I really enjoyed the recent (May 25th 2011) Behind the Bets Podcast on ESPN which featured handicapper Teddy Covers. He talked about many things including how to deal with losing streaks and bankroll management. But the thing that struck me was that he keeps his bankroll as his retirement fund and uses the income from his handicapping service for his living expenses.
This got me thinking about how to manage your bankroll while drawing an income from it. Or in other words how could you become a professional living solely off your winnings and not having any other source of income.
For example imagine you have developed a system which consistently generates positive EV bets so you use the Kelly criterion to size your bets in order to maximise the log of your bankroll growth. However as standard Kelly betting prohibits you from removing anything from your bankroll what happens if you start withdrawing a salary from your bankroll?
My approach is that instead of maximising the growth of your bankroll you would aim to keep your bankroll constant by withdrawing an amount before you bet of the correct size so that the average expected growth of your bankroll due to the bet will return your bankroll to its original level.
The key formula I am using is not the Kelly formula but the next step which is the average bankroll growth caused by a Kelly bet which is
Average bankroll growth per bet is (1+kd)^w*(1-k)^L
Where W and L are the probabilities of winning or losing the bet, k is the Kelly criterion percentage of your bankroll that you bet and D is the (fractional)odds you got on your bet.
For example if you are betting football and your system predicts 55% winners and you are getting odds of -110 (or 10/11 in fractional odds). Kelly says to bet 5.5% of your bankroll on each bet.
From the formula above the average growth per bet is 100.138% (or rather your bankroll will be 0.138% bigger on average after each bet). Therefore you can reduce your bankroll to the reciprocal of 100.138% which is 99.862% and your bankroll will be returned to its original size after the bet (on average). Therefore you can remove 0.138% of your bankroll as salary without leading to a long-term reduction.
This is 1/726th of your bankroll. So if you want to live off your bankroll with this system it depends how many bets you have a year what salary you want and what your bankroll is. If you have 50 bets a year your income is 50/726 of your bankroll so for a salary of 50,000 a year you need a bankroll of 726,000.
This seems like a very high bankroll requirement( although it is an annual income of 7.5%). So the key is to get a better system with a higher success rate and/or have more bets per year. For example If you had a system that generated twice as many bets your bankroll requirement would be halved for a given size of required salary.
If you had a system that won 56% of bets you could withdraw 1/380th of your bankroll per bet or if you expected to win 57% 1/233rd. Even a super duper 60% system allows only 1/86th withdrawal per bet.
So it looks extremely difficult to make a living solely of your bankroll unless you have a very good system with a lot of bets per year.
Are there any flaws in my logic? have I underestimated the risk of ruin? Has all of this been done before?