Barbara borrows $4,500 at 12 percent annually compounded interest to be repaid in four equal annual installments. The actual end-of-year payment is
A) $942
B) $1,125
C) $1,482
D) $2,641

Question

Quizplus · Accepted Answer

To solve this problem, we can use the formula for the present value of an annuity:

PV = PMT × [1 − 1 / (1 + i)n] / i

where PV is the present value (in this case, the amount of the loan), PMT is the equal annual payment, i is the interest rate per period (in this case, the annual interest rate divided by four), and n is the number of periods (in this case, four).

Plugging in the values given in the problem, we get:

PV = $4,500
i = 0.12 / 4 = 0.03
n = 4

PMT = PV × i / [1 − 1 / (1 + i)n]
PMT = $4,500 × 0.03 / [1 − 1 / (1 + 0.03)4]
PMT = $1,481.78

So the equal annual payment should be $1,481.78. However, the question asks for the actual end-of-year payment, which will be slightly different due to rounding.

At the end of the first year, Barbara will owe interest of 12% on the original loan amount of $4,500, which is $540. This will be subtracted from her first payment of $1,481.78, leaving a principal payment of $941.78.

At the end of the second year, Barbara will owe interest of 12% on the remaining principal of $3,558.22, which is $426.98. This will be subtracted from her second payment of $1,481.78, leaving a principal payment of $1,054.80.

At the end of the third year, Barbara will owe interest of 12% on the remaining principal of $2,503.42, which is $300.41. This will be subtracted from her third payment of $1,481.78, leaving a principal payment of $1,181.37.

Finally, at the end of the fourth year, Barbara will owe interest of 12% on the remaining principal of $1,322.05, which is $158.65. This will be subtracted from her fourth payment of $1,481.78, leaving a principal payment of $1,323.13.

Therefore, the actual end-of-year payment is $1,323.13, which is closest to option C ($1,482).

Barbara Borrows $4,500 at 12 Percent Annually Compounded Interest to Be