Problem $4[15$ points$]$ Emma & Arthur

Two years ago, you purchase a house of $$500,000.$ You borrow a mortgage with $80\%$ of LTV $($loan to value ratio$).$ The interest rate on the mortgage is $6\mathrm{.}25\%.$ Payment terms are being made monthly to amortize the loan over $30$ years. You have found another lender who will refinance the current outstanding loan balance plus all the costs associated with the new loan at $5\%$ with monthly payments for $30$ years. Suppose that the new lender will charge $2\mathrm{.}5$ discount points on the new loan and other refinancing costs will equal $$10,000.$

What is the new loan amount if you choose to refinance?

Should you refinance if you hold the loan for $30$ years?

If you choose to refinance, at least how many years should you stay in the house $($do not prepay$)?$ Why?