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Effective Interest Rate (Effective Yield) - YouTube
Channel: Mathispower4u
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- WELCOME TO A LESSON ON THE
EFFECTIVE RATE OF INTEREST,
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OR EFFECTIVE YIELD.
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THE GOALS OF THIS VIDEO
ARE TO KNOW
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THE EFFECTIVE RATE
OF INTEREST FORMULA
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FOR COMPOUNDED
AND CONTINUOUS INTEREST,
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AND ALSO TO DETERMINE
EFFECTIVE RATE OF INTEREST.
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THE EFFECTIVE RATE OF INTEREST
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FOR COMPOUNDED
AND CONTINUOUS INTEREST
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IS THE EQUIVALENT ANNUAL
SIMPLE INTEREST RATE
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THAT WOULD YIELD
THE SAME RETURN AFTER 1 YEAR.
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REMEMBER SIMPLE INTEREST
PAYS INTEREST ONCE A YEAR.
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SO WE SHOULD ALREADY
BE FAMILIAR
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WITH THE COMPOUNDED INTEREST
FORMULA GIVEN HERE,
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AND THE CONTINUOUS INTEREST
FORMULA GIVEN HERE.
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SO TO DETERMINE
THE EFFECTIVE INTEREST RATE,
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WE HAVE A FORMULA
FOR COMPOUNDED INTEREST
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AS WELL AS CONTINUOUS
INTEREST.
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THE EFFECTIVE INTEREST RATE
IDENTIFIED BY R SUB B
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FOR COMPOUNDED INTEREST
= THE QUANTITY 1 + R
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DIVIDED BY N
RAISED TO THE NTH POWER - 1,
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AND FOR CONTINUOUS INTEREST WE
HAVE R SUB E = E TO THE R - 1.
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AND BEFORE WE ACTUALLY USE
THESE FORMULAS,
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I DO WANT TO SHOW
WHERE THEY COME FROM.
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SO FOR THE COMPOUNDED
EFFECTIVE INTEREST RATE
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WE WANT TO KNOW
THE SIMPLE INTEREST RATE
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THAT WOULD YIELD
THE SAME RETURN AFTER 1 YEAR
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AS AN ACCOUNT PAYING
COMPOUNDED INTEREST.
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SO IF WE SET THE SIMPLE
INTEREST RATE FORMULA HERE
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EQUAL TO THE COMPOUNDED
INTEREST FORMULA,
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WE WANT TO SOLVE
FOR THE SIMPLE INTEREST RATE,
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WHICH WOULD BE R SUB E.
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BUT REMEMBER, R SUB E
IS THE SIMPLE INTEREST RATE
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AFTER 1 YEAR,
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SO WE KNOW T WOULD HAVE
TO BE 1.
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SO IF T = 1
WE CAN REWRITE THIS
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AS P x THE QUANTITY 1 + R
SUB E RAISED TO THE 1st POWER
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= P x THE QUANTITY 1 + R
DIVIDED BY N
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RAISED TO THE N POWER,
SINCE T = 1.
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AND NOW TO SOLVE FOR R SUB E
WE CAN DIVIDE BOTH SIDES BY P.
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THIS SIMPLIFIES TO 1,
AND SO DOES THIS.
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SO NOW WE HAVE 1 + R SUB E
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= 1 + R SUB N
RAISED TO THE POWER OF N.
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AND NOW WE JUST SUBTRACT 1
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FROM BOTH SIDES
OF THE EQUATION,
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AND WE HAVE OUR FORMULA
FOR EFFECTIVE INTEREST RATE
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FOR COMPOUNDED INTEREST.
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OUR SUB E = THE QUANTITY 1 + R
DIVIDED BY N
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RAISED TO THE POWER OF N - 1.
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THIS IS THE SAME FORMULA THAT
WE SAW ON THE PREVIOUS SCREEN.
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LET'S GO
AND DERIVE THE FORMULA
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FOR THE EFFECTIVE INTEREST
RATE FOR CONTINUOUS INTEREST.
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AGAIN WE SET THE SIMPLE
INTEREST FORMULA
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EQUAL TO THE CONTINUOUS
INTEREST FORMULA,
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AND WE WANT TO SOLVE
FOR OUR SUB E.
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AGAIN WE KNOW T = 1,
OR 1 YEAR,
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SO WE CAN START
BY WRITING THIS
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AS P x THE QUANTITY
1 + R SUB E = P x E
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RAISED TO THE POWER OF R.
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NOW WE'LL DIVIDE
BOTH SIDES BY P.
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SO ON THE LEFT WE HAVE 1 + R
SUB E = E TO THE R.
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AGAIN SUBTRACT 1
ON BOTH SIDES OF THE EQUATION,
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AND WE HAVE THE EFFECTIVE
INTEREST RATE FORMULA
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FOR CONTINUOUS INTEREST.
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WE HAVE E TO THE R - 1.
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NOW THAT WE KNOW WHERE THESE
FORMULAS COME FROM,
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LET'S TAKE A LOOK
AT TWO EXAMPLES.
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WE WANT TO DETERMINE
THE EFFECTIVE INTEREST RATE
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FOR AN ACCOUNT PAYING 4%
QUARTERLY COMPOUNDED INTEREST.
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WELL, THE INTEREST RATE IS 4%,
SO R = 4%.
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EXPRESSED AS A DECIMAL
IT WOULD BE 0.04.
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AND BECAUSE THE INTEREST
IS PAID QUARTERLY,
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N IS THE NUMBER OF COMPOUNDS
PER YEAR, SO N =4,
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SINCE THERE ARE 4 QUARTERS
PER YEAR.
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THIS IS ALL WE NEED
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TO DETERMINE THE EFFECTIVE
INTEREST RATE.
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WE HAVE R SUB E = QUANTITY 1
+ R DIVIDED BY N, OR 0.04,
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DIVIDED BY 4 RAISED TO THE
POWER OF N, WHICH IS 4, - 1.
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NOW WE'LL GO AHEAD
AND GO TO THE CALCULATOR
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AND GET A DECIMAL
APPROXIMATION FOR THIS.
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SO WE HAVE 1 + 0.04
DIVIDED BY 4.
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WE WANT TO RAISE THIS
TO THE FOURTH POWER
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AND THEN SUBTRACT 1.
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WE'LL GO AHEAD AND ROUND THIS
TO FOUR DECIMAL PLACES,
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SO IT'S APPROXIMATELY 0.0406.
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TO CONVERT THIS TO A
PERCENTAGE WE MULTIPLY BY 100.
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THIS IS GOING TO BE EQUAL
TO 4.06%.
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SO THIS IS THE ANNUAL SIMPLE
INTEREST RATE
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THAT WOULD YIELD
THE SAME RETURN
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AS THIS ACCOUNT PAYING 4%
INTEREST COMPOUNDED QUARTERLY
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AFTER 1 YEAR.
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NOW LET'S TAKE A LOOK
AT OUR SECOND EXAMPLE.
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WE WANT TO DETERMINE
THE EFFECTIVE INTEREST RATE
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FOR AN ACCOUNT PAYING 4%
CONTINUOUS INTEREST.
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SO WE'LL NOTICE THAT R
IS STILL 4%, OR 0.04.
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SO WE HAVE R SUB E = E
RAISED TO THE POWER OF R,
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WHICH IS 0.04, - 1.
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SO NOW WE'LL GO BACK
TO THE CALCULATOR
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AND GET A DECIMAL
APPROXIMATION.
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SO IF WE PRESS
SECOND NATURAL LOG,
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THAT BRINGS UP E RAISED
TO THE POWER OF 0.04 - 1.
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AND AGAIN,
TO FOUR DECIMAL PLACES,
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WE HAVE APPROXIMATELY 0.0408.
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AS A PERCENTAGE
IT WOULD BE 4.08%.
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AND HOPEFULLY IT MAKES SENSE
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THAT THE EFFECTIVE INTEREST
RATE FOR CONTINUOUS INTEREST
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AT THE SAME RATE OF RETURN
WOULD BE SLIGHTLY HIGHER
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THAN AN ACCOUNT THAT PAYS
COMPOUNDED INTEREST,
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BECAUSE WE KNOW
THE MORE COMPOUNDS WE HAVE,
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THE BETTER RETURN
WE WOULD HAVE.
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OKAY. I HOPE YOU FOUND
THIS HELPFUL.
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