A Simple Rule of Thumb for Kelly Betting

There are at least two things to get right to be a great investor. First, you must choose the right investments. But second, you must decide to bet the right amount on each investment. This is called investment sizing or bet sizing and is possibly just as important as picking the right investments in the first place.

Consider an example where you can find investments that you believe will double your money with a 60% probability. If you decided to bet all of your money on these investments each time, eventually, you would go bankrupt because some of them won’t work out.

So the question is, what is the optimal fraction of your holdings that you should bet in order to maximise the growth of your investment over time? Mathematically, what you are maximising is the rate of compounding growth. This is equivalent to maximising the logarithm of your wealth over time. When you maximize this, you come up with a formula for sizing your bets or investments that is called the Kelly criterion.

The Kelly Criterion: Betting Proportionately to Your Edge or Confidence

In simple terms, the larger the edge you feel you have over the market, the more you should be betting as a fraction of your total wealth. If you have high confidence in the outcome of an investment relative to others, then you should be betting a higher fraction. Whereas, if you feel you only have a marginal edge over the market, you should be betting a small fraction of your total wealth.

Let me give a concrete example of this with respect to betting markets like Polymarket, where you can bet on binary outcomes, such as the outcome of an election, and you make a bet on either yes or no.

Let’s take an example where the current market odds an event (e.g. person X winning an election) are 40%, and you believe the true odds should be 50%. In that case, the formula for your Kelly bet size is given as follows:

Kelly fraction = (p - p_m) / (1 - p_m)

Where:

• p is your perceived probability of success

• p_m is the market’s implied probability of success

Plugging in the numbers shows that you should bet: (0.5 – 0.4) / (1 – 0.4) = 0.1 / 0.6 = 0.1667 or 16.67% of your wealth in this case. So if you have 100 Euro, you should bet 16.67 Euro.

Practical Considerations: Betting Less Than Full Kelly

In practice, betting the full Kelly fraction is not so common because it will still result in volatile ups and downs of your wealth. And so some people will tend to bet about half of the Kelly bet or maybe even a bit less to be a bit more conservative with the swings they will see in their wealth.

Simplified Kelly Bet for Low Probability Events

Now, in the special case where you are considering bets on scenarios where the probabilities are low, the formula (p - p_m) / (1 - p_m) simplifies to just p - p_m. For example, if the market thinks the probability is 5% of a president being elected, whereas you see the odds as a 10% chance, you can just take 10% - 5% to get your (full) Kelly bet of 5% of wealth.

Now, if the market-implied probability, p_m, of the event is not small, then the denominator in the fraction above becomes more important. And so you cannot simply use this p minus p_m formula in your analysis.

And that’s the Kelly bet and the approximation for low likelihood events in binary markets.

Finishing example:

• the market thinks the odds are 5%. You think the odds are 6%

• Your bankroll is $100

So, your Kelly bet is 6% - 5% = 1% of $100 = $1.

---

P.S. The full Kelly bet formula is:

It’s just that in a binary market, the market pays $1 to the winner, and so - if you win - you get back (1 / p_m) dollars. This means your net profit is 1/p_m - 1, or b = (1 - p_m)/p_m. Plugging that into the full formula for b, you’ll find that the kelly bet simplifies to the (p - p_m) / (1 - p_m) formula from earlier.

P.P.S. In practise, many investments aren’t binary, i.e. they have multiple outcomes and calculating the Kelly bet becomes complicated. The most useful part of Kelly is just the directional notion that - the more of an edge/confidence you have, the more you can bet as a percent of your bankroll. Possibly the inverse lesson (i.e. don’t bet much if you have low edge/confidence) is even more important.