P$7.$ You need financing for a $$500,000$ investment in an apartment building. The NOI in the first year is projected to be $$35,000.$ You found a bank willing to underwrite a fully$-$amortizing, $30-$year, fixed rate mortgage with constant monthly payments at an interest rate of $6$ percent, compounded monthly.

$($a$)$ What is the debt service coverage ratio if the bank underwrites a mortgage at a loan$-$to$-$value ratio of $75$ percent?

$($b$)$ The bank agrees to finance this investment if the loan$-$to$-$value ratio of the mortgage is less than or equal to $75$ percent and the debt service ratio is greater than or equal to $1\mathrm{.}5.$ What is the largest loan you can get?

Assume that all other contract terms $($e$.$g$.$ APR, full amortization, monthly

compounding$)$ remain the same.