- WELCOME TO A LESSON ON CONTINUOUS INTEREST.
THE GOALS OF THIS VIDEO ARE TO DESCRIBE
WHERE THE CONTINUOUS INTEREST FORMULA COMES FROM,
AND ALSO TO SOLVE PROBLEMS
USING THE CONTINUOUS INTEREST FORMULA.
THE CONTINUOUS INTEREST FORMULA IS THE SAME FORMULA
USED FOR EXPONENTIAL GROWTH,
EXCEPT IT'S OFTEN STATED USING DIFFERENT VARIABLES.
THE FORMULA FOR CONTINUOUS INTEREST CAN BE DERIVED
FROM THE COMPOUNDED INTEREST FORMULA,
WHICH WE'LL DO IN JUST A MINUTE.
BUT FIRST WE NEED SOME BACKGROUND INFORMATION.
E IS A VERY SPECIAL NUMBER, VERY SIMILAR TO PI.
IT'S AN IRRATIONAL NUMBER,
WHICH MEANS IT'S NOT- TERMINATING AND NON-REPEATING.
E IS OFTEN CALLED EULER'S NUMBER,
AND ONE DEFINITION OF E IS THE LIMIT
AS X APPROACHES INFINITY OF THE QUANTITY 1 + 1
DIVIDED BY X RAISED TO THE POWER OF X.
A DECIMAL APPROXIMATION FOR E IS APPROXIMATELY 2.718.
NOW MAYBE YOU HAVEN'T SEEN THIS LIMIT NOTATION HERE
BEFORE.
REALLY WHAT IT MEANS IS AS X APPROACHES INFINITY,
OR X GETS BIGGER AND BIGGER,
THIS EXPRESSION HERE IS APPROACHING THE VALUE OF E.
AND TO GET A BETTER FEEL FOR THIS
WE'LL USE A GRAPHING CALCULATOR TO ILLUSTRATE THIS.
TO SAVE A LITTLE BIT OF TIME,
I'VE ALREADY TYPED IN THE EXPRESSION 1 + 1 DIVIDED BY X
RAISED TO THE POWER OF X INTO Y1.
AND NOW I'M GOING TO GO TO THE TABLE FEATURE,
AND I HAVE THE TABLE FEATURE ON ASK,
WHICH WILL ALLOW US TO TYPE IN ANY X VALUE
THAT WE WANT TO DETERMINE THE VALUE OF Y.
SO IF WE TYPE IN X = 1,
NOTICE HOW THE EXPRESSION IS EQUAL TO 2,
AND AS WE INCREASE X WE'RE GOING TO SEE
THAT THE Y VALUES WILL APPROACH 2.718.
IF WE TYPE IN X = 10, 100, AND 1000,
WE CAN SEE THIS EXPRESSION IS QUICKLY APPROACHING A VALUE
VERY CLOSE TO E.
IF WE TYPE IN 10,000,
NOTICE HOW IT'S VERY CLOSE TO THE VALUE OF E.
AND SO WITH THIS DEFINITION,
AND THE COMPOUNDED INTEREST FORMULA GIVEN HERE,
WE CAN DERIVE THE CONTINUOUS INTEREST FORMULA
GIVEN HERE AT THE BOTTOM.
REMEMBER N IS THE NUMBER OF COMPOUNDS PER YEAR,
AND SO AS N APPROACHES INFINITY
WE WOULD BE APPROACHING THE CONTINUOUS INTEREST FORMULA.
SO TO SHOW THIS,
WE'RE GOING TO MANIPULATE THE RIGHT SIDE OF THIS EQUATION.
THE FIRST THING WE'RE GOING TO DO IS REWRITE R DIVIDED BY N
AS 1 DIVIDED BY THE RECIPROCAL OF THIS,
OR 1 DIVIDED BY N/R.
SO WE'D HAVE "A" = P x 1 + 1 DIVIDED BY N/R.
AND NOW LET'S STOP HERE FOR A MOMENT TO VERIFY THIS IS TRUE.
WE HAVE 1 DIVIDED N/R.
REMEMBER THIS MEANS 1 DIVIDED BY N/R,
WHICH WE CAN WRITE AS A MULTIPLICATION PROBLEM
AS 1 x R/N,
WHICH DOES EQUAL R/N, WHICH IS WHAT WE STARTED WITH.
OKAY, NOW THE NEXT THING WE'RE GOING TO DO
IS PLAY AROUND WITH SOME PROPERTIES OF EXPONENTS.
INSTEAD OF RAISING THIS TO THE N x T POWER,
WE'RE GOING TO RAISE THIS TO THE N DIVIDED BY R POWER,
AND THEN WE'RE GOING TO RAISE THIS TO THE POWER OF R x T.
REMEMBER WHEN WE HAVE POWERS TO POWERS WE MULTIPLY THEM.
SO LET'S MAKE SURE THIS PRODUCT HERE WOULD BE N x T.
SO IF WE HAVE N DIVIDED BY R,
WE MULTIPLY THIS BY RT OR RT/1.
NOTICE HOW THE Rs WOULD SIMPLIFY OUT,
LEAVING US WITH NT, OUR INITIAL EXPONENT.
SO THE RIGHT SIDES OF THESE EQUATIONS
ARE EQUAL TO EACH OTHER.
NOW NOTICE WE HAVE N DIVIDED BY R HERE,
AND N DIVIDED BY R HERE.
WE'RE GOING TO LET X = N DIVIDED BY R
AND PERFORM A SUBSTITUTION.
IF WE DO THAT WE CAN WRITE THIS
AS "A" = P x THE QUANTITY 1 + 1/X.
NOTICE HOW THE EXPONENT HERE WOULD ALSO BE X
RAISED TO THE POWER OF R x T.
NOW AT THE BEGINNING WE SAID AS N APPROACHES INFINITY
THIS FORMULA HERE SHOULD APPROACH
OUR CONTINUOUS INTEREST FORMULA.
SO AS N APPROACHES INFINITY,
SINCE R IS A FIXED INTEREST RATE,
X WOULD ALSO APPROACH INFINITY.
AND NOTICE IF X APPROACHES INFINITY THIS EXPRESSION HERE
IS GOING TO APPROACH THE VALUE OF E
THAT WE SAW ON THE PREVIOUS SCREEN,
WHICH WOULD GIVE US OUR CONTINUOUS INTEREST FORMULA.
WE WOULD HAVE "A" = P.
ALL OF THIS IS EQUAL TO E, AS X APPROACHES INFINITY.
SO WE HAVE JUST E RAISED TO THE POWER OF R x T,
WHICH IS OUR CONTINUOUS INTEREST FORMULA.
NOW THAT WE KNOW WHERE THIS FORMULA COMES FROM,
LET'S TAKE A LOOK AT A COUPLE OF EXAMPLES.
IF YOU INVEST $3,200 FOR 4 YEARS
AT 6% CONTINUOUS INTEREST,
WHAT IS THE ENDING BALANCE?
SO "A", THE AMOUNT AFTER, IS EQUAL TO P,
THE PRINCIPAL OR STARTING AMOUNT,
$3200 x E RAISED TO THE POWER OF R x T.
R IS THE INTEREST RATE, AND T IS THE TIME IN YEARS.
SO R = 6%, WHICH MUST BE EXPRESSED AS A DECIMAL,
WHICH WOULD BE 0.06.
IF WE DROP THE PERCENT SIGN AND DIVIDE BY 100,
THIS WOULD BE OUR DECIMAL.
SO WE HAVE 0.06 x T.
T = 4, FOR FOUR YEARS.
AND NOW WE GO TO THE CALCULATOR
TO GET A DECIMAL APPROXIMATION.
WE HAVE 3200.
WE PRESS SECOND LN OR SECOND NATURAL LOG.
THAT BRINGS UP E RAISED TO THE POWER OF--WE HAVE 0.06 x 4.
IF WE ROUND THIS TO THE NEAREST PENNY
WE WOULD HAVE APPROXIMATELY $4068.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
HOW MUCH DO YOU NEED TO INVEST TODAY
TO HAVE $5000 IN 3 YEARS,
IF THE ACCOUNT PAYS 8% CONTINUOUS INTEREST?
SO NOW WE'RE GIVEN THE AMOUNT WE WANT AFTER 3 YEARS, OR "A".
WE WANT TO DETERMINE THE STARTING AMOUNT, OR P.
SO FOR THIS PROBLEM WE WOULD HAVE 5000 = P x E
RAISED TO THE POWER OF R x T.
R = 8%, OR AS A DECIMAL, 0.08, AND THE TIME IN YEARS IS 3.
SO WE'D HAVE 5000 = P x E RAISED TO THE POWER OF 0.24.
SO WE'LL DIVIDE BOTH SIDES BY E TO THE POWER OF 0.24
NOTICE ON THE RIGHT SIDE THIS SIMPLIFIES TO 1,
SO P, OR THE STARTING AMOUNT,
MUST BE EQUAL TO THIS QUOTIENT HERE.
SO WE'LL GO BACK TO THE CALCULATOR.
5000 DIVIDED BY E RAISED TO THE POWER OF 0.24,
SO YOU WOULD NEED TO INVEST $3,933.14 TODAY
TO HAVE $5,000 IN THIS ACCOUNT IN 3 YEARS.
OKAY. I HOPE YOU FOUND THIS LESSON HELPFUL.