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Publié parMehdi Maziane Modifié depuis plus de 2 années

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**Behavioral Finance, Racetrack Betting and Options and Futures Trading**

William T. Ziemba Alumni Professor of Financial Modeling and Stochastic Optimization Sauder School of Business University of British Columbia, Vancouver, Canada Mathematical Finance Seminar Stanford University January 30, 2004

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**Investing in traditional financial markets has many parallels with racetrack & lottery betting**

Behavioral anomalies such as the favorite-longshot bias are pervasive and also exist and are exploitable in the S&P500 and FTSE100 futures and equity puts and calls options markets. Biases there favor buying high probability favorites and selling low probability longshots. In complex exotic wagers such as the Pick 6, the bias is to overbet the favorite so one must include other value wagers in the betting program. Information such as breeding is important and is useful for the Kentucky Derby and Belmont Stakes. Lotteries have a possible advantage from the unpopular numbers, which have substantial advantages but low probability of success on each play and a long time to have high confidence of success. So the optimal bets are very tiny. Futures trading in the turn of the year effect and some racing bets can have both a high expected return and a high chance of success so the optimal bets very are large and are tempered by risk control considerations.

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**Main points to learn from this lecture**

Means are by far the most important aspect of any portfolio problem. You must have the mean right to have good performance. If you have the mean right and do not overbet you should do well. In levered bets, it’s the left tail that can lead to trouble so you must not overbet or you can have a large disaster occurring without warning. Behavioral and other anomalies can yield strategies that have positive means. Studies of racetrack bias date at least to the late 1940s These biases yield ideas that yield profitable positive mean strategies in racing, sports betting and options markets. The capital growth or Kelly criterion strategy yields the most wealth in the long run and dominates all other essentially different strategies. But in the short run, the expected log criterion with its essentially zero Arrow-Pratt risk aversion index is very risky and can have substantial losses. The most you should ever bet is the log optimal amount; betting more is suboptimal and betting double yields a zero growth rate. Negative power utility, which blends cash with the expected log maximizing portfolio provides more security but has less long run growth. These fractional Kelly strategies are attractive for many investment situations; determination of what fraction to use depends on constrained optimization models.

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**If you bet on a horse, that’s gambling.**

If you bet you can make three spades, that’s entertainment. If you bet cotton will go up three points, that’s business. See the difference? Blackie Sherrod Unbridled at Claiborne Farms, Paris, Kentucky

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**Effect of data input errors on portfolio performance**

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**Mean Percentage Cash Equivalent Loss Due to Errors in Inputs**

Conclusion: spend your money getting good mean estimates and use historical variances and covariances Reference: Chopra and Ziemba (1993), Journal of Portfolio Management, reprinted in Ziemba-Mulvey (1998) Worldwide asset and liability management, Cambridge University Press Results similar in multiple period models and the sensitivity is especially high in continuous time models. See examples in AIMR, 2003.

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Horseracing market in miniature fundamental and technical systems returns and odds are determined by 1) participants -- like stock market, unlike roulette 2) transaction costs -- track take (17%), breakage bet to 1) win -- must be 1st 2) place -- must be 1st or 2nd 3) show -- must be 1st, 2nd or 3rd

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**{ n horses Win market Wi = public’s win bet on horse i=1,…,n**

W = Wi = public’s win pool Q = track’s payback fraction (83%) Return on horse i = QW/Wi if horse i is 1st otherwise {

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**How do you calculate actual returns on win bets**

Experiment: do bettors like high odds horses or low odds horses? Low odds horses are the best ones, high odds horses are the worst ones. Would you rather bet on a 2 to 5 shot and receive 40% profit if you win or a 20 to 1 shot where you receive 2000% profit if you win? The public prefers the latter but the expected returns are much higher for the favorites. The favorite-longshot bias has persisted for more than the last half century.

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**Running up sand dunes - it is difficult to win at the races**

Typical behavior of the betting fortune of the average bettor Probability of being even or ahead for the typical win bettor in California

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**Favorite-Longshot bias at racetracks and in other gambling events**

Behavioral finance Kahneman-Tversky (1979) low probability events are overestimated high probability events are underestimated More bragging rights from picking longshots than from favorites 50-1 wow, was I smart 2-5 easy pick Transactions costs bet $50 to win $10 it’s hardly worth the effort a 1-10 horse having more than a 90% chance of winning has an expected value of about $1.03 (for every $1 bet) a horse has only has an expected value of about 14 cents per dollar invested. The fair odds are about not Early literature goes back to at least 1949 US, British, Asian, etc papers reprinted in Hausch, Lo and Ziemba (1994), Efficiency of racetrack betting markets (Academic Press) Out of print, Amazon/eBay have had originals at high prices; I have low price copies available. Contact me at for any of my racing, lotto and investment books and research papers. Updated in Hausch and Ziemba (2004) Handbook of Sports and Lotto Investments, North Holland

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**Griffith 1949 study - 1386 races in 1947, Churchill Downs, Belmont and Hialeah 1934 data similar**

Number of entries, winners and winners times odds for every odds group Odds (subjective) vs percent winners (objective) Reprinted in HLZ (1994)

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**Favorite-longshot bias at racetrack and in other gambling events**

The effective track payback less breakage for various odds levels in California and New York, more than 300,000 races over various years and tracks, Ziemba and Hausch (1986) Betting at the Racetrack.

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Ziemba (1994) and

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**The bookies know about this.**

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Expected return per dollar bet with and without the track take deducted for different odds levels in the Kentucky Derby and in 35,285 races run during , from data in Snyder (1978) The better races have flatter biases; see references in Tompkins, Ziemba and Hodges (2003) The favorite/longshot bias in S&P500 and FTSE100 index futures options: the return to bets and the cost of insurance.

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**Has the bias in the US changed with rebates and betting exchanges? Yes.**

It seems to be more flat. Data for essentially all North American races for the last 7+ years, about 2 million horses in about 300,000 races.

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We have a good data set: essentially all of the S&P500 futures puts and calls, all years, all strikes.

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**Do the buyers of puts and calls on the Stock Index options behave like racetrack bettors?**

Who buys and sells them At the racetrack there are only buyers, but there are sellers on betting exchanges. In options, futures and stock markets there are buyers and sellers. Demand for options come from both hedging needs and speculative investing (gambling). Hedge funds are active shorters in purely speculative ways or in complex hedging strategies. For the put options the primary use of options is for hedging, and only on a secondary basis is the demand for speculative investing. Hedging demand for puts implies that the expected return is negative and more so for deep out of the money options; cost of insurance. For the call options, the most obvious hedging demand is for those selling call options against existing holdings of equity. This strategy would tend to depress the price of (especially out-of-the-money) call options. If this were the sole mechanism for dealing in call options, this should result in an increase in the expected return for out-of-the-money call options, which we do not observe. Much more likely is that the expected loss from the purchase of out-of-the-money call options is due to some speculative activity that appears to be similar to that for long-shot horse race bets. We have a good data set: essentially all of the S&P500 futures puts and calls, all years, all strikes from 1985 current.

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Methodology Monthly and Quarterly data was used instead of daily data to ensure independence of observations and final outcomes. We used all expiration dates (March, June, September, and December) for the quarterly expiration cycle and all the available monthly expirations (e.g. January options on a March futures). We recorded the final settlement price of the futures on the expiration of the option contracts, the futures contract with exactly three months to expiration and all available one-month and three-month option prices on the futures contract. For the first date in the analysis, the current expiring futures contract was not used and for the final data point, we examined the final futures price in September 2002. All options prices traded at the minimum level at the relevant market or allowed a butterfly arbitrage were excluded [see Jackwerth and Rubinstein (1996)]. The interest rate inputs were obtained from the British Bankers Association (US Dollar or British Pound LIBOR). With seventeen years of quarterly data, we had 68 observations and an average of 39.1 available strike prices per observation for the options on the S&P 500, 69 observations with an average of 30.8 available strike prices for the options on the FTSE 100, and 60 observations with an average of 17.8 available strike prices for the options on the British Pound / US Dollar. With seventeen years of monthly data, we had 187 observations and an average of 39.0 available strike prices per observation for the options on the S&P 500 and 124 observations with an average of 28.6 available strike prices per observation for the options on the FTSE 100.

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**Methodology What would we expect?**

To obtain results comparable with horseracing which also aggregates into categories,, we aggregated the initial odds [N(d2) and N(-d2)] into bands of 5%. We pooled all the options in these twenty ranges and averaged the outcomes for the same 1$ initial investment. What would we expect? When risk premium exist (for example in equity markets), the expected return for the investment in options will differ from the $1 investment. We examined call and put options using the Black Scholes (1973) formula with no risk premium and risk premia of 2%, 4% and 6%. This is done by inputting –2%, -4% and –6%, as the continuous dividend rate [using the Merton (1973) dividend adjustment] in the Black Scholes formula. The ratio of the option prices are determined and plotted as a function of the moneyness. The calls lie above the $1 investment and the puts lie below the $1 investment. Note: 1.0 is deep in the money and 0.0 is deep out of the money

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**Average return per dollar bet vs**

Average return per dollar bet vs. odds levels: 3-month stock index calls, The favorite deep in the money calls have positive expected value like the favorites at the racetrack The longshots used for covered calls and other strategies in high demand have large expected losses Deep out calls have low expected value; 1.23 cents per dollar for .00 to .05. The shape is broadly similar to the racetrack graphs

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**Average return per dollar bet vs**

Average return per dollar bet vs. odds levels: 1-month stock index calls, The call advantage is gone; they all break even. The deep outs still have low expected values.

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**All puts lose, the cost of insurance is high.**

This is essentially all the data on the S&P500 futures options; the graph is smooth with a few kinks. Possible peso effect: you are paying for an event that has not yet occurred but could occur. But this data includes October 1987, October 1989 and crashes, so has a lot in it already.

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**With more data, the graph is smoother.**

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**Trading ideas and the results of one hedged strategy**

You sell the options with negative expected value and buy those with positive expected value and have various levels of hedging using S&P500 futures. 20 years of actual trading with real money and extensive simulation and other research suggests which ones and when to sell/buy/hedge in the various strategies to provide good, steady gains with minimal risk. Bias strategy in various forms. Hedged currencies Positive mean Optimal betting Risk control - must protect against extreme events, bad scenarios Reference: W. T. Ziemba (December 2003) Stochastic Programming approach to Asset-Liability and Wealth Management, AIMR, text plus separate appendix.

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**The results of one hedged strategy (simulation)**

The geometric mean is about 45% net for top graph which uses a short term overpriced market indicator to eliminate five deemed risky of the 70 plays; you win in 60 of 65. The bottom graph has all 70 plays, including the Oct 1987 crash and July 2002 which the measure eliminated.

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**Tests with real money in two small futures accounts of WTZ**

Jan 2002 to present Oct 2003 to present Long euro, long canadian hedged also in this account Mean > 0 in my view weak dollar Small positions to weather storms, still results very good Watch entry and exit and rollups carefully

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**Place market in horseracing**

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.

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The Idea 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 Breiman (1961) So in the long run, the log utility investor dominates.

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**Growth versus security**

Security rises if the bet sizes are decreased with fractional Kelly (negative power) but growth decreases as well. Half Kelly is -w-1. Fractional Kelly f=1/1-a, a<0, in awa is exact for lognormal, approx otherwise. See MacLean-Ziemba, Time to Wealth (2002). Half Kelly is half the Kelly wager plus cash. Right graph has dosage filter; do not bet on horses that cannot run 1 1/4 miles on the first Saturday in May.

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**Horses with high dosage cannot really win the Kentucky Derby or Belmont Stakes**

See Bain, Hausch and Ziemba (2004) An application of expert information to win betting on the Kentucky Derby, ; merging of prices (odds) with expert opinion (breeding measured by dosage).

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**Some Kelly bettors: Keynes, Buffett, Thorp and Benter**

The first two are “stock market” people and are closer to full Kelly. See text in AIMR, 2003. Keynes: Graph of the performance of the Chest Fund, in AIMR, 2003 Growth of assets, various high performing funds in AIMR, 2003 See Thorp (1998) for analysis showing Buffett is close to full Kelly. Buffett’s Sharpe is below Ford’s but if you delete upside, it’s better, see AIMR, 2003.

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**Thorp and Benter - the “gamblers” like smooth wealth paths using fractional Kelly.**

Princeton Newport Partners, L.P., Cumulative Results, November 1968-December 1988 in AIMR, 2003 Hong Kong racing syndicate to 1994, see Benter’s paper in Hausch, Lo, Ziemba (1994)

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**Does it work? Update a terrific, smooth record.**

Hong Kong racing syndicate to about 2001.

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**You can have 700 independent bets all with a 14% advantage and still lose 98% of your fortune**

Half Kelly loses a lot of the growth much of the time in exchange for a smoother wealth path.

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Overbetting 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 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.

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**What is the optimal fractional Kelly?**

MacLean, Sanegre, Zhao and I, JEDC (2004) solve this in a continuous time model where you check discretely and use a Var type criterion on the wealth path. With Kelly, the better the bet is the more you bet and thus the more you lose when you lose so it’s very hard to stay above a wealth path. Theory developed under restrictive assumptions; calculations: algorithm exists but computations lengthy.

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**Guide to Capital Growth Theory and Kelly Criterion Literature**

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**Place/Show system in theory - the place payoff**

Pi = public’s place bet on i=1,…n P = Pi If i and j are the first two horses, gross return/dollar bet on i to place is: Si = public’s show bet on i=1,…n, S = Si If i, j and k are the first 3 horses, gross return/dollar bet on i to show is: Hausch, Ziemba, Rubinstein (1981) Management Science and Ziemba-Hausch books. Academic papers in 1994 Academic Press.

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**Toteboard and place payoffs**

Horse 1 2 3 4 5 6 7 Total Odds 4-5 13-1 6-1 5-2 16-1 11-1 33-1 Win 8293 1009 2116 4212 885 1251 457 18223 Place 2560 660 1386 2610 696 903 339 9214 Show 1570 495 1860 1881 543 712 287 6558 If horses 2 and 3 are the first two horses, the gross payoff/dollar bet to place on horse 2 is: $4.20 (with breakage) returned for each $1 place bet on i.

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**Place probability Efficient win market gives qi = Wi/W.**

place probabilities determined using just public’s win odds. Let qij = probability i is first and j is second Harville (1973) proposed: qij = prob(i first)•prob(j first if i not entered) = qi • Similarly, probability i first, j second, and k third is There are biases in these formulas; see HLZ (1994) - they are ok for this system though as the biases tend to cancel.

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**{ { Expected place payoff**

Expected gross return on a $1 bet to place on horse i = { Probability i and j are 1st & 2nd in either order { Place payoff on i if i & j are 1st & 2nd in either order

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**{ Optimal capital growth model r w.p. q**

If a wager has a positive expected return, how much should be wagered? Example Assume a sequence of independent wagers with the following return per $1 wagered: { r w.p. q 0 w.p. 1-q If rq>1, what is the optimal fraction of wealth to invest each period?

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**Optimal capital growth model (Kelly 1956, Breiman 1961, Algoet and Cover, 1988)**

The expected rate of growth of wealth from wagering fraction f of wealth each period for n periods is: Therefore, maximizing the asymptotic expected rate of growth is achieved by myopic use of a logarithmic utility function, i.e., With one wager, the optimal fraction to bet is f*=edge/odds=p-q.

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**Slew O’ Gold, 1984 Breeders Cup Classic**

f*=64% for place/show; suggests fractional Kelly.

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**Optimal capital growth model, assumes our bet influences the odds and we can bet on multiple horses.**

Non concave program but it seems to converge.

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**Exhibition Park, 1978, typical returns.**

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Aqueduct,

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**Summary Statistics on Dr Z System Bets**

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**Expected value approximation equations**

Notice expected value (and optimal wager) are functions of only four numbers - the totals and the horse in question. These equations approximate the full optimized optimal growth model. Solving the complex NLP: too much work and too much data for most people. This is used in the calculators.

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1983 Kentucky Derby

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1991 Breeders’ Cup Race 5

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**Fractional Kelly place and show bets on the Kentucky Derby**

Axis is max % of wealth you can invest with full Kelly, Unconstrained you bet far too much.. Effect of Arazi in 1992: one race as a 3-year-old, knee operation risky bet. Suggests low Kelly fraction. He lost. No horse has ever won the Kentucky Derby with just one 3-year-old race and very few with two races. As a 2-year-old, Arazi was absolutely brilliant.

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**Does the system still work in 2004?**

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**Yes, it looks pretty good. Bets made with one minute to go.**

Huge amounts of money come in with a lag from the rebate shops at the buzzer. Still the old system does well. Odds of favorite tend to fall at buzzer and that helps us.

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Special case - sub-bet

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Favorites are usually underbet to win; this is useful in situations like the place and show systems. These are high probability low payoff situations. However, the situation is reversed in low probability high payoff situations. Then favorites are overbet because in exotics there are usually multiple races or multiple horses or both so that many tickets are needed to have a good chance of winning. Most bettors pick the favorites and overbet them so their tickets are not too expensive. Then they are most likely not to win and if they win, they get very little because they have picked the favorite so the payoff is very low.

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**This was the behavioral key.**

Filigree, the third choice in the morning line went off at 8-1. The 3rd and 5th races back, he ran faster than the favorite Love at Noon ran in his last two races. So he had a chance to win and he did. Love at Noon went off at 1-5 and had most of the P5 money

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**Top number is final speed number, other numbers are pace within the race.**

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**Approximate Kelly with more money on higher probability wagers.**

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