Bankroll eqn assumes no withdrawals?

[Oxbow Lake]

Dear Forum users.
I’ve a feeling this will have been answered before in which case I’d be grateful for the appropriate link.

I have regularly seen the following formula for calculating bankroll:

Bankroll needed = -(sigma^2/2*hourly win rate)ln(risk of ruin)

Am I right in thinking that this holds if and only if
1. you play for ever
2. you never take any winnings out of your bankroll

(I should have thought if your bankroll didn’t increase that you would always have a 100% chance of going bust if you played forever)

That’s the main question

The second one is:
Are there any formulations for bankroll that assumee that you always withdraw your winnings at each moment you are in profit? This would be a great deal more useful. It would need an extra variable of number of trials (hours, 100hands or whatever) assuming I am correct in thinking that eventually ruin is certain. Ten thousand hours would be fine by me for example

Thankyou.

Ox