Question

Present value with periodic rates. Sam Hinds, a local dentist, is going to remodel the dental reception area and two new workstations. He has contacted A-Dec, and the new equipment and cabinetry will cost $24,000. The purchase will be financed with a(n) 7% loan over 5 years. What will Sam have to pay for this equipment if the loan calls for semiannual payments (2 per year) and monthly payments (12 per year)? Compare the annual cash outflow of the two payments. Why does the monthly payment plan have less total cash outflow each year?

(1) What will Sam have to pay for this equipment if the loan calls for semiannual payments (2 per year)?
(2) What will Sam have to pay for this equipment if the loan calls for monthly payments (12 per year)?
(Round to the nearest cent)
Why does the monthly payment plan have less total cash outflow each year?
a. As more payment are made each year, the (principal) is repaid and thus the interest expense is lower.
b. As more payment are made each year, the (EAR) is repaid and thus the interest expense is lower.
c. As more payment are made each year, the years of the loan are reduced and thus the interest expense is lower.
d. As more payment are made each year, the (APR) becomes smaller and thus the interest expense is lower.

Answer 1

To compare the semiannual and monthly payment amounts for Sam's loan, we must adjust the interest rate and calculate based on the period. Monthly payments result in a lower total annual cash outflow because the principal is repaid faster, thereby reducing the interest expense over the course of the year.

Step-by-step explanation:

To calculate the semiannual payments for the loan at 7% interest over 5 years, we need to convert the annual interest rate to a semiannual rate and then calculate the payments based on the number of semiannual periods. For monthly payments, the same process applies, but with a monthly interest rate and more payment periods.

For semiannual payments, the formula for the payment amount R when the loan amount PV is given by:

PV = R * [1 - (1 + i)^-n] / i

where i is the semiannual interest rate and n is the total number of semiannual payment periods. The monthly payment plan would have a smaller i (since it's a monthly rate rather than semiannual), and more periods.

The reason why option (a) is correct is that as more payments are made each year, the principal is repaid faster and the interest expense is lower because interest is calculated on the remaining balance, which decreases with each payment.