s initial monthly payment.

$($C$)$ Now suppose that after $5$ years, the rate jumps to r$\_1=0\mathrm{.}06.$ Calculate Armando$$s new monthly payment, making sure to account for the payments that he has already made.

$($D$)$ Now suppose that after $10$ years, the rate jumps again to r$\_2=0\mathrm{.}09.$ Note that this would make the loan rather generous by the standards of ARMs, whose rates typically adjust as frequently as every month. Calculate Armando$$s monthly payment now.

$($E$)$ To evaluate whether he made the right decision, Armando considers discounting payments he made under the ARM using the $$xed mortgage rate of r $=0\mathrm{.}04.$ By this metric, was it a good idea to choose the ARM?

$($F$)$ Assessing that the rate adjustments re$$ected $$uctuations in the market rate over the life of the loan, Armando now considers discounting the payments he would have made under the $$xed rate mortgage using the ARM rates. By this metric, was the ARM the right choice?