A family plans to purchase a 225000 house The entire purchas

A family plans to purchase a $225,000 house. The entire purchase price will be financed with a 30-year mortgage at a 4.25% interest rate. Payments will be made monthly. (Assume payments are timely and always the same amount.) What is the monthly payment? How much of the first month\'s payment is interest? How much of the first month\'s payment is applied to the balance? What is the remaining balance on the loan after the first year of payments? How much interest is paid during the first year?

Solution

The formula for computing the monthly repayment installment (A) under a mortgage loan is A = M [r(1 + r)n]/[(1 + r)n - 1], where M is the amount of the loan, r is the rate of interest per month in decimals and n is the period of the loan in months. Here, M = $225000, n = 30*12 = 360, and r = (4.25/100)*1/12 = 0.0425/12. Then P = 225000[0.0425/12 ( 1+0.0425/12)360]/ [ (1+0.0425/12)360-1] = 796.875[ (12.0425/12)360]/ [(12.0425/12)360 -1] = 796.875[ 3.570649477/2.570649477] = 1106.864754 = $ 1106.86 on rounding off to the nearest cent). Thus the monthly payment is $ 1106.86. The amount of interest for the 1st month is 225000*4.25/1200 = $796.875 = $796.88(say) . The part of the payment applied towards the balance is $ 1106.86- $ 796.88 = $ 309.98. The formula for computing the balance B after p months is B = M[(1 + r)n - (1 + r)p]/[(1 + r)n - 1]. Here, p = 12 so that B = 225000[ (1+ 0.0425/12)360 –(1+0.0425/12)12]/ [(1+0.0425/12)360 -1] = 225000[3.570649477- 1.04337716]/ 2.570649477 = 225000( 2.527272317/2.570649477) = $ 221203.35. The total repayment during the 1st year is $ 1106.86*12 = $ 13282.32 . The reduction in the amount of the loan after the 1st year is $ 225000- $ 221203.35 = $ 3796.65. Hence, the amount of interest paid during the 1st year is $ 13282.32 - $ 3796.65 = $ 9485.67.