The Church of Betting

Mixing Arbitrage and Value Betting

Mixing Arbitrage and Value Betting

If you aren’t new to this blog, by now you are probably well aware of the betting strategies called Arbitrage Betting and Value Betting. Both are potentially profitable strategies, each with its own level of potential return, volatility and risks to be aware of. At one time you have probably found yourself wondering which of the two ways to go. Today I want to present to you the options at your disposal. I will pay special attention to the option of mixing Arbitrage and Value Betting.

A hedge in the sharp bookmaker in the case of an arbitrage bet eliminates the risk of the game going the wrong way and guarantees you that your position will return a profit. This comes at a price: long-term you will lose money at the sharp bookmaker, so the hedge will lower your total ROI (return on investment) as compared to a pure Value Betting strategy.

Which one is better?

So, should you do Arbitrage or Value Betting? As I have said before, this depends entirely on your risk tolerance. The most risk-averse punter will exclusively play Arbitrage, while a return maximizing / risk-tolerant one will go for pure Value Betting. Of course, everyone is different, so it is hard to put yourself in one of those two boxes. The good news is you don’t have too. Today I talk about the middle road or rather, the countless middle roads in between Value Betting and Arbitrage. Those offer to every punter the opportunity to find their optimal strategy.

I have already mentioned the opportunity of mixing Arbitrage and Value Betting but never explored it in more depth – a gap I will fill here.

Mixing Arbitrage and Value Betting
Mixing can make it better – Photo by Matan Segev

How does it work?

Disclaimer: Apologies to my readers who are used to British or American odds format. I use European
odds format below as for other odds calculations on my blog. I am looking into options to offer some
kind of an odds converter on my website, but until then I will do calculations using the European
format, since it is the one I am most used to.

A strategy mixing Arbitrage and Value Betting is not harder to execute than a classical Arbitrage betting strategy. I will provide a simplified example.

In a Premier League game between Liverpool and Manchester City, bet365 is offering you odds of 1.8 for o2.5 goals and 2.1 for u2.5 goals.

A scalping software ( such as BetBurger or RebelBetting) or Value Betting software (like Trademate Sports) is scraping the odds and comparing to Pinnacle. It sees that Pinnacle is pricing o2.5 @ 1.95 and u2.5 @ 1.95. Betting o2.5 @ 1.95 in Pinnacle and u2.5 in bet365 @ 2.1 is obviously an arbitrage opportunity for around 1% of sure profit.

Let’s have a closer look

First, we can calculate expected return in each of the two legs of this arb. We will need to remove the overround (a.k.a. the vig/margin) to see what is the probability of the Over and the under happening (we should expect the two probabilities to add up to 1).

It would be fairly reasonable to assume that the Pinnacle odds reflect the true probabilities. The study of Joseph Buchdahl on Wisdom of the Crowds explains the theory behind this phenomenon and provides evidence of it as well. Pinnacle odds are expected to be extremely efficient on huge markets such as Over/Under in the Premier League and less so for markets with lower limits, but we can ignore that for now. We wouldn’t be too far off assuming that the Pinnacle odds reflect the true probabilities of the Over and Under in our game.

Now we need to translate those odds to probabilities. Removing the overround in Pinnacle is trivial in this case, since equal odds means equal portion of the overround attributed to each outcome (otherwise, longer odds carry a higher proportion of the total margin). With both outcomes priced at 1.95, they are obviously seen as equally likely, translating to a probability of 50% for each.

Let us imagine bet365 allows you to bet a maximum of 500 currency units on this outcome. What are your options?

Option 1: Arbitrage Betting – complete risk elimination

Play Under with 500 units in bet365 @ 2.1 and 500 * 2.1 / 1.95 = 538.46 in Pinnacle @ 1.95 for a sure profit of 500 * 2.1 – 500 – 538.46 = 11.53 units. Add a PremiumTradings cash-back for an extra 0.001 * 538.46 = 0.54 units on the sharp leg and you have a sure bet with a profit of 12.07 units.

Option 2: Value Betting – maximum risk for maximum return

We have established that the real probability for the Under to happen in this game is 50%. Betting 500 units on the Under in bet365 @ 2.1 gives you an expected profit of 50% * ((2.1 – 1) * 500) + 50% * (-500) = 25 units of expected profit.

Which way to go?

Morpheus Mixing Arbitrage and Value Betting

The ROI perspective

Sure profit is obviously better than expected profit. However, looking at it from an ROI perspective, in an Arbitrage scenario you are getting a sure ROI of 12.07 / (500 + 538.46) = 1.16%, while in a Value Betting one you are looking at an expected ROI of 25 / 500 = 5%. One reason for the big difference in ROI is the smaller stake in the Value Betting case. You could argue which metric of the two is more important, but seasoned punters tend to say it’s ROI for show and ROBG (rate of bank growth) for dough.

The profit perspective

Looking at it purely from profit perspective, long term you still get more than double Value Betting than Arbing in this case. Is that always so? The striking difference here lies in the arb’s low profit rate of a little over 1%. Therefore, not having to pay the Pinnacle margin adds up to your expected profit significantly. In the Value Betting scenario, on top of the value you get from the soft leg, you benefit from not having to pay the Pinnacle margin.

How does it look like in real life?

In general, it is realistic to expect an average ROI of around 6% if you are betting only the soft leg of your arbs, compared to an average of around 2% in an arbing scenario. However, your arbing turnover is roughly double your value betting turnover. So in terms of total profit, you will normally get only around 50% more expected profit per average bet in a Value Betting scenario.

In Arbing you sacrifice one third of your profit in order to sleep well at night. However, you also lock in two times the capital you would put in in the Value Betting case. A risk-neutral player with an infinite bank would refuse to pay the toll to Pinnacle and just go full Value Betting. However, most people are more or less risk-avoiding, for a very good reason. If you go bankrupt you need quite some time to get back to the table, so going all-in every time you find an edge may not be a very rational approach.

A different point of view

Interestingly, Matthew Trenhaile reports in one of the latest episodes of his ‘Inside Betting’ podcast about a conversation he had with a pro player on staking. The pro player said, that his stakes are not based on his current bank, but on an imaginary figure of a “lifetime bank”. Lifetime bank meaning all the money the punter would have gathered from various activities during his lifetime. Needless to say, this philosophy lead the punter in question to several bankruptcies. On the other hand, once positive variance smiled at him, he made more money that he ever would with more conservative staking. You can listen to it here:

Can’t live with all that risk

Even though (semi-)professional punters tend to be more risk-seeking than the average person, for me the above strategy is a bit crazy. Normally, I would be looking for close to optimal returns, with the volatility reduced in so far that I don’t feel the urge to jump off a cliff after the first serious drawdown hits my bank. Luckily, there is something that can be done.

Option 3: Let’s mix it up

Taking the example from above, let us say we have a bank of EUR 5000. We are allowed to bet a maximum of EUR 500. Risking 10% of our bank on an edge of 5% would be way too high by most staking plans. Betting according to the Kelly criterion (itself too volatile for most people), we would bet 4.55%* of our bank here or EUR 227.27. This is still a rather aggressive staking, since in reality most punters will go for fractional Kelly of sorts, but let’s take it for the sake of the example.

*In practice, calculating your edge for Kelly staking wouldn't be very intuitive. It depends on margin size 
and margin distribution. Most true odds calculators on the Internet disregard the disproportional 
distribution of margin among shorter/longer odds. Furthermore, by the time you have made a precise 
calculation the value odds will be gone. It is better to quickly calculate an estimation in your head and 
work with it. A somewhat correct estimate to use right away is always better than a precise number, the
 calculation of which lasts longer than the lifetime of a value bet. A good rule of thumb is |edge| = 
|arbing %| * 2 + |sharp book's overround|.

Now here comes the key question: Why should you bet only EUR 227.27, when the soft book offers you to take EUR 500 and you can still arbitrage the difference away?

Bet it all, hedge a part

We can just bet what we wish to bet given the edge and size of our bank, and use the rest of the offered limit to turn some sure profit with a sure bet. In this case, assuming we aim for Kelly staking, we would let EUR 227.27 run and cover the remaining EUR 272.73 with an opposite bet of 272.73 * 2.1 / 1.951 = 293.56 at Pinnacle.

That leaves us with an exposure of EUR 227.27 at an edge of 5% for an expected profit of 11.36 units and a sure profit of 272.73 * 2.1 – (272.73 + 293.56) = 6.44 units.

We have the best of both worlds! We have exactly the Value Betting exposure we would like to, but supplement that with a sure arbing profit.

But wouldn’t that lead to faster account closures?

A valid question. After all we have taken the largest possible stake the bookmaker has to offer. Opinions tend to differ on whether that increases the risk of attracting unwanted attention. The answer depends on the bookmaker and the market. In theory, when you beat the closing line you’re getting flagged and restricted, so your stakes shouldn’t matter. But in practice, many bookmakers are far from having that whole process automated. And since there is a manual part to it, a risk manager probably only checks samples of bets. The chance of your bet to ‘make it’ in the sample is therefore probably higher the higher stakes you bet.

Then again, betting higher stakes you would care less about quicker limits, since you will reach any profit level faster and with lower number of bets. So at the end it is up to you to find the balance. The knowledge of how certain bookmakers deal with those situations only comes with experience.

The Big Picture

So let’s compare the 3 options above. For the sake of completeness, I have added a fourth one, a full Value Betting strategy staking according to the Kelly criterion. They are being called:

  • Max Stake Value Betting (VB Max)
  • Kelly Value Betting (VB Kelly)
  • Mixed Strategy: Kelly Value Betting + Arbing the remaining stake (VB/Arb Mix)
  • Max Stake Arbing (Arb Max)

I have simulated a series of 200 bets with the 4 strategies within 4 scenarios – with 90 (pessimistic), 100 (realistic) and 110 (optimistic) winning bets.

Certain things are expected to stay the same in all graphs. The arbitrage strategy will always be a flat upward line. The mixed strategy will always have better results than the Kelly strategy, since it just adds the arbing profits to it. But it is interesting to compare how those fare against each other and how does the MaxVB strategy looks like in the different scenarios.

Best case scenario (110/200 bets won)

In the best case scenario we have 110 won bets or 55%, 5% above the expectation. Here the Max VB strategy is leading you to the highest final bank. However, with it, at some point you lose around half your bank. On the other hand, the Arb/VB Mix captures most of the upside, being kinder to your bank in bad times (which aren’t many in the optimistic scenario). Obviously, Kelly is below the Mix, but its return is still decent. They all outperform the pure Arbing strategy by quite a bit.

Realistic scenario (100/200 bets won)

Now let’s see a realistic scenario – 50% bets won as is expected. You see that the full Value Betting strategy is outperforming at the end, however around the 67th bet you go below zero, meaning you have busted your EUR 5000 bank. In a real world scenario that might mean the end of your betting activities right there. 1000 Monte Carlo simulations of this scenario show a chance of blowing up your bank in 48.5% of the cases with full Value Betting.

The Kelly strategy and the Mix carry you through the bad times by risking less of your bank as it shrinks, but again capture a good share of the upside.

We also see an example for the devastating effect an early drawdown can have on your returns, even with a very high edge of 5%.

Pessimistic scenario (90/200 bets won)

Finally, in a pessimistic scenario, the full Value Betting scenario is obviously a catastrophe, guaranteed to blow your bank. With Kelly you still lose half your bank and the Mixed Strategy will cost you just around a fifth of your bank in this case. Here, have you gone arbing, you would have been grateful for having made that choice, realising huge profits in the sharps and exchanges and losses in the soft books, for a net profit of just above €2000.

Summary statistics

According to the cumulative probability function for a binomial function, a chance for our pessimistic scenario (or worse) to occur is 8.94%. The same applies for the optimistic scenario (or better), since it is a symmetrical case. You can see what the chances are for other scenarios using the Binomial calculator. In general, for the specter of somewhat likely possibilities (80 to 120 winning bets), the probabilities for x wins from 200 tries look as follows:

…with the cumulative probability (the likelihood of that number of wins or worse) looking like this:

What can we learn from this?

First of all, as anyone with enough practical betting experience will know, tail events are bound to happen at some point, so 10% chance of running into our pessimistic scenario is actually quite big. But more importantly, we see that even in the realistic scenario (number of wins in line with expectation), we have around 50% chance to destroy our bank with a full Value Betting staking.

Staking is a topic that deserves its own article and it will probably get one at some point. The simplistic assumptions above are clearly insufficient to lead you to your optimal staking plan. There are whole books written on the topic out there. “Fixed Odds Sports Betting” by Joseph Buchdahl is one I’d recommend.

I only mean to demonstrate that a mixture between Value Betting and Arbitrage to make full use of the offered maximum stake by the bookmaker in a value bet might be the more sensible option that going all-in or hedging all the risk away. How big of an exposure you want to have (a.k.a. your staking plan) depends on many factors, which I don’t aim to fully cover here. But it is worth remembering that you don’t really have to chose between Arbitrage and Value Betting. You can do a mixture of the two, giving any weighting you’d like to each of the two strategies. By tweaking the weighting and testing the results you will soon arrive at your optimal risk/return ratio.

Conclusion

I hope I managed to give you an idea about the opportunities you have mixing Arbitrage and Value Betting. Using this approach, no stake could scare you. You can always extract the maximum from any arbitrage opportunity presented to you. All you need to do is to secure a steady inflow of sure bets (using a software such as RebelBetting, BetBurger or Trademate Sports), decide on your staking plan and train yourself to execute the strategy flawlessly. If you have experience arbing that shouldn’t take to long – you just need to learn calculating what part of the risk you want to hedge.

This is the approach I am using myself when I am able to get a bit more money down than usual. I feel comfortable using it and that is why I also recommend it. It is one of the tools that has helped me greatly improve my total betting profits. I hope it will help you do the same.

Thanks for reading and I would be more than happy to receive your questions and comments below or at my e-mail admin@churchofbetting.com. Also, make sure to subscribe to my newsletter on the top-right corner of this page as well as to my Twitter channel so that you don’t miss out on any new content. Next on the list, I will report on the LoL model I am developing with a friend, which I announced in my article on Rating Models. So I am getting back to work and hope to see you around!

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