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Determining The Value of an Annuity - YouTube
Channel: Mathispower4u
[1]
- WELCOME TO A LESSON
[2]
ON DETERMINING THE VALUE
OF AN ANNUITY.
[4]
THE GOALS OF THIS VIDEO
ARE TO DEFINE AN ANNUITY,
[7]
AND THEN DETERMINE THE VALUE
OF ANNUITY.
[9]
AN ANNUITY IS A SEQUENCE
OF EQUAL PAYMENTS
[12]
MADE AT EQUAL TIME INTERVALS.
[14]
FOR EXAMPLE, AN IRA, OR
INDIVIDUAL RETIREMENT ACCOUNT,
[18]
IS AN ACCOUNT WHERE YOU MAY MAKE
DEPOSITS ON A MONTHLY BASIS
[22]
IN ORDER TO ACCUMULATE
ENOUGH MONEY TO RETIRE ON.
[25]
THE VALUE OF AN ANNUITY IS
THE SUM OF ALL OF THE DEPOSITS
[28]
WITH ALL THE INTEREST EARNED.
[30]
NOW, THIS VIDEO WILL ONLY
ADDRESS ORDINARY ANNUITIES
[33]
WHEN THE PAYMENTS ARE MADE
AT THE END OF EACH PERIOD.
[36]
IF PAYMENTS ARE MADE
AT THE BEGINNING OF EACH PERIOD
[38]
THE FORMULA WOULD BE DIFFERENT
AND IT'S CALLED AN ANNUITY DUE.
[41]
BEFORE WE LOOK AT AN EXAMPLE
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LETS REVIEW
THE THREE BASIC FORMULAS
[45]
USED TO CALCULATE INTEREST.
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FOR SIMPLE INTEREST,
INTEREST IS PAID ONCE PER YEAR.
[51]
FOR A COMPOUNDED INTEREST
[53]
THE INTEREST IS PAID N TIMES
PER YEAR.
[55]
SO IF IT'S COMPOUNDED MONTHLY
N WOULD BE 12,
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QUARTERLY N WOULD BE 4,
AND SO-ON.
[61]
AND THEN FOR CONTINUOUS
INTEREST,
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INTEREST IS PAID CONTINUOUSLY.
[66]
LET'S TAKE A LOOK
AT A BASIC EXAMPLE
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BEFORE WE TAKE A LOOK
AT THE FORMULA USED
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TO DETERMINE THE VALUE
OF AN ANNUITY.
[72]
LET'S SAY YOU DEPOSIT $500 AT
THE END OF EACH YEAR FOR 3 YEARS
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INTO AN ANNUITY THAT PAYS 5%
ANNUAL SIMPLE INTEREST.
[80]
WHAT IS THE VALUE
AT THE END OF THREE YEARS?
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AS WE'RE MAKING THE DEPOSITS
AT THE END OF THE YEAR,
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AT THE END OF THE FIRST YEAR
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WE WOULD MAKE OUR FIRST DEPOSIT
OF $500.
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SO THE VALUE WOULD JUST BE $500.
[93]
NOW, THE END OF THE SECOND YEAR,
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WE WOULD EARN INTEREST
ON THE FIRST $500,
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AND THEN ADD ANOTHER $500
TO THE BALANCE.
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SO WE WOULD HAVE $500
FROM THE FIRST YEAR x 1 + 0.05.
[109]
USING THE SIMPLE INTEREST
FORMULA
[111]
WHERE T WOULD BE EQUAL TO 1
+ THE NEW DEPOSIT OF $500.
[117]
LET'S SEE WHAT THE NEW BALANCE
WOULD BE.
[119]
WE HAVE 500 x THIS WILL BE 1.05
+ THE NEW DEPOSIT OF $500
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AT THE END OF YEAR 2.
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SO WE HAVE $1,025.
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NOW, THROUGHOUT YEAR THREE THE
$1,025 EARNS SIMPLE INTEREST 5%.
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AND THEN AT THE END OF THE YEAR
WE MAKE ANOTHER DEPOSIT OF $500.
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THIS WOULD BE THE VALUE
OF THE ANNUITY
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AT THE END OF THREE YEARS
GIVEN WE USE 5% SIMPLE INTEREST
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WITH DEPOSIT
AT THE END OF EACH YEAR.
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SO WE HAVE $1,756.25.
[181]
SO FROM THIS EXAMPLE YOU CAN SEE
THE PATTERN THAT IS DEVELOPING.
[184]
HOWEVER, IF WE WERE COMPOUNDING
INTEREST MONTHLY
[186]
WITH MONTHLY DEPOSITS,
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YOU CAN SEE THIS METHOD WOULD BE
VERY TIME CONSUMING.
[190]
SO THERE IS A FORMULA
THAT WE CAN USE
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TO DETERMINE THE VALUE
OF AN ANNUITY
[194]
BASED UPON THE NUMBER
OF COMPOUNDS PER YEAR
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AND THE NUMBER OF DEPOSITS
PER YEAR.
[199]
AND HERE IT IS.
[201]
SO R WOULD BE THE ANNUAL
NOMINAL INTEREST RATE,
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T WOULD BE THE NUMBER OF YEARS,
[206]
N IS THE NUMBER OF COMPOUNDS
PER YEAR,
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P IS THE AMOUNT OF EACH DEPOSIT,
[212]
AND "A" WOULD BE THE VALUE
OF THE ANNUITY.
[215]
AND IF WE WERE DEALING
WITH SIMPLE INTEREST
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N WOULD BE EQUAL TO ONE.
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LET'S GO AND TAKE A LOOK
AT A COUPLE OF EXAMPLES.
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AT AGE 30 YOU DEPOSIT $150
AT THE END OF EACH MONTH
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INTO AN IRA THAT PAYS 4%
INTEREST COMPOUNDED MONTHLY.
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AT THE AGE OF 65 WHAT WOULD
THE VALUE OF ANNUITY BE
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AND HOW MUCH INTEREST
DID YOU EARN?
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SO THIS WOULD BE A GOOD EXAMPLE
IF YOU START SAVING EARLY,
[238]
WOULD SAVING $150 A MONTH
BE ENOUGH TO RETIRE ON?
[242]
LET'S SEE IF WE CAN DETERMINE
ALL OF THE VALUES HERE
[244]
IN THE FORMULA.
[245]
SO THE VALUE OF THE ANNUITY
IS GOING TO BE EQUAL TO P,
[248]
WHICH IS OUR MONTHLY DEPOSIT
x THE QUANTITY 1
[256]
+ THE RATE EXPRESSED AS A
DECIMAL, THAT'LL BE 0.04,
[261]
DIVIDED BY N,
[262]
THE NUMBER OF COMPOUNDS PER
YEAR, IT'S MONTHLY SO N IS 12.
[268]
AND RAISE THIS TO THE POWER OF N
x T
[271]
WHERE N IS THE NUMBER OF
COMPOUNDS, THAT'S 12, PER YEAR,
[274]
AND T IS THE NUMBER OF YEARS.
[276]
WELL, FROM 30 TO 65
THAT'LL BE 35 YEARS - 1,
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THERE'S OUR NUMERATOR.
[285]
AND DIVIDE THIS BY R
DIVIDED BY N
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WHERE R IS 0.04
AND N WOULD BE 12.
[294]
SO FROM HERE WE'LL GO
THE CALCULATOR.
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WE'LL START WITH THE PARENTHESIS
FOR THE NUMERATOR OF 150.
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AND THEN WE'LL HAVE
TWO OPEN PARENTHESIS
[306]
1 + 0.04 DIVIDED BY 12.
[311]
AND RAISE THIS TO THE POWER
OF 12 x 35 - 1.
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AND THEN A CLOSED PARENTHESIS
FOR OUR NUMERATOR,
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AND THEN WE'LL DIVIDE THIS
BY A DENOMINATOR OF 0.04
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DIVIDED BY 12,
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$137,059.65 APPROXIMATELY.
[344]
NOW, THE SECOND QUESTION
ASKED US,
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HOW MUCH INTEREST DID WE EARN?
[347]
WELL, THIS IS THE BALANCE
OF THE ACCOUNT,
[350]
BUT TO FIGURE OUT
HOW MUCH INTEREST WE EARN
[351]
WE HAVE TO DETERMINE HOW MUCH
MONEY WE ACTUALLY DEPOSITED
[353]
INTO THE ACCOUNT.
[355]
WE CAN DO THAT PRETTY EASILY.
[356]
WE DEPOSITED $150 x 12
FOR EACH YEAR,
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AND THEN WE DEPOSITED THIS
FOR 35 YEARS,
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SO MULTIPLY THIS BY 35.
[369]
THAT'LL TELL US THE TOTAL AMOUNT
OF OUR DEPOSITS.
[374]
WE DEPOSITED A TOTAL OF $63,000
INTO THIS ANNUITY.
[383]
SO IF WE TAKE THE BALANCE
OF THE ACCOUNT
[385]
AND WE SUBTRACT OUR DEPOSITS,
[389]
THIS WILL GIVE US
THE TOTAL INTEREST EARNED.
[392]
AND THAT'S JUST GOING TO GIVE US
$74,059.64.
[399]
NOW, FOR THE SECOND EXAMPLE,
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WE'RE GOING TO LOOK
AT THE SAME SITUATION
[402]
EXCEPT INSTEAD OF STARTING
AT AGE 30,
[405]
WHAT HAPPENS IF YOU START
AT AGE 40?
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HOW WILL THAT CHANGE
THE BALANCE,
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AS WELL AS THE AMOUNT
OF INTEREST EARNED?
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SO THE FORMULA THAT WILL LOOK
ALMOST EXACTLY THE SAME
[413]
EXCEPT NOW T IS GOING TO BE 25
INSTEAD OF 35
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BECAUSE WE'RE STARTING
SAVE 10 YEARS LATER.
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SO OUR EXPONENT HERE IS GOING TO
BE 12 X 25 INSTEAD OF 12 X 35.
[435]
THE INTEREST IS THE SAME AS WELL
AS BEING COMPOUNDED MONTHLY.
[439]
SO LETS SEE
WHAT THIS WOULD GIVE US.
[441]
SO WE'LL GO THROUGH THE SAME
PROCESS ON OUR CALCULATOR.
[465]
JUST REMEMBER WE NEED
ANOTHER SET OF PARENTHESIS
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FOR THE NUMERATOR
AND DENOMINATOR
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TO KEEP EVERYTHING STRAIGHT.
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SO NOW NOTICE THE BALANCE
DROPPED CONSIDERABLY.
[473]
NOW IT'S ONLY $77,119.42
APPROXIMATELY.
[482]
SO LET'S GO AHEAD AND DETERMINE
THE TOTAL INTEREST EARNED.
[484]
SO THE TOTAL AMOUNT DEPOSITED
WOULD BE 150 PER MONTH,
[488]
SO x 12 FOR 25 YEARS.
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150 x 12 x 25 = $45,000.
[499]
THIS IS THE TOTAL AMOUNT
DEPOSITED OVER THE 25 YEARS.
[504]
SO IF WE TAKE THE BALANCE
AND SUBTRACT OUT THE AMOUNT
[507]
THAT WE DEPOSITED
[508]
THAT'LL TELL US THE TOTAL
INTEREST EARNED.
[511]
WE'RE GOING TO HAVE $32,119.43.
[521]
SO, AGAIN, IF WE SAVE
FOR 25 YEARS
[523]
WE HAVE A BALANCE
OF APPROXIMATELY $77,000.
[526]
WE'VE EARNED ABOUT $32,000
OF INTEREST.
[529]
BUT IF WE START JUST 10 YEARS
EARLIER,
[533]
WE HAVE A MUCH LARGER BALANCE
IN THE ACCOUNT
[535]
OF APPROXIMATELY $137,000.
[538]
IT WOULD BE MORE THAN DOUBLE
THE AMOUNT OF INTEREST EARNED.
[542]
SO I GUESS THAT TELLS US
THAT IF IT'S POSSIBLE,
[544]
WE SHOULD START SAVING FOR
RETIREMENT AS SOON AS WE CAN.
[548]
I HOPE YOU FOUND THIS HELPFUL.
[549]
IN PART 2 WE'LL TAKE A LOOK
[550]
AT HOW WE CAN DETERMINE THESE
VALUES VERY QUICKLY
[553]
ON THE TI-84 GRAPHING
CALCULATOR.
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