SOLUTION: How much you should invest now to get $8,000 annually for ever, given an interest rate of 8% for the first 3 years & 10% for the remaining years

Question 175411: How much you should invest now to get $8,000 annually for ever, given an interest rate of 8% for the first 3 years & 10% for the remaining years
Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
in order for you to get the money forever, you have to invest enough so that the principal never gets below a minimum figure.
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what you will be drawing out each year is 8000.
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assume that you are starting in the 4th year.
your interest rate is 10% forever.
your money that you want to take out each year is $8,000
your equation would be:
p = principal
r = interest rate
i = interest
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p*r = i
solving for p, we get:
p = i/r
if i = 8000 and r = .1, then p = 8000/.1 = 80000
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to see if this is true, then check to see if this holds.
80000 * .1 = 8000 + 80000 = 88000 at the end of the 4th year.
withdraw 8000 and you are left with 80000.
80000 * .1 = 8000 + 80000 = 88000 at the end of the 5th year.
withdraw 8000 and you are left with 80000.
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it's plain to see that if you have 80000 at the beginning of the 4th year you can withdraw 8000 forever assuming the interest rate remains at 10%.
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now to the first 3 years.
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you need to have 80,000 left in the account at the beginning of the 4th year.
your future value is therefore 80,000
you need to withdraw 8000 each year.
your interest rate is .08 per year.
your initial investment is x (unknown)
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this winds up being the future value of an annuity formula.
that formula is so complicated i won't even show it here.
i used the PV formula from excel.
that function takes in arguments as follows:
PV(rate,nper,pmt,fv,type)
where:
rate is the interest rate per payment period = .08 per year.
nper is number of payment periods = 3 years
pmt is 8000 which is the amount you withdraw each period.
fv is future value of the investment = 80,000 cause that's what you need to have at the end of the 3 years.
type is 0 which means the payments are made at the end of each payment period.
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that function gave me the answer.
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the answer is:
initial investment is 84,123.36
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here's how it works:
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the problem:
How much you should invest now to get $8,000 annually for ever, given an interest rate of 8% for the first 3 years & 10% for the remaining years
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the answer:
you invest 84,123.36 now.
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interest rate for the first 3 years is 8% annually.
amount to withdraw each year is 8,000
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end of year 1:
1.08 * 84123.36 - 8000 = 82853.2288
end of year 2:
1.08 * 82853.2288 - 8000 = 81481.4871
end of year 3:
1.08 * 81481.4871 - 8000 = 80000.00607
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now you start at 10% a year interest:
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end of year 4:
1.10 * 80000.00607 - 8000 = 80,000.00668
end of year 5:
1.10 * 80000.00668 - 8000 = 80,000.00735
etc.........
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you will be able to withdraw 8000 forever and the principal will remain the same.
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there is a very slight increase of principal each year due to rounding errors.
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hope this helps.
i tried deriving the formula for the first 3 years but gave up because it was too complicated and was taking too much time.
for problems like this, using a financial calculator, or microsoft excel financial functions, is the best way to go.
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