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Payout Annuity Formula - Part 2 - YouTube
Channel: Mathispower4u
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welcome back for another example on the
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payout annuity formula
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in this example you want to be able to
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withdraw 25 000 each year for 15 years
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and your account pays 5 interest
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a how much do you need in your account
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at the beginning b
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how much total money will you pull out
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of the account and see how much of that
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money is interest so here's the payout
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annuity formula
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let's review what these variables
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represent
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p sub zero is the balance in the account
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at the beginning also called the
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starting amount or principal
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d is the regular withdrawal amount or
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the amount you take out each time period
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r
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is the annual interest rate expressed as
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a decimal
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k is the number of compounding periods
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in one year
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and n is the number of years withdrawals
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are made
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the compounding frequency is not always
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explicitly given
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but is determined by how often you take
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the withdrawals
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so going back to our example let's
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identify all the given information
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well first you want to withdraw 25 000
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each year
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which means d
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would be equal to 25 000
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but also tells us k the number of
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compounds per year equals one since the
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withdrawal is made one time each year so
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k is equal to one
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the withdrawals are made over 15 years
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which means n is equal to 15
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and the account earns 5 interest so r
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is equal to five percent
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which must be expressed as a decimal
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which would be zero point zero five
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to answer part a how much do you need in
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the account at the beginning
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we need to find p sub zero
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so for the next step we'll perform
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substitution
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into the payout annuity formula
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so we'll substitute 25000 for d
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which is here
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we'll substitute one for k
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which is here here and here
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we'll substitute 15 for n
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and finally we'll substitute 0.05
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for r which is here and here
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now if we evaluate this we can determine
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p sub zero how much you need in the
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account at the beginning so now we'll
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grow the calculator
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let's evaluate inside the parentheses
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here in the numerator
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notice how our denominator is just going
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to be .05 because we're dividing by one
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so let's go ahead and evaluate this
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expression here in the numerator
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in the parentheses
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so we'd have one
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minus
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the quantity one
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plus
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again here we're just dividing by one so
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that's going to be plus .05
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parenthesis
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we're going to raise this to the power
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of we can see would be negative 15.
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again when rounding we should include at
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least three significant digits but
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notice here i'm including
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all the decimal places
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so now to find p sub zero we want to
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find this product in the numerator and
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then divide by .05
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so in the numerator we would have 25 000
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which i'll put in parentheses
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times
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.518
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982
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9019
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and we're going to divide this product
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by .05
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to the nearest cent notice that p sub 0
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is 259
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491 dollars and 45 cents
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which is our answer to part a
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how much you need in the account at the
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beginning
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so if you have this much money in the
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account at the beginning
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you'd be able to withdraw 25 000 each
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year for 15 years
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as long as the account earns 5 percent
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part b asked how much total money will
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you pull out of the account well you're
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going to be withdrawing twenty-five
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thousand dollars once a year for fifteen
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years
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so for part b
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we would take twenty-five thousand
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dollars and multiply by fifteen
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which gives a total of three hundred
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seventy-five thousand dollars withdrawn
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from the account over the 15 years
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so this would be part b
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so if we go back to part a just for a
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second
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notice how the starting balance has to
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be this amount but over the 15 years
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you actually withdraw 375 thousand
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dollars so the last part part c
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how much of that money is interest
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well the interest earned would be the
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difference between the amount taken out
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of the account here
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and
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the starting account balance
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so again the interest earned would be
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375 thousand dollars minus
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259 thousand
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491 dollars and 45 cents
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which gives a difference of 115
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508 dollars and 55 cents
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this is how much interest is earned over
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the 15 years
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and that's it for this example i hope
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you found this helpful
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