A $100,000 mortgage is to be retired by equal monthly payments of D dollars interest, co...

A $100,000 mortgage is to be retired by equal monthly payments of D dollars interest, compounded monthly, is charged at the rate of 12% per annum on the input balance; for example, after the first month, the total debt equals

The problem is to determine D such that after a specified time the mortgage is payed in full, leaving a net balance of zero.

(a) To set up the problem, let y[n] denote the unpaid balance after the nth month payment. Assume that the principal is borrowed in month 0 and monthly payments begin in month 1. Show that y[n] satisfies the difference equation

with initial condition

y[0] = $100,000,

where y is a constant. Determine y.

(b) Solve the difference equation of part (a) to determine

(Hint: The particular solution of eq. (P2.63.-1) is a constant Y. Find the value of Y, and express y[n) for n ≥ 1 as the sum of particular and homogeneous solutions. Determine the unknown constant in the homogeneous solution directly calculating y[1] from eq. (P2.63-1) and comparing it to your solution)

(c) If the mortgage is to be retired in 30 years after 360 monthly payments of D dollars, determine the appropriate value of D.

(d) What is the total payment to the bank over the 30-year period?

(e) Why do banks make loans?