$($c$)$ Problem $3.$ Seller Financing. You bought a list of addresses where vacant land owners are delinquent on their property taxes $($and hence in danger of losing their land$)$ from a large county in the Midwest for $$200,$ then sent $$yellow letters$$ to $100$ landowners on this list offering to buy their land at a significant discount. To sweeten the deal, in each letter you also offered to pay cash, to pay off all property taxes and late penalties, and to cover all closing costs. The cost to produce and mail each letter was $$1\mathrm{.}25.$

One landowner calls the next day to agree. You buy his land, which you think is worth about $$100,000,$ for $$20,000$ in cash. You pay off his delinquent property taxes, which have grown with penalties and fees to $$4,950.$ Closing costs for the purchase are $$500.$ You immediately list the vacant land for sale at $$55,000,$ and offer seller financing.

$($a$)$ What will you require as a down payment, if you worry a buyer might not makeany loan payments and your short term objective is to recoup your costs?

$($b$)$ A prospective buyer makes an offer at list price, with a down payment of $15$ percent. You decide to accept, and to finance the outstanding $$46,750$ at a fixed interest rate of $9\%($compounded monthly$)$ over $15$ years, as your primary goal is a quick sale.

$(1)$ What will the buyer pay you each month?

$(2)$ What is the total amount the buyer will pay you?

$(3)$ Total interest paid on the loan?

$(4)$ When will you recoup the costs of your investment?

$(5)$ What is the present value of the loan, assuming your minimum acceptable annual rate of return is $5\%?$

$(6)$ What is the NPV of your investment, assuming your annual discount rate is $5\%?$