Most gamblers or traders want three things:
- to maximise their return,
- to limit drawdowns,
- and to avoid ruin.
This post will walk you through bet sizing with the Kelly Criterion for prediction markets and limit your bankroll's maximum loss with Value-at-Risk. We present a formula for risk management for prediction market gamblers.
A really simple way to explain Kelly Criterion
You want to size your bets in accordance with the returns and risk. The higher the return, the higher the bet. Likewise, the higher the risk, the lower the bet. So, it could be said that bet size is correlated with return and inversely correlated with risk.
We can write it as:
BetSize = Return/Risk.
This is intuitive for most people. We'll expand upon this using the Kelly Criterion.
We can rephrase it as:
BetSize = Edge/MarketOddsOfLosing. Return = Edge and Risk = MarketOddsOfLosing.
Edge can be stated as:
Edge = YourModelProbability - MarketPrice.
And MarketOddsOfLosing can be stated as:
MarketOddsOfLosing = 1 - MarketPrice.
Therefore BetSize = ( YourModelProbability - MarketPrice )/(1 - MarketPrice)
Formula for 1 Bet on a Prediction Market
% of bankroll = ( YourModelProbability - MarketPrice )/(1 - MarketPrice)
Example for 1 Bet
Republican Presidential Nominee 2028 has JD Vance at 39.3% for "yes" and 60.8% for "no". Lets say our model has Vance at 42.3%, this gives us a 3% edge.
% of bankroll = ( .423 - .393)/( 1 - .393) = .03/.607 = .0494
So we should put 4.94% of our bankroll on "yes".
Example of 1 Bet with Fractional Kelly
One flaw of the Kelly Criterion is that the variance is very high. Drawdowns can be very severe and winnings can be very large too. Fun fact, John Kelly, the maker of the Kelly Criterion, smoked 5 packs of cigarettes a day! He might smoked so much, to deal with all the anxiety of going full Kelly!
For people who don't want to experience such highs and lows, a fractional Kelly would be better.
In our previous example, the fraction of bankroll was 4.94%. We can go half Kelly with .0494 * .5, which is 2.47%. Or quarter Kelly with 1.235%. You can even go 8th or 16th Kelly if you wish.
Example with max Value-at-Risk (VAR)
Another issue of the Kelly Criterion is that it leaves one open to long tail risk. Let's say that the model shows a great edge and the risk is low. Following the Kelly Criterion, one would want to make a big bet. But if the model was wrong, this bet would result in a big loss. It is called long tail risk, because the more extreme the model shows the advantage (ie the more long tail it is), the biggest the loss ends up being.
We can limit this with VAR.
For example, let's say that our portfolio's policy has a max VAR of 2%. We would need to diversify our bets across 50 mutually exclusive series. Let's say we went half Kelly and our Kelly fraction is 2.47%. To obey our max VAR, we change the bankroll allocation to this bet to 2%.
Full Walkthrough
- We have a portfolio with 2% max VAR and half Kelly.
- Half Kelly = .5 * ( .423 - .393)/( 1 - .393) = .5 * .03/.607 = .5 * .0494 = 2.47%
- Since #2 is over 2%, we limit it to 2% bankroll.
Conclusion
Using the Kelly Criterion is based off of intuition and you can go fractional Kelly to maximise growth. You can add VAR on top of this to limit long tail risks.