Question

 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?

Answer 1

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.