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?
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.
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.
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).