# A Method Of Making Parimutuel Odds For Thoroughbred Horse Races

Donald A. Swanson

This work describes a subjective method of constructing an odds line from the analysis of a thoroughbred horse race. It uses generic symbols matched up with Fibonacci numbers to solve the problem of computational stabilization and the favorite-longshot bias. Symbols are chosen by the handicapper to represent the race analysis. Fibonacci numbers are used to convert symbol combinations into weighted percentages.

## 1. Symbols - Absolute And Relative Analysis

The leftmost column in figure 1 shows the seven symbols used in the method. There are three one-part symbols and four two-part symbols. The left side symbol part is the "base" (+, N, Ø). The right side symbol parts (+, -) "modify" the base. Word descriptions are names for the symbols. Each symbol is matched up with a Fibonacci number. Symbols are hand-written with a forward slash "/" as a separator, for example: N- / ++ / Ø+. Memorize the symbols and word descriptions by writing them down a few times.

Figure 1.
Symbol Word Description Weight Number
++ double plus 21 7
+ plus 13 6
N+ neutral plus 8 5
N neutral 5 4
N- neutral minus 3 3
Ø+ doubtful plus 2 2
Ø doubtful 1 1

There are two handicapping analysis perspectives affecting symbol selection:

1. Absolute Analysis - Evaluates a single horse in isolation or against todays race conditions. Symbol selection starts with consideration of +, N, Ø before moving up or down to the appropriate symbol. The handicapper should mentally move toward N when uncertain.
2. Relative Analysis - Compares two or more horses against each other. In figure 1 rightmost column the symbols are numbered 1 to 7. The difference between the symbols is the degree of certainty that one horse will outperform the other(s). The range is 0 to 6 degrees. Symbol selection starts with consideration of N versus N before expanding outward. For example: speed ratings of 90 and 92 for two horses might get N, N or N, N+ or N-, N+ with degrees of certainty zero, one, and two. If the speed ratings are 90 and 98 then the two horses might get Ø, +. If the speed ratings are 90, 92, 98 then the three horses might get Ø, Ø+, + with total degrees 1 + 4 = 5.

## 2. Contender Set Percentage (cp) - Factor Symbols

The relative analysis of contenders versus non-contenders is + versus Ø.

Four win contenders in an eight horse field (4 / 8).
+ / + / + / + / Ø / Ø / Ø / Ø
13 + 13 + 13 + 13 + 1 + 1 + 1 + 1 = 52 + 4 = 56
cp = 52 / 56 = .929

nc - number of contenders
fs - field size

cp = (13 * nc) / ((13 * nc) + (fs - nc))

Races with two or more contenders use three factors arranged from left to right in order of importance. Factors can be two-part compounded, for example: class-speed, form-age, or distance-pedigree. Factor names are written in abbreviated form with a separator slash, for example: cls-sp / fm-age / dst-ped.

The N symbol can be extended to NN. The symbols with a neutral base are N+, NN, N-. The factor symbols are the right side symbol parts (+, N, -) matched up with Fibonacci numbers 8, 5, 3 respectively. Figure 2 shows the six combinations of factor symbols and word descriptions which must be memorized. Factor symbols are written plain without the separator slash.

Figure 2.
Symbols Word Description
N  N  N neutral
+  N  N one plus
N  N  - one minus
+  N  - one plus one minus
+  -  - one plus two minus
+  +  - two plus one minus

Symbol combination selection usually starts with consideration of the "one plus" subset: (+ N N), (+ N -), (+ - -).

Factors can also be doubled, for example: speed, speed, distance-pedigree. There are two rules for selecting symbol combinations with doubled factors:

1. The symbol must be the highest weighted one in the combination.
2. The symbol must be duplicated.

The doubled factor subset: (N N N), (N N -), (+ + -).

## 3. Example Race Calculation

Two contenders in an eight horse field (2 / 8).

cp = (13 * 2) / ((13 * 2) + (8 - 2)) = .813

The three factors chosen are class, form-surface, distance.
The (+ N -) combination generally works well.

sf = 8 + 5 + 3 = 16        // sum of factor weights
f1 = 8 / sf = .5
f2 = 5 / sf = .3125
f3 = 3 / sf = .1875
f1pct = f1 * cp = .407     // factor percentage
f2pct = f2 * cp = .254
f3pct = f3 * cp = .152

Each contender gets three symbols one for each factor as shown in figure 3 center column.

Horse Symbols Weights 2 / 8 cls / fm-sf / dst +  N  - #1 +  /  N  /  N 13 / 5 / 5 #2 N- /  N+  / + 3 / 8 / 13

Calculate the weighted percentage then multiply by the factor percentage.

13 + 3 = 16                // factor 1 sum
(13 / 16)  * .407 = .331   // #1 (weight/sum) * f1pct
(3 / 16) * .407 = .076     // #2

5 + 8 = 13                 // factor 2 sum
(5 / 13) * .254 = .098     // #1 (weight/sum) * f2pct
(8 / 13)  * .254 = .156    // #2

5 + 13 = 18                // factor 3 sum
(5 / 18) * .152 = .042     // #1 (weight/sum) * f3pct
(13 / 18) * .152 = .110    // #2

Sum the three percentages for each contender.

.331 + .098 + .042 = .471  // #1
.076 + .156 + .110 = .342  // #2

Sort the final percentages in descending order.

 2 / 8 cls / fm-sf / dst +  N  - #1 +  /  N  /  N .471 #2 N- /  N+  / + .342

## References

1. Cramer M. (1987) The Odds On Your Side: The Logic Of Racetrack Investing, Cynthia Publishing Company
2. McNeill D. & Freiberger P. (1993) Fuzzy Logic: The Discovery Of A Revolutionary Computer Technology And How It Is Changing Our World, Simon & Schuster
3. Mitchell D. (1988) Winning Thoroughbred Strategies: With The Right Strategy, You Can Think Like An Investor, Not A Gambler!, William Morrow & Company
4. Quinn J. (1987) Class Of The Field: New Performance Ratings For Thoroughbreds, William Morrow & Company
5. Quirin W.L. (1979) Winning At The Races: Computer Discoveries In Thoroughbred Handicapping, William Morrow & Company
6. Scott W.L. (1984) How Will Your Horse Run Today?, Amicus Press
7. Scott W.L. (1989) Total Victory At The Track: The Promise And The Performance, Liberty Publishing Company

## Appendix 1. - Objective Factors For Calculating Odds

Scaling percentage to symbol numbers 1 To 7.

pct - percentage from 0 to 1
conv - converted into a 1 to 7 number

conv = int((((pct * 100) + 16.667) / 16.667) + .5)

Win/place percentage.

nc - number of entrants or contenders
st() - number of starts this year plus last year
wp() - number of wins and places this year plus last year
wt() - Fibonacci numbers 1,2,3,5,8,13,21

for i = 1 to nc
wp(i) = wp(i) / st(i)
conv = int((((wp(i) * 100) + 16.667) / 16.667) + .5)
wp(i) = wt(conv)
next i

Earnings per start.

en() - earnings this year plus last year
(earnings are rounded to the nearest thousand)
eps() - earnings per start (capped at USD-180k)
high - highest value found

cap = 180
high = 0
for i = 1 to nc
eps(i) = en(i) / st(i)
if eps(i) > cap then eps(i) = cap
if eps(i) > high then high = eps(i)
next i
for i = 1 to nc
eps(i) = eps(i) / high
conv = int((((eps(i) * 100) + 16.667) / 16.667) + .5)
eps(i) = wt(conv)
next i

Recent representative speed or pace figure.

scale - scaling increment
fg() - speed or pace figure

scale = 2.0
high = 0
for i = 1 to nc
if fg(i) > high then high = fg(i)
next i
offset = (high / scale) - 7
for i = 1 to nc
fg(i) = int(((fg(i) / scale) - offset) + .5)
if fg(i) < 1 then fg(i) = 1
fg(i) = wt(fg(i))
next i

## Appendix 2. - Reverse Reading The Two-Part Symbols

Symbol Word Description
+N positive neutral
-N negative neutral
positive doubtful

Reversing the two-part symbols can be helpful in situations with unknowns or application of historical "angles". A developing horse might be taking a large rise or drop in class or trying a new distance or surface. Symbol selection starts with consideration of base positive or negative. Analysis of available information for todays race will determine whether a positive reverse read should be modified neutral or doubtful.

## Appendix 3. - Converting Percentages Into Truncated Fractional Odds

This ABC function named "truncate" converts a percentage value (pct) into fractional odds. The truncate increment (inc) starts at 20 which rounds down to the nearest 1 / 20th. If none of the tests are passed then the increment is cut in half. The numerator and denominator are calculated in the compound return statement. The function is demonstrated with the "convert to odds" command. The (x / y) = int(x / y) test can substitute for (x mod y = 0).

HOW TO RETURN truncate pct:
PUT (1 / pct) - 1 IN odds
PUT 20, 0 IN inc, dn
WHILE dn = 0:
PUT (floor(odds * inc)) / inc IN odds
SELECT:
odds mod 1 = 0:
PUT 1 IN dn
odds < 5 AND odds mod .5 = 0:
PUT .5 IN dn
odds < 2 AND odds mod .2 = 0:
PUT .2 IN dn
odds < 1 AND odds mod .1 = 0:
PUT .1 IN dn
odds < 1 AND odds mod .25 = 0:
PUT .25 IN dn
odds < .25:
PUT .05 IN dn
ELSE:
PUT floor(inc / 2) IN inc
RETURN odds / dn, 1 / dn

HOW TO CONVERT TO ODDS:
PUT .95 IN pct
WHILE pct > .01:
PUT truncate pct IN n, d
WRITE pct, "  ", n, "/", d/
PUT pct - .05 IN pct

>>> CONVERT TO ODDS
0.95    1 / 20
0.90    1 / 10
0.85    3 / 20
0.80    1 / 4
0.75    3 / 10
0.70    2 / 5
0.65    1 / 2
0.60    3 / 5
0.55    4 / 5
0.50    1 / 1
0.45    6 / 5
0.40    3 / 2
0.35    9 / 5
0.30    2 / 1
0.25    3 / 1
0.20    4 / 1
0.15    5 / 1
0.10    9 / 1
0.05    19 / 1
>>> ?

# A Method Of Measuring Positional Bias In Thoroughbred Horse Races

Donald A. Swanson

This method uses second call positions of race winners to determine positional bias at a given distance. Second call position and field size data are retrieved from the result charts. The four furlong position is used for sprints. The six furlong position is used for routes. Calculate figures for each distance and surface.

repeat
input 2nd call position (c2)
input field size (fs)
list(n) = (c2 / fs) * 10
avg1 = avg1 + list(n)
until done
average the entire list
avg1 = int((avg1 / n) + .5)
sort the list (ascending)
average the low third of the list
lt = int((n / 3) + .5)
avg2 = list item sum from 1 to lt
avg2 = int((avg2 / lt) + .5)
average the high third of the list
ht = (n - lt) + 1
avg3 = list item sum from ht to n
avg3 = int((avg3 / lt) + .5)
display as (avg2,avg3)avg1 for example: (3,8)5

Interpret the figures using the table below.

(+) = early  (N) = neutral  (Ø) = late
Low Third High Third Average
1 (+)
2 (Ø)
3 (ØØ)
4 (++)
5 (+)
6 (Ø)
1 (++)
2 (+)
3 (N+)
4 (N-)
5 (Ø)
6 (ØØ)

## References

1. Brohamer T. (1991) Modern Pace Handicapping, William Morrow & Company.
2. Scott W.L. (1989) Total Victory At The Track: The Promise And The Performance, Liberty Publishing Company.

das60358@comcast.net
Huntsville, Alabama USA
Modified: Nov 30, 2013