Question $24.$ Wilson has just obtained a $$250,000,$ five$-$year, $6\%$ fixed$-$rate mortgage. The mortgage is amortized over $25$ years. The interest rate is compounded semi$-$annually, and Wilson makes bi$-$weekly payments at the end of each month. Immediately after Wilson signed the paperwork, mortgage rates dropped to $5\%.$ The bank has offered him the opportunity to renegotiate the mortgage for a penalty of $$8,500,$ paid immediately.

$(1)$Should Wilson take this opportunity? Assume Wilson$$s opportunity cost $($or market interest rate$)$ equals the mortgage rate. $(5$ marks$)$

$(2)$Assume that Wilson accepts the bank$$s offer and pays mortgage in accordance with new term. How much does Wilson still owe five years after he took out the mortgage $($i$.$e$.,$ immediately after he made the $60$th monthly payment$)?$ What fraction of the $60$th payment represents interest payment? $(8$ marks$)$