A mortgage loan in the amount of $$105,000$ is made at $6$ percent interest for $20$ years. Payments are to be monthly in each part of this problem.

Required:

a$.$ What will monthly payments be if

$(1)$ The loan is fully amortizing?

$(2)$ It is partially amortizing and a balloon payment of $$50,000$ is scheduled at the end of year $20?$

$(3)$ It is a nonamortizing, or $$interest$-$only$$ loan?

$(4)$ It is a negative amortizing loan and the loan balance will be $$155,000$ at the end of year $20?$

b$.$ What will the loan balance be at the end of year $5$ under parts a $(1)$ through a $(4)?$

c$.$ What would be the interest portion of the payment scheduled for payment at the end of month $61$ for each case $(1)$ through $(4)$ above?

d$.$ Assume that the lender charges $4$ points to close the loans in parts a $(1)$ through a $(4).$ What would be the APR for each?

e$.$ Assuming that $4$ points are paid at closing and the $20-$year loan is prepaid at the end of year $5,$ what will be the effective rate of interest for each loan in parts a $(1)$ through a $(4)?$

f$.$ Assume the loan is fully amortizing except that payments will be $$interest only$$ for the first three years $(36$ months$).$ If the loan is to fully amortize over the remaining $17$ years, what must the monthly payments be from year $4$ through year $20?$

g$.$ If this is a negative amortizing loan and the borrower and lender agree that the loan balance of $$155,000$ will be payable at the end of year $20$:

$(1)$ How much total interest will be paid from all payments? How much total principal will be paid?

$(2)$ What will be the loan balance at the end of year $3?$

$(3)$ If the loan is repaid at the end of year $3,$ what will be the effective rate of interest?

$(4)$ If the lender charges $5$ points to make this loan, what will the effective rate of interest be if the loan is repaid at the end of year $3?$