Introduction

Tune in to Racing UK or ATR and the chances are the focus will be on picking the winner of the next race. The Racing Post has pages of form and commentary distilled into selections, naps and tips, typically resulting in one selection per race being made. Tipsters tables contain one selection per race,  Tom Segal’s Pricewise column in the Racing Post usually recommends one and occasionally two selections in a handful of Saturday races. For any gambler the key measure of success is the amount of money made or lost over a reasonable time period, and implicit in the various pieces of advice on offer is that one selection per race is the way to achieve gambling success. It seems obvious – there can only be one winner, I just need to find it! One of the consequences of making one selection per race is that you are maximising the chances of sustaining a long losing run. The volatility of your profits/losses are also maximised, as is the path dependency of your trading strategy. None of these are attractive characteristics.  Apart from the effect on your bank balance, losing runs can lead to self-doubt as methodology and existence of a trading edge are questioned, yet the length of the losing run  may be just noise, in line with what you might expect given the size of your trading edge. So what sort of losing runs might you expect given different degrees of edge over the market?  In this blog post Monte Carlo Simulation (MC) is used to compare losing runs given different degrees of trading edge and at different odds.

Methodology

A ten runner race is set up with a set of book odds where the book sums to a 7% over-round. A rating is attached to each horse, and the true odds of each horse winning is defined to be a function of the book odds and its rating. The function works so that highly rated horses have lower true odds than the book odds and vice versa for lowly rated horses. One of the parameters in the function is the degree to which the ratings have an edge over the market. The greater the edge the more the book odds are adjusted. The approach is Bayesian in nature.  The ratings used are arbitrary – they express in numerical form the the likelihood of a particular horse winning – the results presented here are not specific to the use of rating systems. Implicit in any bet placed by a gambler in a probabilistic setting is a set of underlying decisions based upon preferences or rankings that can be thought of as a set of ratings, even if they aren’t expressed as such.

Monte Carlo methods are used to run the race 30,000 times (defined as one simulation, this is equivalent to betting on 15 races a week for 40 years) using the true odds, as defined earlier, to determine the probability of each horse winning. If the winner coincides with the horse that is also top rated, the gambler wins. The book odds associated with the top rated selection and the level of edge are kept constant per simulation run. The process is repeated so that simulations are run at 4 different book odds and 4 levels of edge, to give 16 simulations in total. The book odds chosen are evens, 3/1, 6/1 and 9/1 and the levels of edge chosen to correspond to differing levels of Return on Capital (RoC) of 10%, 5% , 0% (break-even) and -7%.  The latter case represents someone with no edge whose losses over time equal the book over-round.

Relationship Between Edge,  Book Odds and True Odds

Table 1 below gives the relationship between the odds at which you back and the true odds for given levels of edge. So backing at 6/1 with a 10% edge represents true odds of 5.3/1. At a 5% edge backing a 3/1 shot represents true odds of 2.8/1, and backing an even money shot with a 10% edge has true odds of 4/5. The difference between book and true odds is small and sets the context for the analysis that follows. Whilst not the subject of this blog post, tables such as this can be used to give trigger levels at which bets become interesting for a given level of perceived trading edge.

Book Odds with 7% over-round	10% edge	5% edge	break-even	no edge
evens	0.8	0.9	1.0	1.1
3/1	2.6	2.8	3.0	3.3
6/1	5.3	5.6	6.0	6.5
9/1	8.1	8.5	8.9	9.7

Table 1: Book odds and true odds for differing levels of edge

Relationship Between Edge, Book Odds and Losing Run Length

Table 2 below gives the maximum losing run that from each simulation. The longest losing run experienced from betting at constant odds of evens with a 10% edge was 14 races, at 9/1 with a 10% edge 80 races. The reason it is often a long wait between drinks for top rated selections is the size of the trading edge compared with the odds at which horses are backed. Since the number in Table 2 represent the extreme case of the simulation, the length of losing run that occurs 5% of the time  is reported in Table 3.  Note how the length of losing run changes little with edge. If you typically bet at 6/1 and think you have a 5% edge, and you are on your 17th losing wager, there are no obvious signs from Tables 2 and 3 that you are experiencing anything other than a losing run that occurs one time in twenty. If Pricewise has a 10% edge and gives 3 selections a week all at 9/1, these results suggest that at worst  he could go half a year without a selecting a winner. Note that in practice gamblers will be betting wherever value is perceived regardless of book odds, and the fixing of odds across all simulations is artificial. However it would be straightforward to weight the results to reflect the proportion of bets you typically placed at various odds.

In finance one criteria used to judge the quality of returns delivered by investment managers is the Sharpe Ratio. This penalises returns by the volatility of the return stream. Inspection of table 3 shows that the highest Sharpe Ratio would come from betting even money shots. To emphasise, there is no suggestion that betting even money represents greater value than betting at bigger odds. The simulations are set up so the Return on Capital achieved are the same, and the value inherent in the even money shot is the same as in the 6/1 shot. However the path to terminal wealth followed by betting at evens is inherently less volatile than betting at bigger odds.

Book Odds with 7% over-round	10% edge	5% edge	break-even	no edge
evens	14	14	16	19
3/1	24	27	27	27
6/1	59	59	60	65
9/1	80	80	80	80

Table 2: Book odds and maximum losing runs  for differing levels of edge

Book Odds with 7% over-round	10% edge	5% edge	break-even	no edge
evens	3.2	3.5	3.8	4.2
3/1	8.7	9.3	9.8	10.6
6/1	16.9	17.8	18.8	20.3
9/1	25.2	26.4	27.9	;30.3

Table 3: Book odds and 95% probability losing runs

Relationship Between Edge, Book Odds and Time to Last Cumulative Loss

The results presented so far are unaffected by staking plans. In Table 4 below the number given represents the last race in the simulation at which cumulative profits are negative. This gives a sense of the number of races for the signal inherent in the edge to outweigh the noise. For level stakes betting at 6/1 with a 10% edge , profitability is always positive from the 2,324th race. Note the substantial step up in the wait for cumulative profitability when betting at 6/1 compared with 9/1 at the 5% edge level, and when betting at 3/1 compared with 6/1 at the 10% edge level.  The results highlight the increased path dependency inherent in betting at higher odds. The range of possible outcomes is such that it can take much longer to move into positive cumulative profitability.

Note that employing staking plans such as The Kelly Criterion would potentially improve this level staking result so that the month numbers were lower, particularly for the higher odds results presented, however since the cumulative profits/losses will have meandered around zero the effect on the broad thrust of the conclusion reached is likely to be small.

Book Odds with 7% over-round	10% edge	5% edge
evens	12	432
3/1	628	2,673
6/1	2,324	2,919
9/1	2,923	8,444

Table 4: Book odds and number of races and last breake-ven race

Conclusions

If you choose to bet on the horse that represents your top pick in a race, and you adopt this as a betting approach over many races, you are maximising the total profits you can expect to accrue over time. However this approach has costs associated with it. Whilst maximising expected total profits, you are also maximising both the volatility of your trading profits and exposure to path dependency.

Losing run length is primarily driven by the odds at which you back horses. It is difficult to identify that you have lost your edge in the middle of a losing run because losing run length is primarily driven by the odds at which you bet rather than the size of your betting edge. What may appear to be a loss of ability could merely be an unlucky run that is merely noise.  The reason it is often a long wait between drinks for top rated selections is the size of the trading edge compared with the odds at which horses are backed.

Betting at shorter prices minimises trading profit volatility, path dependency and reduces losing run length. Splitting your stake across more than one selection in a race will (subject to your edge being similar across all runners in a race) increase the probability that your edge will be reflected in your trading profits. These profits will not be as large as if you had made one winning selection, however what you make will be made far more often.  Betting on a number of horses in a race effectively creates one shorter priced aggregate bet. This has a number of attractive features – it reduces losing run length, reduces trading profit volatility and reduces exposure to path dependency. The cost of this approach is that over the long run total profits will be less than betting on one selection only. The trade-off between the two approaches is interesting. Given the associated drawbacks, it is surprising the one selection per race approach appears to be so little questioned and so popular.