Oct 05, 2020 The Kelly Criterion is a formula used to bet a preset fraction of an account. It can seem counterintuitive in real time. The Kelly formula is: Kelly% = W – (1-W)/R where: Kelly% = percentage of capital to be put into a single trade. W = Historical winning percentage of a trading system. R = Historical Average Win/Loss ratio. In probability theory and intertemporal portfolio choice, the Kelly criterion (or Kelly strategy or Kelly bet), also known as the scientific gambling method, is a formula for bet sizing that leads almost surely to higher wealth compared to any other strategy in the long run (i.e. Approaching the limit as the number of bets goes to infinity). Sep 01, 2017 The answer is that the formula commonly known as the Kelly Criterion is not the real Kelly Criterion - it is a simplified form that works when there is only one bet at a time. How to use the 'real' or generalised Kelly Criterion.
Have you ever wondered to yourself how to calculate EXACTLY what % of your account balance to risk on a given trade?
I mean you're sitting there, about to open a trade for what seems to be a good opportunity, but you're not sure what the optimal volume / lot size is. On one hand, you don't want to risk too much in case the market goes against you, but on the other hand, you don't want to risk too littleeither. Sound familiar?
That's what we are going to explore today – EXACTLY what % of your balance is it optimal to risk on a given trade. It's going to be fun!
Let's imagine that we are playing a game: you bet $1 on a coin flip. Rules are:
- On heads I pay you $2;
- On tails you lose your $1
Sounds like an incredible opportunity, right?
From a mathematical perspective the Expected Value (EV) of this investment can be calculated as:
EV = $2 * 0.5 – $1 * 0.5 = $0.50
A positive EV indicates a profitable opportunity. What it means is that in the long run, if you average out all the tosses, then the player (you) will be earning an average of 50 cents per coin toss.
This coin toss game is great analogy for a profitable trading strategy – e.g. where the TakeProfit = 2 x StopLoss, and the frequency of profitable and loss trades is roughly the same. Obviously a great trading strategy!
You are KING!! You cannot lose this one!
Actually, not. Turns out you can lose even in a favourable game like this…… In comes Money Management…
Let's say that you only have $100 to play this game. And because it's such a great opportunity you decide that at any given point in time you will bet 75% of your funds. Let's see how this will play out for you:
First toss:
- you bet 75% x $100 = $75
- it's heads! – you just earned $150
- your balance is now $250
Second toss:
- you bet 75% x $250 = $187.50
- it's tails! – you just lost $187.50
- your balance is now $62.50
So you see now that after only two tosses you are already in the negative. And since Heads and Tails come in with roughly the same frequency, in the long run this pattern will continue and you will lose all your money!! Shocking!
PS: Heads-Tails or Tails-Heads – order doesn't matter, check for yourself.
What did you do wrong? :<
You over-betted!!
Let's see what happens to your account balance if you choose to bet a different % of your balance (I challenge you to check these numbers):
- 10% -> $108
- 20% -> $112
- 30% ->$112
- 40% -> $108
- 50% -> $100 (you break even)
- 60% -> $88
- 70% -> $72
- 80% -> $52
- 90% -> $28
- 100% -> $0
![Kelly Kelly](https://www.assetmacro.com/wp-content/uploads/2015/06/Kelly-Formula.png)
Can you see the pattern? The return increases, drops off, then goes into negative. So that means there has to be a maximum somewhere between 20% and 30%.
This is where the Kelly formula comes into play:
K = ( PxB – (1–P) ) / B
where:
- K – optimal % risk
- P – odds of winning
- B – payout on the bet
This formula determines the optimal % of your account that you can bet to get the most profitable outcome in the long run. Let's calculate K for our scenario:
- B = 2 (payout is 2:1)
- P = 0.5 (50% chance of winning)
Plugging B and P into the formula gives:
K = (0.5*2 – (1 – 0.5)) / 2 = 0.5 / 2 = 0.25
Therefore, the best outcome in the long run is achieved by betting 25% of your account on every coin toss!
- 25% -> $112.50 (after 1 heads and 1 tails)
Indeed, this outcome is better than anything we calculated prior.
Note: this percentage is called one Kelly and for every trading strategy the exact value can be different.
Kelly's criterion is a good start, but it's not the full picture. If you visualize the relationship between balance growth and the % of risk, it will look like this:
From here we can witness the same pattern as we noticed before – to the left of one Kelly return increases as you increase risk. Then return drops off and becomes zero at 2K. After 2K you are bound to lose your investment in the long run.
![Kelly Kelly](https://www.arbcruncher.com/img/kelly.gif)
The best part is that this chart always holds – for any strategy!!
As long as you have calculated your Kelly correctly, you can draw up a chart like this for your specific case and understand how your investment will behave in the long term depending on your selected risk exposure. Let's talk more about that.
We already know that in an ideal scenario you are best taking the approach of betting EXACTLY one Kelly. However, the world is not always ideal, so now we will analyse the Kelly chart and understand the implications of sub-optimal betting (i.e. not one Kelly).
I like to break up the Kelly chart into four sections:
- Yellow: from Zero to 1/2 Kelly is the Conservative risk area
- Orange: 1/2 Kelly to 1 Kelly is the Aggressive risk area
- Red: 1 Kelly to 2 Kelly is the Over-Aggressive risk area
- Black: Anything above 2 Kelly falls in the Insane risk-taking area
Conservative Risk-Taking
Half-Kelly is a surprisingly cool number. The fact of the matter is that sometimes the Kelly formula can give you quite high values, for example 25% risk. Even though theoretically this may be the optimal risk exposure, in practice it may be too high.
Reasons include things like a possibility of a string of consecutive losses which are not adjusted for in the Kelly Criterion as well as balance volatility which we will talk about in another article (I will post the link HERE when I publish it). In short, there are legitimate reasons why you may think your Kelly is too high. That's when we look at the Half-Kelly.
The main reason why the Half-Kelly is so great is because it halves your risk but the long-term return only goes down by 25%. You can even see it from the chart. Other reasons include a reduction in balance volatility by more than 50% and a larger margin of safety in your risk estimate. More on that here.
If you are trading below the Half-Kelly then you are being quite conservative. Which may be a good thing for risk-averse investors or if you are in control of a very large balance.
Aggresive Risk-Taking
Anything between Half-Kelly and a Full Kelly is deemed Aggressive risk-taking.
Your returns are higher, but not much higher. Consider our example with the coin flip game: risking 20% (4/5 Kelly) took our balance to $112, and risking 25% (Full Kelly) took our balance to only $112.50.
That's only an extra 0.5% (50 cents over $100) profit for a 5% increase in risk. Is it really worth it? That's why we associate this area with Aggressive risks.
Over-Aggressive Risk-Taking
For any long-term return that you are going to get by betting in the red area, there is a matching return in the yellow and orange areas. So naturally, why would you EVER set your risks in the red area.
Allow me to clarify..
If you are looking at ONE single trade, then of course, if you win – the more you had risked, the more will be your return. However what we are looking at here is lots and lots of trades – the Kelly Criterion works in the long run.
Therefore, what the Over-Aggressive risk zone is telling us is that in the long run there is no need to risk such a large % of your balance. You can achieve the same outcome by risking less.
Question: 'But what if I don't have a systematic trading strategy? What if all my trades are different and ad-hoc? In that case the Kelly Criterion does not apply??'
Answer: Get a trading strategy. Then come back and read this article.
Insane Risk Taking
The title and the chart speak for themselves. If you are in this zone – get out now!! The only other alternative is a margin call.
A good example of insane risk taking was when we bet 75% of our balance in the coin-tossing game. There 2 x Kelly was equal to 50%. So we were definitely in the insane zone, and you saw for yourself how quickly the balance went down.
Very easy. But you have to have a trading strategy for this to work.
If your trading strategy has a fixed StopLoss and TakeProfit, then you are in luck my friend! For you the value of B equals to TakeProfit / StopLoss (don't forget to subtract / add the spread if it's not included in your TP / SL).
You now have B, all you have left to find out is the value of your P parameter.
To get P you need to look at your history of trading in similar market conditions. I recommend checking the past 100 trades just to be certain.
See what % of trades are winners. That will be your P.
Now that you have both B and P – plug them into the Kelly formula and see what you get. Oooohhh…… Exciting!
![Kelly Criterion Formula Kelly Criterion Formula](https://i1.wp.com/blogs.cfainstitute.org/investor/files/2018/06/Kelly-Criterion-Chart-2.png?resize=640%2C399)
My K is negative. What does this mean?
If your K is negative, then your trading strategy is a bad trading strategy. You will lose in the long run.
Get a new trading system.
Let's look at a trading example.
Say, you have a EURUSD trading strategy that wins approximately 70% of the time. The StopLoss in your strategy is 40 pips and the TakeProfit is 20 pips (spread accounted for).
This means that your B and P parameters are as follows:
- B = 20 pips / 40 pips = 0.5
- P = 70% = 0.7
Let's input these values into Kelly's formula and see what we get:
K = ( PxB – (1–P) ) / B
K = ( 0.7 x 0.5 – (1–0.7) ) / 0.5 = 0.1
![Kelly Criterion Formula Kelly Criterion Formula](https://insights.matchbook.com/wp-content/uploads/sites/2/2016/03/kelly_criterion_equation.png)
Can you see the pattern? The return increases, drops off, then goes into negative. So that means there has to be a maximum somewhere between 20% and 30%.
This is where the Kelly formula comes into play:
K = ( PxB – (1–P) ) / B
where:
- K – optimal % risk
- P – odds of winning
- B – payout on the bet
This formula determines the optimal % of your account that you can bet to get the most profitable outcome in the long run. Let's calculate K for our scenario:
- B = 2 (payout is 2:1)
- P = 0.5 (50% chance of winning)
Plugging B and P into the formula gives:
K = (0.5*2 – (1 – 0.5)) / 2 = 0.5 / 2 = 0.25
Therefore, the best outcome in the long run is achieved by betting 25% of your account on every coin toss!
- 25% -> $112.50 (after 1 heads and 1 tails)
Indeed, this outcome is better than anything we calculated prior.
Note: this percentage is called one Kelly and for every trading strategy the exact value can be different.
Kelly's criterion is a good start, but it's not the full picture. If you visualize the relationship between balance growth and the % of risk, it will look like this:
From here we can witness the same pattern as we noticed before – to the left of one Kelly return increases as you increase risk. Then return drops off and becomes zero at 2K. After 2K you are bound to lose your investment in the long run.
The best part is that this chart always holds – for any strategy!!
As long as you have calculated your Kelly correctly, you can draw up a chart like this for your specific case and understand how your investment will behave in the long term depending on your selected risk exposure. Let's talk more about that.
We already know that in an ideal scenario you are best taking the approach of betting EXACTLY one Kelly. However, the world is not always ideal, so now we will analyse the Kelly chart and understand the implications of sub-optimal betting (i.e. not one Kelly).
I like to break up the Kelly chart into four sections:
- Yellow: from Zero to 1/2 Kelly is the Conservative risk area
- Orange: 1/2 Kelly to 1 Kelly is the Aggressive risk area
- Red: 1 Kelly to 2 Kelly is the Over-Aggressive risk area
- Black: Anything above 2 Kelly falls in the Insane risk-taking area
Conservative Risk-Taking
Half-Kelly is a surprisingly cool number. The fact of the matter is that sometimes the Kelly formula can give you quite high values, for example 25% risk. Even though theoretically this may be the optimal risk exposure, in practice it may be too high.
Reasons include things like a possibility of a string of consecutive losses which are not adjusted for in the Kelly Criterion as well as balance volatility which we will talk about in another article (I will post the link HERE when I publish it). In short, there are legitimate reasons why you may think your Kelly is too high. That's when we look at the Half-Kelly.
The main reason why the Half-Kelly is so great is because it halves your risk but the long-term return only goes down by 25%. You can even see it from the chart. Other reasons include a reduction in balance volatility by more than 50% and a larger margin of safety in your risk estimate. More on that here.
If you are trading below the Half-Kelly then you are being quite conservative. Which may be a good thing for risk-averse investors or if you are in control of a very large balance.
Aggresive Risk-Taking
Anything between Half-Kelly and a Full Kelly is deemed Aggressive risk-taking.
Your returns are higher, but not much higher. Consider our example with the coin flip game: risking 20% (4/5 Kelly) took our balance to $112, and risking 25% (Full Kelly) took our balance to only $112.50.
That's only an extra 0.5% (50 cents over $100) profit for a 5% increase in risk. Is it really worth it? That's why we associate this area with Aggressive risks.
Over-Aggressive Risk-Taking
For any long-term return that you are going to get by betting in the red area, there is a matching return in the yellow and orange areas. So naturally, why would you EVER set your risks in the red area.
Allow me to clarify..
If you are looking at ONE single trade, then of course, if you win – the more you had risked, the more will be your return. However what we are looking at here is lots and lots of trades – the Kelly Criterion works in the long run.
Therefore, what the Over-Aggressive risk zone is telling us is that in the long run there is no need to risk such a large % of your balance. You can achieve the same outcome by risking less.
Question: 'But what if I don't have a systematic trading strategy? What if all my trades are different and ad-hoc? In that case the Kelly Criterion does not apply??'
Answer: Get a trading strategy. Then come back and read this article.
Insane Risk Taking
The title and the chart speak for themselves. If you are in this zone – get out now!! The only other alternative is a margin call.
A good example of insane risk taking was when we bet 75% of our balance in the coin-tossing game. There 2 x Kelly was equal to 50%. So we were definitely in the insane zone, and you saw for yourself how quickly the balance went down.
Very easy. But you have to have a trading strategy for this to work.
If your trading strategy has a fixed StopLoss and TakeProfit, then you are in luck my friend! For you the value of B equals to TakeProfit / StopLoss (don't forget to subtract / add the spread if it's not included in your TP / SL).
You now have B, all you have left to find out is the value of your P parameter.
To get P you need to look at your history of trading in similar market conditions. I recommend checking the past 100 trades just to be certain.
See what % of trades are winners. That will be your P.
Now that you have both B and P – plug them into the Kelly formula and see what you get. Oooohhh…… Exciting!
My K is negative. What does this mean?
If your K is negative, then your trading strategy is a bad trading strategy. You will lose in the long run.
Get a new trading system.
Let's look at a trading example.
Say, you have a EURUSD trading strategy that wins approximately 70% of the time. The StopLoss in your strategy is 40 pips and the TakeProfit is 20 pips (spread accounted for).
This means that your B and P parameters are as follows:
- B = 20 pips / 40 pips = 0.5
- P = 70% = 0.7
Let's input these values into Kelly's formula and see what we get:
K = ( PxB – (1–P) ) / B
K = ( 0.7 x 0.5 – (1–0.7) ) / 0.5 = 0.1
This means that the optimal risk for this trading strategy that will maximize your profits in the long term is 10%.
If you want to be a bit more conservative, then go with the Half-Kelly of 5%.
Whatever you do, don't invest more than 10% per trade – it's pointless.
If you invest more than 20% then you will turn this great strategy into one that will ruin your account.
That's how you apply the Kelly Criterion in practice.
Till soon, my friend!
Kirill
What are you waiting for?START LEARNING FOREX TODAY!
The Kelly Criterion has come to be accepted as one of the most useful staking methods for sharp bettors. While most of us think we have an understanding of the Kelly Criterion and how it works, this is merely a simplified version of the formula. Our latest Guest Contributor has provided an in-depth explanation of the 'real' Kelly Criterion. Read on to learn more.
Anyone who is unfamiliar with how the Kelly Criterion can be used to determine optimal bet sizes should read Dominic Cortis' article on how to use the Kelly Criterion for betting. This approach works well in most cases, however, there are some situations where the Kelly Criterion formula can give some head-scratching results.
Using the examples below, we can examine the potential flaws in using a simplified Kelly Criterion formula.
Example #1 - A soccer game where both a visitor win and draw outcome provide the bettor with an edge:
The Kelly formula would suggest staking 2.5% of bankroll on both the visitor win and the draw, staking a total of 5% of bankroll. Looking at the Handicap odds for the same soccer game changes how we might view the use of the Kelly Criterion.
Example #1A - The same soccer game in example 1 re-stated as a Handicap line:
A bet on the visitor +0.5 at odds of 2.50 is the equivalent to betting half the amount on both the visitor win and draw (both at odds of 5.00). So why does the Kelly formula give a different answer?
The answer is that the formula commonly known as the Kelly Criterion is not the real Kelly Criterion - it is a simplified form that works when there is only one bet at a time.
How to use the 'real' or generalised Kelly Criterion
Below is an explanation of how to apply the generalised Kelly Criterion to betting:
Step - 1: List all possible outcomes for the entire set of bets.
Step - 2: Calculate the probability of each outcome.
Step - 3: For each possible outcome, calculate the ending bankroll for that outcome (starting bankroll plus all wins minus all losses). Leave the bet amounts as variables.
Step - 4: Take the logarithm of each ending bankroll from step 3.
Step - 5: Calculate the weighted average of the logarithms from step 4, weighted by the probabilities from step 2. Call this the 'objective'.
Step - 6: Find the set of bets that maximises the objective from step 5. These are the optimal bets according to the Kelly Criterion.
In order to find the set of bets that maximises the objective, simply use Microsoft Excel's built-in 'solver' module (see below) - this takes care of the complexities of advanced calculus and eliminates a tedious trial-and-error approach.
The result from using these six steps is as follows:
Note that this is identical to the result in Example #1A, where the simplified version of the Kelly Criterion does work. By making two mutually exclusive bets on the same game, the two bets act as a partial hedge for each other – reducing the overall level of risk, which Kelly rewards by increasing the bet amount (compared to the calculation in Example #1).
Additional uses from the generalised Kelly Criterion
We have already seen how this generalised Kelly Criterion can produce completely different results than the simplified Kelly formula that most bettors will use when there are multiple edges in the same game.
There are, of course, occasions when you might have multiple edges on different games, all taking place at the same time. The example below is one such situation:
Example #2 - Betting with an edge on four separate games that are all taking place at the same time.
Now while each of these bets make sense individually, using the simplified Kelly Criterion would result in staking 110% of bankroll - something that clearly doesn't make sense. However, by applying the six steps stated above, we can see how the generalised Kelly Criterion produces a different set of results.
Because the four games are independent, the probability of each outcome can be calculated as the product of the probabilities of each game; for example, the first row probability would be calculated as:
In additional to calculating the optimal staking amount for a bet with multiple edges, the generalised Kelly Criterion can also be used when bettors have a viable hedging opportunity.
Example #3 - Hedging Garbine Muguruza to win Wimbledon in 2015.
Using the 2015 Wimbledon tournament example previously used in this hedging article, we can see how the generalised Kelly Criterion should be applied to a hedging opportunity.
If you had €1,000 starting bankroll, and you bet €10 on Muguruza at 41.00 to win Wimbledon 2015 outright, you would have to decide how much to hedge on Williams at 1.85 in the final. Let's assume that the odds in the final are efficient, that is, they accurately reflect each outcome's probability so that there is no edge on either side.
The more commonly used simplified Kelly formula would provide the following results in the scenario:
However, applying the generalised Kelly procedure as stated above yields the results below:
Using this method shows that optimal strategy would be to bet €183.41 on Williams at 1.85 to beat Murguruza in the final and win the tournament - this will hedge most, but not all, of your open position on Murguruza to win the tournament.
Taking the exponential of the objective gives an interesting number, called the 'certainty equivalent'. In Example #3 above, the certainty equivalent is exp(7.072341) = 1,178.90. This means that, from a Kelly Criterion perspective, the bettor would be indifferent between having the listed set of bets and having €1,178.90 in risk-free cash.
Kelly Criterion Formula
If we remove the hedge bet, we are left with the following:
So, the effect of the hedge bet is to raise the certainty equivalent from €1,166.87 to €1,178.90. So even though the hedge bet itself has a negative expected value, the resulting reduction in risk is so beneficial that from a Kelly perspective, it has created added value that's equivalent to €12.03 in cash.
Some other applications of Generalised Kelly
Kelly Criterion Formula For Excel
- Finding optimal bet sizes for a set of 'round robin' combinations of parlays or teasers;
- Finding optimal bet sizes for a set of futures bets on several different teams to win the same division or championship;
- Deciding between different ways to hedge an existing bet (money line, spread, buying/selling points), especially if some options result in a 'middle' opportunity;
- Figuring out how much to add to, or exit from, an existing position after a line move.
Use the real Kelly Criterion to empower your betting - get the lowest margins and highest limits online at Pinnacle.
This Guest Contribution was made by on of our Twitter followers - @PlusEVAnalytics. If you would like to make your own contribution, contact us on Twitter or email us.