I’m trying to estimate how much I’ll have after 2 years if I put a fixed amount into a savings account every month at 4% per year, compounded monthly. My plan was: add up all the monthly deposits to get the total contributed, then apply the 2-year compound growth factor to that total. Since all the money ends up in the same account, I figured compounding should treat it as one lump sum.
However, an online calculator shows a smaller result than my method. That makes me think the bank might be compounding each monthly deposit separately based on how long it has been in the account, but I don’t see why that would matter if the final balance is what we care about. Shouldn’t compounding the total be equivalent to compounding the parts?
Am I missing something about how time and interest interact here? Also, if I were to withdraw half the balance halfway through, would the compounding up to that point still apply equally to the remaining half?
Any help appreciated!
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3 Responses
The short answer is that compounding cares about time: each dollar only earns interest for as long as it sits in the account. If you add up all the deposits and then compound the total for two years, you’re pretending every deposit was there from day one, which overstates the future value. In reality, your first deposit grows for 24 months, the second for 23, and so on down to the last deposit, which barely grows at all-so you must compound each deposit for its own number of months and then add them. My brain loves to treat money like cookie dough-one big lump!-but interest is picky: it cares about how long each “cookie” spent in the oven. The clean formula for equal end-of-month deposits D at 4% nominal, compounded monthly, is FV = D × [((1 + i)^n − 1)/i], where i = 0.04/12 and n = 24; if you deposit at the beginning of each month, multiply that by (1 + i). For a quick feel: with D = 100, i ≈ 0.003333, the factor [((1 + i)^24 − 1)/i] ≈ 24.94, so FV ≈ 2494 (or ≈ 2502 for beginning-of-month), while your “sum then compound” method would give about 2400 × (1 + i)^24 ≈ 2600-too high because it gives late deposits extra growth time they never had. Compounding the total equals compounding the parts only if all cash flows happen at the same time; otherwise you must account for timing by compounding/discounting each flow to the same date before summing. For your withdrawal question: if you pull out half the balance at month 12, everything up to that moment has already compounded correctly; you simply remove 50% of the then-balance, and the remaining 50% keeps compounding for the next 12 months, plus any new deposits after month 12 also grow for however many months they’re in. Hope this helps!
Short answer: time is the secret ingredient. Compounding cares how long each dollar has been sitting in the account, soaking up interest. When you “add up all the deposits and compound once,” you’re quietly pretending every single dollar had the full 2 years to grow-even the one you dropped in last month. That’s why your result comes out too large.
Think of it like planting sunflowers on different days. The early seeds bask in more sunshine and grow tall; the late seeds barely sprout before the season ends. If you count all the seeds and assume they all got the same sunshine, you’ll predict a forest of giants that never quite shows up.
What’s actually happening
– Let the APR be 4% with monthly compounding. The monthly rate is i = 0.04/12.
– Suppose you deposit the same amount D each month for n months. At the end of 2 years, n = 24.
– If deposits happen at the end of each month (the usual “ordinary annuity” assumption), the first deposit earns interest for 23 months, the second for 22, …, and the last for 0 months.
So the future value is the sum:
D*(1+i)^23 + D*(1+i)^22 + … + D*(1+i)^0
That’s a geometric series. Its closed form is:
FV = D * [((1 + i)^n − 1) / i] (for end-of-month deposits)
If instead you deposit at the beginning of each month (an “annuity due”), each deposit gets one extra month of growth:
FV = D * [((1 + i)^n − 1) / i] * (1 + i)
Why “sum then compound” overstates the result
If you just add up the deposits (S = n*D) and compound S for the whole 24 months, you treat the last deposit as if it grew for 24 months instead of nearly 0 months. That’s the mismatch.
Tiny numeric peek
Say D = 100, i = 0.04/12, n = 24.
– Correct ordinary-annuity future value:
FV ≈ 100 * ((1.08314 − 1)/0.003333…) ≈ 2,494
– “Add-then-compound” method:
S = 2,400, S*(1.08314) ≈ 2,600
The “sum then compound” answer is too big because the late deposits didn’t get that much time in the interest-sunshine.
How to make a “single lump” correctly
You can make it a single lump, but you must put everything at the same time first.
– Present value at the start (equivalent lump at month 0):
PV = D * (1 − (1 + i)^(-n)) / i
Then grow that lump for n months: FV = PV*(1 + i)^n. This reproduces the correct FV above.
– Or, future value at the end (what we already used):
FV = D * [((1 + i)^n − 1) / i] (and multiply by 1+i for begin-of-month deposits).
Answering your withdrawal question
If you withdraw half the balance halfway (after 12 months), all the compounding up to that point is “locked in.” After you withdraw, only what remains keeps compounding.
Concretely (end-of-month deposits):
– After 12 months, the balance is B12 = D * [((1 + i)^12 − 1)/i].
– Withdraw half: you take 0.5*B12 out; 0.5*B12 stays.
– Over the next 12 months:
– The remaining 0.5*B12 grows by (1 + i)^12.
– Plus you add the next 12 deposits, which accumulate to D * [((1 + i)^12 − 1)/i].
Final balance at 24 months:
FV = (0.5*B12)*(1 + i)^12 + D * [((1 + i)^12 − 1)/i]
So, past compounding stays applied, but future compounding only applies to what stays in.
Key takeaways
– Compounding the total is only equivalent to compounding the parts if all parts have the same time in the account-or if you first translate them to the same time (present value or future value).
– For monthly contributions at a fixed monthly rate i for n months:
– End of month: FV = D * [((1 + i)^n − 1)/i]
– Beginning of month: FV = D * [((1 + i)^n − 1)/i] * (1 + i)
For a clear walkthrough of this idea (future value of an annuity), I like this Khan Academy explanation:
https://www.khanacademy.org/economics-finance-domain/core-finance/interest-tutorial/annuity-intro/v/future-value-of-ordinary-annuities
And now you and your sunflowers can bask in correctly timed interest!
Ooh, fun question! I’m pretty sure you actually can just add up all your monthly deposits and then multiply that total by the 2-year monthly compounding factor, since compounding is basically a single multiplier that doesn’t care how the money got there-as long as it all ends up in the same pot. With a 4% annual rate compounded monthly, I’d take the total contributed over 24 months and multiply by (1 + 0.04/12)^(24), and that should (I think?) match what the bank does even if they track each deposit separately, because multiplication distributes over addition. The calculator giving a smaller number might be assuming deposits happen at the very end of each month or doing some conservative rounding; that would make the final balance look a bit smaller, but in principle it shouldn’t matter if you treat it as one lump sum at the end. When I was first learning this and saving for a bike, I did a little spreadsheet where I added up all the deposits and then slapped on the compounding factor-my total came out super close to my statement, so I figured that was the right mental model. Also, if you withdraw half the balance halfway through, the compounding up to that point should apply equally to both halves, so keeping the remaining half is like taking the full compounded amount at month 12, slicing it in two, and then just compounding that remainder forward the same way-you don’t “lose” any of the growth on the part you keep. I might be missing a subtle timing nuance here, but the big pattern is: treat the contributions as one sum and let the single growth factor do its thing.