15. $ 0,000 is borrowed at 6% compound annual interest, with the loan to be repa
ID: 2618130 • Letter: 1
Question
15. $ 0,000 is borrowed at 6% compound annual interest, with the loan to be repaid with 10 equal annual payments. a. If the first payment is made one year after receiving the $100,000, how b. If the first payment is made four years after receiving the $100,000, how c. If the first payment is made four years after receiving the $100,000, how much of the third payment will be an interest payment? much of the first payment will be an interest payment? much of the last payment will be an interest payment?Explanation / Answer
a. We can use following Present Value of an Annuity formula to calculate the periodic annual payments with 6% interest rate
PV of loan = PMT * [1-(1+i) ^-n)]/i
Where,
Present value of loan (PV) = $100,000
Annual payments PMT =?
Number of annual payments n =10
Annual interest rate I =6%
Therefore
$100,000 = PMT * [1- (1+6%) ^-10]/6%
Or PMT = $13,586.80
Now prepare an amortization chart based on this PMT to know the amount of interest in third annual payment.
Amortization Chart
Year
Interest Payment @ 6% of new balance of previous period
Total Payment (PMT on Loan)
New Balance (Previous balance + Interest - Payment)
0
$0.00
$0.00
$100,000.00
1
$6,000.00
$13,586.80
$92,413.20
2
$5,544.79
$13,586.80
$84,371.20
3
$5,062.27
$13,586.80
$75,846.68
The amount of interest in third annual payment is $5,062.27.
b. As the first payment is made four years after receiving fund, therefore first we have to calculate future value of loan at the end of year 3 after that PMT of loan is calculated
FV = PV * (1+i) ^n
Where FV=?
PV =$100,000
Annual percentage interest rate i = 6% per annum
Time period n =3 years
Therefore,
FV = $100,000* (1+6%) ^3
= $119,101.60
Now this future value of loan will become present value in our annuity calculation for periodic annual payments
PV of loan = PMT * [1-(1+i) ^-n)]/i
Where,
Present value of loan at the end of year 3 (PV) = $119,101.60
Annual payments PMT =?
Number of annual payments n =10
Annual interest rate I =6%
Therefore
$119,101.60 = PMT * [1- (1+6%) ^-10]/6%
Or PMT = $16,182.09
Now prepare an amortization chart based on this PMT to know the amount of interest in third annual payment.
Loan amount after 3 years
$119,101.6
Annual payment =
$16,182.09
Interest rate =
6%
Time period
10 years
Amortization Chart
Year
Interest Payment @ 6% of new balance of previous period
Total Payment (PMT on Loan)
New Balance (Previous balance + Interest - Payment)
0
$0.00
$0.00
$119,101.60
1
$7,146.10
$16,182.09
$110,065.60
2
$6,603.94
$16,182.09
$100,487.45
3
$6,029.25
$16,182.09
$90,334.61
4
$5,420.08
$16,182.09
$79,572.59
5
$4,774.36
$16,182.09
$68,164.85
6
$4,089.89
$16,182.09
$56,072.66
7
$3,364.36
$16,182.09
$43,254.92
8
$2,595.30
$16,182.09
$29,668.13
9
$1,780.09
$16,182.09
$15,266.12
10
$915.97
$16,182.09
$0.00
The amount of interest in first annual payment is $7,146.10
c. The amount of interest in last annual payment is $915.97 (refer the above amortization chart)
Amortization Chart
Year
Interest Payment @ 6% of new balance of previous period
Total Payment (PMT on Loan)
New Balance (Previous balance + Interest - Payment)
0
$0.00
$0.00
$100,000.00
1
$6,000.00
$13,586.80
$92,413.20
2
$5,544.79
$13,586.80
$84,371.20
3
$5,062.27
$13,586.80
$75,846.68
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