William T. Ziemba

Alumni Professor of Financial Modeling and Stochastic OptimizationSauder School of Business

University of British Columbia, Vancouver, Canada

Mathematical Finance SeminarStanford University

January 30, 2004

Behavioral Finance, Racetrack Betting and

Options and Futures Trading

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.

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.

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

Effect of data input errors on portfolio performance

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.

€

t =1

RA

• market in miniature • fundamental and technical systems • returns and odds are determined by1) participants -- like stock market, unlike roulette2) transaction costs -- track take (17%), breakage 

• bet to1) win -- must be 1st2) place -- must be 1st or 2nd3) show -- must be 1st, 2nd or 3rd

Horseracing

Win market

• n horses

• 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

0 otherwise{

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.

Typical behavior of the betting fortune of the average bettor

Probability of being even or ahead for the typical win bettor in California

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

Favorite-Longshot bias at racetracks and in other gambling events

• Behavioral financeKahneman-Tversky (1979)low probability events are overestimatedhigh probability events are underestimated

• More bragging rights from picking longshots than from favorites50-1 wow, was I smart2-5 easy pick

• Transactions costsbet $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 100-1 horse has only has an expected value of about 14 cents per dollar invested. The fair odds are about 700-1 not 100-1.

• Early literature goes back to at least 1949US, 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 wtzimi@mac.com 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

Number of entries, winners and winners times odds for every odds group

Odds (subjective) vs percent winners (objective)

Reprinted in HLZ (1994)

Griffith 1949 study - 1386 races in 1947, Churchill Downs, Belmont and Hialeah1934 data similar

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.

Ziemba (1994) and

The bookies know about this.

Expected return per dollar bet with and without the track take deducted for different odds levels in the Kentucky Derby 1903-1986 and in 35,285 races run during 1947-1975, 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.

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.

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

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.

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.

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

Methodology

• 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.

Average return per dollar bet vs. odds levels: 3-month stock index calls, 1985-2002

• 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

Average return per dollar bet vs. odds levels: 1-month stock index calls, 1985-2002

1 Month Stock Index Futures Call Options Wealth Relatives

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

0.00.10.20.30.40.50.60.70.80.91.0

FTSE

S&P

• The call advantage is gone; they all break even.

• The deep outs still have low expected values.

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 2000-2002 crashes, so has a lot in it already.

With more data, the graph is smoother.

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.

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.

Tests with real money in two small futures accounts of WTZ

• 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

Jan 2002 to present Oct 2003 to present

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

Place market in horseracing

Bettors do not like place and show bets.

The Idea

1. Use data in a simple market (win) to generate probabilities of outcomes

2. Then use those in a complex market (place and show) to find positive expectation bets

3. 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.

€

lim

T → ∞wealth log investorT

wealth any other essentially different investorT→ ∞

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- in w 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.

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, 1981-2001; merging of prices (odds) with expert opinion (breeding measured by dosage).€

Dosage = Speed

Stamina

The first two are “stock market” people and are closer to full Kelly. See text in AIMR, 2003.

Some Kelly bettors: Keynes, Buffett, Thorp and Benter

Keynes: Graph of the performance of the Chest Fund, 1927-1945 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.

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)

Does it work? Update a terrific, smooth record.

Hong Kong racing syndicate to about 2001.

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.

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.

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.

Guide to Capital Growth Theory and Kelly Criterion Literature

1956 Kelly heuristic paper, original idea (Latane, 1957, also)

1961 Breiman, original correct proofs

1969 Thorp original application to sports betting

1981 Hausch-Ziemba-Rubinstein, application to place and show system, books later –1984, 86, 87

1988 Algoet and Cover most general proofs

1994 Hausch-Lo-Ziemba reprints many key articles

1995 Hakansson & Ziemba survey finance view point in Finance Handbook, reviewsHakansson’s work

1998 Janacek MSc Thesis, Charles Univ. creative student

1998 Thorp brilliant math analysis

1999 MacLean-Ziemba, fractional Kelly examples (series of papers 1986+)

2002* MacLean-Ziemba theory of targets rather than time, Time to Wealth

2003 Ziemba AIMR, more simply written, many references

2004 MacLean-Ziemba et al how to calculate the “optimal” fractional Kelly;controversial constrained optimization, JEDC

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:

1 + P

1

2

)P - P - (QP

i

ji

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:

1 + S

1

3

)SkSjSi - (QS

i

−−

Place/Show system in theory - the place payoff

Hausch, Ziemba, Rubinstein (1981) Management Science and Ziemba-Hausch books. Academic papers in 1994 Academic Press.

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

Toteboard and place payoffs

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.

$4.24 = 1 + 660

1

2

1386) - 660 - 4)((.83)(921

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 •

Efficient win market gives qi = Wi/W.

place probabilities determined using just public’s win odds.

Similarly, probability i first, j second, and k third is

)q - q - )(1q - (1

qqq = q

jii

kjiijk

Place probability

There are biases in these formulas; see HLZ (1994) - they are ok for this system though as the biases tend to cancel.

Expected place payoff

1) + P2

P - P - QP)(

)q-(1

qq +

)q-(1

qq(

i

ji

j

ji

i

jiij n,1,...,=j ≠

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

Optimal capital growth model

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?

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:

€

R( f ,n) = q(1+ fr)1/ n + (1−q)(1− f )1/ n[ ]

n

€

limn→∞R( f ,n) = (1+ fr)q ⋅(1− f )1−q

)f1ln()q1()fr1ln(qe −−++=

)f1ln()q1()fr1ln(qmaxf

−−++

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.

Slew O’ Gold, 1984 Breeders Cup Classic

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

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.

Exhibition Park, 1978, typical returns.

Aqueduct, 1981-82

Summary Statistics on Dr Z System Bets

€

Ex Place i = 0.319 + 0.559wi /w

pi / p

⎛

⎝ ⎜

⎞

⎠ ⎟

Ex Show i = 0.543+ 0.369wi /w

si /s

⎛

⎝ ⎜

⎞

⎠ ⎟

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.

1983 Kentucky Derby

1991 Breeders’ Cup Race 5

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.

0

5000

10000

15000

20000

25000

30000

35000

19811982198319841985198619871988198919901991199219931994199519961997199819992000

Years

Wealth

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Does the system still work in 2004?

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.

Special case - sub-bet

• 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.

• 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

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

Approximate Kelly with more money on higher probability wagers.