Question

a) "Clay borrowed $32,000 from a bank at an interest rate of 11.16% compounded monthly. The loan will be repaid in 72 monthly installments over 6 years. Immediately after his 48th payment, Clay desires to pay the remainder of the loan in a single payment. Compute the total amount he must pay."

b) "Suppose that $5,000 is placed in a bank account at the end of each quarter over the next 7 years. What is the future worth at the end of 7 years when the interest rate is 8.2% compounded monthly?"

Answer 1

a). The formula for calculating 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 = 32000, r = 11.16/1200 = 0.0093 and n = 72. Then P = 32000*0.0093(1.0093)⁷²/[(1.0093)⁷² -1] = 297.60( 1.947420217)/(0.947420217) = $611.72 ( on rounding off to the nearest cent). Clay has made 48 monthly payments amounting to 48 *611.72= $ 29362.56.

Further, the formula for calculating the remaining loan balance (B) in respect of a fixed payment loan of $ L after p months is

B = L[(1 + r)ⁿ - (1 + r)^p]/[(1 + r)ⁿ - 1].

Here, p = 48 so that B = 32000[(1.0093)⁷² –(1.0093)⁴⁸]/ [(1.0093)⁷² -1] = 32000*(1.947420217-1.559455966 )/( 0.947420217) = $ 13103.85 ( on rounding off to the nearest cent).

Thus, Clay must pay $ 29362.56+ $ 13103.85 = $ 42466.41.

Please post part b) again separately.