Kelly Math Weirdness

We started talking about Kelly criterion a couple weeks ago. As you play with the ideas yourself, I’ll point out 2 subtleties. One here and another below in the Masochism section.

Edge/Odds

I posted a couple ways to express the Kelly formula. Because it’s easy to remember, I prefer the simple expression edge/odds.

If you use this version too, let me offer some user notes.

• It only works when there’s a possibility of loss
  
  This is a technicality but consider the following bet:
  
  A stock is $100 and you believe it is 90% to worth $100 and 10% to be worth $300.
  
  The expected arithmetic return is therefore 20% (.90 x 100 + .10 x $300 minus your $100 investment)
  
  The odds or percent return when you win is 200%
  f* = Edge/odds = 20/200 = 10%
  
  With this version of the formula…

…you get a divide by zero error. Which is nature’s way of saying “Bruh, you can’t lose with this proposition you should bet 100% why you asking a calculator.”

• The second user note for using edge/odds is noticing a a counterintuitive idea:
  
  For a given level of edge, the optimal Kelly fraction to bet decreases as you get better odds (ie the denominator increases).
  
  Kelly has a preference for high win rates, an attribute that always arrives with negative skew.
  
  We’ll address this in the next section.

Bias towards negatively skewed bets

Consider 2 bets:

• A 90% chance of getting paid 22.22%, 10% chance of losing my bet
  
  The expectancy is straightforward. If you start with $10 and play 10x betting $1 on each trial you will lose $9, and your last dollar will get you paid $10 leaving you with $11 total. A 10% total return or 10% arithmetic expectancy.
  
  Using the spreadsheet:

The prescribed Kelly fraction is to bet 1% of your capital on this proposition.

This is a positively skewed bet. You lose most of the time, but win a large amount occasionally.

Let’s look at a negatively skewed bet with the same 10% expectancy.

• A 90% chance of getting paid 22.22%, 10% chance of losing my bet

Again, we start with $10 and bet $1 each time. You will earn $.22 9x or $2 and lose a dollar on the 10th trial. Once again you’re net profit is $1 or a 10% expected return. But look what calculator spits out:

The expectancy is the same but now Kelly wants you to bet nearly 1/2 your bankroll.

My intuition is that Kelly conclusions are loaded on volatility as opposed to higher order moments of a distribution. I’ve discussed this many times but to find the links I asked MoontowerGPT:

The first link of the responses is the most relevant (it’s embedded in the second link as well):

🔗Lessons From A Skewed Coin

Kelly’s bias towards negatively skewed bets is already understood:

And here you have Euan’s adjustment:

🔗The Kelly Criterion and Option Trading

[Euan needs no boost from me but I’ll add that his book Positional Option Trading was terrific. My notes here]

In real-life, almost nobody is aggressive enough to bet full Kelly (at least amongst those who would consider using Kelly in the first place). Half or quarter Kelly is more common and Euan’s adjustment will lower the prescribed full Kelly amount even further in the presence of strong negative skew.

This bit from Fortune’s Formula is instructive:

A Kelly’s bettor’s wealth is more volatile than the Dow or S&P 500 have historically been. In an infinite series of serial Kelly bets, the chance of your bankroll ever dipping down to half its original size is 50%.

A similar rule holds for any fraction 1/n. The chance of ever dipping to 1/3 of your original bankroll is 1/3. The chance of being reduced to 1% of your bankroll is 1%.

Any way you slice it the Kelly bettor spends a lot of time being less wealthy than he was.

A Kelly bettor has a 1/3 chance of halving the bankroll before doubling it. – The half Kelly bettor has only a 1/9 chance of halving before doubling.

The half Kelly bettor halves risk but cuts expected return by one 1/4.

• If you have gotten this far, you’ll probably enjoy these poll questions which strike at a lack of strict risk ordering and transitivity in comparing propositions.
• I’m done writing about Kelly and my current take is when faced with a bet whose properties lend themselves to the formula I’d like to see what it prescribes to get a ballpark for the upper bound of how much to bet. The ultimate choice of sizing would incorporate my instincts about the shape of the payoff and personal comfort.
• I’ve shared my summary of the Haghani bet sizing study and the overwhelming conclusion is people, including economists and grad students, instincts are quite poor on bet sizing. Just acquiring the knowledge that Kelly exists would help a reader recruit their “System 2 thinking” even if the details are foggy.
  
  This was a widely read post:
  
  🔗Bet Sizing Is Not Intuitive