Based on your financial expertise, provide your response or viewpoint on the given financial question or topic. The response format is open.

Frequency of investments to maximise returns (and minimise fees)
Wow, this turns out to be a much more difficult problem than I thought from first looking at it. Let's recast some of the variables to simplify the equations a bit. Let rb be the growth rate of money in your bank for one period. By "growth rate" I mean the amount you will have after one period. So if the interest rate is 3% per year paid monthly, then the interest for one month is 3/12 of 1% = .25%, so after one month you have 1.0025 times as much money as you started with. Similarly, let si be the growth rate of the investment. Then after you make a deposit the amount you have in the bank is pb = s. After another deposit you've collected interest on the first, so you have pb = s * rb + s. That is, the first deposit with one period's growth plus the second deposit. One more deposit and you have pb = ((s * rb) + s) * rb + s = s + s * rb + s * rb^2. Etc. So after n deposits you have pb = s + s * rb + s * rb^2 + s * rb^3 + ... + s * rb^(n-1). This simplifies to pb = s * (rb^n - 1)/(rb - 1). Similarly for the amount you would get by depositing to the investment, let's call that pi, except you must also subtract the amount of the broker fee, b. So you want to make deposits when pb>pi, or s*(ri^n-1)/(ri-1) - b > s*(rb^n-1)/(rb-1) Then just solve for n and you're done! Except ... maybe someone who's better at algebra than me could solve that for n, but I don't see how to do it. Further complicating this is that banks normally pay interest monthly, while stocks go up or down every day. If a calculation said to withdraw after 3.9 months, it might really be better to wait for 4.0 months to collect one additional month's interest. But let's see if we can approximate. If the growth rates and the number of periods are relatively small, the compounding of growth should also be relatively small. So an approximate solution would be when the difference between the interest rates, times the amount of each deposit, summed over the number of deposits, is greater than the fee. That is, say the investment pays 10% per month more than your bank account (wildly optimistic but just for example), the broker fee is $10, and the amount of each deposit is $200. Then if you delay making the investment by one month you're losing  10% of $200 = $20. This is more than the broker fee, so you should invest immediately. Okay, suppose more realistically that the investment pays 1% more per month than the bank account. Then the first month you're losing 1% of $200 = $2. The second month you have $400 in the bank, so you're losing $4, total loss for two months = $6. The third month you have $600 in the bank so you lose an additional $6, total loss = $12. Etc. So you should transfer the money to the investment about the third month. Compounding would mean that losses on transferring to the investment are a little higher than this, so you'd want to bias to transferring a little earlier. Or, you could set up a spreadsheet to do the compounding calculations month by month, and then just look down the column for when the investment total minus the bank total is greater than the broker fee. Sorry I'm not giving you a definitive answer, but maybe this helps.