The Kelly Criterion

Kelly Criterion

One formula that helps avoid Gambler’s Ruin (losing it all), while betting the optimum amount is the Kelly Criterion, first described in 1956 by J.L. Kelly. It is used by some traders and professional gamblers, and I have started considering it when deciding the right amount to bet on any single lawsuit.

This formula comes complete with a mathematical proof and tells a trader, gambler, or investor how much one should optimally bet, based on the probability of winning the bet and the payoff amount if the bet wins.  The formula assumes no ties.

Shown below is the general formula:

Here is an example where you have a chance to bet $1, and collect $2 if you win ($ 1 original bet plus the $1 profit [the variable labeled “b” in the formula]). I call this a “2X” bet. In this example you have a 60% probability of doubling your money; the Kelly Criterion says that you should bet 20% of your bankroll (or your entire net worth, if your net worth were all in the form of cash).

This formula demonstrates quite logically that:

          1) for a 2X bet you should not bet anything unless the probability of winning is above 50%. In fact, if the probability is below 50%, you should try to find a way to take the other side of the bet or trade (go short).

          2) At a 51% probability of winning, Kelly suggests that you should bet 2% of your bankroll.

          3) At an 80% probability of winning, Kelly suggests that you should bet 60% of your bankroll.

Kelly’s original paper made the point that the criterion is only valid when a series of bets are made. Even though the formula says that one should bet 98% of your bankroll when you have a 99% chance of winning a 2X bet, that still leaves you with a 1% chance of going broke—too high for me, at this stage in my life.

Although I am comfortable taking risks and making substantial bets, the Kelly Formula feels more aggressive than my inner voice approves. Obviously, for a lawsuit investment I have to adjust the formula based on the probabilistic estimate of when the bet will pay off, and then adjust it based on my assumed discount rate (the amount that reflects the discount for getting paid later.) And a minor detail is that I never know the exact odds. But even if I did know the exact odds, I can’t conceive of putting 60% of my net worth on the line, even if I had a marvelous 80% probability of winning. So maybe I need to take my own advice and get a little specialized psychotherapy—so I can de-wussify and get on board with making bigger bets.

All-In

Many successful entrepreneurs, traders, and investors take measured and intelligent risks, trying to avoid at almost any cost the all-or-nothing bet. This is after all, the ultimate risk of Gambler’s Ruin. The younger you are the less reckless it is violate this aspect of the rule because you have on average a smaller bankroll and more time to recover if you lose it all. If you are poor, you are likely to have no alternative to the all-in bet. This is why I encourage young people to get started on their entrepreneurial ventures sooner rather than later. I love it when a 14-year-old puts every last dollar he owns into a lawn mower to start his landscaping business—as long as he keeps enough capital to fill up the gas tank.

And the Survey Says…

I wanted to know how my friends and family would answer questions related to the Kelly Criterion. I asked them to complete the following survey:

I have three questions for you about how much of your bankroll you would wager in three different circumstances.

I start with the premise that you are playing an absolutely fair, honest and random game.  For example it is the picking a chip out of a hat.  Some of the chips say “win” and the rest say “you lose”. If you win, you double the amount of your bet; if you lose then your bankroll shrinks by the size of your bet.

You know in advance exactly what the chances are of winning.  In every case, you are 30 years old (please try to adjust your thinking to how you would have played or will play this game at this age.) You have a bankroll of $100,000 which is all the assets you have in this world and your bankroll is all you have to get you have to work with. You have no home, no IRA, no retirement, no job. The random contest you are playing is always in your favor but sometimes it is more in your favor than other times. The three conditions we will consider are 60%, 75% and 90%.  In other words you have either a 60%, 75%, or a 90% chance of doubling the money you bet. Another way of looking at it is that you have a 40%, 25%, or a 10% chance of losing the amount of your bet.

For each of these percentages, what percentage of your bankroll would you bet?  (the choices are between 0% and 100% in 10% increments.) Let’s go.

Question 1) At a 60% probability of doubling your bet, what percentage of your bankroll would you bet?

Question 2) At a 75% probability of doubling your bet, what percentage of your bankroll would you bet?

Question 3) At a 90% probability of doubling your bet, what percentage of your bankroll would you bet?

Discussion of the Survey Results

·       I asked everyone to imagine they were 30 years old, and had a $100,000 cash bankroll that comprised their entire net worth.  I wanted to minimize the bias that as people get older they generally shy away from bigger risks. I was intentionally silent about if the participant had children or “big responsibilities”.

·       Sometimes what people say they will do differs from what they will actually do, when the rubber meets the road.  If I had about $5-10 million dollars I was willing to invest in a real live experiment, I am sure I could have found 50 30 year olds to take the test with real money.  It was not that important to me.

·       For each of these questions, the Kelly Criterion calculates the optimum answer (based on one bet in a long series of bets).  If you know exactly what the probability of winning and the exact amount of the payoff, the Kelly Criterion tells you the optimum amount to bet.  Theoretically, if you always bet this amount, over many opportunities, on average after many bets, you have the greatest chance of having the biggest ending bankroll.

·       If you bet less than the Kelly Criterion amount, you are being “risk averse”, to your detriment.  If you are betting more than this amount, you are taking greater risks that optimum, also to your detriment, on average in the long run.  If you bet zero, when you have a 90% chance of doubling your bet, you are “ultra risk averse”.  If you bet 100% of your stack, you are going “all-in”.

·       I was surprised that the survey results didn’t show a more risk averse average response from my friends and family—maybe I hang around some “weird” folks.

·       The survey results show the “Wisdom of the Crowds”.  Although the responses varied greatly, the average results for each question are reasonably close to the Kelly Criterion—although, some participants were very risk averse and others quite willing to lay it all on the line.  I declare the aggregate results rational, although I might be saying this because my answers were close to the average results.

·       The results showed that on average, the more the bet approached “all-in” or 100% of one’s bankroll, the more conservative the response, relative to the Kelly Criterion.  I consider this reasonable.  The question, as framed, never promised that the participant was going to get to make the decision 1,000 days in a row.

·        The biggest diversity of answers came on the third question.  We had three respondents that said they would not bet a penny (even though they had a 90%) chance of doubling their money, and four that said they would bet their entire bankroll. Slightly different risk tolerances.

Here is a summary of the results:

1. Probability of Win	60%	70%	90%	Total
   Kelly Criterion Percentage	20%	50%	80%
   Survey Average Amount Bet	25.8%	39.26%	59.26%
   # of  Risk Averse	28	49	56	133
   # @ Kelly Percentage	15	14	9	38
   # Greater than Kelly Percentage	38	18	16	72

The End of the University as We Know It

Very well written piece by Nathan Harden.  


I loved the enthusiasm and agree that this is where we should be headed.  But because of all the subsidies for the existing system and parents students don't pay enough attention to the amount of debt they are picking up. I think that change will be far slower than in the music industry.

Under New Management

The sign that really confounds me is “Under New Management  – In other words, “Those jerks that were running this place before are long gone. And the new management can’t possibly by as bad as the prior management.”  

Hustling to make a little money!

I really get a kick out of Mark Cuban. If I was going to start a new business I would give my right eye (perhaps an exaggeration) to get a few hours of his time to pick his brain. But alas in today's world you can get some of it free at blogmaverick.com.

This small piece shows how he thinks about making money with a very small investment is not the kind of idea one would hear from Obama or other career politicians.

Hate Crime versus the First Amendment

Hate crime. These crimes occur when a perpetrator targets a victim because of his or her perceived membership in a social group. This as opposed to a polictically correct jerk that beats me up for no other reason than he can.

First, I defy you to find one case in the US when hate crime laws were enforced when a minority assaulted a white man because he was white or heterosexual. Prime example of selective prosecution.

More importantly it is an infringement of the first amendment. I can hate you and I can tell the world that I hate you and that is “generally” protected by the first amendment. But combining the first amendment protected statement that “I hate you because you are a Libertarian” with the violent assault is double trouble.

If I assault you for any reason I can be convicted of a felony and perhaps go to jail for a few years. Let’s say two years at the state pen.

But if I tell you that I hate you (because of the group you belong to) while I assault you then instead of going to prison for two years then I might be sent there for four to five years.

So in essence we are sending the politically correct assaulter to prison for two years and sending non-politically correct assaulter to prison for four years. The only difference is that the outspoken assaulter told his victim that he hated the victim (because the victim was a Libertarian) while he was beating him up.

If I am a victim I really don’t care what your motivations are when you are assaulting me. Don’t do it for any reason. And if you do beat me up then I don’t want you to get any special time off because you are politically correct.

If you choose to assault me because I am a white heterosexual male, then I think you should go to prison just as long as if you beat me up because I am rooting for the San Francisco Giants against the Dodgers in Los Angeles.

Great story about science, math and engineering

"Freeman Hrabowski says the United States is not producing enough scientists and engineers - professions critical to creating more jobs."

Our unemployment insurance system sends the wrong signals.

I have heard liberals argue recently in favor of growing and extending our unemployment insurance programs. But I have real problems with how unemployment insurance as it is currently structured and financed today.

Let’s consider an example of Bill, an employer who is considering hiring Joe. If there were no employee taxes, no unemployment insurance, no benefits and no other employer costs, it would be simple to calculate the cost at say $10 an hour. And as long as Joe can generate at least $11 in increased revenue for Bill then Bill is ahead as an employer by $1 an hour. But it is not that simple. Bill must also pay perhaps $.50 an hour for unemployment insurance, $.75 an hour for social security and Medicare employer contributions, $1 an hour for workers compensation insurance and $2 an hour in health care benefits. All of a sudden Joe’s cost to Bill is $14.25 an hour (not to mention increased liability insurance, capital costs, management time and other overhead costs). So if Joe can still only produce $11 in marginal income, Bill lays Joe off. And then you know what happens? Bill’s cost for unemployment insurance goes up even higher (say to $.75 an hour) because Bill has a history of laying employees off so he is deemed to have increased the cost of providing this insurance.

This last characteristic about employer’s cost of unemployment insurance is another disincentive to hire anyone in the first place. Why hire anyone on the margin? If Bill guesses wrong, he not only loses on his underlying bet but his future cost of business will increase along with his higher unemployment insurance premiums. Our incentives should encourage, long-term employment, mid-term employment and short-term employment. At least when Joe is working in what might be a short-term job, he is learning, producing and positioning himself far better to land his next gig than sitting at home collecting unemployment insurance.

I am fine with charging an employer for workers compensation insurance. Because that is a real cost attributable to more work and is roughly priced according to how dangerous the work is. It reflects approximately the marginal cost of each additional hour of work. All things being equal, we would rather encourage safer work than less safe work. This charge to Bill reflects real costs associated with the industry in which Bill operates, and does not subsidize dangerous jobs by making safer jobs less competitive.

But it is counterproductive to charge employers for the cost of unemployment insurance. It makes marginally profitable employees and prospective employees less profitable and sometimes not profitable at all. When the later happens people lose their jobs and go onto unemployment insurance, raising unemployment insurance premiums and increasing the subsidies from government.

I am all for a limited safety net provided by government. But government ends up subsidizing unemployment insurance benefits and recently has extended them to up to three years.

We should be simplifying our safety net. If you don’t have enough money for food, basic housing and medical care I support helping these folks. But the system would be far more productive if we streamlined it and separated it from the issue of whether the individual just lost his job.

I have liberal friends that argue that unemployment benefits are a great way to pour money into the economy. And it might do some of that. But this trick might have had more positive benefits when most of what these unemployed people consume goods that were manufactured in the US. That is rarely the case today. The government is subsidizing unemployment benefits today and creating jobs, but most of them are in China.