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Determining the Effective Yield of an Investment - YouTube
Channel: Mathispower4u
[1]
- WELCOME TO A LESSON
[2]
ON DETERMINING EFFECTIVE
ANNUAL YIELD.
[5]
LET'S START BY REVIEWING THE
THREE DIFFERENT WAYS THAT YIELD
[7]
OR INTEREST CAN BE CALCULATED.
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THE MOST BASIC FORMULA
IS SIMPLE INTEREST,
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WHERE INTEREST IS PAID
ONCE A YEAR.
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THE MOST COMMON INTEREST
IS COMPOUND INTEREST,
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WHERE THE INTEREST IS PAID
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A CERTAIN NUMBER OF TIMES
PER YEAR,
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IN THIS CASE N TIMES PER YEAR.
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AND THERE'S ALSO SOMETHING
CALLED CONTINUOUS INTEREST,
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WHERE THE INTEREST IS PAID
CONTINUOUSLY
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THROUGHOUT THE YEAR.
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AND YOU SHOULD ALREADY
BE FAMILIAR
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WITH ALL THREE
OF THESE FORMULAS.
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THE EFFECTIVE ANNUAL YIELD,
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SOMETIMES CALLED
EFFECTIVE ANNUAL RATE
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OR ANNUAL EQUIVALENT RATE,
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IS THE SIMPLE INTEREST RATE THAT
WOULD PRODUCE THE SAME RETURN
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AT THE END OF ONE YEAR
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AS AN ACCOUNT THAT PAID
COMPOUNDED INTEREST.
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AND THE COMPOUNDED INTEREST RATE
IS CALLED THE NOMINAL RATE.
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SO THE EFFECTIVE ANNUAL YIELD
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WOULD BE THE SIMPLE
INTEREST RATE
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THAT PRODUCES THE SAME RETURN
AS THE NOMINAL RATE.
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AND THE EFFECTIVE ANNUAL YIELD
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IS A WAY OF COMPARING
DIFFERENT INVESTMENT ACCOUNTS
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THAT PAY DIFFERENT
COMPOUNDED INTEREST.
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AND IT CAN ALSO BE USED
FOR LOANS,
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SINCE A LOAN IS AN INVESTMENT
FOR THE PERSON OR COMPANY
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MAKING THE LOAN.
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AND WE SHOULD NOTE
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THAT SOME INSTITUTIONS CALCULATE
AER DIFFERENTLY
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BY INCLUDING OR NOT INCLUDING
CERTAIN FEES.
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SO IT'S ALWAYS IMPORTANT
TO READ THE FINE PRINT.
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BEFORE WE TAKE A LOOK
AT THE FORMULA
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FOR EFFECTIVE ANNUAL YIELD,
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LETS LOOK AT AN EXAMPLE.
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DETERMINE THE FUTURE VALUE
OF AN INVESTMENT OF $2,000
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THAT PAYS 6% INTEREST
COMPOUNDED MONTHLY FOR ONE YEAR.
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SO HERE'S OUR COMPOUNDED
INTEREST FORMULA.
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LET'S GO AHEAD AND ANSWER
THIS FIRST QUESTION.
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SO THE AMOUNT AFTER THE ONE YEAR
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IS GOING TO BE EQUAL
TO THE PRINCIPAL,
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WHICH IS $2,000 x THE QUANTITY 1
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+ THE RATE
EXPRESSED AS A DECIMAL.
[111]
SO 6% WOULD BE .06 DIVIDED BY N,
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THE NUMBER OF COMPOUNDS PER YEAR
THAT'S PAID MONTHLY.
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SO N WOULD BE 12.
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WE RAISE THIS TO THE POWER
OF N x T.
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AGAIN, N IS THE NUMBER OF
COMPOUNDS, SO THAT WOULD BE 12.
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AND THE TIME IN YEARS IS 1 YEAR.
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SO LET'S GO AHEAD AND FIGURE OUT
WHAT THIS FUTURE VALUE WOULD BE.
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SO WE'D HAVE 2,000,
OPEN PARENTHESIS,
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1 + .06 DIVIDED BY 12,
CLOSE PARENTHESES,
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RAISED TO THE 12th POWER.
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SO THE ENDING BALANCE WOULD BE
$2,123.36.
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THE NEXT SENTENCE SAYS THEN USE
THE SIMPLE INTEREST FORMULA
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TO DETERMINE WHAT THE SIMPLE
INTEREST RATE WOULD NEED TO BE
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TO HAVE THE SAME RETURN.
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AND THE SIMPLE INTEREST RATE
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WOULD BE THE EFFECTIVE
ANNUAL YIELD.
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SO WE WANT TO USE THIS
SIMPLE INTEREST FORMULA
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TO DETERMINE WHAT R WOULD BE
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IF WE KNOW THAT THE AMOUNT AFTER
ONE YEAR IS EQUAL TO $2,123.36.
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SO THE PRINCIPAL WAS $2,000.
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1 + OUR INTEREST RATE,
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THAT'S GOING TO BE
OUR EFFECTIVE ANNUAL YIELD,
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RAISED TO THE POWER OF T,
BUT T = 1.
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SO IF WE SOLVE THIS FOR R, WE'LL
HAVE THE EFFECTIVE ANNUAL RATE.
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SO WE'LL DIVIDE BOTH SIDES
BY 2,000.
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THAT'S GOING TO GIVE US 1.06168.
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AND WE'LL SUBTRACT 1
FROM BOTH SIDES,
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AND SO THE SIMPLE INTEREST RATE
WITH THE SAME RETURN
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WOULD BE .06168,
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WHICH WILL CONVERT TO 6.17%.
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SO 6% WOULD BE
THE NOMINAL INTEREST RATE.
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6.17% WOULD BE THE EFFECTIVE
ANNUAL RATE,
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OR EFFECTIVE ANNUAL YIELD.
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NOW LET'S GO AHEAD AND TAKE
A LOOK AT THE SHORTCUT FORMULA
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TO DETERMINE THE EFFECTIVE
ANNUAL YIELD.
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SO HERE'S THE FORMULA
THAT WE CAN USE
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TO DETERMINE
THE EFFECTIVE ANNUAL RATE,
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WHERE Y WOULD BE THE EFFECTIVE
ANNUAL RATE,
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R IS THE INTEREST RATE
AS A DECIMAL,
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AND N IS THE NUMBER OF COMPOUNDS
PER YEAR.
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BEFORE WE TAKE A LOOK
AT SOME ADDITIONAL EXAMPLES,
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LETS TAKE A LOOK AT
WHERE THIS FORMULA CAME FROM.
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REMEMBER ON OUR FIRST EXAMPLE
WE DETERMINED THE AMOUNT, "A,"
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USING COMPOUNDED INTEREST,
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AND THEN WE USED THAT AMOUNT
TO DETERMINE R
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IN THE SIMPLE INTEREST FORMULA.
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SO IF WE REPLACE "A"
IN THE SIMPLE INTEREST FORMULA
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WITH THE COMPOUNDED
INTEREST FORMULA,
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AND THEN SOLVE FOR R
IN THE SIMPLE INTEREST FORMULA,
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IT SHOULD GIVE US THE FORMULA
FOR EFFECTIVE ANNUAL YIELD.
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LET'S GO AHEAD
AND GIVE IT A TRY.
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REMEMBER, T = 1, SO WE'D HAVE
P x THE QUANTITY 1
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+ R/N TO THE POWER OF N
= THE SIMPLE INTEREST FORMULA P
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x THE QUANTITY 1
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+ AND THEN INSTEAD OF USING
LITTLE R WE'LL USE BIG Y.
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RAISED TO THE POWER OF T,
BUT T = 1.
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SO IF WE SOLVE THIS EQUATION
FOR Y,
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WE SHOULD HAVE OUR EFFECTIVE
ANNUAL YIELD FORMULA.
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WE CAN DIVIDE BOTH SIDES BY P.
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THAT WOULD SIMPLIFY OUT.
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SO WE'D HAVE 1 + R/N RAISED
TO THE POWER OF N = Y + 1.
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AND IF WE SUBTRACT 1 ON
BOTH SIDES WE HAVE OUR FORMULA.
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Y = THE QUANTITY 1 + R/N
TO THE NTH POWER - 1.
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LET'S GO AHEAD AND TAKE A LOOK
AT TWO MORE EXAMPLES.
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HERE IT SAYS TO DETERMINE
THE EFFECTIVE ANNUAL YIELD
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OF THE FOLLOWING TWO ACCOUNTS
IF YOU INVEST $5,000.
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AND NOTICE THAT TO DETERMINE
THE EFFECTIVE ANNUAL RATE
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IT'S NOT REQUIRED THAT WE KNOW
THE AMOUNT OF THE INVESTMENT.
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SO WE HAVE Y = 1 + .049.
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NOW, THAT'S COMPOUNDED MONTHLY,
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SO N = 12 RAISED
TO THE 12th POWER - 1.
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AND THEN FOR THE SECOND EXAMPLE
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WE'D HAVE Y = THE QUANTITY 1
+ .08.
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NOW, THIS IS COMPOUNDED
QUARTERLY,
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SO THERE'S FOUR QUARTERS
IN ONE YEAR, SO N = 4.
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LET'S GO AHEAD AND SEE
WHAT THIS GIVES US.
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HERE'S OUR FIRST
EFFECTIVE ANNUAL YIELD.
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LET'S GO AHEAD AND DETERMINE
THE SECOND ONE WHILE WE'RE HERE.
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SO LET'S GO AHEAD AND CONVERT
THESE TO PERCENTAGES.
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REMEMBER, WE CONVERT A DECIMAL
TO A PERCENT
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BY MULTIPLYING BY 100.
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SO THE FIRST EFFECTIVE
ANNUAL YIELD
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WILL BE APPROXIMATELY 5.01%,
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AND THE SECOND EFFECTIVE
ANNUAL YIELD
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WILL BE APPROXIMATELY 8.24%.
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REMEMBER, WHAT THIS TELLS US
IS THAT
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IN ORDER TO GET THE SAME RETURN
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AS A 4.9% RETURN COMPOUNDED
MONTHLY,
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IT WOULD TAKE A SIMPLE
INTEREST RATE OF 5.01%.
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AND FOR 8% COMPOUNDED QUARTERLY,
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IT WOULD TAKE A SIMPLE INTEREST
OF 8.24% FOR THE SAME RETURN.
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I HOPE YOU HAVE FOUND
THIS HELPFUL.
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IN THE NEXT VIDEO WE'LL TAKE A
LOOK AT HOW WE CAN USE A TI-84
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TO DETERMINE THE EFFECTIVE
ANNUAL YIELD VERY QUICKLY.
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THANK YOU FOR WATCHING.
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