Consider an n-year fixed rate mortgage with principal P. The borrower makes a fixed repayment of M at the end of each month and the remaining balance is charged at the constant monthly compounded rate r over the next month. a. If R, is the remaining balance after j months, show that R = P. R = P(1+1)-M, R = P(1+i) - M (1+(1+i)). R = P(1+)-M (1+(1+1)+(1+1)*).... where i=r/12 b. Write down the general expression for R, in the form given above, and deduce that R,= P(1+1)-M((1+1)-1)/ c. Given that R2=0, find the monthly repayment M and hence deduce that (1-(1+i) 1-(1+i) 2 d. Determine the interest component, 1,, of the repayment at the end of month j. e. For a ten year fixed rate mortgage with principal $1,000,000 and fixed rate 2%, for each month determine the interest component as a percentage of the monthly repayment M and plot the percentages as a function of the month index j.