Most efficient balance for earning interest?
May 26, 2009 8:46 AM   Subscribe

Mathematically speaking, what's the optimal amount to keep in an account that earns a high rate below below a certain sum and a very low one above it?

So I have an account that pays 4% APY on every dollar balance below $10,000 and 1% for every dollar beyond. To maximize the interest I earn, it seems like I should pack it up with the full $10K, right? But then my annual interest dollars are only getting 1%, so they won't compound as much.

Second question: imagine I have two such accounts, with equal terms, and $10K to distribute between them. Am I right to think the best option is to put $9615 into the first account, $385 into the second, and at the end of the year move the first account's new $385 into the second account (which then has a balance of $385+15+385, given its own interest)? Or am I missing something?

(The account is AmericaNet Bank's rewards checking offering, if you're curious. This question is mostly something I'm curious about rather than something I'd actually do, since earning the 4% on two accounts would require twenty debit card purchases a month, and who knows whether the terms will stay the same for more than a year?).
posted by electric_counterpoint to Work & Money (6 answers total)
If you have two accounts, it doesn't matter how you divvy up the cash as long as you can keep both of them under $10k. You could put $5k in each and then just forget about it until they both hit the maximum simultaneously; the result would be the same as splitting it $9000 and $1000 and moving money from one to the other when the first hit $10k.
posted by Kadin2048 at 8:55 AM on May 26, 2009

If you have one account, we need to know the transaction cost to remove money from the account (if there is one) and we need to know what the alternative to keeping money in the account is (i.e. if you move the money out of the account, what interest rate will you earn on it?)
posted by ssg at 9:05 AM on May 26, 2009

If you have two such accounts and $10,000 to invest, then any combination that never pushes your average balance over $10,000 is equally good.

But I think you're really asking, "What's the maximum I can keep in the account without tripping the 1% reward?"

Or am I missing something?

Compounding happens once per statement cycle, based on your average balance. So the problem you want to solve is, "how much money should I pull out of my account each month to keep my average account balance at exactly $10000"

The answer is roughly: Pull $32.74 out of your account on the first day of the statement period.

(nominal rate = 3.928%, applied to $10000 with 12 compounding periods= $32.74/month interest)
posted by qxntpqbbbqxl at 9:11 AM on May 26, 2009 [1 favorite]

'Optimal' could mean a lot of different things but assuming you have a lot of money just laying around not doing anything else, your only goal is to maximize interest earned, the 4% interest is you best available interest rate, and you have a different account that earns more than 1% but less than 4%:

The simple answer is, keep $10,000 in the high/low interest account and transfer every dollar above $10,000 into the different account (earning more than 1%) as soon as possible.

That keeps the maximal amount of money earning 4% at all times and the minimal amount earning 1%.

As to the two accounts question: its doesn't make a bit of difference how the money is split between the two accounts as long as neither goes above $10,000. So 9000/1000 or 5000/5000 or just whatever. 4% is 4% and that doesn't change whether you have 9999/1 or 5000/5000 or any other combination. The only necessary objective there is to keep both accounts below $10,000.

A to the practical question of whether you should spend time transferring money around when your account gets above $10,000: Practically speaking, 1% and 4% are both such small rates, and any amount that will accrue above $10,000 is so small in relative terms, that it really doesn't make much difference if your account goes a little above $10,000 for a few months or even a year. 4% of $10,000 is only $400.

1% of $400 is $4 whereas 4% of $400 is $16.

Point is: whether that $400 is earning the 4% interest rate or the 1% interest rate is only a difference of $12/year.

It's a little different if you are assuming "compound interest" but in that case we need to known whether it is compounded month, weekly, daily, continuously, or what.

And regardless of which type of compound interest you are working with, the difference between the compound and simple interest answers is very, very small over the course of a few months or a year. (Compound interest is powerful when compared with simple interest over the course of 20 or 30 years but not very powerful over the course of a few months or a year.)

So in practical terms if you were doing this with a real bank account you would put in the $10,000 and then transfer the excess out every 6 months or year or whatever interval is convenient--realizing that you are not absolutely maximizing earning compared to what you could if you transferred the excess out every day, but that that the amount you are losing compared with following the absolutely optimal strategy is very, very small.
posted by flug at 10:28 AM on May 26, 2009

flug has it. As long as you take out excess money every year you will be missing out on at most $12 annually.

Compound interest improves matters by a factor of about 2.7x (specifically, e times). Doesn't make a difference at this scale.
posted by katrielalex at 4:56 PM on May 26, 2009

Keep $5k in each account. If you don't do anything it'll take 17 years or so for them to hit $10k, by which point they'll probably change the rules anyway.
posted by madcaptenor at 5:59 PM on May 26, 2009

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