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# Future value with integrals

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A retiree is paid \$1700 per month by an annuity. If the income is invested in an account that earns 9% interest compounded continuously, what is the future value of the income after ten years? (Round your answer to two decimal places.)

Oct 9, 2019

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How long will the annuity last for?

Oct 9, 2019
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Since the maximum compounding is continuous compounding, therefore 9% compounded continuously is equal to:e^0.09 =1.0941742837 - 1 =0.0941742837 - which is the effective annual rate.
But, in order to convert it to a monthly compound, we must take the 12th root of: 1.0941742837^(1/12) =1.0075281954 - 1 x 1200 =9.0338345334% compounded monthly =9.41742837 which is 9% compounded continuously.
FV = PMT x {[[1 + 0.090338345334/12]^(10*12) - 1] / (0.090338345334/12)}
FV =\$1,700 x{[[ 1.0075281954 ]^120 - 1] / (0.0075281954)}
FV =\$1,700 x                   193.88485999672
FV =\$329,604.26 - FV of the income after 10 years.

Note: You could also use:0.09/12 =0.0075 =e^0.0075 =1.0075281954 - which is the same as above.

Oct 9, 2019
edited by Guest  Oct 9, 2019