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My wife and I currently have three retirement accounts from different jobs. The only one currently getting paid into is my TSP account. Would we realize better gains if we combined the other two accounts into the TSP? Say they are all earning 10% annually. From what I've seen on charts the compound interest curve grows exponentially. It seems like I'd have three accounts growing slower or one growing faster. Is that the case?

  • Does the TSP account offer tiered interest rates (dependent on the size of the account?) – unutbu Jun 18 '18 at 20:44
  • TSP is free to use and maintain. True of your other accounts too? Does any account perform significantly better than the others? Not really "interest" at a fixed rate like savings acct, but rate of return from the various investments & index funds in the respective accounts. How do the returns stack up to each other & to your goals? Did you approach them all in the same way, or did your contributions/allocations differ? e.g. TSP "2050" fund is higher-risk than a near-term "2020" fund for someone retiring sooner. Similar allocations for all of your accounts? Maybe revisit those choices too. – mc01 Jun 18 '18 at 22:44
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No. The compounding is a multiplier, and multiplication is distributive over addition. So

1.1 * (x + y) = (1.1 * x) + (1.1 * y).

That is assuming that the accounts are large enough that no individual payment gets lost as a rounding error (three accounts earning 1.4c each will pay you 3c, combined they will pay you 4c). So long as the accounts are at least a few hundred dollars, even if the 10% is paid daily, you shouldn't have a problem.

Also assuming there are no fixed costs per account. Obviously those can be reduced by combining accounts.

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    Its worth emphasising the per-account costs. If they are all earning 10%, but you are also paying $1,000 in fixed fees per account then combining the accounts will add $2,000 to your annual savings. However if all three accounts charge 1% then you are earning 9% (10% growth minus the 1% charge) on all three accounts and combining them will make no difference. – Paul Johnson Jun 18 '18 at 10:18
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    Surely the interest is a multiplier, and compounding is an exponent to that multiplier, but the crucial point is that it operates on the multiplier, not the result and so is also distributed over addition in the multiplicand, i.e. 1.1^5 * (x + y) = (1.1^5 * x) + (1.1^5 * y) – Will Jun 18 '18 at 13:42
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    @Will that's true, but you only need to consider a single year to demonstrate "compounding interest" would be better than the shorthand "compounding" in the answer though – Chris H Jun 18 '18 at 14:25
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    @Will I think a simpler way to say that is that the compounding is just repeated multiplication - a^5 is just short-hand for a × a × a × a × a - so doesn't change the operators in the sum. Crucially, that's different from an exponential like 5^a, which wouldn't be distributive: 5^1.1 + 5^1.1 ≠ 5^2.2 – IMSoP Jun 18 '18 at 16:24
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    Combining accounts should reduce the number of periodic statements you receive, and the number of things for your to keep track of. – Nayuki Jun 18 '18 at 16:58
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Rupert's got the formal answer. I'd like to chip in with a everyday sort of example.

Imagine you had $200 dollars, and you had the choice between investing it in a single account that earned 10% in a year, or two accounts that each earn 10% in a year.

If you chose the first, and put $200 in, you would expect $220 by the end of the first year. If you chose the second, you'd have two accounts with $100, that you'd expect to have $110 by the end of the year. $110+$110 = $220.

After the second year, the single account would have $242. And the two accounts would each have $121.

It's true that the growth is exponential - but it's the same exponential no matter how much you start with. In the 10% example, you're going to double your money in roughly 7 years. It doesn't matter if you've got $10, $10k, or $10 million, in 7 years you'll have doubled the amount.

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Often, investment managers will charge a smaller % fee for accounts funded greater than some threshold.

If consolidating your accounts gets you over or close to such a threshold, then it would be beneficial to do so. Otherwise, the same actual compounding process is applied mathematically regardless of how many pots your money is divided into (assuming each returns the same interest rate).

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