How do I calculate what my monthly contribution should be to an interest bearing account, knowing how much I'll need to have in the account in a given number of years?
For Example,
I need $8,203.03 in an account in 25 years. Money in the account garners 4% interest compounded monthly. My principle investment in the account is $1,000. What is the formula that can tell me how much will I need to contribute monthly in order to have $8,203.03 in the account in 25 years?
You can re-arrange the future value of an annuity equation to solve for payment (Pmt) needed instead of Future Value (FV):
FV = Pmt * (((1 + r) ^ n) – 1)/r)
Pmt = FV / (((1 + r) ^ n) – 1)/r)
That will get you payment needed to hit a given future value, but it won't account for the future value of the the starting balance, that should be removed from the desired future value after interest is calculated:
$1,000 at 4% for 25 years with monthly compounding:
n = number of periods = 12 months x 25 years = 300
r = rate per period = .04/12 = 0.003333
FV = Future Value = 8203.03
FV = PV(1+r)^n
FV = 1000 * (1+0.003333)^300
FV = 1000 * (1.003333)^300
FV = 1000 * 2.7135
FV = $2,713.5
So your $1,000 will be worth $2,713.5 without any additional contribution, remove that from your goal of $8,203.03 and your target from the monthly contributions is: 5,489.53
So to solve for monthly contribution (Pmt) we'll use:
n = number of periods = 12 months x 25 years = 300
r = rate per period = .04/12 = 0.003333
FV = Future Value = 5489.53
Pmt = FV / (((1 + r) ^ n) – 1)/r)
Pmt = 5489.53 / (((1 + .003333) ^ 300) – 1)/.003333)
Pmt = 5489.53 / ((1.003333^300) – 1)/.003333)
Pmt = 5489.53 / (1.7138)/.003333)
Pmt = 5489.53 / 514.19
Pmt = 10.68
You need to contribute $10.68 per month to hit your goal. May be off a few pennies due to rounding.
Even easier would be to leverage an online calculator, this one seems accurate and comes up with $10.64. Also could solve in a spreadsheet which would enable you to handle rounding more flexibly.
You need the balance on month N to equal Bf. You make regular contributions of b to the account and get a monthly percentage yield of r (expressed as a real number multiplier. i.e., r = 1.005 is 0.5% monthly). Let B(n) be the balance at the beginning of month n; after the old balance has interest applied and the new contribution is added. Then
B(0) = B0
B(1) = B0r + b
B(2) = B0r^2 + br + b
...
B(n) = B0r^n + b(r^(n-1)+ ... + 1)
= B0r^n + b((r^n)-1)/(r-1)
We can solve for b by using B(N) = Bf:
Bf = B0r^N + b(r^N-1)/(r-1)
b = (Bf - B0r^N) / [(r^N-1)/(r-1)]
Your answer comes out to something like $10.88. This gives you the contribution to reach the goal at the beginning of the last month in the 25-year period; to get the amount at the end, you can re-run the math for n = N + 1 and subtract the contribution from the N+1st month (leaving only the last month's interest). Or, you could rearrange the system and solve for end-of-month balances directly. That might be easier. The key is that understanding the derivation allows you to reason about the details of what your numbers mean.
