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:
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
|
