Sam Hinds, a local dentist, is going to remodel the dental reception area and add two new workstations. He has contacted A-Dec, and the new equipment and cabinetry will cost $24,000. The purchase will be financed with an interest rate of 7.5% loan over 6 years. What will Sam have to pay for this equipment if the loan calls for semiannual payments (2 per year) and weekly payments (52 per year)? Compare the annual cash outflows of the two payments. Why does the weekly payment plan have less total cash outflow each year?
SOLUTION :
The values provided in the question are as follows :
Principal (P ) = $24,000
Interest rate =7.5 %
Nos of years = 6 years
For semiannual payments,
Number of payments (T )= 6 years * 2 = 12 payments, (semiannual payments =2 per year)
( r ) semiannual interest rate = 7.5%/2 = 3.75% or 0.0375
For weekly payments,
Number of payments (T )= 6 years * 52 = 312 payments, (weekly payments =52 per year)
( r ) semiannual interest rate = 7.5%/52 = 0.1442% or 0.001442
The formula to calculate the Semiannual payments is as follows:
Semiannual payments = rP/ 1- (1+r )-T
Semiannual payments= 0.0375 * $ 24,000/ 1- ( 1+ 0.0375 )-12
Semiannual payments= $900/1- (1.0375 )-12
Semiannual payments= $900 / 1-0.64289897802
Semiannual payments= $900 / 0.35710102198
Semiannual payments= $2520.295
Semiannual payments= $2520.30
Total cash outflow = $2520.3*12 = $30,243.6
Annual cash outflow (Semiannual payments ) = $2520.3 * 2 = $ 5,040.6
The formula to calculate the weekly payments is as follows:
Weekly payments = rP/ 1- (1+r )-T
Weekly payments = 0.001442 * $ 24,000/ 1- ( 1+ 0.001442 )-312
Weekly payments = $34.608 / 1- (1.001442 )-312
Weekly payments = $ 34.608 / 1-0.63789605588
Weekly payments = $ 34.608 / 0.36210394412
Weekly payments = $ 95.574766754
Weekly payments= $ 95.57
Total cash outflow = $95.57 * 312 = $ 29,817.84
Annual cash outflow (Weekly payments ) = $95.57 * 52 = $ 4,969.64
With weekly payment, principle is reduced faster as compared to the semiannual payment .interest is calculated on the balance amount .when balance value is less , interest amount is also less . so total cash outflow each year will less in the case of weekly payments.
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