- WELCOME TO A LESSON
ON DETERMINING THE VALUE OF AN ANNUITY.
THE GOALS OF THIS VIDEO ARE TO DEFINE AN ANNUITY,
AND THEN DETERMINE THE VALUE OF ANNUITY.
AN ANNUITY IS A SEQUENCE OF EQUAL PAYMENTS
MADE AT EQUAL TIME INTERVALS.
FOR EXAMPLE, AN IRA, OR INDIVIDUAL RETIREMENT ACCOUNT,
IS AN ACCOUNT WHERE YOU MAY MAKE DEPOSITS ON A MONTHLY BASIS
IN ORDER TO ACCUMULATE ENOUGH MONEY TO RETIRE ON.
THE VALUE OF AN ANNUITY IS THE SUM OF ALL OF THE DEPOSITS
WITH ALL THE INTEREST EARNED.
NOW, THIS VIDEO WILL ONLY ADDRESS ORDINARY ANNUITIES
WHEN THE PAYMENTS ARE MADE AT THE END OF EACH PERIOD.
IF PAYMENTS ARE MADE AT THE BEGINNING OF EACH PERIOD
THE FORMULA WOULD BE DIFFERENT AND IT'S CALLED AN ANNUITY DUE.
BEFORE WE LOOK AT AN EXAMPLE
LETS REVIEW THE THREE BASIC FORMULAS
USED TO CALCULATE INTEREST.
FOR SIMPLE INTEREST, INTEREST IS PAID ONCE PER YEAR.
FOR A COMPOUNDED INTEREST
THE INTEREST IS PAID N TIMES PER YEAR.
SO IF IT'S COMPOUNDED MONTHLY N WOULD BE 12,
QUARTERLY N WOULD BE 4, AND SO-ON.
AND THEN FOR CONTINUOUS INTEREST,
INTEREST IS PAID CONTINUOUSLY.
LET'S TAKE A LOOK AT A BASIC EXAMPLE
BEFORE WE TAKE A LOOK AT THE FORMULA USED
TO DETERMINE THE VALUE OF AN ANNUITY.
LET'S SAY YOU DEPOSIT $500 AT THE END OF EACH YEAR FOR 3 YEARS
INTO AN ANNUITY THAT PAYS 5% ANNUAL SIMPLE INTEREST.
WHAT IS THE VALUE AT THE END OF THREE YEARS?
AS WE'RE MAKING THE DEPOSITS AT THE END OF THE YEAR,
AT THE END OF THE FIRST YEAR
WE WOULD MAKE OUR FIRST DEPOSIT OF $500.
SO THE VALUE WOULD JUST BE $500.
NOW, THE END OF THE SECOND YEAR,
WE WOULD EARN INTEREST ON THE FIRST $500,
AND THEN ADD ANOTHER $500 TO THE BALANCE.
SO WE WOULD HAVE $500 FROM THE FIRST YEAR x 1 + 0.05.
USING THE SIMPLE INTEREST FORMULA
WHERE T WOULD BE EQUAL TO 1 + THE NEW DEPOSIT OF $500.
LET'S SEE WHAT THE NEW BALANCE WOULD BE.
WE HAVE 500 x THIS WILL BE 1.05 + THE NEW DEPOSIT OF $500
AT THE END OF YEAR 2.
SO WE HAVE $1,025.
NOW, THROUGHOUT YEAR THREE THE $1,025 EARNS SIMPLE INTEREST 5%.
AND THEN AT THE END OF THE YEAR WE MAKE ANOTHER DEPOSIT OF $500.
THIS WOULD BE THE VALUE OF THE ANNUITY
AT THE END OF THREE YEARS GIVEN WE USE 5% SIMPLE INTEREST
WITH DEPOSIT AT THE END OF EACH YEAR.
SO WE HAVE $1,756.25.
SO FROM THIS EXAMPLE YOU CAN SEE THE PATTERN THAT IS DEVELOPING.
HOWEVER, IF WE WERE COMPOUNDING INTEREST MONTHLY
WITH MONTHLY DEPOSITS,
YOU CAN SEE THIS METHOD WOULD BE VERY TIME CONSUMING.
SO THERE IS A FORMULA THAT WE CAN USE
TO DETERMINE THE VALUE OF AN ANNUITY
BASED UPON THE NUMBER OF COMPOUNDS PER YEAR
AND THE NUMBER OF DEPOSITS PER YEAR.
AND HERE IT IS.
SO R WOULD BE THE ANNUAL NOMINAL INTEREST RATE,
T WOULD BE THE NUMBER OF YEARS,
N IS THE NUMBER OF COMPOUNDS PER YEAR,
P IS THE AMOUNT OF EACH DEPOSIT,
AND "A" WOULD BE THE VALUE OF THE ANNUITY.
AND IF WE WERE DEALING WITH SIMPLE INTEREST
N WOULD BE EQUAL TO ONE.
LET'S GO AND TAKE A LOOK AT A COUPLE OF EXAMPLES.
AT AGE 30 YOU DEPOSIT $150 AT THE END OF EACH MONTH
INTO AN IRA THAT PAYS 4% INTEREST COMPOUNDED MONTHLY.
AT THE AGE OF 65 WHAT WOULD THE VALUE OF ANNUITY BE
AND HOW MUCH INTEREST DID YOU EARN?
SO THIS WOULD BE A GOOD EXAMPLE IF YOU START SAVING EARLY,
WOULD SAVING $150 A MONTH BE ENOUGH TO RETIRE ON?
LET'S SEE IF WE CAN DETERMINE ALL OF THE VALUES HERE
IN THE FORMULA.
SO THE VALUE OF THE ANNUITY IS GOING TO BE EQUAL TO P,
WHICH IS OUR MONTHLY DEPOSIT x THE QUANTITY 1
+ THE RATE EXPRESSED AS A DECIMAL, THAT'LL BE 0.04,
DIVIDED BY N,
THE NUMBER OF COMPOUNDS PER YEAR, IT'S MONTHLY SO N IS 12.
AND RAISE THIS TO THE POWER OF N x T
WHERE N IS THE NUMBER OF COMPOUNDS, THAT'S 12, PER YEAR,
AND T IS THE NUMBER OF YEARS.
WELL, FROM 30 TO 65 THAT'LL BE 35 YEARS - 1,
THERE'S OUR NUMERATOR.
AND DIVIDE THIS BY R DIVIDED BY N
WHERE R IS 0.04 AND N WOULD BE 12.
SO FROM HERE WE'LL GO THE CALCULATOR.
WE'LL START WITH THE PARENTHESIS FOR THE NUMERATOR OF 150.
AND THEN WE'LL HAVE TWO OPEN PARENTHESIS
1 + 0.04 DIVIDED BY 12.
AND RAISE THIS TO THE POWER OF 12 x 35 - 1.
AND THEN A CLOSED PARENTHESIS FOR OUR NUMERATOR,
AND THEN WE'LL DIVIDE THIS BY A DENOMINATOR OF 0.04
DIVIDED BY 12,
$137,059.65 APPROXIMATELY.
NOW, THE SECOND QUESTION ASKED US,
HOW MUCH INTEREST DID WE EARN?
WELL, THIS IS THE BALANCE OF THE ACCOUNT,
BUT TO FIGURE OUT HOW MUCH INTEREST WE EARN
WE HAVE TO DETERMINE HOW MUCH MONEY WE ACTUALLY DEPOSITED
INTO THE ACCOUNT.
WE CAN DO THAT PRETTY EASILY.
WE DEPOSITED $150 x 12 FOR EACH YEAR,
AND THEN WE DEPOSITED THIS FOR 35 YEARS,
SO MULTIPLY THIS BY 35.
THAT'LL TELL US THE TOTAL AMOUNT OF OUR DEPOSITS.
WE DEPOSITED A TOTAL OF $63,000 INTO THIS ANNUITY.
SO IF WE TAKE THE BALANCE OF THE ACCOUNT
AND WE SUBTRACT OUR DEPOSITS,
THIS WILL GIVE US THE TOTAL INTEREST EARNED.
AND THAT'S JUST GOING TO GIVE US $74,059.64.
NOW, FOR THE SECOND EXAMPLE,
WE'RE GOING TO LOOK AT THE SAME SITUATION
EXCEPT INSTEAD OF STARTING AT AGE 30,
WHAT HAPPENS IF YOU START AT AGE 40?
HOW WILL THAT CHANGE THE BALANCE,
AS WELL AS THE AMOUNT OF INTEREST EARNED?
SO THE FORMULA THAT WILL LOOK ALMOST EXACTLY THE SAME
EXCEPT NOW T IS GOING TO BE 25 INSTEAD OF 35
BECAUSE WE'RE STARTING SAVE 10 YEARS LATER.
SO OUR EXPONENT HERE IS GOING TO BE 12 X 25 INSTEAD OF 12 X 35.
THE INTEREST IS THE SAME AS WELL AS BEING COMPOUNDED MONTHLY.
SO LETS SEE WHAT THIS WOULD GIVE US.
SO WE'LL GO THROUGH THE SAME PROCESS ON OUR CALCULATOR.
JUST REMEMBER WE NEED ANOTHER SET OF PARENTHESIS
FOR THE NUMERATOR AND DENOMINATOR
TO KEEP EVERYTHING STRAIGHT.
SO NOW NOTICE THE BALANCE DROPPED CONSIDERABLY.
NOW IT'S ONLY $77,119.42 APPROXIMATELY.
SO LET'S GO AHEAD AND DETERMINE THE TOTAL INTEREST EARNED.
SO THE TOTAL AMOUNT DEPOSITED WOULD BE 150 PER MONTH,
SO x 12 FOR 25 YEARS.
150 x 12 x 25 = $45,000.
THIS IS THE TOTAL AMOUNT DEPOSITED OVER THE 25 YEARS.
SO IF WE TAKE THE BALANCE AND SUBTRACT OUT THE AMOUNT
THAT WE DEPOSITED
THAT'LL TELL US THE TOTAL INTEREST EARNED.
WE'RE GOING TO HAVE $32,119.43.
SO, AGAIN, IF WE SAVE FOR 25 YEARS
WE HAVE A BALANCE OF APPROXIMATELY $77,000.
WE'VE EARNED ABOUT $32,000 OF INTEREST.
BUT IF WE START JUST 10 YEARS EARLIER,
WE HAVE A MUCH LARGER BALANCE IN THE ACCOUNT
OF APPROXIMATELY $137,000.
IT WOULD BE MORE THAN DOUBLE THE AMOUNT OF INTEREST EARNED.
SO I GUESS THAT TELLS US THAT IF IT'S POSSIBLE,
WE SHOULD START SAVING FOR RETIREMENT AS SOON AS WE CAN.
I HOPE YOU FOUND THIS HELPFUL.
IN PART 2 WE'LL TAKE A LOOK
AT HOW WE CAN DETERMINE THESE VALUES VERY QUICKLY
ON THE TI-84 GRAPHING CALCULATOR.