Payout Annuity Formula - Part 1 - YouTube

- WELCOME TO A LESSON ON THE PAYOUT ANNUITY FORMULA.

IN AN ANNUITY YOU START WITH NOTHING,

PUT MONEY INTO AN ACCOUNT ON A REGULAR BASIS,

ENDING UP WITH MONEY IN YOUR ACCOUNT.

WE WILL NOW LEARN ABOUT A VARIATION

CALLED A PAYOUT ANNUITY.

WITH A PAYOUT ANNUITY YOU START WITH THE MONEY IN THE ACCOUNT

AND PULL MONEY OUT OF THE ACCOUNT ON A REGULAR BASIS.

ANY REMAINING MONEY IN THE ACCOUNT EARNS INTEREST.

AFTER A FIXED AMOUNT OF TIME THE ACCOUNT WILL END UP EMPTY.

PAYOUT ANNUITIES ARE TYPICALLY USED FOR RETIREMENT.

PERHAPS YOU HAVE SAVED $500,000 FOR RETIREMENT

AND WANTED TO TAKE MONEY OUT OF THE ACCOUNT EACH MONTH

TO LIVE ON.

YOU MAY WANT THE MONEY TO LAST 20 YEARS OR MORE.

THIS IS A PAYOUT ANNUITY.

SO HERE'S THE PAYOUT ANNUITY FORMULA

WHERE P SUB 0 IS THE BALANCE IN THE ACCOUNT IN THE BEGINNING,

ALSO CALLED THE STARTING AMOUNT OR PRINCIPLE.

D HERE IS THE REGULAR WITHDRAWAL

FOR THE AMOUNT YOU TAKE OUT EACH TIME PERIOD.

R IS THE ANNUAL INTEREST RATE EXPRESSED AS A DECIMAL,

WHICH IS HERE AND HERE.

FOR EXAMPLE, 5% WOULD BE 0.05.

K IS THE NUMBER OF COMPOUNDING PERIODS IN ONE YEAR,

WHICH IS HERE AND HERE.

AND FINALLY, N IS THE NUMBER OF YEARS WITHDRAWALS ARE MADE.

N IS HERE ON THE FORMULA.

NOW, THE COMPOUNDING FREQUENCY IS NOT ALWAYS EXPLICITLY GIVEN,

BUT IS DETERMINED BY HOW OFTEN YOU TAKE THE WITHDRAWALS.

AND THIS FORMULA CAN ALSO BE USED FOR LOANS,

WHICH WE'LL SEE IN A FUTURE LESSON.

BEFORE WE TAKE A LOOK AT SOME EXAMPLES

LETS TALK ABOUT ROUNDING.

IT IS IMPORTANT TO BE VERY CAREFUL ABOUT ROUNDING

WHEN PERFORMING CALCULATIONS WITH EXPONENTS.

IN GENERAL, YOU WANT TO KEEP AS MANY DECIMALS--

IN GENERAL, YOU WANT TO KEEP AS MANY DECIMALS

DURING THE CALCULATIONS AS YOU CAN.

BE SURE TO KEEP AT LEAST THREE SIGNIFICANT DIGITS,

WHICH MEANS WE INCLUDE THREE NUMBERS AFTER ANY LEADING ZEROS.

FOR EXAMPLE, IF WE WERE ROUNDING 0.00012345,

WE WOULD ROUND TO 0.000123.

THIS NUMBER HAS THREE SIGNIFICANT DIGITS.

AND THIS WILL USUALLY GIVE A CLOSE ENOUGH ANSWER,

BUT KEEPING MORE DIGITS IS ALWAYS BETTER.

NOW LET'S TAKE A LOOK AT TWO TYPES OF EXAMPLES.

AFTER RETIRING YOU WANT TO BE ABLE TO WITHDRAW $1,800

EVERY MONTH FOR A TOTAL OF 20 YEARS.

IF YOUR RETIREMENT ACCOUNT EARNS 3% INTEREST,

HOW MUCH WILL YOU NEED IN YOUR ACCOUNT BEFORE YOU RETIRE?

LET'S BEGIN BY IDENTIFYING ALL OF THE IMPORTANT INFORMATION.

SINCE YOU WANT TO WITHDRAW $1,800 EVERY MONTH,

THIS TELLS US THAT D, THE REGULAR WITHDRAW AMOUNT,

IS 1,800.

AND BECAUSE THE WITHDRAWS ARE MONTHLY,

K, NUMBER OF COMPOUNDS PER YEAR, WOULD BE 12.

YOU WANT TO MAKE WITHDRAWS FOR A TOTAL OF 20 YEARS, SO N IS 20.

AND THE ACCOUNT EARNS 3% INTEREST

WHERE R WOULD BE 3% AS A DECIMAL, AND THEREFORE R = 0.03.

AND OUR GOAL HERE IS FIND P SUB ZERO,

THE NECESSARY BALANCE IN THE ACCOUNT.

SO NOW WE'LL PERFORM SUBSTITUTION INTO THE FORMULA

AND FIND P SUB ZERO.

SO D = 1,800, K = 12, WHERE K IS HERE, HERE, AND HERE.

N = 20 AND R = 0.03, WHERE R IS HERE AND HERE.

AND NOW WE NEED TO EVALUATE THIS TO FIND P SUB ZERO.

AND IT'S HARD TO EVALUATE THIS ALL AT ONE TIME

IN THE CALCULATOR,

SO BEGIN BY EVALUATING THIS QUANTITY HERE IN THE NUMERATOR,

AS WELL AS THIS QUANTITY HERE IN THE DENOMINATOR.

SO EVALUATING THIS EXPRESSION INSIDE THE PARENTHESIS

IN THE NUMERATOR WE WOULD HAVE

(1 - THE QUANTITY 1 + 0.03 DIVIDED BY 12),

CLOSE PARENTHESIS.

WE'RE GOING TO RAISE THIS TO THE POWER OF -20 x 12.

AND NOTICE HOW I INCLUDED ALL 10 DECIMAL PLACES

HERE IN THE SECOND STEP.

WE COULD ROUND TO THREE SIGNIFICANT DIGITS,

BUT, AGAIN, THE MORE DECIMAL PLACES WE USE

THE MORE ACCURATE OUR ANSWER.

AND THE DENOMINATOR'S GOING TO BE 0.03 DIVIDED BY 12,

WHICH IS 0.0025.

SO NOW WE'LL FIND THE PRODUCT OF THE NUMERATOR,

AND THEN DIVIDE BY THE DENOMINATOR,

SO IN THE NUMERATOR WE HAVE 1,800 X 0.450777286.

AND WE'RE GOING TO DIVIDE THIS BY 0.0025,

WHICH WILL GIVE US P SUB ZERO.

ROUNDING TO THE NEAREST CENT,

NOTICE THAT P SUB ZERO IS $324,559.65.

SO THIS IS THE STARTING AMOUNT

OR BALANCE YOU WOULD NEED IN YOUR ACCOUNT TO BEGIN WITH

IN ORDER TO WITHDRAW $1,800 EVERY MONTH FOR 20 YEARS

IF YOUR ACCOUNT EARNS 3% INTEREST.

NOW LETS TAKE A LOOK AT A SECOND EXAMPLE.

HERE YOU ALREADY KNOW YOU HAVE $300,000 SAVED FOR RETIREMENT,

YOUR ACCOUNT EARNS 4% INTEREST.

YOU WANT TO KNOW HOW MUCH YOU'LL BE ABLE TO PULL OUT EACH MONTH

IF YOU WANT TO BE ABLE TO TAKE WITHDRAWS FOR 15 YEARS.

LETS BEGIN BY IDENTIFYING THE IMPORTANT INFORMATION.

SO $300,000 IS P SUB ZERO, THE STARTING AMOUNT OR PRINCIPLE.

THE ACCOUNT EARNS 4% INTEREST,

AND THEREFORE R = 0.04, 4% AS A DECIMAL.

YOU WANT TO WITHDRAW EVERY MONTH, SO K IS GOING TO BE 12.

AND YOU WANT TO MAKE WITHDRAWS FOR 15 YEARS, SO N = 15.

SO FOR THIS PROBLEM WE WANT TO FIND D,

THE WITHDRAW AMOUNT PER MONTH.

SO WE'LL SUBSTITUTE 300,000 FOR P SUB ZERO,

0.04 FOR R, 12 FOR K,

WHICH IS HERE, HERE, AND HERE, AND 15 FOR N.

NOTICE HERE WE'RE SOLVING FOR D.

SO WE'LL FIRST EVALUATE THE PARENTHESIS

IN THE NUMERATOR, AND DENOMINATOR,

AND THEN SOLVE FOR D.

SO STARTING WITH THE NUMERATOR,

WE'D HAVE (1 - THE QUANTITY 1 + 0.04 DIVIDED BY 12),

CLOSE PARENTHESIS, RAISED TO THE POWER OF -15 x 12,

WHICH GIVES US THIS DECIMAL HERE.

AND OUR DENOMINATOR'S GOING TO BE 0.04 DIVIDED BY 12,

WHICH WE SEE EXPRESSED AS A DECIMAL HERE.

NOW FOR THE NEXT STEP WE WANT TO FIND THE COEFFICIENT OF D,

SO WE'LL FIND THIS QUOTIENT, AND THEN MULTIPLY BY D.

SO 0.4506404955 DIVIDED BY 0.0033333333

GIVES US OUR COEFFICIENT HERE FOR D.

SO NOW WE HAVE THE EQUATION 300,000 = 135.19215D.

SO TO SOLVE FOR D

WE'LL NOW DIVIDE BOTH SIDES BY THE COEFFICIENT OF D.

SO THIS SIMPLIFIES TO 1, SO WE HAVE D ON THE RIGHT SIDE,

AND THIS QUOTIENT WILL GIVE US THE VALUE OF D.

SO WE HAVE 300,000 DIVIDED BY 135.19215,

ROUNDING TO THE NEAREST CENT D = $2,219.06,

WHICH MEANS UNDER THESE CONDITIONS

YOU WOULD BE ABLE TO WITHDRAW $2,219.06 EVERY MONTH

FOR 15 YEARS.

I DO WANT TO GO OVER ONE MORE EXAMPLE

BUT WE'LL TAKE A LOOK AT THAT EXAMPLE IN PART 2.

I HOPE YOU FOUND THIS HELPFUL.