Suppose a client is going to contribute $X per month. A trade could be done every month, incurring a transaction fee, or every other month, or every third month, etc. When is the optimal time?

It depends on the spread of expected return of the asset class to be invested in over the expected return on cash (that isn’t really right, because they actually ought to be risk adjusted returns, but I am going to ignore that for simplicity), the amount of cash, the expected next inflow and amount, and the transaction cost.

For example, assume a client is going to add $1,000 to their account each month, the cost of the trade is $10, the expected return on cash is zero and on the prospective investment 8%.

The first month there is $1,000 in cash and investing it will cost $10. That is a 100 bps cost but the expected return (monthly) is just 64 bps [1.08^(1/12)-1] so that is a bad deal. The next month there is $2,000 and investing it costs only 50 bps, but the return for a month is still 64 bps so now it should be invested.

Here is the formula:

((1+I)/(1+C))^(T/12)-(T/D)-1

Where:

- D is the dollar amount of the proposed trade
- T is the transaction cost of the trade in dollars
- C is the expected annual return on cash
- I is the expected annual return on the proposed investment
- T is the time in months until there will be additions to the account or a rebalancing, etc. (I am assuming these changes will be of significant size. Obviously, if the upcoming deposit is $1 it wouldn’t be worth waiting for.)

If the solution to the formula below is positive, do the trade, if negative, don’t.