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