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Carrot Dicing A Model for the Tote Price Distribution for Horseracing Charles Magri ( [email protected] ) 2 December, 2002 Roadmap for Presentation The Model Full Description The Evidence Empirical Distribution Fit Exploitation – A profitable strategy

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Carrot dicing l.jpg

Carrot Dicing

A Model for the Tote Price Distribution

for Horseracing

Charles Magri

( [email protected] )

2 December, 2002


Roadmap for presentation l.jpg
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


The recipe l.jpg
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


Some statistics a l.jpg
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


Some statistics b l.jpg
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)


Some statistics c l.jpg
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


Some statistics d l.jpg
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 +  )

{

(

)


Evidence i a l.jpg
Evidence I(a)

  • Distribution of Favourites ( 11 horse race )


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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 )


Evidence i b l.jpg
Evidence I(b)

  • Distribution of Favourite ( 11 horse race )


Evidence i c l.jpg
Evidence I(c)

  • Distribution of 2nd Favourite ( 11 horse race )


Evidence i d l.jpg
Evidence I(d)

  • Distribution of 3rd Favourite ( 11 horse race )


Evidence i e l.jpg
Evidence I(e)

  • Distribution of 4th - 7th Favourites ( 11 horse race )


Evidence i f l.jpg
Evidence I(f)

  • Distribution of 8th - 11th Favourites ( 11 horse race )


A strategy l.jpg
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



Evidence ii b l.jpg
Evidence II(b)

  • Long Favourite - minimum 20% outlay



Evidence ii d l.jpg
Evidence II(d)

  • Long Favourite - minimum 20% outlay


Evidence ii e l.jpg
Evidence II(e)

  • Long Favourite - minimum 30% outlay


Evidence ii f l.jpg
Evidence II(f)

  • Long Favourite - minimum 10% outlay


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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


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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


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