Question

$300,000 loan

6% per year APR

30 year armotization

Payable monthly

Paid exactly 5 years

What is the loan balance?

Please show standard factor notation.

Answer should be: $279,163.07

Answer 1

The formula to find the monthly payment for any amortized loan is

P = L[c(1 + c)^n]/[(1 + c)^n - 1]

Note: "amortized" is a fancy way of saying "spread out the payments to be paid back over time"

where,

P = monthly payment
L = loan amount (how much the bank gives you). This is the principal
c = monthly interest rate
n = number of payments (each payment occurring monthly)

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Monthly interest rate: c = (yearly rate)/(12 months per year) = r/12 = 0.06/12 = 0.005

Number of payments: n = (number of years)*(12 months per year) = 30*12 = 360
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So in this case,

P = unknown (we're solving for this)
L = 30,0000
c = 0.005
n = 360

Plug the values L = 300000, c = 0.005 and n = 360 into the formula to get

P = 300000[0.005(1 + 0.005)^360]/[(1 + 0.005)^360 - 1]


P = 300000[0.005(1.005)^360]/[(1.005)^360 - 1]


P = 300000[0.005(5.4581902)]/[5.4581902 - 1]


P = 300000(0.027290951)/(5.4581902 - 1)


P = 300000(0.027290951)/(4.4581902)


P = 300000(0.0059955) approximately

P = 1798.65

p=1798.65

So your monthly payment is $1798.65

You will make 360 of these payments (same payment each time), so you will pay a total of 360*1798.65 = 647514 dollars

That figure of 647514 includes both the principal of $30,0000 and extra interest you pay back.

Yearly Amortization Schedule

PAYMENTS	YEARLY TOTAL	PRINCIPAL PAID	INTEREST PAID	BALANCE
Year 1 (1-12)	$21,583.82	$3,684.04	$17,899.78	$296,315.96
Year 2 (13-24)	$21,583.82	$3,911.26	$17,672.56	$292,404.71
Year 3 (25-36)	$21,583.82	$4,152.50	$17,431.32	$288,252.21
Year 4 (37-48)	$21,583.82	$4,408.61	$17,175.21	$283,843.60
Year 5 (49-60)	$21,583.82	$4,680.53	$16,903.29	$279,163.07
Year 6 (61-72)	$21,583.82	$4,969.21	$16,614.61	$274,193.86
Year 7 (73-84)	$21,583.82	$5,275.70	$16,308.12	$268,918.16
Year 8 (85-96)	$21,583.82	$5,601.10	$15,982.72	$263,317.06
Year 9 (97-108)	$21,583.82	$5,946.56	$15,637.26	$257,370.50
Year 10 (109-120)	$21,583.82	$6,313.33	$15,270.49	$251,057.17
Year 11 (121-132)	$21,583.82	$6,702.72	$14,881.10	$244,354.45
Year 12 (133-144)	$21,583.82	$7,116.13	$14,467.69	$237,238.32
Year 13 (145-156)	$21,583.82	$7,555.04	$14,028.78	$229,683.28
Year 14 (157-168)	$21,583.82	$8,021.02	$13,562.80	$221,662.27
Year 15 (169-180)	$21,583.82	$8,515.74	$13,068.08	$213,146.53
Year 16 (181-192)	$21,583.82	$9,040.97	$12,542.85	$204,105.57
Year 17 (193-204)	$21,583.82	$9,598.59	$11,985.22	$194,506.97
Year 18 (205-216)	$21,583.82	$10,190.61	$11,393.20	$184,316.36
Year 19 (217-228)	$21,583.82	$10,819.15	$10,764.67	$173,497.21
Year 20 (229-240)	$21,583.82	$11,486.45	$10,097.37	$162,010.76
Year 21 (241-252)	$21,583.82	$12,194.91	$9,388.91	$149,815.85
Year 22 (253-264)	$21,583.82	$12,947.06	$8,636.75	$136,868.78
Year 23 (265-276)	$21,583.82	$13,745.61	$7,838.21	$123,123.17
Year 24 (277-288)	$21,583.82	$14,593.41	$6,990.41	$108,529.76
Year 25 (289-300)	$21,583.82	$15,493.50	$6,090.32	$93,036.26
Year 26 (301-312)	$21,583.82	$16,449.11	$5,134.71	$76,587.16
Year 27 (313-324)	$21,583.82	$17,463.65	$4,120.17	$59,123.51
Year 28 (325-336)	$21,583.82	$18,540.77	$3,043.05	$40,582.73
Year 29 (337-348)	$21,583.82	$19,684.32	$1,899.49	$20,898.41
Year 30 (349-360)	$21,583.82	$20,898.41	$685.41	$0.00
	$647,514.57	$300,000.00	$347,514.57

so loan balance after 5 years = $279163.07 (check in the above table )