SOLUTION: which of the following investments is larger after 29 years? a) $100 is deposited monthly and earns 8.75% interest compounded monthly. b) 1250 is deposited annually and earns 8.7

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Question 166535: which of the following investments is larger after 29 years?
a) $100 is deposited monthly and earns 8.75% interest compounded monthly.
b) 1250 is deposited annually and earns 8.75% interest compounded annually.
Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
formula for future value of an annuity is:
FV+=+M+%2A+%28%28%281%2Bi%29%5En-1%29%2Fi%29%2A%281%2Bi%29
FV = future value
M = money invested each time period
i = interest rate each time period
n = number of time periods.
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the assumption in this formula is that the money is invested at the beginning of the time period and the interest is calculated at the end of the time period.
as an example:
let n = 1
let M = 1000
let i = .10
FV = 100* ((1.10)^1-1)/.10)*(1.10)
this becomes
100 * (.10/.10)*(1.10)
this becomes
100*1.10 = 110.
the money is invested at the beginning of the time period and the interest is accrued as the end of the time period.
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if n = 2, the following happens:
FV = 100 * (1.10)^2 - 1)/.10)*(1.10)
this becomes ((100 * .21)/.10)*(1.10)
this becomes 231.
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here's what happens.
100 is invested in the beginning of the first year and becomes 110 by the end of the first year.
at the beginning of the second year another 100 is invested. this combines with 110 to become 210 total investment at the beginning of the second year. this earns 10% interest to become 231 by the end of the second year.
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back to your problems.
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i'll do option b first then do option a.
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option b) 1250 is deposited annually and earns 8.75% annual interest compounded annually.
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formula to use is: FV+=+M+%2A+%28%28%281%2Bi%29%5En-1%29%2Fi%29%2A%281%2Bi%29
M = 1250
i = .0875
n = 28
application of the formula yields:
FV = 147150.4663
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i'll do option a next.
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option a) $100 is deposited monthly and earns 8.75% annual interest compounded monthly.
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in order to do this one, some adjustments need to be made.
since the money is to be compounded monthly, we have to calculate the monthly interest rate and the monthly number of terms.
monthly interest rate = annual interest rate / 12 = .0875 / 12 = .007291667
number of monthly terms = number of annual terms * 12 = 28 * 12 = 336.
same formula is used.
that formula is: FV+=+M+%2A+%28%28%281%2Bi%29%5En-1%29%2Fi%29%2A%281%2Bi%29
M = 100
i = .0875 / 12 = .007291667
n = 28 * 12 = 336
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FV = 144853.7427
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1250 invested annual becomes 147150.4663
100 invested monthly becomes 144853.7427
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assuming that the investment is made at the beginning of the time period and the interest is accrued at the end of the time period, the 1250 investment earning interest yearly makes more money than the 100 investment earning interest monthly.
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HOWEVER, .............
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assuming that the investment is made at the end of the time period and the interest is accrued at the end of the time period, the Future Value of the investment is changed as follows:
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the formula becomes FV+=+M+%2A+%28%28%281%2Bi%29%5En-1%29%2Fi%29, rather than FV+=+M+%2A+%28%28%281%2Bi%29%5En-1%29%2Fi%29%2A%281%2Bi%29
that little multiplication by %281%2Bi%29 at the end is taken away.
using the adjusted formula, the Future Values become as follows:
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1250 invested annually and compounded annually becomes 135310.7736
100 invested monthly and compounded monthly becomes 143805.1733
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the 100 invested monthly makes more money than the 1250 invested annually if the investment is made at the end of each time period rather than the beginning of each time period.
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the answer depends on whether the money is invested at the beginning of each time period of at the end of each time period.
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you may wish to check which they were assuming.
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