You 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.

Tutors, sign in to answer this question.

Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...

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.

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

Already have an account? Log in

By signing up, I agree to Wyzant’s terms of use and privacy policy.

Or

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Your Facebook email address is associated with a Wyzant tutor account. Please use a different email address to create a new student account.

Good news! It looks like you already have an account registered with the email address **you provided**.

It looks like this is your first time here. Welcome!

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Please try again, our system had a problem processing your request.

Michael D.

Math/Accounting/Business/Social Studies and Test Prep Tutor

$11.25 per 15 min

View Profile >

Ron H.

Patient tutor (CPA) with tons of business experience

$11.25 per 15 min

View Profile >

Karen H.

Securities FINRA Test prep series 7,63,65,66 more+Insurance license.

$15 per 15 min

View Profile >