Kelly Formula
Understanding the Kelly Criterion. Investors often face a tough decision when trying to decide how much money to allocate, as staking either too much or too little will result in a large impact either way. The Kelly criterion is a money-management formula that calculates the optimal amount to ensure the greatest chance of success. A Kelly Strategy Calculator Introduction. J.L.Kelly, in his seminal paper A New Interpretation of Information Rate (Bell System Technical Journal, 35, 917-926 see below) asked the interesting question: how much of my bankroll should I stake on a bet if the odds are in my favor? This is the same question that a business owner, investor, or speculator has to ask themself: what proportion of my. The Kelly Formula was created to help calculate the optimal fraction of capital to allocate on a favorable bet. It doesn't mean that the formula is the be-all-end-all solution, but it.
Using the Kelly Criterion with Your Portfolio. Extending Kelly a bit further (like Ed Thorp, author of two math bibles for the investor/bettor Beat the Dealer and Beat the Market, has done) we can do a bit of hand-waving and make it work for the stock market.Some derivations of 'Stock Market Kelly' involve using back-looking numbers such beta to approximate the continuous returns of securities. Avoid investing more than 20% to 25% of your money into any one stock or security, even if the Kelly formula tells you otherwise. The Bottom Line. Even though it worked for the gamblers doesn’t necessarily mean the Kelly criterion will work for you. While it’s good to use many different money management formulas, don’t rely on only one.
As a speculator – there’s three important steps to ensure maximizing your wealth overtime. . .
First – find situations where you have an edge (an attractive investment you have an advantage in).
Second – invest your bankroll properly (position sizing).
And third – never blow yourself up.
I’ve written many times before about the ‘mental tools’ that help speculators find an edge. Like using Expected Value Analysis (EVA) to make better contrarian bets. Or why ‘second-level thinking’ is important. Or how to spot and exploit favorable optionality.
I’ve also written many times before about the hazards of blowing up. And why this must be avoided at all cost (i.e. avoid negatively asymmetric scenarios – where risks far outweigh any potential gain).
But – I haven’t touched much on investing your bankroll properly (position sizing). . .
Putting it simply – position sizing refers to allocating the optimal amount of capital in each investment (bet).
For instance: two hedge funds with the same amount of capital and same list of stocks can generate very different results – simply based on how they invest their capital.
Great speculators know that it’s not about just finding an attractive play. But how can they make the most money from it.
Like Charlie Munger – considered Warren Buffet’s smarter-half – said: “The wise ones bet heavily when the world offers them that opportunity. They bet big when they have the odds. And the rest of the time, they don’t. It’s just that simple…”
Now – a speculator shouldn’t put all their money in a single play (even if there’s a large edge). But they also shouldn’t under-invest. Or worse – over-diversify.
What to do?
Well – that’s where the Kelly-Criterion (aka the Kelly Formula) comes in. . .
Long story short: in 1956, a former Bell Labs (now owned by Nokia) employee – John Kelly – used ‘information theory’ to create an ideal betting formula.
Kelly wanted to calculate the optimal amount of capital to allocate on a favorable bet given fixed odds.
There are many ways to express the Kelly Formula. But a simplified version of it from value investor Mohnish Pabrai’s book – ‘The Dhandho Investor’ – is:
Edge / Odds = Fraction of Capital That Should Be Allocated (aka E/O = F)
What does all this mean?
An Edge is the total expected value of the bet.
And the Odds are what the market’s offering if you win.
Then – once you calculate the edge and the market’s odds – you determine how much of your bankroll to invest.
Here’s an example – imagine having $1,000 dollars in your pocket and you’re walking around a Las Vegas casino.
You spot an empty table that’s playing a coin game where each flip offers two dollars ($2) for heads. But with each tail you lose one dollar (-$1).
It’s not hard to see the opportunity here. Meaning – each toss has a 50-50 chance of heads or tail, but the market’s offering 2-to-1 odds for landing on heads.
(Remember – many believe that markets are totally efficient – thus there’s no edge and the odds perfectly represent the probabilities of success. Meaning – in the market’s eyes – there’s zero expected value).
So – using Kelly’s Formula – you find the edge is $0.50 cents (aka the expected value; the math for this is 50% x $2 + 50% x -$1).
And the odds are $2 (the amount the market’s offering each time you land heads).
Thus – the fraction (amount) of your capital that should be allocated with each coin toss is 25%. (Mathematically it’s $0.50/$2.00 = 25%).
Or – said otherwise – it’s best to bet $250 with each coin flip (since you have a $1,000 bankroll).
And according to Kelly – as long as the edge and odds stay the same – this is the optimal strategy for maximizing wealth in the long-term while playing this game.
Now – at the time – not many traders adopted the Kelly Formula (most likely because efficient market hypothesis was very big back then).
But – one trader discovered this formula. And used it to beat the market by 20% for three decades. . .
I’m talking about Edward O. Thorp. (He was the original ‘quant’ trader and created his own option-pricing model long before the Black-Scholes-Merton option model existed. His auto-biography – ‘A Man of All Markets’ – is a must read in the Speculators Anonymous Reading List).
Here’s what Thorp had to say about the Kelly Formula in the book ‘Hedge Fund Market Wizards’:
“… The Kelly criterion is the bet size that will produce the greatest expected growth rate in the long term. If you can calculate the probability of winning on each bet or trade and the ratio of the average win to average loss, then the Kelly criterion will give you the optimal fraction to bet so that your long-term growth rate is maximized…”
Now – although the Kelly Formula’s a very important tool (one that I use constantly). It still has some issues.
For starters – it was created for gambling. Not the markets.
Secondly – in the market – it’s up to the speculator to determine what they believe the edge is. (The best they can do is estimate).
Thus – for market trading – here’s another version of the Kelly Formula that works well (according to Macro-Ops). . .
Also – like Thorp – the speculator should use the Kelly Formula as a reference point.
For example – if the formula states to invest 20% of your bankroll – you can use ‘half-Kelly’ or ‘quarter-Kelly’. (Meaning cut the 20% in half, or by a quarter, etc).
This helps speculators who are uncertain about their edge. Or want to be more conservative.
Thus – in summary – I believe the Kelly Formula’s very useful. And that it belongs in a speculator’s ‘mental toolbox’.
It isn’t perfect (the formula’s only as good as its inputs) – but it does have practical use in the markets (as Ed Thorp’s proven).
Position sizing’s a very important (yet often not talked about) part when placing trades. Especially when the speculator has an edge.
Kelly Formula Example
For instance – finding a positively asymmetric (low risk – high reward) opportunity is only half the task. The other half is figuring out how to size the position properly (over-investing can be ruinous, but under-investing won’t maximize portfolio returns).
This is why the Kelly-Formula’s so important. And why I always use it.
Also – for those that aren’t the best mathematicians – don’t worry. Here’s an easy to use Kelly-Calculator. All that’s needed are the inputs.
Thus – keep the Kelly Formula in mind next time you’re about to place a trade or a bet.
PS – here’s a must-read piece about the Kelly Formula written by Michael Mauboussin (one of my favorite economists).
PPS – there’s an entire book written about the history and use of the Kelly Formula. It’s called “Fortune’s Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street’ – by William Poundstone. I highly recommend it.
Introduction
J.L.Kelly, in his seminal paper A New Interpretation of Information Rate (Bell System Technical Journal, 35, 917-926 see below) asked the interesting question: how much of my bankroll should I stake on a bet if the odds are in my favor? This is the same question that a business owner, investor, or speculator has to ask themself: what proportion of my capital should I stake on a risky venture?
Kelly did not, of course, use those precise words — the paper being written in terms of an imaginary scenario involving bookies, noisy telephone lines, and wiretaps so that it could be published by the prestigious Bell System Technical journal.
Assuming that your criterion is the same as Kelly's criterion — maximizing the long term growth rate of your fortune — the answer Kelly gives is to stake the fraction of your gambling or investment bankroll which exactly equals your advantage. The form below allows you to determine what that amount is.
Disclaimer
- The Kelly Strategy Bet Calculator is intended for interest only.
- We don't recommend that you gamble.
- We don't recommend that you place any bets based upon the results displayed here.
- We don't guarantee the results.
- Use entirely at your own risk.
Kelly Formula Options
Kelly Strategy Bet Calculator
Results
- The odds are in your favor, but read the following carefully:
- According to the Kelly criterion your optimal bet is about 5.71% of your capital, or $57.00.
- On 40.0% of similar occasions, you would expect to gain $99.75 in addition to your stake of $57.00 being returned.
- But on those occasions when you lose, you will lose your stake of $57.00.
- Your fortune will grow, on average, by about 0.28% on each bet.
- Bets have been rounded down to the nearest multiple of $1.00.
- If you do not bet exactly $57.00, you should bet less than $57.00.
- The outcome of this bet is assumed to have no relationship to any other bet you make.
- The Kelly criterion is maximally aggressive — it seeks to increase capital at the maximum rate possible. Professional gamblers typically take a less aggressive approach, and generally will not bet more than about 2.5% of their bankroll on any wager. In this case that would be $25.00.
- A common strategy (see discussion below) is to wager half the Kelly amount, which in this case would be $28.00.
- If your estimated probability of 40.0% is too high, you will bet too much and lose over time. Make sure you are using a conservative (low) estimate.
- Please read the disclaimer below.
More Information
The BJ Math site used to contain a great collection of papers on Kelly betting, including the original Kelly Bell Technical System Journal paper. Unfortunately it is now defunct, and only contains adverts for an online casino. However, you can find much of the content through the Wayback Machine archive. The Internet Archive also contains a copy of Kelly's original paper which appeared as A New Interpretation of Information Rate, Bell System Technical Journal, Vol. 35, pp917-926, July 1956. (If this link breaks — as it has done several time since this page was written — try searching for the article title).
We based the above calculations on the description given in the book Taking Chances: Winning With Probability by John Haigh, which is an excellent introduction to the mathematics of probability. (Note that there is a misprint in the formula for approximating average growth rate on p359 (2nd edition) and the approximation also assumes that your advantage is small. There is a short list of corrections which can be found through John Haigh's web page).
Note that although the Kelly Criterion provides an upper bound on the amount that should be risked, there are sound arguments for risking less. In particular, the Kelly fraction assumes an infinitely long sequence of wagers — but in the long run we are all dead. It can be shown that a Kelly bettor has a 1/3 chance of halving a bankroll before doubling it, and that you have a 1/n chance or reducing your bankroll to 1/n at some point in the future. For comparison, a “half kelly” bettor only has a 1/9 chance of halving their bankroll before doubling it. There's an interesting discussion of this (not aimed at a mathematical reader) in Part 4 of the book Fortune's Forumla which gives some of the history of the Kelly criterion, along with some of its notable successes and failures.
Jeffrey Ma was one of the members of the MIT Blackjack Team, a team which developed a system based on the Kelly criterion, card counting, and team play to beat casinos at Blackjack. He has written an interesting book The House Advantage, which examines what he learned about managing risk from playing blackjack. (He also covers some of the measures put in place by casinos to prevent the team winning!)
Kelly Formula Online
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