You have managed to save $$50,000$ and are buying your first house for $$250,000.$ You are planning to borrow $$200,000$ today $($t$=0)$ by taking out a $15-$year mortgage that requires monthly payments at an APR of $9\%.$

a$)$ What are your monthly payments on the mortgage?

After $5$ years pass, you are considering refinancing your old mortgage with a new $10-$year mortgage. The annual percentage rate on the new mortgage is $8\%.$ In order to refinance, you must close the old mortgage by paying the outstanding principal.

b$)$ After $5$ years, what is the outstanding balance of the old mortgage?

If you choose to refinance, you have to pay refinancing fees of $$1,500.$ Assume that you do not have the $$1,500$ but that you can borrow the $$1,500$ as part of your new mortgage $($so the starting balance of the new mortgage is the sum of the outstanding balance of the old mortgage at t$=5$ and the refinancing fees of $$1,500).$

c$)$ What are your monthly payments on the new mortgage?

d$)$ Is it advantageous to refinance under those conditions? If so$,$ by how much are you better off by refinancing? If not, by how much are you worse off by refinancing?