You are probably aware of the new type of promotions that sportsbooks have been known to give out recently. Just a few years ago these free bets were basically non-existant, but now they have taken over almost fully for the sportsbook bonuses.
This is obviously a bad thing for us as customers and bettors as there where more value in the bonuses that you could start to play with immediately and didn’t dissappear when the bet were resolved. But that is how the reality is now for those of us that are part of the business.
But there are still plenty of value to be found in these free bets that are offered as signup bonuses and reload offers. Sure, the amounts are often smaller, but you can still squeek out plenty of expected value of this money you recieve if you think a little further than simply arbitrage betting.
We will share some math on free bets and afterwards give some ideas as to how you can increase the expected value you can receive from these.
So you receive a free bet from your favorite bookmaker worth $100. The common thing is to place this money on one or more bets that you like and just hope to net a win.
We will use an example of bets that are offered on the NFL match between Baltimore Ravens at the Cleveland Browns. Let us assume that the spread is set at 0, and the moneyline is 1.91 on both of the teams for ease of this example.
It is also assumed that this line is set relatively fairly (see our article on implied win probabilites) and both of the teams have a 50% chance of winning.
Let us just illustrate our example here where we compare a $100 free bet to a $100 cash bonus. We will show what the expected value of these two options is to your average sports bettor.
First let us say that you are given the $100 we started this section with. If you place your money on the Ravens to win, your expected value of this free bet would be:
EV[Free bet wager] = 0,50 * ($100 * 0,91) + 0,50 * (0) = $45,5
So your expected value of a free bet, assuming that you do not have to rollover the win amount, is about $45,5 if you simply place a random wager.
If you are a winning bettor you will probably squeeze out some extra value of this bet, but then you aren’t likely to be offered much in the form of free bets I reckon.
This bet is quite a bit worse than if you had received that as bonus money that you could wager with. And that even if you had a terrible rollover rate of, let us say 6 times the bonus.
Just see how the expected value is on a straight wager on the above example:
EV[Straight wager] = 0,50 * ($100 * 0,91) + 0,50 * (-$100) = $-4,5
For every $100 you are wagering, you expect to lose $4,5. So if we extrapolate this to cover our rollover requirement we end up with:
EV[$600 worth of wagers] = 6 * [0,50 * ($100 * 0,91) + 0,50 * (-$100)] = $-27
This ends up with this bonus giving us a net positive value added of $73, quite a bit better than the $45,5 offered by the free bet.
If we relax the assumptions and assume that this is a winning sports bettor, or close to it, then the cash bonus becomes even better as our expected value of it would be the full cash bonus, or even more, while the maximum EV we can get from this simple wager with the free bet is around $50.
This looks like quite dismal reading for those of us who wanted to get that extra edge in online betting. But fear not. The discrepancy between the free bet and cash bonuses in terms of expected value is not as large as we have shown in the example above. Let us show what we mean by giving another one.
We keep looking for what we feel are good bets and come across the moneyline on Minnesota Vikings visiting the Green Bay Packers. They are understandably big underdogs, and we are offered 4.00 odds by backing the Vikings (Packers have the odds of 1.25).
After some pondering we decide to place our wager on the Vikings instead, hoping for a big payout. Let us see what happens to our expected value:
EV[Free bet wager with higher odds] = 0,2381 * ($100 * 3,00) + 0,50 * (0) = $71,43
Say what? Did our value suddenly increase by betting on another team? Both no and yes. No in the sense that it doesn’t matter what the teams there are, but yes because now the odds have changed.
The reason that we get this increase is because we have placed a wager on an event that will happen less likely, but have much higher odds. And this is good as we only realize the value of a free bet when we lose, not when we win. By betting $100, losing the bet and getting that $100 is like getting a free $100. But if you place a bet of $100 and win, you will only receive the money that you won, and lose the initial wager.
This is why it is in our interest to increase our chances of a potential loss, non-intuitive as that might be, as that will also increase the potential payout if we should happen to win. How big of a chance you would like to give yourself (and in turn EV) is largely up to you, but as you probably guessed, the higher the odds, the higher the EV.
An alternative to just use the free bet to bet straight up on matches is to make use of an arbitrage to secure some value from the free bet. This is quite common and used by many sports bettors, and although this will leave less value on the table, I have no problem with, do it myself sometimes and in some cases it can even be the way to optimal bankroll growth (or expected growth).
An example of this would be to bet on both sides of the example we first posted with the match between the Baltimore Ravens and the Cleveland Browns. Let us say we place $100 on the Ravens to win with our free bet and then place $47,65 (we will explain this obscure amount in a moment):
EV[Free bet wager] = 0,50 * ($100 * 0,91) + 0,50 * (0) = $45,5 EV[Straight wager] = 0,50 * ($47,65 * 0,91) + 0,50 * (-$47,65) = $-2,15 EV[Total] = $45,5 - $2,15 = $43,35
Notice that the expected value of using arbitrage on this wager will net you a somewhat smaller expected value, but you are now guaranteed to receive this amount no matter the outcome of the match that is being held in Cleveland.
If Baltimore wins: Win $91 and lose $47,65 for a net of: $43,35 If Cleveland wins: Win $43,35 and lose $0 for a net of: $43,35
You see this ensures you that you make something of this free bet, which is liberating to some. There is possible to increase the EV even more you can bet on the other example we used earlier. Place a bet of $100 on the Minnesota Vikings and $240 on the Green Bay Packers:
EV[Free bet wager] = 0,2381 * ($100 * 3,00) + 0,7619 * (0) = $71,43 EV[Straight wager] = 0,7619 * ($240 * 0,91) + 0,2381 * (-$240) = $-11,43 EV[Total] = $71,43 - $11,43 = $60,00
This ends up showing an even bigger profit as you will win $60 no matter the outcome. A better proposition I would say.
Note though that there is a slight problem with this method of arbitrage. Some of the sportsbooks may have you clear the winnings of a free bet by imposing rollover requirements on your potential winnigs. This could complicate things and make betting on such low probability events worse, but you should take this into account when you are calculating this yourself.
We have now shown a couple of different methods that can be used to play with your free bets. If anything you should have learned from this article, it should be that it is generally a good idea to place the free bet on an underdog. The bigger the better (usually). Whether you decide to use arbitrage, individual bets or even several bets is up to your personal preferences, but if you would like to look further into what is optimal usage of free bets we can give you some further food for thought.
As investors of any kind the goal of our endeavour is to acquire as much wealth as possible. A formula that can help us understand how we can easier achieve this is the Kelly criterion. This is a formula that stipulates how much of our bankroll we should wager on a bet in order to maximize the growth of our bankroll.
If you want to learn more about the Kelly criterion and maximizing growth, check out our advanced bankroll management article.
Let us take our first example of the bet to check how much that should be. Note that it is obvious that we should use the whole free bet, but how much should be use on arbitrage betting and how much should we let ride?
First we need to calculate some basic numbers to fit into the Kelly formula. We should think of the straight wager as an alternative to the arbitrage bet. This means that we should measure our straigth wager up against the EV we are missing out on, not the merits of the bet itself for Kelly purposes.
The odds and Kelly fraction is thus calculated as:
Odds = 91 / 43,35 ≈ 2,10 f = [(1,1 * 0,5) - 0,5] / 1,1 = 4,55%
The result here means that you should wager about 4,55% of your bankroll on this free bet if you have the chance. If you have $2000 to bet with, you should place a wager of about $91.
Since the free bet is for $100 simply means that you will only bet a small amount on the other side in order to “safe” in some of the bet, but not all. If your bankroll would have been $5000 for example, you should bet about $227, which is way more than the amount of the free bet. This means you should let all of it ride.
The conclusion of this is that if you have a smaller bankroll, using arbitrage with the free bet is a smart move in order to faster grow your bankroll. If you have a bigger one you aren’t all that afraid of variance and can afford to put more on the line to squeek out more EV.
Note also that a common practice is to only use a fractional Kelly when betting. Full Kelly (which we have used in this example) is very aggressive and not often used by practicioners. Half-Kelly or even quarter-Kelly is much more common, and as you would guess, the bankroll needed to let the bet ride increases by quite a bit then. So the correct thing for those out there with limited funds is to secure the money you can get from an arbitrage.
