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The Kelly criterion and its variants: theory and practice in sports, lottery, futures & options trading .
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Professor William T ZiembaAlumni Professor of Financial Modeling and Stochastic Optimization (Emeritus), UBCICMA Financial Markets Centre, University of ReadingVisiting Professor Mathematical Institute, Oxford University andPresident, William T Ziemba Investment Management Inc
Kaist Lecture Program
PIWM 2007 to present
I=excess mean return
i=leveraging factor for long market exposure
I argue that getting the mean right is themost important ingredient in winning strategies.
Risk tolerance is the reciprocal of risk aversion.
When RA is very low such as with log u, then the errors in means become 100 times as important.
Conclusion: spend your money getting good mean estimates and use historical variances and covariances
f*=64% for place/show; suggests fractional Kelly.
Kelly and fractional Kelly - explaining the overbetting that leads to hedge fund disasters: you cannot ever bet more than full Kelly and usually you should bet less
Hedge fund manager won bidding for 2008 lunch with Warren Buffett for $600K+
Stewart Enterprises, Payoff ≤ 24 months
Prob Net Return
0.01 lose all investment
Pabrai bet 10% of his fund
What’s the full Kelly bet?
f* =0.975; half Kelly 0.3875; quarter Kelly=0.24375
Other opportunities: must compute against all options (nonlinear or stochastic optimization) for the available wealth
Risk tolerance: what fractional Kelly to use?
Black Swans: we call them bad scenarios
Long vs short run planning
These were independent
166 times out of 1000 the wealth is more than 100 times the initial wealth with full Kelly but only once with half Kelly does the investor gain this much
But probability of being ahead is higher with half Kelly, 87% vs 95.4%
Min wealth is 18 and only 145 with half Kelly
700 bets all independent with a 14% edge: the result, you can still lose over 98% of your fortune with bad scenarios
With half Kelly, lose half of wealth only 1% of the time but is is 8.40% with full Kelly
So even after 700 plays, the strategy is still risky
See Simulation paper for more on this and two more examples
Probability of doubling and quadrupling before halving and relative growth rates versus fraction of wealth wagered for Blackjack (2% advantage, p=0.51 and q=0.49
Should you ever be above 0.02 that is positive power utility like
It is growth-security dominated.
Betting more than the Kelly bet is non-optimal as risk increases and growth decreases; betting double the Kelly leads to a growth rate of zero plus the riskfree asset.
LTCM was at this level or more, see AIMR, 2003.
Several similar blowouts are discussed in Ziemba and Ziemba (2007) including Amaranth and Niederhoffer.
f=1/(1- ) = fraction (Kelly) in log optimal portfolio, rest in cash
=0 f=1 full Kelly
=-1 f=1/2 1/2 Kelly
=--3 f=1/4 1/4 Kelly futures trading down here
This is exact with log normality and approximate otherwise but it can be way off.
Correspondence: Nov 16, 2005 to Elwyn Berlekamp
Dec 13, 2006 to WTZ
Samuelson postulated three investors, all risk averse and concave
WTZ adds two more: Ida (the most risk averse) and Victor (the most risk accepting)
Cash return zero, Stock 50-50: $4 or $0.25 for $1 bet (for every period)
Inefficiencies are possible since:
1) more complex wager
2) prob(horse places) > prob(horse wins) ==> favorites may be good bets
To investigate place bets we need:
1) determine place payoffs
2) their likelihood
3) expected place payoffs
4) betting strategy, if expected payoffs are positive
Bettors do not like place and show bets.
Use data in a simple market (win) to generate probabilities of outcomes
Then use those in a complex market (place and show) to find positive expectation bets
Then bet on them following the capital growth theory to maximize long run wealth
Non concave program but it seems to converge.
In practice, adjust q’s to replicate biases.
Victor Lo research on this in his thesis and Hausch, Lo, Ziema 1994, 2008 books
What we do in the system is to reduce the non-convex log optimization problem down to four numbers: Wi,, W, and Si, S or Pi, P,
Thousands of race results regress the expected value and the optimal Kelly bet as a function of these four variables.
Hence, you just find horses where the relative amount bet to place or show is below the bet in the win pool.
The calculator tells you when the expected value is say 1.10 or better and calculates the optimal Kelly bet.
So this can be done in say 15 seconds.
Real results April 2005-March 2006
Up ~ 36,000 ~ 2% on bets ~ 1.5 M,
System -7%, rebate ~ 9%, edge ~ +2%
We keep doing this searching for good bets at 80 tracks
Small cap stocks have outperformed large cap stocks in January on a regular basis since 1926
Average excess returns of smallest minus largest decile of US stocks, 1926-93, Source: Ibbotson Associates
January effect, 1926-1995. January size premium = R(10th)-R(1st).
Futures play with anticipation, mid December to mid January, this is a typical year in the mid 90s, Value Line versus S&P, 1992-3
Probability of reaching $10 million before ruin for Kelly, half Kelly and quarter Kelly strategies
Relative growth rate and probability of doubling, tripling or tenfolding before halving for various Kelly strategies
Futures markets - much more violent
Russell 2000 - has more volume than Value Line
Effect moved into December
Textbooks and finance experts say effect is not there
Graphs in Hensel-Ziemba paper in Keim-Ziemba (2000) Worldwide security market imperfections, Cambridge University Press.
Doing this trade is like driving a dynamite truck smoking a cigar. You do it carefully.
Rendon-Ziemba (2007) update to 2005 turn of the year Value Line/S&P500 and Russell 2000/S&P500 spread trades
Ziemba (2011) new paper updating to 2011, showing the trade still worked in 2009/10 and 2010/11
The paper Investing in the Turn-of-the-Year hasgraphs and tables of the trades for various years
The next slide has four years of VL/S&P trades and the following slide has four years of the R2000/S&P trades
Each year was a little different but it was possible to win each year.
Finally, the third set of graphs after the monthly R2000-S&P500 futures spread, 1993-2009 has 2009/2010 and 2010/11 where we entered on the dots and exited on the squares as the market turned.
The downside of the analysis is that the expected time to win a lot is in the millions of years.
To stay above a wealth path using a Kelly strategy is very difficult
Kelly fractions and path achievement
MSZZ (2004) using a continuous time lognormally distributed asset model calculate that function to stay above a path at various points in time to stay with a high exogenously specified value at risk probability.
Convex case like Geyer-Ziemba (2008) Siemens Vienna pension model - can do on a computer; in MacLean, Zhao, Ziemba (2009)
Bernoulli (1732) translated in 1954; original idea of log utility; marginal utility, current wealth proportion to St Petersburg Paradox
MacLean, Thorp, Ziemba (2009) book with main articles reprinted plus new ones
See Ziemba and Hausch (1996), Aucamp (1993), Browne (1997) and MacLean, Thorp, Ziemba (2010) for more on this.
Essentially all of the material in this talk is in the following books plus the papers listed above
Ziemba, The Stochastic Programming Approach to Asset Liability Management, AIMR, 2003
Ziemba-Hausch, Dr Z’s Beat the Racetrack, William Morrow, 1987 (has UK betting system)
Hausch-Lo-Ziemba, Efficiency of Racetrack Betting Systems, Academic Press, 1994. Classic new and reprinted articles, bible for Hong Kong professional betting teams. Originals sell for huge prices as high as $12,000 I am told, I sold one for $1400. 2008 2nd edition from World Scientific in Singapore at a low price.
Ziemba-Vickson, Stochastic Optimization Models in Finance, Academic Press, 1975. Classic articles, new articles, huge collection of portfolio theory, problems.Reprinted by World Scientific, Singapore, 2006.
Ziemba et al, 6/49 Lotto Guidebook, 1986
Ziemba-Hausch, Betting at the Racetrack, 1986, exotic bet pricing
Ziemba and Ziemba (2007) Scenarios for Risk Management and Global Invetment Strategies, Wiley
MacLean, L.C., E. O. Thorp, Ziemba, W.T., Eds., The Kelly Capital Growth Criterion: Theory and Practice, World Scientific
Books all available, [email protected] for information. Amazon has them at low prices.