How to determine the optimal Kelly fraction

math_formulas___ps_brush_set_by_soulrebel83-d15vjl7As we have discussed previously, the Kelly criterion is mathematically the optimal method of money management when betting. It takes into consideration the probability, and the odds in working out what percentage of your bank one should bet. The higher the overlay (advantage that you have over the bookmaker) and the shorter the odds, the larger the bet size. Whilst the smaller the overlay and bigger the outsider, the smaller the bet.

But can we ever be sure that the probabilities that Sportpunter’s models produce are accurate? No they aren’t. They are never the same as what the actual exact probability of a team to win actually is. This is because there are many things that the Sportpunter’s models cannot take into consideration. Perhaps a player recently got married, or had a family member pass away. There are countless things that make the probability not exactly correct.

However, it doesn’t need to be, and it is impossible for that to happen. What we require is that the probabilities are more accurate than the odds that the bookmakers set. Then we have an overlay and an advantage.

If you were playing certain card games, you may be able to work out the exact probability given certain circumstances, and then could use the full Kelly criterion for betting, but considering that the probability that we produce may not be 100% accurate, how do we use the Kelly criterion for betting? If we use the full Kelly criterion, surely we would be over betting?

And this is the case. A method to reduce this over betting is to use a fractional Kelly – basically decreasing your bet size by a certain factor.

I’ve always said that it is unwise to bet more on any one bet than 5% of your bank, but how do we determine the optimal Kelly fractional for each model?

Let’s have a look at some data and determine this. We are going to look at the MLB Totals model since it went live on 7/5/14, to look at what perhaps might be the optimal Kelly fraction for next year.

Since going live online, the baseball totals model made 7.24% ROI from 719 bets, and the average overlay during this time was 10.7%. This means that the optimal Kelly fractional would be 10.7/7.24 = 1.482.

This at first seems quite low, but that is credit to how well the Sportpunter MLB model went. If in hindsight, you were to use this Kelly fractional during the 2014 season, then you would have turned a $10,000 bank into $53,943. A very nice investment indeed!

However on closer examination, some of the bet sizes might be too large. The maximum bet size suggested using the 1.48 Kelly fractional is 27%. I think a lot of people would feel a bit cautious about betting $2,700 out of their $10,000 bank. However the average bet size is only 7.5%.

If we decide to keep with the previously stated rule, never to bet more than 5% of one’s bank, then we can adjust the optimal Kelly fractional to 0.27/.05 * 1.482 = 8.

This means that the optimal Kelly fraction to bet, so much so that we don’t bet any more than 5% of the bank would be 8. In this scenario, we managed to increase our bank from $10,000 to $19,152, almost doubling the bank in one year.

The maximum bet of the bank was indeed 5%, and the average bet size was 1.4%. Betting an average $140 into a $10,000 bank may not be risky enough for some, and one could adjust as so they desire. Perhaps you would chose a max 7.5% of bank bet which equals Kelly fractional of 5.3, and an end of year bank of $25,145.
Either way, the above calculations, whilst complex for many, show that there is a way of working out the optimal Kelly fraction. If it has all gone over your head, not to worry; just set a Kelly fraction that you feel comfortable betting.

Keep in mind that this includes one massive assumption: that the overlays and profits as seen in the past will continue in the future. Perhaps 719 bets might not be enough to accept this with absolute certainty, and in fact no amount of bets ever will. It’s always better to lean on the side of being conservative, which, for many punters, is a very hard thing to do.

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