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Ex: Compounded Interest with Different Compounding - YouTube
Channel: Mathispower4u
[1]
-IN THIS LESSON,
WE'LL BE COMPARING
[2]
HOW THE NUMBER OF COMPOUNDS PER
YEAR AFFECTS THE BALANCE
[5]
OF AN ACCOUNT
FOR A COMPOUNDED INTEREST.
[9]
SO IF YOU DEPOSIT $2,500
INTO A BANK
[11]
THAT PAYS 4% ANNUAL INTEREST,
[14]
THE THREE QUESTIONS ARE IF THE
INTEREST IN COMPOUNDED ANNUALLY
[18]
WHAT IS THE ACCOUNT BALANCE
AFTER FIVE YEARS,
[20]
IF THE INTEREST IS COMPOUNDED
QUARTERLY,
[23]
WHAT IS THE ACCOUNT BALANCE
AFTER FIVE YEARS,
[25]
AND FINALLY, IF THE INTEREST IS
COMPOUNDED MONTHLY
[28]
WHAT IS THE ACCOUNT BALANCE
AFTER FIVE YEARS.
[31]
SO HERE'S THE FORMULA USED FOR
A COMPOUNDED INTEREST WHERE "A"
[35]
IS THE AMOUNT AFTER TIME T,
[38]
P IS THE PRINCIPLE
OR STARTING AMOUNT,
[41]
R IS THE ANNUAL RATE
EXPRESSED AS A DECIMAL,
[44]
N IS THE NUMBER OF COMPOUNDS PER
YEAR, AND T IS TIME IN YEARS.
[51]
SO THE ONLY THING
THAT'S CHANGING
[52]
FOR THESE THREE QUESTIONS
[53]
IS N THE NUMBER
OF COMPOUNDS PER YEAR.
[57]
WHICH MEANS FOR ALL THREE OF
THESE QUESTIONS
[59]
WE CAN USE THE FORMULA "A" = THE
PRINCIPLE
[63]
OR STARTING AMOUNT IS 2,500 x
THE QUANTITY OF 1 +
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THE ANNUAL INTEREST RATE
EXPRESSED AS A DECIMAL.
[73]
WELL 4% AS A DECIMAL IS 0.04.
[78]
WE'LL DIVIDE THIS BY N, THE
NUMBER OF COMPOUNDS PER YEAR
[82]
AND RAISE IT TO THE POWER OF N x
T WHERE AGAIN,
[86]
N IS THE NUMBER OF COMPOUNDS
WHICH WILL CHANGE
[88]
BUT T IS THE TIME IN YEARS AND
IN EACH CASE,
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T IS EQUAL TO 5,
[92]
BECAUSE IT'S 5 YEARS.
[93]
SO OUR EXPONENT
IS GOING TO BE 5N.
[97]
NOW LOOKING AT THE FIRST
QUESTION,
[99]
WE'RE TOLD THE INTEREST IS
COMPOUNDED ANNUALLY
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WHICH MEANS IT'S ONLY COMPOUNDED
ONCE A YEAR
[103]
SO N IS EQUAL TO 1.
[106]
ON THE SECOND QUESTION IF THE
INTEREST IS COMPOUNDED QUARTERLY
[109]
AND SINCE THERE ARE 4 QUARTERS
PER YEAR N = 4,
[114]
AND THEN FOR THE LAST QUESTION
[116]
IF THE INTEREST IS COMPOUNDED
MONTHLY
[118]
AND SINCE THERE ARE 12 MONTHS
PER YEAR N = 12.
[123]
SO WHEN N = 1 WE WOULD HAVE "A"
= 2,500 x THE QUANTITY 1 PLUS--
[132]
IF N IS 1 THIS WOULD JUST BE
0.04 RAISED TO THE 5 x 1 POWER
[140]
WHICH IS JUST GOING TO BE 5.
[142]
LET'S SET UP EACH OF THESE AND
THEN WE'LL EVALUATE THEM
[143]
ON THE CALCULATOR
AT THE SAME TIME.
[146]
SO FOR THE SECOND QUESTION
[147]
BECAUSE THE INTEREST IS
COMPOUNDED QUARTERLY AND IS 4,
[150]
SO WE WOULD HAVE "A" = 2,500 x
THE QUANTITY 1 + 0.04
[159]
DIVIDED BY N WHICH IS NOW 4
[162]
RAISED TO THE POWER OF 5 x N
OR 5 x 4.
[168]
AND THEN FOR THE LAST QUESTION
[169]
WHEN THE INTEREST WAS COMPOUNDED
MONTHLY, N IS 12
[173]
SO WE HAVE "A" = 2,500 x 1 +
0.04 DIVIDED BY 12
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RAISED TO THE POWER OF 5 x 12.
[188]
NOTICE IN THIS EXAMPLE HERE,
[190]
WHEN WE DIVIDE THE INTEREST RATE
BY 4
[192]
WE'RE GETTING A QUARTERLY
INTEREST RATE
[194]
AND THEN 5 x 4 REPRESENTS THE
NUMBER OF QUARTERS IN 5 YEARS.
[199]
SO WE'RE REALLY JUST GETTING
COMMON UNITS
[200]
BY DIVIDING THE RATE BY 4
[203]
AND MULTIPLYING
THE NUMBER OF YEARS BY 4.
[205]
AND THE SAME THING
FOR THIS EQUATION,
[207]
DIVIDING THE INTEREST RATE
BY 12
[209]
GIVES US A MONTHLY INTEREST
RATE AND THEN 5 x 12
[212]
GIVES US THE NUMBER OF MONTHS
IN 5 YEARS.
[216]
SO NOW WE'LL GO TO THE
CALCULATOR AND DETERMINE
[218]
THE ACCOUNT BALANCE FOR EACH
TYPE OF COMPOUNDED INTEREST.
[222]
SO FIRST WE'LL HAVE 2,500 x THE
QUANTITY 1 + 0.04
[231]
AND RAISE THIS TO THE 5th POWER.
[236]
SO IF THE INTEREST
IS COMPOUNDED ANNUALLY,
[238]
THE ACCOUNT BALANCE WILL BE
$3,041.63.
[250]
IF THE INTEREST IS COMPOUNDED
QUARTERLY OR WHEN N = 4
[254]
WE WOULD HAVE 2,500 x THE
QUANTITY 1 + 0.04
[260]
DIVIDED BY 4 RAISED
TO THE POWER OF 5 x 4
[266]
WHICH WOULD BE
TO THE 20th POWER.
[270]
NOW THE ACCOUNT BALANCE IS
$3,050.48.
[283]
AND THEN FINALLY, WHEN THE
INTEREST IS COMPOUNDED MONTHLY
[287]
WE'D HAVE 2,500 x THE QUANTITY 1
+ 0.04 DIVIDED BY 12
[296]
AND ALL THIS IS RAISED TO THE
POWER OF 5 x 12
[300]
WHICH WOULD BE
TO THE 60th POWER.
[304]
AND NOW NOTICE THE BALANCE
IS $3,052.49.
[316]
SO AS WE CAN SEE
FROM THIS EXAMPLE,
[318]
THE MORE COMPOUNDS PER YEAR THE
HIGHER THE BALANCE WOULD BE.
[327]
OKAY, I HOPE THIS WAS HELPFUL.
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