reduced kelly bet sizing (received via PM)

[Ganchrow]

Ganchrow,

i was playing around w/ your kelly calc, and noticed something strange:

suppose my winrate is 59%, all at -113 odds. at full kelly, the median bankroll (if i start w/ $10k) after 600 bets is $731k.

i've heard that if one bets half kelly, one gets (in theory) 75% of the full kelly's return (based on median bankroll, and not average BR). similarly, if one bets 70% of full kelly, one's (median) return is ~90% of full kelly. but when i input 0.7 and 0.5 for 70% and half kelly (respectively) into the calc, the results are very different.

instead of expecting a median BR (after 600 bets) of 658k and 548k for 70% and 50% kelly betting (respectively), the calculator gives me a median BR of 498k (for 70% kelly) and a median bankroll of 251k (for half-kelly).

what's the explanation for this? have i been misinformed about the expected median return for half-kelly?

As a general rule it is indeed true that with the edges and odds one is likely to encounter in sports betting, the expected growth rate of half-Kelly and 70%-Kelly correspond to approximately 75% and 90% that of full-Kelly (respectively). You should note that these approximations will break down drastically at the extremes.

So why the discrepancy after 600 bets?

The two approximations you've noted apply to (geometric) average growth rates, while the median bankroll is a compounded growth figure. As Albert Einstein allegedly quipped, “The most powerful force in the universe is compound interest.”

Now while Einstein's authorship of the above statement is dubious, there's no question that the effect of compound interest over a large number of trials can be substantial.

So let's look at your example above:

• US Odds: -113
• Win Prob: 59%
• Bankroll: $10,000.000
• Trials: 600

At full-Kelly:

• Stake: $1,267.000
• Expected Growth: $71.807 (71.807/10,000 = 0.71807%)
• Median bankroll after 600 trials: $731,869.998 ≈ (1+0.71807%)⁶⁰⁰ (slight difference due to rounding)

At 70%-Kelly:

• Stake: $891.031
• Expected Growth: $65.375 (65.375/10,000 = 0.65375%)
• Median bankroll after 600 trials: $498,855.372 ≈ (1+0.65375%)⁶⁰⁰ (slight difference due to rounding)

At 50%-Kelly:

• Stake: $638.125
• Expected Growth: $53.905 (53.905/10,000 = 0.53905%)
• Median bankroll after 600 trials: $251,696.177 ≈ (1+0.53905%)⁶⁰⁰ (slight difference due to rounding)

So indeed what we see is that 50%-Kelly expected growth is about 75% of full Kelly growth (0.53905% / 0.71807% = 75.069%), and that 70%-Kelly growth is about 90% of full Kelly growth (0.65375% / 0.71807% = 91.043%).