A lender is considering what terms to allow on a loan. Current market terms are $9$ percent interest for $25$ years for a fully amortizing loan. The borrower, Rich, has requested a loan of $$100,000.$ The lender believes that extra credit analysis and careful loan control will have to be exercised because Rich has never borrowed such a large sum before. In addition, the lender expects that market rates will move upward very soon, perhaps even before the loan is closed. To be on the safe side, the lender decides to extend to Rich a CPM loan commitment for $$95,000$ at $9$ percent interest for $25$ years; however, the lender wants to charge a loan origination fee to make the mortgage loan yield $10$ percent.

Required:

a$.$ What origination fee should the lender charge?

b$.$ What fee should be charged if it is expected that the loan will be repaid after $10$ years?