Carrot Dicing

1. Carrot Dicing A Model for the Tote Price Distribution for Horseracing Charles Magri ( cmagri@mbox.com.au ) 2 December, 2002

2. Roadmap for Presentation • The Model • Full Description • The Evidence • Empirical Distribution Fit • Exploitation – A profitable strategy • Generalisation of Application • Share markets, Portfolio Selection • Parimutuel Financial markets • Deutsche & Goldman options on economic statistics • Further Work

3. The Recipe • Ingredients • 8 different sized pots • A case of whiskey, rum, brandy or vodka • A meat axe • About 10,000 carrots • Method • Consume about half a bottle of the liquid and stand for about 15 to 20 minutes • Select a carrot. Make 7 random chops into carrot with axe • Place the biggest piece in the biggest pot, the second biggest in the second biggest pot, etc. • Repeat last two steps taking two swigs of the liquid between carrots making sure not to spill any of the liquid onto the food

4. Some Statistics (a) U1, U2, ..., Un i.i.d U(0,1) Cutting locations sort ascending U(1), U(2), ... U(n)Ordered Statistics successive differences S1, S2, ..., Sn, Sn+1Sample Spacings where Sj = U(j) - U(j-1), U(0) = 0, U(n+1) = 1 sort (descending) S(1), S(2), ..., S(n), S(n+1)Ordered Spacings CARROT DICING MODEL

5. Some Statistics (b) Order Statistics • The density function for U(k) is given by :- n k-1 n-k gk (x) = (k-1, 1, n-k) . x (1 -x)

6. Some Statistics (c) Order Spacings The density function for S(k)is given by :- n + 1 hk (x) = (k-1, 1, n-k+1) . n! .  ... xj x: j=1, ... k-1dx1 dx2...dxn xjx , j=k, ..., n, n xj = 1 - x

7. Some Statistics (d)   • Favourite in a 3-horse race (n=2, k=1) • hk(x) = 6. du dv = 6. min(x, 1-x) du u,v  x max(0, 1 - 2x) u+v = 1-x 0 if x < 1/3 = 6 ( 3x -1 ) 1/3  x < 1/2 6 ( 1 - x ) if 1/2  x < 1 • E(Fav price) = E. 1 . = 6 ln(4/3) x ( 1 +  ) ( 1 +  )  1.73 / ( 1 +  ) { ( )

8. Evidence I(a) • Distribution of Favourites ( 11 horse race )

9. The Data • Saturday races • for the major metropolitan centres, SR, MR, BR, AR • from 9 March 02 to 2 November 02 ( 8 months ) • Tote prices are Final Dividends from NSWTAB • 1110 races in all • 11-horse race - most frequent (161 races )

10. Evidence I(b) • Distribution of Favourite ( 11 horse race )

11. Evidence I(c) • Distribution of 2nd Favourite ( 11 horse race )

12. Evidence I(d) • Distribution of 3rd Favourite ( 11 horse race )

13. Evidence I(e) • Distribution of 4th - 7th Favourites ( 11 horse race )

14. Evidence I(f) • Distribution of 8th - 11th Favourites ( 11 horse race )

15. A Strategy • Define n sig =  i = 1( carrot i - fav i)2 • Target : • proportionate to sig • inv proportionate to n Tgt = 100 x sig x 8 / n • Bet only if favourites are long • Bet on horses that are long but have short place prices investing in win bets so as to make the target Tgt • Abandon race if just betting on very long-odds horses

16. Evidence II(a)

17. Evidence II(b) • Long Favourite - minimum 20% outlay

18. Evidence II(c) Further Testing

19. Evidence II(d) • Long Favourite - minimum 20% outlay

20. Evidence II(e) • Long Favourite - minimum 30% outlay

21. Evidence II(f) • Long Favourite - minimum 10% outlay

22. Other Financial Market Applications • Portfolio Selection • allocation of funds amongst n competing assets • Parimutuel Markets • Deutshe & Goldman Sachs options on economic data using Longitude’s PDCA (Parimutuel Digital Call Auction) technology

23. Further Work • Investigate “Local Exchanges (Totes)” for BR and AR • Look at mid-week races also • Find more exploitation strategies • Research further the “short favourite” instances and relate to “crowd behaviour” models or ising models in physics • Investigate Tote distributions for overseas horse races • Tidy up analytic work on order spacings • Investigate other applications in finance • Investigate price slippage in last minute impact on results