Is there a concept similar to "continuous compounding" but related to dollar cost averaging when buying (e.g.) shares or bitcoins?

Question

I know that dollar cost averaging (DCA) is "drip feeding" a total amount to be invested over regular intervals like daily, weekly or monthly, so that each share purchase (or whatever the investment is) is at the market rate at the time.

Is there a concept similar to continuous compounding, but related to DCA? as in, the "theoretical" effect of continuously streaming in (with infinitely small intervals) parts of the total sum to be invested?

I am taking continuous compounding to mean the theoretical maximum limit of compound interest, where daily interest compounds more quickly than monthly etc, and c.c. is "what would happen if interest was 'streamed' in infinitely small intervals".

(Assume there are no fees on any of the transactions, or the fee is a fixed % of the transaction size)

Have tried searching but either this info isn't out there (or the concept doesn't exist), or my search terms were just rubbish!

Edited to add what I am trying to achieve - I don't think this will change my investment strategy, but I would like to understand "theoretically" if a concept similar to c.c. applies to dollar cost averaging, in terms of whether it's meaningful to think about what would happen when "drip feeding" money in over smaller and smaller intervals. If so what is the name of this concept, if not then why does it not apply?

For example if I wanted to feed in that investment over 6 months. I'd "capture" the market price only 6 times if I invested monthly, 26 times if investing weekly, 182 times if investing daily, etc. It seems intuitive to me that the smaller the time interval, the closer the 'accumulated' purchase price tracks the actual market (and in the theoretical 'continuous investment' case, that it would track market prices exactly), but I can't quite translate that in my head into what the actual effect is of "tracking the market price" by buying in smaller intervals.

Answer 1

Can you clarify what you are trying to achieve?

Keep in mind, if you were getting 100% annual interest at a bank, a yearly CD, no compounding, returns $200 (duh). Monthly compounding jumps to $261, daily, $271.46. But "continuous", $271.82 (a multiple of the number 'e').

The concept of continuous compounding doesn't really apply when we talk about stocks, bitcoin, or mortgages, for that matter.

EDIT - I understand a bit more now. My investments only match the market during the time they are invested. i.e. The assets that were there on Dec 31 are matching the market YTD. A working person who invests with weekly (or payperiod) deposits actually sees a different final result. Separate from paychecks, I suppose you'd see a bit better return, on average, than by investing quarterly, or monthly, if you make those purchases as soon as funds are available. In the old days, we'd talk a bit about transaction costs, which have all but faded away. I don't quite know how to capture the phrase you are looking for, but would maintain that the results of weekly investing will be close to daily over the long term.

Answer 2

There won't be a concept of a dollar cost averaging equivalent for continuous compounding unless there is a corresponding concept of a continuous function for the market price of the bitcoins that can be agreed upon by buyer and seller. With a mortgage it's easy. We know the equations that are behind mortgages. With a market, it is harder. The "current value" at any instant is a more complicated thing -- the kind of thing high velocity trades are made of.

You could get close to this effect with a stochastic buying process. You could choose to make 100 or 1000 purchases at random times unformly distributed within the interval. With such a purchase, the expectation of the value of your purchases will be that of the continuous process and the standard deviation can be driven as small as you please by making more purchases. But that's as close to a continuous process as I think you can get.