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 ​$22 comma 000. The purchase will be financed with an interest rate of 8​% loan over 8 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 outflows of the two payments. Why does the monthly payment plan have less total cash outflow each​ year? What will Sam have to pay for this equipment if the loan calls for semiannual payments ​(2 per​ year)?

Answer 1

Monthly payment and annual cash outflows are:

Monthly payment	=	[P * R * (1+R)^N ] / [(1+R)^N -1]
Using the formula:
Loan amount	P	22,000.00
Rate of interest per period:
Annual rate of interest		8.000%
Frequency of payment	=	Once in 1 month period
Numer of payments in a year	=	12/1 =	12
Rate of interest per period	R	0.08 /12 =	0.6667%
Total number of payments:
Frequency of payment	=	Once in 1 month period
Number of years of loan repayment	=	8
Total number of payments	N	8*12 =	96
Period payment using the formula	=	[ 22000*0.00667*(1+0.00667)^96] / [(1+0.00667 ^96 -1]
Monthly payment	=	311.01
Annual cash out flow		3,732.08

Semi annual payment and annual cash outflows are:

Semi annual payment	=	[P * R * (1+R)^N ] / [(1+R)^N -1]
Using the formula:
Loan amount	P	22,000.00
Rate of interest per period:
Annual rate of interest		8.000%
Frequency of payment	=	Once in 6 month period
Numer of payments in a year	=	12/6 =	2
Rate of interest per period	R	0.08 /2 =	4.0000%
Total number of payments:
Frequency of payment	=	Once in 6 month period
Number of years of loan repayment	=	8
Total number of payments	N	8*2 =	16
Period payment using the formula	=	[ 22000*0.04*(1+0.04)^16] / [(1+0.04 ^16 -1]
Semi annual payment	=	1,888.04
Annual cash out flow		3,776.08

semi annual payment results in higher annual cash outflow because increase in time to payment increases interest payment. Monthly payment results in lesser annual cash outflow because earlier the amount paid back lesser the interest on outstanding loan amount.

Total payments under semi-annual payments = $1,888.04 X 16 = $30,208.64

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