Introduction to Hedging and Arbitrage
maxwellAI
19/05/2016
Introduction
There are already some really great articles ([1], [2], [3]) which explain (sports bet) hedging and arbitrage in detail, and the point of this post is not to re-write these articles, but rather to:
briefly introduce the concept/s and provide example/s of opportunities I have had for hedging in my own bet placement this year
demonstrate what opportunities have existed (for hedging) in the 2016 NRL season to date and how to identify them
calculate (maximum) theoretical profit which could have been made by taking advantage of hedging
This post will focus more so on hedging over arbitrage, however the calculations/concepts are very similar.
Sports bet Hedging and Arbitrage
Hedging
Definition
Sports bet hedging is the process of placing multiple non-similtanuoes bets in the opposite direction on dependent events to guarantee profits (or mitigate loss) [4]. Opportunity for hedging usually arises in the event of odds shifting over time. An example is initially placing a bet on a team to win an NRL match at long odds, and after an odds shift, subsequently placing a bet on the opposing team to also win. Dependent on the size of the odds shift hedging can guarantee a profit no matter the outcome because at least one of the teams must win.
The main point of difference to arbitrage is that odds shifts are assumed or unknown and an initial stake may already be placed on one outcome. In arbitrage similtaneous or near similtaneous stakes are placed to guarantee profits.
Real world example
The most basic (and real) example follows:
In round 9 of the 2016 NRL season, I put a $100 stake on St George (at 2.45 odds) to defeat the Warriors (who were $1.65 favorites). At the time I placed the bet it was about four days before the actual match was to take place. Two days before the match took place, the Warriors coach announced that he was dropping six players from the squad and replacements would only be announced immediately before the match. This caused the odds for Warriors to shift dramatically and the odds (just before the game commenced) were 1.65 for St George and 2.39 for the Warriors. So in this case the odds almost completely flipped and St George became favorites.
This presented me with a hedging opportunity. This is because if I now placed a specified stake on the Warriors to win I would be gaurenteed profit no matter who actually won.
Lets just do the quick calculation. First I staked $100 on St George to win at 2.45 odds. Now I see that just before the match, the Warriors are paying $2.39. I know this is a hedging opportunity and so lets just say I also stake $100 on Warriors to win. This means we have staked $200 in total.
The return I will get if St George win = odds x initial stake - total stake
Which is: 2.45 x $100 - $200 = $45
The return I will get if Warriors win is = 2.39 x $100 - $200 = $39
So you can see if I did this I would have been guaranteed to either get a $45 profit if St George won, or a $39 profit if Warriors won!!
Of course the actual stake I put on the Warriors does matter which is why I said we need to put a specified stake on the Warriors. If you want to guarantee you get the exact same profit no matter who wins, the calculation to determine how much I should stake (called an unbiased stake) on Warriors is:
Stake on St George x (odds for St George/odds for Warriors )
100 x (2.45/2.39) = $102.51
If we just re-do the return calculation/s above using a stake of $102.51 for the Warriors and $202.51 as the total stake you will notice that the result is $42.49 for each. So now I have a gaurenteed profit of $42.49 no matter who wins!!
You might think this is pretty incredible and a no brainier to take the guaranteed profit, however in this case I did not!. Again you need to way up the risk vs reward. In this example my predictive model indicated St George had a 73% chance of winning, Warriors were down 6 players and I stood to profit $145 as opposed to the gaurenteed profit of $42.49. What would have you done???
Determining hedging oppertunities and formulas
I thought I would start with a real world example before I jumped into the formulas, but if you want to determine if an opportunity for hedging exists, we need to determine the margin and/or profit margin of the outcome/s. Intuitively if the profit margin is positive then a hedging opportunity exists and additionally indicates how much guaranteed profit you will make.
Given:
o1 = Decimal odds of initial stake
o2 = Decimal odds of the dependent outcome to compare
The margin (m) = 1/o1 + 1/o2
The profit margin (p) = 1/m - 1
If the margin is less than 1 or (more intuitively), if the profit margin is greater than zero, then a hedging opportunity exists. The profit margin is the amount of profit you will receive if you place an unbiased stake on the dependent outcome. We can check this by again using our real world example.
o1 = 2.45 (odds of initial stake on St George)
o2 = 2.39 (odds of Warriors at odds close (IE just before the game))
m = 1/2.45 + 1/2.39 = 0.83
p = 1/0.83 - 1 = 0.21
We observe that the margin is less than 1 and the profit margin is positive. We observe that a hedging opportunity exists and we expect a 21% profit on our total stake if we choose to take the hedging opportunity. Lets confirm with our real example:
Recall if I stake $100 on St George at 2.45 and 102.51 on Warriors at 2.39 I stand to profit $42.49 no matter the outcome. $42.49 as a percentage of the total stake (t) ($202.51) = 0.21 which is the same as our calculated profit margin!
For completeness the formula for determining the amount to stake on the dependent outcome is:
Given
s1 = initial stake
s2 = amount to stake on the dependent outcome
o1 = Decimal odds of initial stake
o2 = Decimal odds of the dependent outcome to compare
s2 = s1 x (o1/o2)
In our real world example:
s1 = initial stake = $100
s2 = amount to stake on the Warriors
o1 = 2.45 (odds of initial stake on St George)
o2 = 2.39 (odds of Warriors at odds close (IE just before the game))
s2 = 100*(2.45/2.39) = 102.51
Hedging oppertunities in the 2016 NRL season to date
Before I go on I must stress that hedging opportunities are not known (although might be predicted) at the time of initial stake. This is the point of difference to arbitrage. If an odds difference exists in which you can simultaneously place bets (say at different bookmakers) to guarantee profit this is arbitrage bet placement. If the opportunity to guarantee profit arises due to odds shift over time by placing opposing bets (for example placing a bet that both team/s will win) on dependent events, then this is hedging. Therefore when I determine the hedging opportunities which have existed in the NRL this year, it doesn’t mean that we were aware of these opportunities, it just highlights how many have existed and how large the opportunities were.
If you are placing a bet on a match in the hope that you will get a hedging opportunity then you still have to have some way of forecasting (predicting) the probability that this will happen!!
OK, so lets take a look at the hedging opportunities (and their profit margin size) which have existed this year. To do this we need to have a data set which contains odds shift details (such as min max, open close odds) of each match.
You can compile this yourself (from historic odds websites) or refer to some pre-processed generally available data sets. I am going to utilize a pretty decent (and generally maintained) data set which is posted on Aussportsbetting.com. In accordance to the terms of use of this data-set I can only provide a link to the website and cannot provide or supply this data-set directly.
The data-set contains basic match details such as the home/away team, the scores, and importantly (averaged) opening and closing odds, and min/max odds. The odds details are mainly compiled from the OddsPortal.
We are reviewing matches up to round 9 which represents 72 matches.
To use this data set what I am going to do is assume a couple of scenarios:
The first scenario assumes that we have staked on a team as soon as odds have opened up and see how often a hedging opportunity has come up at odds close (just before match start). This represents my real world example.
The second scenario takes the best (maximum) odds for a team and compares the opposing teams odds at close. This represents a scenario when you have been ‘lucky’ enough to grab the odds for a team when they have been highest and should represent the theoretical maximum hedging profit margin.
To analyse these scenarios we need to:
Determine the margin and profit margin of the opening odds of home teams and the closing odds of away teams
Determine the margin and profit margin of the opening odds of away teams and the closing odds of home teams
Subset the data on any matches where the profit margin was >0
Identify the best historic hedging opportunities and determine the total potential profit
Complete the same exercise using maximum odds
There is one final scenario which would potentially present more hedging opportunities - that is odds movement during live matches. Since I don’t have the details of in-game odds movement for all matches I could not present these results but suspect that there may be more (and higher profit margin) opportunities.
OK, lets take a look at all matches where the profit margin was positive when comparing home team opening odds against away team closing odds (ordered by highest to lowest profit margin):
| Date | Home.Team | Away.Team | Home.Score | Away.Score | Home.Odds.Open | Away.Odds.Close | Profit.Margin |
|---|---|---|---|---|---|---|---|
| 2016-03-12 | Canberra Raiders | Sydney Roosters | 21 | 20 | 2.03 | 2.15 | 0.044 |
| 2016-03-19 | Gold Coast Titans | Wests Tigers | 30 | 18 | 2.16 | 2.02 | 0.044 |
| 2016-03-06 | Sydney Roosters | South Sydney Rabbitohs | 10 | 42 | 2.16 | 1.95 | 0.025 |
| 2016-03-12 | Parramatta Eels | North QLD Cowboys | 20 | 16 | 2.47 | 1.74 | 0.021 |
| 2016-05-01 | Gold Coast Titans | Melbourne Storm | 0 | 38 | 2.31 | 1.81 | 0.015 |
| 2016-04-02 | Wests Tigers | Cronulla Sharks | 26 | 34 | 2.30 | 1.81 | 0.013 |
| 2016-03-26 | Sydney Roosters | Manly Sea Eagles | 20 | 22 | 2.40 | 1.75 | 0.012 |
| 2016-03-06 | Gold Coast Titans | Newcastle Knights | 30 | 12 | 2.10 | 1.95 | 0.011 |
| 2016-03-14 | Wests Tigers | Manly Sea Eagles | 36 | 22 | 2.58 | 1.66 | 0.010 |
| 2016-04-28 | South Sydney Rabbitohs | Wests Tigers | 22 | 30 | 1.40 | 3.52 | 0.002 |
We observe that there are 10 instances where if we placed a stake on the home team when odds first opened we would have been presented with a hedging opportunity. We notice though that aside from the top two hedging opportunists (4.4%), the majority of present a potential profit margin of less than 2% or close to break even.
Using the top match (Canberra v Roosters) as an example, the way to read this result is; If I placed a stake on Canberra (home) to win when the odds first opened at 2.03, and then placed a subsequent specified stake on the Roosters at odds close (2.15), I could guarantee myself a profit of 4.4%.
Lets quickly check the math. I know the profit margin is 4.4% and so this is the amount I expect on top of my total stake (t):
The profit margin (p) = 1/m - 1
= 1/(1/2.03+ 1/2.15) - 1
=0.044
To guarantee the 4.4% profit I need to determine the amount to stake on Roosters. Lets assume I have already staked $100 on Canberra, the amount I will need to stake on Roosters to get the 4.4% profit margin is $94.42:
s2 = s1 x (o1/o2)
= 100 * (2.03/2.15)
= $94.42
Now now matter if Canberra or Roosters win I get a return of $8.58:
If Canberra win: 2.03*100 - 194.42= 8.58
If Roosters win: 2.15*94.42 - 194.42 = 8.58
8.58 as a percentage of 194.42 (8.58/194.42) equals 4.4%
Now lets take a look at all matches where the profit margin was positive comparing away team opening odds and home team closing odds.
| Date | Home.Team | Away.Team | Home.Score | Away.Score | Away.Odds.Open | Home.Odds.Close | Profit.Margin |
|---|---|---|---|---|---|---|---|
| 2016-05-01 | New Zealand Warriors | St George Dragons | 26 | 10 | 2.31 | 2.39 | 0.175 |
| 2016-04-29 | Parramatta Eels | Canterbury Bulldogs | 20 | 12 | 1.98 | 2.36 | 0.077 |
| 2016-04-17 | Wests Tigers | Melbourne Storm | 18 | 19 | 1.77 | 2.74 | 0.075 |
| 2016-03-20 | New Zealand Warriors | Melbourne Storm | 14 | 21 | 2.05 | 2.11 | 0.040 |
| 2016-04-04 | Canterbury Bulldogs | Canberra Raiders | 8 | 22 | 3.03 | 1.56 | 0.030 |
| 2016-03-13 | Melbourne Storm | Gold Coast Titans | 34 | 16 | 5.33 | 1.26 | 0.019 |
| 2016-04-17 | Canberra Raiders | Cronulla Sharks | 16 | 40 | 2.09 | 1.97 | 0.014 |
| 2016-03-13 | Cronulla Sharks | St George Dragons | 30 | 2 | 2.63 | 1.65 | 0.014 |
| 2016-04-30 | Penrith Panthers | Canberra Raiders | 19 | 18 | 2.58 | 1.67 | 0.014 |
| 2016-04-09 | New Zealand Warriors | Manly Sea Eagles | 18 | 34 | 3.02 | 1.52 | 0.011 |
| 2016-03-26 | Canberra Raiders | Gold Coast Titans | 20 | 24 | 2.97 | 1.53 | 0.010 |
| 2016-03-27 | St George Dragons | Penrith Panthers | 14 | 12 | 2.07 | 1.97 | 0.009 |
| 2016-04-03 | Parramatta Eels | Penrith Panthers | 18 | 20 | 2.79 | 1.58 | 0.009 |
| 2016-04-25 | Newcastle Knights | Manly Sea Eagles | 10 | 26 | 1.50 | 3.08 | 0.009 |
| 2016-04-02 | Melbourne Storm | Newcastle Knights | 18 | 14 | 6.31 | 1.20 | 0.008 |
| 2016-04-09 | Penrith Panthers | North QLD Cowboys | 18 | 23 | 1.68 | 2.52 | 0.008 |
| 2016-04-30 | Manly Sea Eagles | North QLD Cowboys | 18 | 34 | 1.34 | 4.05 | 0.007 |
| 2016-04-10 | Newcastle Knights | Wests Tigers | 18 | 16 | 1.81 | 2.25 | 0.003 |
| 2016-04-02 | North QLD Cowboys | St George Dragons | 36 | 0 | 4.59 | 1.28 | 0.001 |
This list of opportunities is a bit longer (19) purely because away teams tend to start a longer odds. You can see that The St George v Warriors game is by far and away the largest hedging opportunity and has been for the entire season. The profit margin (17.5%) is slightly lower than in my real world example simply because I had higher odds at time of stake, where-as the odds in the table represent the average across bookmakers. We can also see a couple of matches in would have returned 7.7% and 7.5% respective while the remainder would have returned < 4%.
Finally lets just have a quick look at the maximum theoretical hedging opportunity/s. To do this we are just running the same exercise as above but we are looking at the maximum odds of a home/away team as compared to the closing closing odds of the home/away team. This represents our maximum hedging opportunity assuming we are going to hedge on a game just before it starts:
First the maximum odds of a home team vs the close odds of an away team:
| Date | Home.Team | Away.Team | Home.Score | Away.Score | Home.Odds.Max | Away.Odds.Close | Profit.Margin |
|---|---|---|---|---|---|---|---|
| 2016-04-29 | Parramatta Eels | Canterbury Bulldogs | 20 | 12 | 2.79 | 1.67 | 0.045 |
| 2016-03-12 | Canberra Raiders | Sydney Roosters | 21 | 20 | 2.03 | 2.15 | 0.044 |
| 2016-03-19 | Gold Coast Titans | Wests Tigers | 30 | 18 | 2.16 | 2.02 | 0.044 |
| 2016-03-31 | Manly Sea Eagles | South Sydney Rabbitohs | 12 | 16 | 2.98 | 1.59 | 0.037 |
| 2016-04-17 | Canberra Raiders | Cronulla Sharks | 16 | 40 | 2.24 | 1.93 | 0.037 |
| 2016-05-01 | Gold Coast Titans | Melbourne Storm | 0 | 38 | 2.42 | 1.81 | 0.036 |
| 2016-03-19 | Newcastle Knights | Canberra Raiders | 24 | 24 | 2.67 | 1.69 | 0.035 |
| 2016-03-20 | St George Dragons | South Sydney Rabbitohs | 8 | 6 | 3.45 | 1.47 | 0.031 |
| 2016-03-12 | Parramatta Eels | North QLD Cowboys | 20 | 16 | 2.50 | 1.74 | 0.026 |
| 2016-03-06 | Sydney Roosters | South Sydney Rabbitohs | 10 | 42 | 2.16 | 1.95 | 0.025 |
| 2016-03-26 | Sydney Roosters | Manly Sea Eagles | 20 | 22 | 2.45 | 1.75 | 0.021 |
| 2016-03-05 | North QLD Cowboys | Cronulla Sharks | 20 | 14 | 1.47 | 3.31 | 0.018 |
| 2016-03-21 | Manly Sea Eagles | Cronulla Sharks | 22 | 12 | 2.11 | 1.96 | 0.016 |
| 2016-04-07 | Brisbane Broncos | St George Dragons | 26 | 0 | 1.20 | 6.52 | 0.013 |
| 2016-03-10 | Penrith Panthers | Canterbury Bulldogs | 16 | 18 | 2.53 | 1.69 | 0.013 |
| 2016-04-02 | Wests Tigers | Cronulla Sharks | 26 | 34 | 2.30 | 1.81 | 0.013 |
| 2016-03-04 | Manly Sea Eagles | Canterbury Bulldogs | 6 | 28 | 1.67 | 2.57 | 0.012 |
| 2016-04-28 | South Sydney Rabbitohs | Wests Tigers | 22 | 30 | 1.42 | 3.52 | 0.012 |
| 2016-03-06 | Gold Coast Titans | Newcastle Knights | 30 | 12 | 2.10 | 1.95 | 0.011 |
| 2016-03-14 | Wests Tigers | Manly Sea Eagles | 36 | 22 | 2.58 | 1.66 | 0.010 |
| 2016-04-08 | South Sydney Rabbitohs | Sydney Roosters | 10 | 17 | 1.54 | 2.93 | 0.009 |
| 2016-04-17 | Wests Tigers | Melbourne Storm | 18 | 19 | 2.97 | 1.52 | 0.005 |
And finally the maximum odds of an away team vs the closing odds of a home team:
| Date | Home.Team | Away.Team | Home.Score | Away.Score | Away.Odds.Max | Home.Odds.Close | Profit.Margin |
|---|---|---|---|---|---|---|---|
| 2016-05-01 | New Zealand Warriors | St George Dragons | 26 | 10 | 2.33 | 2.39 | 0.180 |
| 2016-04-29 | Parramatta Eels | Canterbury Bulldogs | 20 | 12 | 2.05 | 2.36 | 0.097 |
| 2016-04-17 | Wests Tigers | Melbourne Storm | 18 | 19 | 1.77 | 2.74 | 0.075 |
| 2016-03-25 | South Sydney Rabbitohs | Canterbury Bulldogs | 12 | 42 | 1.75 | 2.78 | 0.074 |
| 2016-03-27 | St George Dragons | Penrith Panthers | 14 | 12 | 2.32 | 1.97 | 0.065 |
| 2016-04-10 | Newcastle Knights | Wests Tigers | 18 | 16 | 2.01 | 2.25 | 0.062 |
| 2016-03-20 | New Zealand Warriors | Melbourne Storm | 14 | 21 | 2.11 | 2.11 | 0.055 |
| 2016-04-25 | Newcastle Knights | Manly Sea Eagles | 10 | 26 | 1.58 | 3.08 | 0.044 |
| 2016-04-09 | Penrith Panthers | North QLD Cowboys | 18 | 23 | 1.77 | 2.52 | 0.040 |
| 2016-03-20 | St George Dragons | South Sydney Rabbitohs | 8 | 6 | 1.61 | 2.91 | 0.037 |
| 2016-04-25 | Melbourne Storm | New Zealand Warriors | 42 | 0 | 2.63 | 1.70 | 0.033 |
| 2016-04-04 | Canterbury Bulldogs | Canberra Raiders | 8 | 22 | 3.03 | 1.56 | 0.030 |
| 2016-04-30 | Manly Sea Eagles | North QLD Cowboys | 18 | 34 | 1.38 | 4.05 | 0.029 |
| 2016-04-23 | Canberra Raiders | Wests Tigers | 60 | 6 | 3.35 | 1.48 | 0.027 |
| 2016-04-30 | Penrith Panthers | Canberra Raiders | 19 | 18 | 2.66 | 1.67 | 0.026 |
| 2016-03-17 | North QLD Cowboys | Sydney Roosters | 40 | 0 | 5.02 | 1.28 | 0.020 |
| 2016-03-13 | Cronulla Sharks | St George Dragons | 30 | 2 | 2.67 | 1.65 | 0.020 |
| 2016-03-13 | Melbourne Storm | Gold Coast Titans | 34 | 16 | 5.33 | 1.26 | 0.019 |
| 2016-04-17 | Canberra Raiders | Cronulla Sharks | 16 | 40 | 2.11 | 1.97 | 0.019 |
| 2016-03-28 | Wests Tigers | Parramatta Eels | 0 | 8 | 1.70 | 2.54 | 0.018 |
| 2016-03-18 | Canterbury Bulldogs | Parramatta Eels | 6 | 20 | 2.66 | 1.65 | 0.018 |
| 2016-04-09 | New Zealand Warriors | Manly Sea Eagles | 18 | 34 | 3.04 | 1.52 | 0.013 |
| 2016-04-01 | Gold Coast Titans | Brisbane Broncos | 16 | 24 | 1.26 | 5.16 | 0.013 |
| 2016-03-26 | Canberra Raiders | Gold Coast Titans | 20 | 24 | 2.97 | 1.53 | 0.010 |
| 2016-03-06 | Gold Coast Titans | Newcastle Knights | 30 | 12 | 2.09 | 1.95 | 0.009 |
| 2016-04-03 | Parramatta Eels | Penrith Panthers | 18 | 20 | 2.79 | 1.58 | 0.009 |
| 2016-04-02 | Melbourne Storm | Newcastle Knights | 18 | 14 | 6.31 | 1.20 | 0.008 |
| 2016-04-24 | Cronulla Sharks | Penrith Panthers | 20 | 18 | 3.05 | 1.50 | 0.005 |
| 2016-04-02 | North QLD Cowboys | St George Dragons | 36 | 0 | 4.65 | 1.28 | 0.004 |
| 2016-03-28 | Cronulla Sharks | Melbourne Storm | 14 | 6 | 2.27 | 1.79 | 0.001 |
As expected the hedging opportunities are a little more prevalent when we compare maximum odds against closing odds, and observe that maximum away team odds vs closing home team odds present the most opportunities (30).
Conclusion
Now that we have looked at the different scenarios we can conclude that:
Of 72 matches to date, the maximum theoretical number of (positive profit margin) hedging opportunities which were available was 30 (41% of matches).
The number of hedging opportunities which have arisen from comparing opening odds of away teams against closing odds of home teams is 19 (26% of matches).
Hedging opportunists rarely return more than 4% profit
St George vs Warriors has been the best hedging opportunity of the season at 18% profit margin (using bookmakers averages)
Hedging is opportunistic and while the opportunities do arise, they may still be unpredictable (but you can keep an eye out for them!)
Tables and graphs to assist in intepreting hedging oppertuities
OK, so now we have identified the opportunities which have existed and we understand how to calculate the potential profit margins; how do you easily keep an eye on any hedging opportunities after you have put down a stake?
Probably the easiest way to do this is to visualize in a graph and/or refer to a table rather than scribbling down calculations all the time. If you are code eccentric, I’ll show you how to quickly whip these visualizations up in R programming language (otherwise just keep my table/graph handy).
The first table/graph we will look at is a break even hedging opportunity. This is our baseline. A break even hedging opportunity is one where; if you stake a specified amount on the opposing team you are gaurenteed to break even. You might want to do this if you fear that the team you placed (maybe a large) bet on really doesn’t stand a chance to win and so you are looking to ‘opt out’ without loosing money (if given the opportunity)
Lets quickly review a few formulas so we know how to calculate this. What we want to calculate is the o2 under a break even scenario (margin = 1, profit margin = 0)
We know that given:
o1 = Decimal odds of initial stake
o2 = Decimal odds of the dependent outcome to compare
The margin (m) = 1/o1 + 1/o2 and the profit margin (pm) = 1/m - 1
Therefore to calculate o2 for m=1 we re-arrange the formula/s so that:
o2 = o1/(o1*m-1)
Or if we want to know o2 given a desired profit margin:
o2 = o1/((1/pm+1)-1)
So for example if we have put down a stake at odds of 1.90 we would need to wait until the opposing teams odds are at least 2.11 to be able to hedge to break even
o2 = 1.90/(1.90*1-1) = 2.11
We can quickly and easily generate a reference table using this knowledge. All we will do is generate a table showing the odds ‘o1’ of the team we may have bet on and the corresponding value of the the opposing odds ‘o2’. The value of ‘o2’ is what the odds must be on the opposing team in order to break even when hedging:
| o1 | o2 | m | pm |
|---|---|---|---|
| 1.3 | 4.33 | 1 | 0 |
| 1.4 | 3.50 | 1 | 0 |
| 1.5 | 3.00 | 1 | 0 |
| 1.6 | 2.67 | 1 | 0 |
| 1.7 | 2.43 | 1 | 0 |
| 1.8 | 2.25 | 1 | 0 |
| 1.9 | 2.11 | 1 | 0 |
| 2.0 | 2.00 | 1 | 0 |
| 2.1 | 1.91 | 1 | 0 |
| 2.2 | 1.83 | 1 | 0 |
| 2.3 | 1.77 | 1 | 0 |
| 2.4 | 1.71 | 1 | 0 |
| 2.5 | 1.67 | 1 | 0 |
| 2.6 | 1.62 | 1 | 0 |
| 2.7 | 1.59 | 1 | 0 |
| 2.8 | 1.56 | 1 | 0 |
| 2.9 | 1.53 | 1 | 0 |
| 3.0 | 1.50 | 1 | 0 |
| 3.1 | 1.48 | 1 | 0 |
| 3.2 | 1.45 | 1 | 0 |
| 3.3 | 1.43 | 1 | 0 |
| 3.4 | 1.42 | 1 | 0 |
| 3.5 | 1.40 | 1 | 0 |
| 3.6 | 1.38 | 1 | 0 |
| 3.7 | 1.37 | 1 | 0 |
| 3.8 | 1.36 | 1 | 0 |
| 3.9 | 1.34 | 1 | 0 |
| 4.0 | 1.33 | 1 | 0 |
| 4.1 | 1.32 | 1 | 0 |
| 4.2 | 1.31 | 1 | 0 |
| 4.3 | 1.30 | 1 | 0 |
| 4.4 | 1.29 | 1 | 0 |
| 4.5 | 1.29 | 1 | 0 |
| 4.6 | 1.28 | 1 | 0 |
| 4.7 | 1.27 | 1 | 0 |
| 4.8 | 1.26 | 1 | 0 |
| 4.9 | 1.26 | 1 | 0 |
| 5.0 | 1.25 | 1 | 0 |
The way to read the table is pretty simple. o1 represents the odds of the initial stake and o2 is the odds that the opposing team must reach in order to break even in hedging. So you can see that when o1 is 1.3 the opposing team odds must reach 4.33 before you have a hedging opportunity and so on. We also include ‘m’ which is the margin, and ‘p’ which is the profit margin. Since this is the break even odds, ’p is obviously 0.
An alternative way to visualize this is in a graph. You could plot o2 against o1 and read off the o2 value required directly from the graph:
As an example you can read from the graph that when ‘o1’ is 1.5, ‘o2’ must be 3.00:
Since we might want to do better than just break even, we can expand on the above and generate a table and graph with multiple options. Lets create a table showing the odds that o2 have to be in order to generate profit margins from 0-30%
| o1 | o2pm00 | o2pm05 | o2pm10 | o2pm15 | o2pm20 | o2pm25 | o2pm30 |
|---|---|---|---|---|---|---|---|
| 1.3 | 4.33 | 5.46 | 7.15 | 9.97 | 15.60 | 32.50 | Inf |
| 1.4 | 3.50 | 4.20 | 5.13 | 6.44 | 8.40 | 11.67 | 18.20 |
| 1.5 | 3.00 | 3.50 | 4.13 | 4.93 | 6.00 | 7.50 | 9.75 |
| 1.6 | 2.67 | 3.05 | 3.52 | 4.09 | 4.80 | 5.71 | 6.93 |
| 1.7 | 2.43 | 2.75 | 3.12 | 3.55 | 4.08 | 4.72 | 5.53 |
| 1.8 | 2.25 | 2.52 | 2.83 | 3.18 | 3.60 | 4.09 | 4.68 |
| 1.9 | 2.11 | 2.35 | 2.61 | 2.91 | 3.26 | 3.65 | 4.12 |
| 2.0 | 2.00 | 2.21 | 2.44 | 2.71 | 3.00 | 3.33 | 3.71 |
| 2.1 | 1.91 | 2.10 | 2.31 | 2.54 | 2.80 | 3.09 | 3.41 |
| 2.2 | 1.83 | 2.01 | 2.20 | 2.41 | 2.64 | 2.89 | 3.18 |
| 2.3 | 1.77 | 1.93 | 2.11 | 2.30 | 2.51 | 2.74 | 2.99 |
| 2.4 | 1.71 | 1.87 | 2.03 | 2.21 | 2.40 | 2.61 | 2.84 |
| 2.5 | 1.67 | 1.81 | 1.96 | 2.13 | 2.31 | 2.50 | 2.71 |
| 2.6 | 1.62 | 1.76 | 1.91 | 2.06 | 2.23 | 2.41 | 2.60 |
| 2.7 | 1.59 | 1.72 | 1.86 | 2.00 | 2.16 | 2.33 | 2.51 |
| 2.8 | 1.56 | 1.68 | 1.81 | 1.95 | 2.10 | 2.26 | 2.43 |
| 2.9 | 1.53 | 1.65 | 1.77 | 1.91 | 2.05 | 2.20 | 2.36 |
| 3.0 | 1.50 | 1.62 | 1.74 | 1.86 | 2.00 | 2.14 | 2.29 |
| 3.1 | 1.48 | 1.59 | 1.70 | 1.83 | 1.96 | 2.09 | 2.24 |
| 3.2 | 1.45 | 1.56 | 1.68 | 1.80 | 1.92 | 2.05 | 2.19 |
| 3.3 | 1.43 | 1.54 | 1.65 | 1.77 | 1.89 | 2.01 | 2.15 |
| 3.4 | 1.42 | 1.52 | 1.63 | 1.74 | 1.85 | 1.98 | 2.10 |
| 3.5 | 1.40 | 1.50 | 1.60 | 1.71 | 1.83 | 1.94 | 2.07 |
| 3.6 | 1.38 | 1.48 | 1.58 | 1.69 | 1.80 | 1.91 | 2.03 |
| 3.7 | 1.37 | 1.47 | 1.57 | 1.67 | 1.78 | 1.89 | 2.00 |
| 3.8 | 1.36 | 1.45 | 1.55 | 1.65 | 1.75 | 1.86 | 1.98 |
| 3.9 | 1.34 | 1.44 | 1.53 | 1.63 | 1.73 | 1.84 | 1.95 |
| 4.0 | 1.33 | 1.42 | 1.52 | 1.61 | 1.71 | 1.82 | 1.93 |
We observe that when our staking odds are 1.3 that to break even the odds of o2 have to be 4.33 to break even. To generate a profit margin of 30% the odds of o2 have to be impossibly high (infinite). For a more reasonable comparison, using starting odds (o1) of 2.0, we see that the odds range for a profit margin of 0-30 are 2.00-3.71.
Finally we can plot this table data if we want a visual graph to pick values from:
Again using o1 of 2.00, we can read across the y-axis and see that to break even we need an o2 of 2.00, to generate a profit of 20% we need an o2 of 3.00 etc.
Arbitrage
As I indicated in the introduction, this article focus’ more on Hedging, and if you want more detail on arbitrage I encourage you to read any of the articles I reference in the introduction. So briefly:
Arbitrage is very similar in concept to Hedging (and in fact you can determine the margin and profit margin of an arbitrage opportunity in the exact same way), however the difference is that arbitrage opportunities occur similtaniuosly or in very close space of time. As an example:
Lets just say that you observe that a bookmaker ‘A’ has the following (even) odds for two teams:
o1: 1.9
o2: 1.9
You also see that at the same time bookmaker ‘B’ has different odds for the same team/s:
o1: 1.6
o2: 2.35
Using the Hedging opportunity table, you see that when o1 is 1.9, then you can generate a profit margin of 5% when o2 = 2.35.
Knowing this, you will observe that if I placed a stake on ‘o1’ with bookmaker ‘A’ (at 1.9 odds), and immidiatly after/ similtaneuosly also stake on ‘o2’ at Bookmaker ‘B’ then I will guarantee myself a 5% profit. I have exploited the differing odds that bookmakers have offered and guaranteed myself a profit no matter the outcome of the match!
You can double check the math:
Assume a $100 stake on o1 at Bookmaker A.
Now calculate the stake I need to place on o2 at bookmaker ‘B’ to guarantee my 5% profit
Stake 2 = $100*(1.9/2.35) = $80.85
Now determine the guaranteed profit amount:
If o1 win: 1.9*100 - 180.85 = $9.15
If o2 win: 2.35*80.85 - 180.85 = $9.15
9.15 as a percentage of 180.85 (9.15/180.85) equals 5%
So we can see that the calculations are the same as for Hedging, and you can use the tables/graphs to also identify potential arbitrage opportunities. Fair warning however, Arbitrage opportunities are even less frequent then hedging opportunities and last for a very brief period of time. In order to take advantage of Arbitrage opportunities you need accounts with very many bookmakers, be able to place bets very quickly and additionally would need some sort of ‘automated’ notification system to alert you of any opportunity which arose.
Final words
This is by no means a comprehensive article on hedging/arbitrage, and as I alluded to in the introduction presents the most basic example of hedging/arbitrage. Specifically I have not explored hedging/arbitrage on events with multiple potential outcomes (such as horse racing where any one of 10+ horses can win) and have only given simplistic examples of head to head betting where two outcomes are assumed. Also for the purposes of simplicity I have ignored (assumed it is negligible) the potential for draws in NRL matches - and one reason is that draw odds are rarely offered for head to head NRL matches. So in reality hedging on NRL games is not absolute guaranteed profit because in the event of a draw you (at best) may get your money back (or at worst lose both bets). So make sure you check the terms of your bookmaker!! Generally if a draw event is not offered for a match, then you will get your money back in the event of a draw (dead heat). So be careful if your bookmaker actually offers odds for a draw occurring.