Question

A couple who borrow $50,000 for 20 years at 6%, compounded monthly, must make monthly payments of $358.22. (Round your answers to the nearest cent.) (a) Find their unpaid balance after 1 year. $ (b) During that first year, how much interest do they pay?

Answer 1

The formula used to calculate the fixed monthly payment (P) required to fully amortize a loan of $ L over a term of n months at a monthly interest rate of r is P = L[r(1 + r)ⁿ]/[(1 + r)ⁿ - 1]. Here, L = 50000, r = 6/1200 = 0.005 and n = 20*12 = 240. Therefore, P = 50000*0.005(1.005)²⁴⁰/ [(1.005)²⁴⁰-1] = 250*3.310204476/2.310204476 = $ 358.22 (on rounding off to the nearest cent).

(a). The formula used to calculate the remaining loan balance (B) of a fixed payment loan of $ L over a term of n months at a monthly interest rate of r, after p months is B = L[(1 + r)ⁿ - (1 + r)^p]/[(1 + r)ⁿ - 1]. Here, p = 12 so that B = 50000*[(1.005)²⁴⁰-(1.005)¹²]/[(1.005)²⁴⁰-1] = 50000*(3.310204476-1.061677812)/ 2.310204476 = $ 48665.10(on rounding off to the nearest cent).

(b). The reduction in principal amount in the 1^st year is $ 50000- $ 48665.10 = $ 1334.90. The amount repaid by the couple is 12*$358.22 = $ 4298.64. Therefore, the interest paid by the couple is $ 4298.64- $ 1334.90 = $ 2963.74(on rounding off to the nearest cent).