الرئيسية [World Scientific Handbook in Financial Economics Series] The Kelly Capital Growth Investment...
أبلغ عن مشكلةThis book has a different problem? Report it to us
إختار نعم إذا كان إختار نعم إذا كان إختار نعم إذا كان إختار نعم إذا كان
نجحت في فتح الملف
يحتوي الملف على كتاب (يُسمح أيضًا بالرسوم الهزلية)
محتوى الكتاب مقبول
يتوافق الإسم و المؤلف و لغة الملف مع وصف الكتاب. تجاهل الخانات الأخرى لأنها ثانوية!
إختار لا إذا كان إختار لا إذا كان إختار لا إذا كان إختار لا إذا كان
- الملف تالف
- الملف محمي بواسطة الوسائل التقنية لحماية حق المؤلف DRM.
- الملف ليس كتاب (علي سبيل المثال، xls, html, xml)
- الملف عبارة عن مقال
- الملف مقتطف من كتاب
- الملف عبارة عن مجلة
- الملف نموذج إختبار
- الملف سبام
هل تعتقد أن محتوى الكتاب غير مناسب ويجب حظره
لا يتطابق الإسم أو المؤلف أو لغة الملف مع وصف الكتاب. تجاهل الخانات الأخرى.
Change your answer
Efficiency of Racetrack Betting Markets, 567- 574. World Scientifi c (2008) 695 48 The Dr.Z Betting System in England t William T . Ziemba and Donald B. Hausch The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Managemen/ Science Division, Faculty of Commerce, University of British Columbia, Vancouver Be, Canada V6T lZZ School of Business, University of WISconsin, Madison, WISconsin 53706 Abstract The betting strategy proposed in Hausch, Ziemba and Rubinstein (1981) and Ziemba and Hausch (1984,1987) has had considerable some success in North American place and show pools. The place pool is England is very different. This paper applies a similar strategy with appropriate modifications for places bet at British racetracks . The system or minor modifications also applies in a number of other countries such as Singapore with similar betting rules. The system appears to provide positive expectation wagers. However, with the higher track take it is not known how often profitable wagers will exist or what the long run performance might be. I. IntroductIon At North American racetracks, the parimutuel system of betting utilizing electric totalizator boards in the dominant method of betting. Las Vegas and the other legal sports books may set odds on particular betting situations, but these fixed odds are not available at racetracks. In England and in other Commonwealth countries, such as Italy and France, odds betting against bookies is the dominant betting scheme. This fixed-odds system is introduced in Hausch, Lo and Ziemba (1994). They also indicate a tendency of higher (lower) returns for lower (higher) odds ranges. Thus, the favorite-Iongshot bias appears to exist in England (see e.g. Ali (1977), Busche and Hall (1988» . This paper applies the betting system proposed by Hausch, Ziemba and Rubinstein (1981) and Hausch and Ziemba (1985) in England. This strategy is also called the Dr.Z system in the trade books Ziemba ; and Hausch (1984,1987) who discuss it more fully. The strategy utilizes the Kelly criterion (Kelly (1956» which maximizes the expected logarithm of wealth. The Kelly criterion has several advantages. First, it maximizes the capital growth asymptotically. Second, it prevents bankruptcy. Third, the expected time to reach a specified goal is minimum when the goal increases. Fourth, it is superior to any different strategy in the long mn. These properties were proved in Breiman (1961). See McLean, Ziemba and Blazenko (1992) for discussion of these properties. n. The System Instead of the North American parimutuel system of win, place, and show, the bets in England are to win and place. By "place" the British mean "finish in the money.· This is what North Americans call show except for one important difference. The number of horses that can place in a particular race is dependent on the number of starters. t Modified from an Appendix in Ziemba and Hausch (1987). Efficiency of Racetrac k Belling Markets Copyright © 1994 by Academi c Press. Inc. All rights of reproduction in any form reserved . 567 696 W T. Ziemba and D. B. Hausch W. T. ZIEMBA AND D. B. HAUSCH 568 The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Table 1. Relationship between number of horses that place and number of starters Number of Horses that Place Number of Starters one: the winner four or less two: winner and second five, six, or seven three: winner, second, and third eight to fifteen four: winner, second, third, and fourth sixteen or more The place pools are not shown on the tote board, but the current payoffs for place bets for each horse are flashed on the screen. Bookies, on the othet hand, simply pay a percentage of the win odds, as shown in Table 2. There are many types of exotic bets as well. The tote jackpot corresponds to what North Americans call the pick six or sweep six. The tote placepot bet has no analogue in North America. The average rates of return on various bets on and off course against a bookmaker or the tote are listed in Table 3. The track take is 5% larger in the place pool than in the win pool. The tote take is larger than what the bookies make on average, and on-course betting takes are much less than off-course takes. The races in England are on the turf for distances of generally at least a mile, except for some shorter races for two-year·-olds. The season in southern England is unique in that races are run for about three days at each race course. The jockeys, trainers, and so forth then move on to a new course. After a month or so they return to the same course. Handicapping is very sophisticated in England. It has to be, with little infonnation easily accessible (they have no analogue of the Daily Racing Fonn, although some past performances are available in newspapers) and all that moving from course to course. The method of computing the place payoffs in England differs from that used in North America. In both locales, the net pool is the total amount wagered minus the track take. In North America, the cost of the winning in-the-money tickets is first subtracted to form the profit. This profit is then shared equally among the in-tbe-money horses. Holders of winning tickets receive a payoff consisting of the Original stake plus their proportionate share of the horse's profits. This means that the amount of money wagered on the other horses in the money greatly affects the payoff. In England, the total net pool is divided equally among the horses that finish in the money. This means that the payoff on a particular horse depends upon how much is bet on this horse to place but not on how much is bet on the other horses. Since the minimum payoff is £1 per £1 wager, management is able to keep a control on betting for particular favorites. Once this minimum level is reached, it does not pay to wager on a given horse. This occurs whenever the percentage of the place pool that is bet on a given horse becomes as large as Q,. which is the track take for place, divided by m, which is the number of in-the-money horses. In a race with 8-15 starters, if Q, is about 0.735, and m=3, thejust-get-your-money-back point is reached when the bet on a particular horse to place becomes 24.5~ of the total place pool : 0.735/3 = 24 . 5~. Hence in England you will often see horses whose place payoffs are £1 or just slightly higher. This method of sharing the place pool tends to favor longer-priced horses at the expense of the favorites. 697 The Dr. Z Betting System in England THE DR. Z BETIING SYSTEM IN ENGLAND 569 Table 2. Bookmakers payoff for place bets ~umbcr of R",nncn T),pe of Il>cc Two 10 five Six or seven Any The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Eighl or more fnctioa of Win Odds Paid on Hones Place Element Rcprded . No place betting Twch'e 10 tifteen Anyexcepl handicaps involving IWdve or morr runnen IUndlcaps Sixteen 10 twent}··one Handlcaps Twenty·two or more Handicaps l Finland ~ond 1 Fint, second. and third : fint, second. and third Fint, second. third, and fourth fint, second, third. and fourth S,wm : Roduchi'c (197S). Table 3. Rates of return on different types of betS in England on thoroughbred and greyhound racing J\.\te of ReNm Type of Bcl <" ) On·course bookmaker Olf·course bookmaker Single bello " 'in " 'i th olr·course bookmaker Double bello " 'in ...ith oll'·course bookmaker Treble bet 10 "'in " 'ith off·course bookmaker lTV x"en bello ....in ...·ith olf·course bookmaker Computer slraighl forecasl v";lh olf·course bookmaker Greyhound forecasl with olf·course bookmaker Greyhound forecast double ...·ith olf·course bookmaker Place clement of e ..ch· ..."')" with oll'·course bookmaker ."nle·post betting ...-Ith off·coutR bookmaker Horse race 10le ...·in pool (on course) Horse I'3ce 10le ...·in pool (oll' course) Horse race tOle place pool (on course) Horse race tOle place pool (off course) Horse raCe tOle daily double pool (on course) Horse race 10le daily double pool (off course) Horse race 10le da.i.l~· treble pool (on coune:) Horse racc 10le da.i.ly treble pool (olf course) Horse race tote da.i.l~· forecast pool (on course) Horse race tOle da.i.lr forecasl pool (olf course) Horse raCe lote jackpOI pool (on course) Horse raCe 10le Jackpol pool (off course) Horse race 10le pbccpol pool (on course) Horse race 10le pbccpol pool (oft' course) Greyhound lote pool betting. average 90 81 85 S......,: Ro,h"hild (l9i6). 78 72 70-75 65 76 58 80 96 80 i7 75 -:'2 7. 71 70 67 70 67 "0 67 70 67 83.5 698 W T. Ziemba and D. B. Hausch W. T. ZIEMBA AND D. B. HAUSCH 570 The current track take to win is about 20.6% and to place is 26.5%. and the breakage is of lOc variety. or more properly lOp. for pence2. These track takes are much higher than those in North America. Since the track paybacks to win and place are different. we call the former. Q..=0.794. and the latter. Q,=0.735. It is easy to apply the Dr.Z system in Great Britain, although with its much higher track takes, there may not be many Dr.Z system bets. see Mordin (1992) for a discussion of this. We utilize the substitution that q, =QJO" where 0, are the odds to win on the horse under consideration. The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. The expected value per pound bet to place on horse i is EXPlace - (probability of placing) (place odds) - (Prob) (PO,). (1) In (1). PO, refers to the odds to place on horse i. Prob. the probability of placing is determined as follows.' • -:We can calculate these track takes as follows: The payoff on horse i if it wins is Qw W/W, where Qw is the track payback to win. and W. and W are the bet amount of horse i and the total bet amount. respectively. So let q, = W,tW, the efficient-market assumption. Let B be the average breakage, namely, 4.5p. Since breakage can be 0.1.2 ...... 9 pence, its average is 4.5p. Then the payoff on i is Q,/q,-B. which equals the odds 0,. since the odds are based on total return (not return plus original stake as in North America) . So q, = QJ(B+O,). Summing over all n horses gives 1 A :E - Q.. ; .. 1 (B + 0)' I since some horse must win. Hence 1 Q... - - . - - - For place. there are one. two. three. or four horses that are in the money. depending upon the number of starters. So m Q, - --'--- where m=1.2.3 or 4. With the above formulas. Qw '" 0.794 and Q,. '" 0.745. 'These equations were developed using the 1981-1982 Aqueduct data to relate probability of in-themoney finishes to q. the probability of winning and n, the number of horses. Equations (2), (3) and (4) had R2 of 0.991,0.993 and 0.998, respectively. These equations are valid when q ranges from 0 to 0.6 The Dr. Z Betting System in England THE DR. Z BETTING SYSTEM IN ENGLAND 699 571 With n=5 to 7 horses, the first 2 horses place and Prob - 0.0667 +2 .37q -1.61q2 -0.OO97n. (2) With n=8 to 15 horses, the first 3 horses place and Prob - 0.0665 + 3.44q - 3.47q 2 - 0.0049n. (3) The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. With n= 16 or more horses, the first 4 horses place and Prob - 0.0371 +4.47q -6.29q2 -O.OOI64n. (4) Figures 1,2, and 3 determine Prob directly using only 0" the win odds on the horse in question. Figure 1 applies when there are five, six or seven horses . Figure 2 corresponds to equation (3) and applies when there are eight to fifteen horses. Finally, Figure 3 corresponds to equation (4) and applies when there are sixteen or more horses. The optimal Kelly criterion bet is to wager (prob POi-I)/(POi-l) percent of your betting wealth'. We can determine the optimal fraction of your wealth to bet indicated by equation (5) using Figure 4 . for (2), from 0 to 0.45 for (3), and 0 to 0.3 for (4), which should be the case in most instances. However, Figures 1,2 and 3 are valid for any q. 'In a race with n=2,3, or 4 horses, only one horse places, the winner. Such races are rare. Also, it is unlikely that the win and place pools would then become so unbalanced as to yield a Dr.Z system bet. However, one would occur when PO/Oj was at least 1.44, for a track payback of 0 .794 and an expectedvalue cutoff of 1.14, since 1.14/0.794 is 1.44. In such a case, one would have a good bet. 'We have assumed that your bets will be small and hence will not affect the odds very much . Thus to determine the optimal bet b for betting wealth wo, you maximize Prob log(wo + (PO;- I)b] + (1 Prob)log(woO)' whose solution is equation (5). W T. Ziemba and D. B. Hausch 700 W. T. ZIEMBA AND D. B. HAUSCH 572 The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. 1.1)r-------------------, I) , 2 9 ? 3 Odds 10 Win on Ho,.. I Figure 1. Probabilities of placing for different odds horses when the race has five to seven starters o 3 ? 8 9 10 Figure 3. Probabilities of placing for different odds horses when the race has eight to fifteen starters The Dr. Z Betting System in England 701 573 The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. THE DR. Z BETTING SYSTEM IN ENGLAND 0.25 o 3 s 7 6 Odds to Win on Horu 8 9 J Figure 3. Probabilities of placing (or different odds horses when the race has sixteen or more starters 1.0 1 ~ 0.91 ~ 1 ~ 1 Proc • 0.95 Prob - 0.90 O.S 0.7 .§ 0.6 j: Prob - 0.80 i Prob - 0.70 Prob - 0.60 j Proo • O.SO .. 0.2 ~ Prob .E e ~ 0.40 0.1 Prob - 0.30 0.0 Place Odds, PO, :ure 4. Optimal bets when the probability of placing is Prob anel the place odds of the horse in question ; PO j 702 574 W. T. Ziemba and D. B . H ausch W. T. ZIEMBA AND D. B. HAUSCH References Breiman,L. (1961) ·Optimal gambling systems for favorable games .· in Proceedings of the Fourth Berkeley Symposium . Mathematical Statistics and Probability I, 65-78. University of California Press. Figgis,E.L. (1974) "Rates ofrerurn from flat race betting in England in 1973. · Sponing Life 11 (March). Hausch,D.B., Ziemba,W.T. and Rubinstein,M. (1981) "Efficiency of the market for racetrack betting. ' The Kelly Capital Growth Investment Criterion Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/14/17. For personal use only. Management Science 27, 1435-1452. Hausch,D.B. and Ziemba,W .T . (1985) "Transactions costs, extent of inefficiencies, entries and multiple wagers in a racetrack betting model." Management Science 31, 381-394. Kelly ,J .L. (1956) "A new interpretation of information rate. " Bell System Technica/Journa/ 35, 917 -926. MacLean,L.C. , Ziemba,W.T. and Blazenko,G. (1992) "Growth versus security in dynamic investment analysis.· Management Science 38, 1562-1585. Mordin,N. (1992) "Grab your place in the pool!" Sponing Life. Rothschild, Lord (1978) Royal commission on gambling, Vols I and II. Presented to parliament by Command of Her Majesty (July) . Ziemba,W.T. and Hausch ,D.B. (1984) Beat the racetrack. Harcourt Brace Jovanovich, San Diego . Ziemba,W .T. and Hausch ,D.H . (1987) Dr.Z's beat the racetrack . Revised ed ition. William Morrow and Co. Inc. New York.