The Kelly criterion formula was primarily created to find out theoretically the optimum amount required to resize the amount of investment or bet for optimal gain. As in gambling, when the gambler knows the odd, using the kelly criterion formula can help maximize the profit.
Primarily the idea was created to find the optimum size of bets when the odds are known. Later, this formula was practically demonstrated in gambling. Earlier it was believed that winning a bet depends on the odds and the situation. But the importance of betting amount was not given much importance. This formula first brought the idea of scientific gambling. The Kelly criterion showed that the optimum amount of betting size is important to increase the winning probability in the long run.
Later, it was found that the same idea can be replicated in stock market investments. This also gave us the idea of optimum investment size, proper diversification and scientific management of risk. Today, we find fund managers and big funding houses are using this criterion to practically manage their investment size. It is known that even big names like Warren Buffet also used this criterion to organize his investments.
Background of Kelly criterion
As discussed earlier, the Kelly criterion was applied to manage the betting amount in gambling. People tend to gamble to earn extra. Therefore as the gambling interest rose, more and more research was being carried out to make the betting size more scientific. From one such research, it was found that there is a good relation between investment size and profit.
The Kelly criterion established that an optimum investment amount gives you positive returns in the long run. Soon it was accepted by all and found that betting can be scientific too. Downsizing the betting amount in gambling proved to be a correct method if one wanted to bet more for maximum profit.
Later, this formula proved to be useful in stock market investment also.
Who created the Kelly criterion for the first time?
Back in 1956, a Bell Lab scientist discovered this formula. John Larry Kelly Jr. was a Bell Lab scientist and a researcher. He has been working on Claude Shanon’s Information theory when he came up with this mathematical formula. Since then, the Kelly criterion found numerous applications in the field of betting, investment and asset management.
From then on, the application of the Kelly criterion formula has been a part of mainstream value investing in all kinds of assets. In gambling, sports betting, in the stock market – this method has been in use successfully.
What is the Kelly Criterion? – The Definition
The Kelly criterion is a purely mathematical formula that helps to determine the size of a bet. In other words, the formula helps to increase the bet amount when odds are in favor and downsize the bet amount when the odds are against the gambler.
It is a mathematical formula that enhances capital growth. The formula gives output in decimals. When converted into a percentage, the actual amount of investment can be found.
Here, in the diagram, we can see the Kelly criterion formula. It has been charted in respect of risk vs return.
The formula is –
f = Bp-Bq = edge odds
or K = p * B (1 – p) / B
where,
B = fractional odds i.e. ratio of risk over reward or win over the loss,
f = portion of wealth wagered or % of investment or bet wagered for maximum profit
p = probability of wins against odds
and q == 1-p (probable losses)
From this diagram, it is easily understood that as a general thumb rule the return increases with risk. For a conservative investor, the return is low. When the investor turns aggressive, the return increases. But for an over-aggressive gambler, the return drastically comes down, because the risk increases too much.
For the aggressive trades, as the diagram shows, the return is maximum, But over-aggressive trades are described as insane here. The return on bet decreases sharply. At some point the risk becomes suicidal. Those are very risky bets. The return goes below the zero line. And the bet is a highly losing bet when the curve touches the lowermost point. At such times there is a risk of losing the whole betting amount.
Hence it is the optimum investment amount that gives the highest return over time when the odds are manageable. Increasing bet size or increasing, odds will only end up losing money in the long run. This optimum investment amount is found to be nearly 20% of capital when odds are constant. If the odds increases, the risk of losing the bet increases too. Over a long time, one gets to win more if the Kelly criterion is followed judiciously.
What is a good Kelly criterion?
The graph above shows betting fractions and how they perform. The Kelly ratio gives us guidance on value investing, What should be the ideal percentage of investment that can give maximum profit over longer periods.
Here we have used the win probability ratio in a coin flip is 60%. The optimum amount of bet for this is known as the Kelly bet. It is found to be 20% of total investment which gives optimum profit. With time the profit increases. Each round will give a profit of 2.03% over time. With less than 20% bet sizing, the return decreases. A 10% bet size gives a 1.5% return over time. But when bet size increases, the losing chances increase. Bet size is more than 38% will increase the chance of losing by a great deal. The Kelly Bet is found to be nearly 20% which gives maximum return over time.
Betting more than 20% decreases the chance of profit over a long time. This is how the Kelly criterion gives a successful investment strategy that will give maximum profit over a longer time. This is how value investing is done. The idea of diversified fund allocation also comes from this. The fund managers use this idea to their advantage to a great extent.
How is the Kelly Criterion calculated?
The Kelly criterion has a purely mathematical calculation. This is known as the Kelly criterion formula. The Kelly criterion can be applied both to Gambling and in-stock investments.
The Kelly criterion formula in Gambling
This formula was created by scientist John Keller for gambling. The idea was to create an ideal betting size that would give a maximum return in long run. The formula is,
f* = p – q/b = p + (p-1)/ b
Here, f* is the current bet size with respect to total capital, p is the winning probability, q is the losing probability (q = 1-p) and b is the proportion of bet to be gained with every win.
Say, a gambler bets with 2:1 odds in favour. If the gambler plays with Rs 10, then a win gives the gambler Rs 30. It gives a gain of Rs 20 on an investment of Rs 10. Here b = 20/10 = 2.0
Say, the gambler plays for a bet with a winning probability of 60%.
Under that condition, p = 0.60 and q = 0.40 and the gambler is given 1 to 1 odds, the winning bet b = 1.
Therefore, the gambler should bet 20% of the total capital on each bet.
Because, f* = 0.6 – 0.4/1 = .20. Hence 20% of the betting amount is to be used with every bet to make a profit that can be sustained over a long period.
If, b = q/p, then the gambler has zero edges and the Kelly formula recommends betting nothing in that condition.
If b < q/p, then the gambler has a negative edge, the Kelly formula will give a negative result. In that case, the gambler should take the other side of the bet to get a positive edge.
The Kelly criterion formula for investing
The Kelly criterion formula allows for partial losses. A stock market investor knows he has to allow for partial losses to gain in the long run. This is not meant for a one-time investment. So the formula is used in more of a generalized form.
Here f* = fraction of capital investment for a particular investment,
p = the probability of growing the capital invested
q = the probability of decreasing the value of capital investment (q = 1-p)
a = the fraction of capital that is lost in case of a negative outcome. If, in the end, capital is eroded by 10%, then a = 0.1
b = the fraction of capital that is gained from a positive outcome. If the investor gains a net value of 10%, then b = 0.1
We should also consider that the Kelly criterion formula works only when the outcome is known. But in the stock market, it is not the case. The investor can guess the outcome but doesn’t know about the outcome beforehand. For this reason, it is not recommended for a conservative investor to use the Kelly criterion in investment.
It must also be noted that this formula can give the value of Kelly fractions being higher than 1, or K > 1. Therefore, as per this formula using leverage in stock market investment theoretically favors the investor.
Does Warren Buffet use the Kelly criterion?
From the start of this millennium, the Kelly criterion formula has made its place in the mainstream stock investment theory.
It was also known that well-known investors like Warren Buffet and Bill Gross have made good use of this theory in their investments. This news made the application of the Kelly criterion very popular with other stock market investors.
Detailed Explanation of Kelly Criterion
This formula is a somewhat simple equation to understand the idea of having an edge. This expression means having an advantage over others.
As in gambling, the bettor works with the winning edge to win bets. The odds mean having a chance of winning.
Hence a good bet means having an edge when odds are in favor.
But all these terms we discussed here, in this section, are terms used in the betting circle. A bettor is given odds in a betting game. It means the chance of winning the bet. When odds come in favor, the bettor gets an edge over others.
These understood, now we move to stock market investment terms. The edge can be explained as the expected value. This is the term we always deal with when it comes to investing. We refer to the expected value as the value of the stock when all the factors are considered. Then a probable outcome is calculated with respect to time.
An investor invests in the stock market for the appreciation of invested capital. But investors find it very hard to decide on the amount of capital to invest. Such confusion is common. Even if there’s no cap on capital investment, how much to invest in one stock is a critical question. Some, to enhance the profit, over diversify the capital investment. In such cases, the investor misses the opportunity to gain more when odds are in favour.
Such confusion results in not getting the expected return. Downsizing the capital investment within a prescribed limit and limiting diversification to some extent may give you the expected return in such a scenario.
The Kelly criterion formula comes into play very effectively in such conditions. Kelly formula recommends the investor identify the fractions of capital to invest. Generally, on expected returns, this amount of investment per asset amounts to approximately 20% of the investment. We get this calculation by transforming the fraction into a percentage.
So, it is evident that unless the investor knows how to limit investment it is hard to make a profit in the long run. The investment amount should vary from case to case. But to protect the invested capital in the long run and enjoy capital appreciation consistently, an investor must know when to invest more and when to decrease/downsize invested capital.
The Kelly criterion formula helps investors to understand this basic object of investment method for long-time capital appreciation.
Key Takeaways from Kelly Criterion
For investors, who are ready to invest with a view to position sizing, the Kelly criterion is ideal for them. The trader must keep in mind the following aspects when using the Kelly criterion formula.
This is not an ideal method for all kinds of investment scenarios. Kelly criterion formula works best when used in long time frames.
When, under some conditions, the mathematical calculation gives a negative number, the trader should avoid investing in those trades. Because the calculation is not in their favour.
If the Kelly formula is being used in futures and options trades and the formula comes up with a negative fraction, the trader should invest in the opposite derivatives, e.g. Puts instead of Calls. The negative number can be interpreted to take opposite positions.
It has always been the hardest part for a trader to follow the thin line between risk and reward and maintain the balance. According to the Kelly criterion, position resizing helps the investor to turn risk into reward. Following this formula helps the investor to downsize investment where the odds are high and the investor doesn’t have any edges.
On the other hand, following this formula, makes the investor invest more in less risky bets. Thus, in the long run, the investor gains from using the Kelly formula by managing risk.
It is important to use correct inputs while using the Kelly criterion. The formula is as good as the accuracy of the inputs in the calculation. If one of the inputs is incorrect, the whole outcome throws the investment advantage off balance, making investment chaos.
At times, Kelly formula comes up with numbers that can be seen as overly aggressive. Because of this the risk increases. Therefore, it is recommended that, under no condition, the investment amount should not cross 15% to 20% of the total investment capital.
The black swan events can not be predicted through the Kelly criterion. Investors should be aware of that.
What does the Kelly criterion maximize?
The kelly criterion suggests that a good investment should be for long-term growth and the capital sizing has to be maintained. If one follows the Kelly criterion in long-term investments, the formula will help to maximize profit in the long run.
Few Examples of Kelly Criterion
As we have been discussing, the Kelly criterion has found numerous uses in different fields. Here, we are going to discuss some of the examples of implementation of the Kelly criterion.
Kelly Criterion in Stock Market
Though this formula was discovered by Kelly back in 1956, the application of the Kelly criterion in the stock market has been started very recently. From 2000 onwards, big investors like Warren Buffet, George Soros, and Bill Gross started showing interest in the Kelly criterion. From then on other investors also showed interest in this application. The application of the Kelly criterion in the stock market grew.
Here we are going to see a scenario where the condition is ripe for use of the Kelly criterion.
Here, an investor is trying to enhance his profit. So he uses the Kelly criterion formula. Let’s assume his risk over reward ratio is 1 to 1. Which, in betting terms, he receives a 1 to 1 odd. That means, the investor can gain the same amount he can lose potentially for this investment or bet.
Based on existing data, his trade setup will allow him to win 55% of the times he invests/ trades.
Hence,
f = Bp – qB = edge odds,
or, K = p x B (1-p)/B
B = 1
p = 55% or ,55
q = (1-p) = 1- .55 = .45
putting these values in the formula, we get
f = 1 x .55 – 1 x .45 = .55 – .45 = .1 = 10%
Hence, the Kelly criterion formula recommends that the investor should invest 10% of his capital in this setup for long-term gain.
Kelly Criterion in Sports Betting
Sports betting is a popular place for betting. Unlike Roulette or Blackjack, sports betting is another arena that attracts a lot of attention from all phases of the global population. These are not pure gambling but more guesswork. But whether you invest or bet, it becomes a number game. And in number games, mathematics plays a big role. We find the successful application of the Kelly criterion formula in this betting arena.
Let us assume, a person is betting in EPL (English Premier League). Say, the return league match between Liverpool and Manchester City of 2021 being replayed. Man City is favoured to win the match. Bettor is given 2:3 odds. Bet costs say $50. The potential payout is $100 with 2:1 risk/reward ratio. The Kelly criterion formula, in this scenario, would be,
f = Bp – qB = edge odds,
or
K = p x B (1 – p) / B
B = 2:1 or 2
p = 0.67 (the bettor receives 2:3 odds, .33333 x 2 = .66666 = .67)
q = 1 – p = 1 – 0.67 = 0.33
Let’s put the values obtained in the formula. We get,
K = {0.67 x 2 – .33 )}/2
K = (1.34 – .33)/2
K = 1.01/2 = .505 = 50% (approx)
According to the Kelly formula, the bettor should wager 50% of the allocated amount on this bet. Assuming that the bettor has a high risk to reward ratio which is 2:1 in this case, the kelly formula has come up with this conclusion. As the bettor has a 67% chance of winning, the Kelly criterion formula suggested a high portion of the allocated amount. But betting more than the usual amount may turn risk as there is a great amount of risk involved.
As has already been mentioned, the two cases mentioned above are theoretically true and purely hypothetical. But the investor or the bettor would adjust accordingly as per the prevalent situation. The inputs will be different and, of course, there will be a different outcome. The examples shown here give a broad guideline on the probable outcome.
How Investors Use Kelly Criterion Formula?
The investor may work with the Kelly criterion in the way discussed here. Practically one can work with the Kelly criterion and create own Kelly strategy. Here as we can see, we have used the Kelly formula in a slightly different way. Instead of p, B and q we have used W and R, though the implication is similar.
As we can see in the diagram, W stands for the win percentage according to historical data. W will stand for the success rate of any strategy in case of trade or investment in the stock market. And R represents the trader’s historical win/ loss ratio. K is the Kelly fraction. The value of K is the recommended percentage of capital to invest in a particular asset.
So we have two key components in this formula that determine the outcome. The first one is W.
W = winning probability or probability of a strategy that gives a positive return. For a strategy, we can get that from the strategy success rate. Say a combination of MACD and EMA crossover (50 and 13 period) gives a success rate of 63% for a daily time frame in cash equity trading. Then W will be .67.
R is the second key component in this formula. Let us assume that the trader has a historically low success rate of 38% with this strategy.
Therefore the Kelly fraction for that particular trader using the same strategy will be as follows.
K = W – (1 – W)/R = .67 – (1 – .67)/ .38 = .67 – .33/ .38 = .67 – .87 = – .20 = – 20%
Hence, from this equation, we can see that the Kelly fraction is negative. The Kelly formula suggests that the investor should not invest in this asset. We can otherwise conclude that, as the investor has a low success rate using this strategy, the investor should refrain from investing or refrain from using this strategy.
Application of Kelly Formula in Algo Trading
Algorithmic trading, popularly known as algo trading, makes good use of the Kelly formula. In the previous example, we saw that a particular investor has a 38% success rate in adopting a strategy. Therefore the investor’s poor success rate is reflected in the outcome of the calculation resulting in a negative recommendation. An automatic question arises, why one should have a poor success rate in adopting a successful strategy?
The answer lies in the emotional state, age factor and other components of a trader’s background. As the trader is the principal stakeholder in this whole process of trading or investing, these factors matter a lot.
Algorithmic trading or automatic trading omits these factors quite successfully. That’s why more and more traders are opting for automatic trading.
In algo trades, position sizing for a particular trade is very important. The role of the Kelly criterion comes into play here.
The algo trading system today employs the Kelly strategy for the size of investment per asset. Now let us follow the formula given earlier.
As per the formula, K = W – (1 – W)/R. Successful algo systems have a high success rate on average. Let us consider one such algo strategy that has 81% success according to the past three-year annual data. So, W = 0.81.
Let us assume that R = 90% = o.90, The execution success rate is considered 90%, because there may be a systemic lapse, data input and other lapses.
Therefore, K = W – (1 – W)/R = .81 – (1 – .81)/ .90 = .81 – .19/ .90 = .81 – .21 = .60 = 60%
As per the input data, the Kelly formula recommends investing 60% of capital in the same strategy. It also shows that as the success rate is high, the Kelly formula suggests investing more in this algo strategy. If the success rate was lower, the Kelly factor would be low and the recommendation would also be different, as we saw in the previous example.
Kelly Criterion in Portfolio management
Fund managers have also been using the Kelly criterion in portfolio management to get maximum positive output from invested capital. It needs to be understood that portfolio management or fund management is not equivalent to a one-time investment in a particular asset class.
The Kelly criterion runs on inputs. Portfolio management requires the management of multiple investments in multiple asset classes. Secondly, due to the limitations of this formula which favors the high-risk trades if odds are in favor, continuous rebalancing is necessary to make the application more efficient.
Hence there is a need for repeated calculations for multiple allocations. In addition, to maintain the continuous rebalancing practice, as a rule of thumb, daily updation of Kelly allocations is necessary. Also, due to varying market scenarios and changes in risk/ reward metrics, recalculations of the Kelly criterion should be carried out periodically and resulting reallocations of funds are necessary to extract maximum gain.
The Kelly criterion suggests using leverage when returns are good. The value of K goes above 1. That is not unusual for a favorable investment condition. The higher the value of K, if K>1, then the higher the capital requirement. When K goes above 1, the formula recommends trading with high leverage sometimes. In those cases, when high leverages are not allowed, the value of K needs to be adjusted in practice. Fund managers have to do that in practice. Otherwise, the application of the Kelly criterion in portfolio management helps to grow funds in long run.
A Real-Time Analysis of Kelly Criterion
In the following part, a hypothetical example is presented here.
Let us assume, there is a stock X that has a mean annual return (m) of 10.7%. Its annual std. dev. (standard deviation D) is 12.4%. The investor invests in the stock for long period. The investor can borrow capital @ r = 3.0%, which is a risk-free interest.
Therefore, the mean excess return is, M = m – r = (10.7 – 3.0)% = 7.7%,
Hence, in this case, the Sharpe ratio S = mean excess return/ std. dev. = M/D = 7.7%/ 12.4% = 0.077/0.124 = 0.62
Let us now calculate the Kelly leverage, f* = S/D = 0.62/0.124 = 5.00.
It signifies that the capital should be 500% which needs to be invested. That means the investor should invest with leveraged capital.
Here the Kelly criterion recommended taking (5 – 1) = 4 times leverage on the investor’s capital. Our market dynamics do not let us borrow that much capital even if the market condition is in favor.
Therefore, in practice, the investor would adjust the Kelly criterion factor and invest accordingly.
In addition, using the Sharpe ratio and risk-free interest rate, we can calculate a long time compounded growth rate (CAGR) or CGR that can be obtained from a particular strategy.
The Kelly Calculator
For the bettors and gamblers, there is a kelly calculator. This calculator helps the bettor to calculate the betting capital if the odds are known. Though the website has also added a disclaimer with it stating that the website doesn’t encourage bettors.
This is what the Kelly calculator looks like. The details of this calculator can be found on this website.
Before we go more into the result found for this particular situation, it must be mentioned that all the parameters are customizable as shown on the calculator screen under the green tick. The customizable fields here are,
a) Gambling Bankroll
b) Odds offered
c) Your estimate of your probability of winning percentage
d) Bets must be multiples of
e) The minimum bet allowed.
A person interested to calculate before betting would fill up the customizable fields known and the calculator will come up with answers.
Here, it is assumed that a bettor has $1000 as the betting capital. The odds offered to the bettor os 7 to 4. The estimated probability to win is 40%. Bets can be placed in multiples of $1. And the minimum bet allowed is $10.
After filling up the customizable spaces in the way mentioned earlier, the calculator came up with the following results.
As per the Kelly criterion formula, K = .0571. So, the investor should use 5.71% of capital. The betting bankroll was $1000. So, the investor should invest $57 for each bet.
The investor is expected to win 40 times out of every 100 times. Every win will fetch $ 99.75 excluding the stake capital of $57, which will be returned with each win.
But every time the bettor loses, the stake money is lost.
Under these conditions, the capital grows on an average of 0.28% with each bet.
Each calculation outcome is rounded to the nearest multiples of $1.
If every time, the bet is not exactly $57, the bettor should never go above $57.
The outcome of this bet is not related to any other bets.
The Kelly criterion is a maximally aggressive formula. The formula by itself is targeted to increase capital growth to the maximum rate possible, in the long run. Professional gamblers do not bet so aggressively. Usually, the gamblers do not bet more than 2.5% of their bankroll on each bet. Here it should have been $25 for each bet.
Some bettors prefer the half kelly criterion. For them, the ideal betting stake is $28 on each bet, in this case.
Limitations of Kelly criterion
The Kelly criterion is an efficient mathematical formula-based calculative approach to investment. The Kelly factor recommends the investor about the percentage of investment. This formula is targeted to guide the investor positively. Being an aggressive model, it recommends the investor to grow capital aggressively in the long run.
Though it is known to be efficient, it has some inherent limitations. When the odds are in favor of the investor, the Kelly criterion assumes to take a very aggressive path to grow capital rapidly. This leads the investor to a very risky path where risk is high. So a loss can wipe out the invested capital.
At times, the Kelly factor recommends to invest with high leverage. In such a situation, a loss can even wipe out the reserve capital too. Investors using the Kelly criterion should be aware of this and adjust the calculations accordingly.
These high-risk factors lead to the development of the half kelly criterion, where investors use 50% of the Kelly factor and invest accordingly.
Most investors prefer to invest not more than 20% to 25% of capital at the most, in each investment ignoring the recommendations of Kelly formula.
The Kelly strategy does not consider the importance of diversification, which is very important for an investor.
The fractional Kelly is also developed to avoid the over-estimation done by the Kelly formula. In this method, the investor invests a fraction of Kelly recommended percentage for investment
The Kelly criterion does not consider any cap on investment capital. The investor needs to keep much capital in reserve to make up losses. But in the long run, the Kelly strategy provides positive capital growth on average. Also, the success rate is high.
The success of the Kelly formula depends entirely on the input data. Hence the input data must be completely honest and transparent.
The Kelly criterion can not foresee the Black Swan events, which affect all kinds of investments.
The performance of the Kelly criterion has to depend on the individual investment constraints of the investor in the long run. Also, there is no time-bound output. So the investor must not have any time-bound goal to achieve. The Kelly criterion does not specify the maximum time limit for which the investment needs to be kept.
What is better than Kelly criterion?
Arguably, the Kelly criterion shows the best investing method of investment for an investor in the long term perspective. But there are other similar investment strategies used by the investors.
Also, besides the Kelly criterion formula, there are other mathematical models used by investors. Some of them are the Black-Scholes model and the Kalman filter.
There is also another equivalent technique like the “mean-variance optimal portfolio” given by WolframAlpha. In addition, there is also Harry Markowitz’s mean/variance optimization method. It is also known as the Markowitz Model.
How Are the Black-Scholes Model, the Kelly Criterion, and the Kalman Filter Related?
All of these three concepts, the Kelly criterion, the Black-Scholes Model and the Kalman filter are mathematical models that are used to increase investment worth.
We have been discussing that the Kelly criterion is a purely mathematical formula that is used to enhance capital growth through investment strategy.
The Black-Scholes Model is more of an asset valuation mathematical model. Practically this mathematical model is used to calculate the theoretical value of options contracts as per maturity time and other factors.
The Kalman filter is another mathematical model used for the valuation of unknown variables in a dynamic state. It is a non-linear digital filter that is used in time series analysis. This method is used in market price forecasting.
Pitfalls to avoid by an investor using the Kelly criterion
The Kelly criterion is not entirely a foolproof method. It has flaws and those have been pointed out earlier in this article. An investor using this formula in money management should always keep that in mind and should not blindly depend on the outcome. The over-aggressive nature of this formula tends to over-estimate and asks for high leverage. At such times, the investor may use the fractional Kelly or half-Kelly method to curb the over-aggressive investment pattern.
This mathematical formula, like all other mathematical formulae, is valid only if the input data is correct. So, care should be taken when doing the Kelly calculation.
Whether in betting or investing in the stock market, it should be kept in mind that this formula is not designed to provide investors/ bettors with a pot of gold. Rather this formula helps the investor to grow capital in the long run. Therefore, momentary losses, even unexpectedly high, should not be considered as any discouragement for using the Kelly formula. Rather, the investor should fill up the void in capital and keep investing in the way recommended by this mathematical formula. Because we all know that the formula always holds in the long run.
