# Using Contingent Probabilities and the Kelly Criterion for Trade Management

The Kelly Criterion is truly a hidden gem, largely because it directly contradicts the mean-variance utility framework of Modern Portfolio Theory (MPT). The high priests of modern finance have suppressed this practical decision tool to protect their own devices, which is good –because it gives you an edge on the competition.

The purpose of the Kelly Criterion is to maximize the compound return, while leaving the probability of going broke equal to zero. Initially, it was used by blackjack players to determine how much to bet when they counted cards. When the count was high, they would have a higher expectation and would bet more. But the amount to bet was critical, because in actuality, the edge that a card counter has is actually quite small. It can vary between 52/48 for the player to 55/45 when the deck is loaded with tens and aces (depending on the # of decks, BJ odds etc). Ironically the odds in the stock market for daily follow through (betting on an up day after a down day and vice versa) are nearly identical, while the payoff is much smaller than doubling your bet every day. The Kelly Criterion is as follows:

For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds, the Kelly bet is (from Wikpedia http://en.wikipedia.org/wiki/Kelly_criterion) :

where

*f** is the fraction of the current bankroll to wager;*b*is the net odds received on the wager (that is, odds are usually quoted as “*b*to 1″)*p*is the probability of winning;*q*is the probability of losing, which is 1 −*p*.

Note that one of the important hitches in applying the kelly criterion to the market is that it was designed assuming that if you lose the entire wager is lost. From this perspective, it can be applied directly to options, but it must be modified for practical use. You can theoretically never lose all your money by investing in the S&P500 for the next day. Before i get into contingent probabilities, lets discuss how I like to do this modification. First i divide the market into two states: 1) a bull market 2) a bear market. I establish this by determining whether the 252-day average is rising, that is if the 252-day average of SPY prices is greater than the day before, we are in a bull market. If the average is falling we are in a bear market. I compute the average daily price return and the probability of winning versus losing for each state. Here are the results on the SPY for the last 4000 bars:

% Positive | ADR* | |

Bull Market (rising 1-year MA) | 53.50% | 0.05% |

Bear Market (falling 1-year MA) | 51.76% | -0.02% |

*average daily return |

I assume that in bear markets you will be short as a baseline, and long in bull markets. Note that in bear markets, you need to take (1-51.76%)=48.24% as your winning percentage and, -1*.02%= .02% as your average profit. Now we need to take a look at avg wins versus average losses:

Avg Win | Avg Loss | Net Odds | |

Bull Market (rising 1-year MA) | 0.70% | -0.70% | 1.00 |

Bear Market (falling 1-year MA) | 1.34% | -1.21% | 1.11 |

From here we can start to calculate a modified Kelly for the stock market– lets call it MDVK because i am known for my judicious use/abuse of acronyms.

* B*=

*x*

**T***20/(the current vix level)*

** MDVK**= (B x 1/2 x (100/(1-Kelly Criterion)))

*max(300%)*

*where T= 100% trend bet going long in bull markets/short in bear markets*

*where B= baseline bet going long in bull markets/short in bear markets*

Our standard opening bet * T * is basically 100% in the direction of the long term trend times the relative VIX (volatility index) level. This is designed to normalize our bets under the assumption of a long-term VIX of 20. It also forces us to bet less in bear markets, because they are inherently more volatile. In October 2008 when the VIX was 80, you would be betting only 25% of your portfolio (20/80). Essentially

**prevents us from getting into trouble by 1) forcing us to bet with the long-term trend, and therefore avoiding the fat-tails 2) it normalizes our bet-sizing so that we can use the same “wager” which minimizes the effects of any one bad bet.**

*B*The formula in the MDVK can be explained as follows: 1/2 represents the fact that we are betting 50% of the normal kelly recommendation. This is due to the fact that the true probabilities in the stock market are unknown, whereas in poker or blackjack they are know with certainty. This prevents the problem of overbetting, and the use a 1/2 adjustments tends to be near optimal out of sample. This system is very conservative, but it basically is designed to minimize bet sizes when strong edges are not present. Returning to the equation,the 100/(1-kelly) represents the fact that we need to adjust the betting amount upwards given that we will not lose 100% of our wager in one bet. Note that the MDVK was adjusted, and the maximum allocation is 300%—which in my opinion should not be exceeded to avoid crippling the account via a rare event. In the above example, assuming no other information, the MDVK would be calculated as follows:

Scenario 1) Assuming a bear market with a VIX of 30:

for the kelly criterion: net odds=1.1 p(winning)=48% p(losing)=52%

f = (1.1*.48-.52)/1.1 = .73%

MDVK= 20/30*.5*(100%-.73%) or .6666*.5*.96= **31%**

Therefore, our optimal bet is 31% of capital.

for bull markets:

Scenario 2) Assuming a bull market with a VIX of 20:

for the kelly criterion: net odds=1 p(winning)=53.5% p(losing)=46.5%

f= (1*53.5%-47.5%)/1= 6%

MDVK=20/20*.5*(100%-6%) or .5*(94%)= 47%

Therefore in bull markets under scenario 1 our optimal allocation is 47%– this is a very conservative allocation, a full kelly would dicate being 94% invested, but that gives little leeway to increasing bet sizes under the prescence of strong additional edges. This is the subject of enhancement using contingent analysis, and will be completed during the weekend.

“The Kelly Criterion is truly a hidden gem, largely because it directly contradicts the mean-variance utility framework of Modern Portfolio Theory (MPT)” – I’ll have to disagree. In most cases Kelly optimal portfolios are (close to) mean variance optimal, unless there is very large skew and kurtosis in the returns. Note that the mean-variance framework only gives you a proportional portfolio where the leverage depends on your risk preference. With log-utility the mean variance framework produces (close to) Kelly optimal portfolios.

Well what you are saying is somewhat true within the context of long-term asset allocation. However, once we get deeper and discuss contingent probabilities—ie specific case examples, there is really no application of the MPT to this. Optimization is just too unstable—really what is needed is an estimate of bet size. Nonethless your comments are splitting hairs—-most traders do not have the resources or knowledge to perform classic MPT principles, the kelly criterion is far more practical.

David, you are a quant machine!!! When you do part II of this study, please incorporate your ideas of using the Kelly Criterion to weight or score different trading systems.

Let’s say, I have 5 trading systems that trade long and short the S&P. Is there a clever way to use the Kelly Criterion to weight these signals to determine my overall long or short direction of the trade and the overall % to bet based on this calculation?

Thanks for sharing your thoughts!

John

Well i do plan to cover that in part 2 this evening……but not entirely for multiple systems. But i will give suggestions. Over time, i will discuss a more formal process—but i don’t want to get overly technical right off the bat!

thanks for the kind words.

The fecundity and thoroughness of this nascent blog continues to astound!

Love the adjectives! you should consider writing for the New Yorker!

thanks for the kind words.

dv

Its possible to factor drawdowns into Kelly and and significantly lower volatility of returns.

David, this is very interesting, but I think there might be a problem.

You defined the amount you will bet as

MDVK= (B x 1/2 x (100/(1-Kelly Criterion)))max(300%)

but in the calculations you used

MDVK= 20/30*.5*(100%-.73%)

which does not equal 31%, it equals 33.09%. On the other hand, your formula would seem to yield MDVK= 20/30*0.5*(100%/(100%-0.73%)) = 33.57%, which also differs.

For the Bull case, the value matches the formula MDVK=20/20*.5*(100%-6%), but using your formula one would get MDVK=20/20*.5*(100%/(100%-6%)) = 53.19%.

Regardless, once you remove the effect of the decrease in vol in Bear market, you would get 50.37% as the amount to bet in Bear vs 53.19% in Bull (if you use the version of the formula in the end of the text the numbers are 49.63% in Bear and 47% in Bull). So your use of Kelly criterion essentially tells you that you should bet similar amounts in Bull or Bear; it is the vol that matters the most to decide how much to gamble. If you use the formula in the end of the text it just doesn’t make sense; Kelly tells you to bet a lot more in Bull than in Bear (it tells you to bet over 8 times more), but MDVK (wihtou the vol adjustment) tells you to bet more in Bear.

Now in the first MDVK formula, you introduce a non-linearity which increases the amount to bet quite slowly if Kelly is close to zero, and quite sharply if Kelly is close to one. I think that might be one reason why the MDVK formula makes you bet similar amounts in this example (disregarding the vol): a ratio of 8:1 (6% to 0.73%) is transformed by 100/(1-Kelly) to a ratio of 1.0683:1.0073 or 1.05:1. So it might be interesting to revise the formula and let the Kelly criterion determine the wager.

Another idea is to just go ahead and use the approach in Thorp’s paper, which is based on a continuoous approximation of increments in prices and explicitly takes into account the volatility to determine the amount to bet. You could measure time-varying volatility and means (either with a model or with simple regimes as you do here) and input those directly into the formula, which is basically

Kelly = (m-r)/s^2

where m is the asset’s mean, r is the risk-free rate and s is the volatility.