The speed which shows recoup is dependent on many factors. With A Chorus Line for instance they went to San Francisco and played a large theatre to sell-out business for 6 weeks. They came into town partially recouped, went to sell-out and so the payoff was rather quick.
Other shows muddle along breaking even weekly but never paying off. Tommy comes to mind although there are many more.
Back in the day, producer Morton Gottlieb was known to over-capitalize. Same Time, Next Year which was not a big show, opened out of town and made money there. When it came in, the reserve fund was unused and the investors were paid off in a matter of a few weeks. The rest was gravy.
Gottlieb's production of Tribute with a big star (Jack Lemmon) used the same strategy and paid back the investors 100% on opening night in NY. This was a major publicity coup.
"If my life weren't funny, it would just be true. And that would be unacceptable."
--Carrie Fisher