The Kelly criterion is
the fraction of capital to wager to maximize compounded growth of capital. Even
when there is an edge, beyond some threshold, larger bets will result in lower
compounded return because of the adverse impact of volatility. The Kelly
criterion defines this threshold. The Kelly criterion indicates that the
fraction that should be wagered to maximize compounded return over the long run
equals:
F = PW – (PL/W)
where
F = Kelly criterion
fraction of capital to bet
W = Dollars won per
dollar wagered (i.e., win size divided by loss size)
PW =
Probability of winning
PL =
Probability of losing
When win size and loss
size are equal, the formula reduces to:
F = PW – PL
For example, if a
trader loses $1,000 on losing trades and gains $1,000 on winning trades, and 60
percent of all trades are winning trades, the Kelly criterion indicates an
optimal trade size equal to 20 percent (0.60 − 0.40 = 0.20).
As another example, if
a trader wins $2,000 on winning trades and loses $1,000 on losing trades, and
the probability of winning and losing are both equal to 50 percent, the Kelly
criterion indicates an optimal trade size equal to 25 percent of capital: 0.50
− (0.50/2) = 0.25.
Proportional
overbetting is more harmful than underbetting. For example, betting half the
Kelly criterion will reduce compounded return by 25 percent, while betting
double the Kelly criterion will eliminate 100 percent of the gain. Betting more
than double the Kelly criterion will result in an expected negative compounded
return, regardless of the edge on any individual bet. The Kelly criterion
implicitly assumes that there is no minimum bet size. This assumption prevents
the possibility of total loss. If there is a minimum trade size, as is the case
in most practical investment and trading situations, then ruin is possible if
the amount falls below the minimum possible bet size.
[Thorp]: The Kelly criterion of what fraction of your
capital to bet seemed like the best strategy over the long run. When I say long
run, a week playing blackjack in Vegas might not sound very long. But long run
refers to the number of bets that are placed, and I would be placing thousands
of bets in a week. I would get to the long run pretty fast in a casino. In the
stock market, it’s not the same thing. A year of placing trades in the stock
market will not be a long run. But there are situations in the stock market
where you get to the long run pretty fast—for example, statistical arbitrage.
In statistical arbitrage, you would place tens or hundreds of thousands of
trades in a year. 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.
The Kelly criterion will give you a long-term growth trend.
The percentage deviations around that trend will decline as the number of bets
increases. It’s like the law of large numbers. For example, if you flip a coin
10 times, the deviation from the expected value of five will by definition be
small—it can’t be more than five—but in percentage terms, the deviations can be
huge. If you flip a coin 1 million times, the deviation in absolute terms will
be much larger, but in percentage terms, it will be very small. The same thing
happens with the Kelly criterion: in percentage terms, the results tend to
converge on the long-term growth trend. If you use any other criterion to
determine bet size, the long-term growth rate will be smaller than for the
Kelly criterion. For betting in casinos, I chose the Kelly criterion because I
wanted the highest long-term growth rate. There are, however, safer paths that
have smaller drawdowns and a lower probability of ruin.
…if you bet half the Kelly amount, you get about
three-quarters of the return with half the volatility. So it is much more
comfortable to trade. I believe that betting half Kelly is psychologically much
better.
[Schwager]: Say I am
playing casino blackjack, and I know what the odds are. Do I bet full Kelly?
[Thorp]: Probably not quite. Why? Because sometimes the
dealer will cheat me. So the probabilities are a little different from what I
calculated because there may be something else going on in the game that is
outside my calculations. Now go to Wall Street. We are not able to calculate
exact probabilities in the first place. In addition, there are things that are
going on that are not part of one’s knowledge at the time that affect the
probabilities. So you need to scale back to a certain extent because
overbetting is really punishing—you get both a lower growth rate and much higher
variability. Therefore, something like half Kelly is probably a prudent
starting point. Then you might increase from there if you are more certain
about the probabilities and decrease if you are less sure about the
probabilities.
[Schwager]: In
practice, did you end up gravitating to half Kelly?
[Thorp]: I was never forced to make that decision because
there were so many trade opportunities that I usually couldn’t put on more than
a moderate fraction of Kelly on any single trade. Once in a while, there would
be an exceptional situation, and I would hit it pretty hard.
…there are no zero-risk trades.
[Schwager]: Do you
want to expound?
[Thorp]: There was some remote possibility that we
overlooked something. There is always the possibility that there is some
unknown factor.
A couple of important things to remember about the Kelly
criterion are that it is really only useful when a large enough number of bets
can be made (i.e. you need repeatability) and over-betting will eventually lead
to ruin. It is hard to apply to investing because you almost never know the
exact odds or the exact payoffs. But I do believe it can be generalized to
investing.
If you can do enough work so that you have extremely high confidence
that PW is greater than PL and where the amount you make
if you are right is greater than the amount you lose if you are wrong, then the
Kelly criterion will say it is a bet worth taking. Yes, there are situations
where the payoffs can be high enough where the Kelly criterion would say to
make the bet even if PW is less than PL, but I think
you can build in an extra margin of safety by only focusing on situations where
your diligence leads you to believe that PW > PL AND
$W > $L.
Then you next need to decide what size to make the position.
You want to have a big enough position size to make a difference, but you have
to make sure not to make your
position sizes too big, which I think may happen as a result of having too much
confidence when you put in a lot of work to understand something, or from not
doing enough work and thus not understanding the odds and payoffs well
enough....so either from overconfidence or lack of effort.
And as all of these decisions still depend on judgment and
putting in the work to able to make good judgments, I think this Charlie Munger
quote might be most fitting: "It's not supposed to be easy. Anyone who
finds it easy is stupid."
