Arbing without a Hedge: The Insanity of Betting Variance

The question whether it wouldn’t be better to skip hedging in Sports Arbitrage has been asked before. Covering your arb in a sharp book is a sort of an insurance premium. You sacrifice the money you are going to lose in the sharp books in order to have a stable and predictable profit from each bet. If you do not insure yourself you would, at least theoretically, end up with a higher return on turnover in the long run, but you would have to endure some crazy betting variance along the way. Anyone who has tried some value betting strategies with real money will know what I am talking about.

So at the end it all comes down to whether you think the price of getting rid of this variance is worth it. It is hard to give an intuitive answer to this question if you don’t have a good feeling for how much variance you should expect. I have run some calculations to show you just that.

Statistics

A simplified example

So I want to simulate a strategy of placing arbitrage bets on soft books without covering them in sharps. It would be similar to a value betting strategy with a certain edge.

Bank

I start with a bank of 100 units, placing 1 unit per bet.

Odds

For simplicity, it is assumed that all bets are placed at odds of 2. Since odds of 2 imply a probability of 50%, this must be the average odds of your bets in the long run.

Edge

I have assumed you are always getting an edge of 1%. In my experience this is approximately your average return on turnover when doing sports arbitrage, so I consider it to be a realistic number. This means for every bet with odds 2 and implied probability of 50%, your real chance of winning it will in fact be 1% higher, or 50.5%. Your chance of losing will be 49.5%, respectively.

Number of bets

I have run three simulations: one with 100 bets, one with 1000 bets and one with 5000 bets.
Using the above preconditions, I have run a 1000 Monte Carlo simulations for each number of bets to find out what is the chance to end up with a loss and what is the chance to blow your bank.

A “Monte Carlo simulation” might sound scary to those of you who haven’t dealt with it. In fact it is quite simple. I just simulate a thousand instances of 100/1000/5000 consecutive bets under the conditions described above, using the Excel random number generator (the RAND() function). Here is what I have found out.

100 bets

You cannot blow your bank in 100 bets under the above conditions, as for this you would need to lose all 100 of them, which is extremely unlikely. However, what is the chance to end up with a zero or negative return after the 100 bets even though you have an edge of 1%? Higher then you might have thought, it turns out. When simulating 1000 instances of 100 bets and looking at how many of those instances failed to show a profit, the average of all simulations seems to lie around 50% and I very rarely get anything below 48%.

In fact, to get the exact probability of failing to turn a profit we don’t need a Monte Carlo simulation. Since the profits of such series would follow a binomial distribution we might simply calculate it. The actual formula is:

…but since I’m way too lazy for that, I just go to this binomial calculator, input p=0.505, number of trials = 100, number of successes = 51 (the minimum number required for a profit) and hit the button. It turns out in exactly 49.99% of the cases, 100 bets with an edge of 1% will fail to turn a profit.

Think about that for a second. In a single arbing session you might place anywhere from 20 to 100 bets. But if you have a few bad days, don’t have access to that many bookmakers and you are arbing only on the weekends 100 arbs might be your total for a couple of weeks. So you might go for a couple of weeks of arbing without hedging and there will still be a 50% chance that you will have nothing to show for it! That would be very disappointing and very likely too.

Let’s see what happens when we increase the number of bets to 1000.

1000 bets

As the number of bets increases, we would expect our edge to finally materialize in a nice profit. Indeed, the percentage of negative profits for series of 1000 bets looks better. Not much better though. Again using the binomial calculator, we find there is a 38.79% chance to end your 1000 bet adventure without a profit.

1000 bets can extend to one or a few months of arbing. You must keep in mind that in the meantime your money is locked and you need to invest in transaction fees and software subscription, which are not accounted for in the equation. And at the end of it, if you don’t hedge in sharps, you have a 38.79% chance to have absolutely no profit.

What’s more, there is even the slight chance that you blow your bank. The Monte Carlo simulation of 1000 instances shows a destroyed bank or two. The chance for this to happen seems to revolve around 0.1%. So what if we increase the number of bets even further to a 5000?

5000 bets

That must be much better, right? Well… kind of. The binomial calculator says there is still a 24.41% possibility that you fail to turn a profit. That is 1 out of 4, not exactly negligible.

5000 bets is easily half a year of arbing. 27 arbs per day if you arb every single day. Considering that summers are usually quite weak and you will have to go out of your cave to communicate with real people now and then, that probably translates to at least a full year of arbing. And you have a 1 in 4 chance to fail to make any profit from it. On top of the wasted time you will need to cover a year of arbitrage software fees. Ouch.

But it doesn’t end here. With such a long time span you get the added benefit of around 5% chance of blowing your entire bank, as the Monte Carlo simulations show.

I have run a simulation on a series of 10000 bets whereby the chance to blow your bank increases to around 10% and the chance of finishing in negative territory is at 16.10%. You can play with the numbers some more with the calculator linked above.

Varying the edge

The numbers above are all for an edge of 1%, which I find to be a realistic estimate for most arbers. I have also run the calculations for an edge of 2% and 3%. Such a high edge is rare, but certainly possible to obtain. Intuitively, with a higher edge it must be less likely to blow your bank or finish in negative zone. The question is, how much less likely?

Below I have summarized the results for the different sample sizes and edges. You will find the probability of negative or zero return first and the probability for losing the whole bank second.

Fail to profit / Lose entire bank

 100 bets1000 bets5000 bets
1% edge49.99% / 0%38.79% / ~0.1%24.41% / ~4.5%
2% edge45.99% / 0%27.39% / ~0.01%8.07% / ~1.25%
3% edge42.04% / 0%17.94% / ~0%1.75% / ~0.2%

The Verdict

Given the above, should you skip hedging your arbitrage bets? Well, if you are fine with arbing for a year or two without turning a profit you could … I guess. Unfortunately I don’t know many people who would wait for that long. Moreover, you need to pay your arbitrage software and transaction fees in the meantime and they don’t really depend on how well your arbitrage business is running.

So for me, it’s a clear no. I always hedge my arbs and I believe this to be the superior strategy from a risk/reward perspective. But since everyone has their own risk appetite, feel free to decide for yourself. Just make sure to keep the numbers above in mind.

By the way, this simplified example can also give you an idea about the inherent variance in most value betting strategies. An edge of around 1% is what most successful value bettors and tipsters work with. You might make good money, but you would be in for a wild ride.

If anyone has some related experience, I would love to read from you in the comments. If you want to run your own Monte Carlo simulations you can check out this video or drop me an email at admin@churchofbetting.com and I will give you my Excel sheet.

Thanks for reading and see you around!

4 thoughts on “Arbing without a Hedge: The Insanity of Betting Variance

  1. Well a 1% edge would indicate a negative arb.

    Pinnacle
    home 1.92
    Away 2.00

    You get odds 2.00 on the home side at a soft book
    Thats allready a 2% edge with a 0% arb. 1% arb is roughly 4% edge.

    If you have a decent sized bankroll betting the Pinnacle side is just pissing money away.

    • Good point. After all, the reason to skip hedging is increasing your edge and you certainly do so as you show in your example.

      However, I think your edge calculation is a bit too optimistic. You assume that Pinnacle always prices the event perfectly. First, I think normally (and especially when an arbitrage occurs) Pinnacle does not price perfectly, but just better then the soft book. And second, sometimes (though rarely), the soft book prices better. I have had many cases where I place the bet in the soft book only to find that the Pinnacle price has drifted against me in the meantime and the arb is gone.

      That aside, I absolutely agree that skipping the hedging part adds to the edge. It is a totally legitimate strategy, provided that you keep your stakes reasonably small. It really comes down to your risk aversion. You can also take the middle road by hedging only part of the exposure. I would personally rather collect my profits sooner than later and am ready to pay some price for that.

  2. Hi!

    Thank you for very interesting read.

    First I must say I was surprised how big chance it was to end on negative not to mention broke after a decent size sample. I understand your article is basically a numbers game and it isn’t meant to be a practical guide or such. But here’s nevertheless a few observations I made that must be taken into account in order to take the given theory closer to the reality.

    First, when I did arbs years ago, I naturally wanted to maximize the monetary profit per arb. If possible, I wanted to go all in with my whole bankroll. Okay that was never possible and there’s other factors to take account too but for the sake of an argument I’ll not go there any deeper. I would’ve never imagined betting 10% or 30% of my bankroll without closing the arb (i.e. hedging). I might have done that with 1% or even 3% but that’s about it. So if one operates with a small bankroll (as I did), this is something to consider. 10% x Bankroll x 1% arb value is more than 1% Bankroll x 4% predicted value.

    Second, soft bookmakers are really fast in closing unwanted customers accounts. Arbers are in the top of the unwanted list. Whether one only seeks value (without closing the arb) doesn’t matter in this sense. The numbers of arbs you presented as being the ‘standard’ seems extremely high for me. Is 5000 arb bets really something one could achieve per half a year? I highly question this number (Even 500 seems really high) as one would very quickly run out of available soft bookmakers worth of using.

    Third, if one would choose to maximize value without arbing, it would definitely be wise to eliminate all the smaller leagues where your model has the biggest margins of error. This translates to (much) smaller number of bets for the value model. Lower number of bets converts to longer time period. And as you mentioned, it might already take several years with your numbers to make profit and most people are not willing to wait that long. They surely aren’t patient enough to wait double, treble or even longer time.

    • Hi Antti,

      It is indeed smart to vary the size of your stake depending on the edge you get. That is why most people go with Kelly stakes for value betting strategies.

      As I wrote in the article I consider 5000 bets to be realistically equivalent to at least a full year of arbing. That is if you have access to enough soft books and your accounts with them last long enough, which of course depends on many additional factors. But I agree that getting 5000 bets in a year would be challenging in any case. At the very least, you would need to arb full-time to achieve such target.

      Among everything else small leagues will surely get you limited faster. Focusing only on certain sports or leagues will decrease the amount of arbs one can get significantly.

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