Parviz F. answered • 09/16/13

Mathematics professor at Community Colleges

Tajh C.

asked • 09/15/13You plan to retire in 30 years and decide to save $10,000 per year. If the interest rate is 6% compounded monthly, how much will you have in 30 years? Assume that each deposit is made at the end of every year.

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Parviz F. answered • 09/16/13

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Andre W. answered • 09/15/13

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This problem asks for the future value of an annuity with monthly deposits.

The standard formula for this amount is

FV =R*((1+i)^{n}-1)/i,

where R is the payment per compounding period, i = the interest rate per compounding period, and n the number of compounding periods. This formula assumes that one payment is made each compounding period, which in your problem is not the case: the compounding period is a month (n=30*12), while a payment R=10,000 is made once year.

We can get an approximation if we assume instead that a payment of R=$10,000/12 is made every month.

With i=0.06/12 we get

FV =(10000/12) *((1+(0.06/12))^{30*12}-1)/(0.06/12)

=$ 837,100

This is an *over*estimate, because it assumes monthly payments with monthly compounding.

We can get an *under*estimate if we assume annual compounding with annual payments. In this case n=30, i=0.06 and

FV=10000 * ((1+0.06)^{30}-1)/0.06

=$ 790,580

The actual answer will lie between these two estimates.

Kirill Z. answered • 09/15/13

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Since the first deposit will be made by the end of the first year, there will be 29 deposits.

The first deposit will grow to be 10,000*(1+0.06/12)^{12*29}; the second deposit grows to be 10,000*(1+0.06/12)^{12*28} and so on. The last deposit will grow to 10,000*(1+0.06/12)^{12}≈$10616.78.

We need to find the sum 10,000*(1+0.06/12)^{12}+10,000*(1+0.06/12)^{12*2}+…+10,000*(1+0.06/12)^{12*29}=

=10000*[(1+0.06/12)^{12}+((1+0.06/12)^{12})^{2}+…+((1+0.06/12)^{12})^{29}]

So we need to sum geometrical progression with the first term (1+0.06/12)^{12}≈1.061678 and the base 1.061678, the total number of terms being 29.

S=q*(q^{N-1}-1)/(q-1), where q is the base of geometrical progression, N is the number of terms.

Plug in the number to get:

S≈80.432448

Final answer is S*10000=804324.48

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