# Risk-Free Bets and Free Bets

Hi everyone,

Wanted to make a post about optimal Risk-Free Bet and Free Bet strategy. because of all of the promotions that are currently going on in the US. I found the following information helpful on how to max EV and wanted to pass it on to the community.

Risk-Free Bets

• Overview - These are probably the most common promotions. They work like this: You make a bet up to a certain amount (\$1,000 is common) and if you win it is treated as a normal \$1000 bet. But if you lose, you get the amount you bet ābackā. Why ābackā? Because sometimes you get your principal amount back in site credit (very good, FanDuel does this) but sometimes you get your principal amount back in āFree Betsā (not as good, but will touch upon how to make these more valuable later). Site credit is treated like account balance, when you bet it and win you get your profit back plus your principal. Free Bets are inferior because when you bet and win you only get the profit into your account.

• Strategy - The best strategy to use for risk free bets is one where you lose with high frequency. That sounds insane but if you think about when you get āfree moneyā from a risk-free bet, itās only when you lose. So you want to target bets with frequent loses and big payouts when you win. Look at the following two examples.

• Bet 1: +100 , True Odds = 50%
If we bet our risk-free bet of \$1000 here, we will win \$2,000 50% of the time (EV \$1000) and we will return \$1,000 the other 50% of the time (EV \$500). Our average return will be \$1500 minus our \$1000 bet for a total EV of +\$500.

• Bet 2: +400, True Odds = 20%
If we use our risk free bet of \$1000 on this long shot, we will win \$5,000 20% of the time (EV \$1,000) and we will return \$1,000 the other 80% of the time (EV \$800). Our average return of \$1800 minus our principal of \$1000 will give us an EV of +\$800 .

Both bets are neutral EV in a vacuum but just by betting longer odds, we can increase the EV of our Free-Bet substantially. I am pretty sure the EV on these actually plateaus at 100% when you get to 10-1. I am sure Matt and Will could tell you why, but I can just tell you that it happens. So long story short, target 10-1 aka +1000 bets with your Risk-Free Bets.

Free Bets

Free bets are like RFB cranked up to 100. You should never even consider betting these on favorites (I did this once and thatās what launched my exploration of the topic). Like I mentioned above, when you win using a āFree Betā you donāt get your principal amount back. So if you bet a \$200 free bet ticket at +100 and win, your account is credited with \$200. If you bet \$200 of principal and win, your account is credited with \$400 (principal back + profits). Here is the math:

• Bet 1 +100, True Odds 50%
You take your \$100 free bet ticket and bet it on a coinflip. You win \$100 50% of the time (EV \$50) and \$0 50% of the time (EV \$0). Our average return on our \$100 free bet ticket is \$50.

• Bet 1 +900, True Odds 10%.
You take your \$100 free bet ticket and bet it on a longshot. You win \$900 10% of the time (EV \$90) and \$0 90% of the time (EV \$0). Our average return on our \$100 free bet ticket is \$90.

This is really important because when you can increase the value of your free bet tickets, you also increase the value of Risk-Free bets that pay losses in free bet tickets. I think free bet ticket EV increases on a log curve approaching but never hitting the ticket amount. So if you took +1900 in our above example, your return on a 0% edge bet would bet \$95. The higher odds you go, the closer you get to \$100, but your gains wont be anywhere as significant as they were when we went from a +100 to a +900 bet.

Anyway, TLDR for Risk-Free bets target +1000 bets and for free bet tickets target bets +1000 or over depending on your risk appetite. I am sure my formatting leaves a lot to be desired and this might be a bit rambly but I found this information incredibly valuable for those of us who didnāt intuitively know this before we started using sportsbook promotions. Cheers!

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It depends on the size of the freebet relative to your bankroll size but on average you want to chase high +EV bets at odds of +500 or higher.

In theory you want to bet on the 200-1 shots on the Korn Ferry Tour with a 50+% EV but in practice Iāve had much more success with 5-1 to 10-1 shots in the Top 10/20 markets with something closer to 15% EV.

The extreme +EV golfers that go off at 50-1 shots or higher almost always seem to miss the cut and very rarely finish even in the top 20.

Awesome post; well-formatted in my opinion. Love this kind of exercise that digs into something (seemingly) simple and finds some deeper insights.

For the risk-free bets, I think the math looks like this:

For a \$1 bet, and defining p as the true probability of winning the bet (and we assume the book offers fair odds) you have expected profit = p * (1/p ā 1) + (1-p) * 0 = p * ((1-p)/p) = 1-p.

So as you said, as p approaches 0 (i.e. American odds approach +Infinity) your expected profit approaches \$1 (or 100%).

From what I can tell āFree Betsā are functionally identical to RFBs, arenāt they? That is, your expected profit on a \$1 credit should again be: p*(1/p ā 1) = 1-p.

But, you are still correct that getting your money back from the risk-free bet in the form of āfree betsā is obviously not as good as getting it back in full (unless you can find a bet with Infinite odds! In which case the expected profit on your āfree betā will be equal to 100%, and so a \$1 free bet is as good them giving you back \$1**).

** only as good if you are an insane risk-neutral person who is indifferent between a guaranteed \$1 or \$1M dollars with probability 1 in 1 million and and \$0 with probability 99.999ā¦%

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Yeah seems like you are exactly right Matt. Thank you for showing us the math on these! I think maybe another interesting promo to take a look at would be profit boosts. They are not as popular as the free bets, but DK offers a lot of them from what I have seen.

Also, thinking about free bets in terms of Kelly is really interesting. I donāt really know how I would start thereā¦I kinda just size mine right around where the curve starts to flatten and expect that is probably fine as the free bet amounts arenāt usually too high.

But if I had a free bet that was like half the size of my bankroll or something extreme like that I am not sure what I would doā¦wondering if you had any thoughts on selecting sizing?

Pick a T10 or T20 with the highest +EV that has a >10% chance of winning

How did you arrive at that answer?

Played dozens of freebets and found what works best in practice

You want to hit these once in a while. Going months without hitting one sux

I think we need to be careful when drawing conclusions from small sample sizes - gambling is a tricky beast that likes to deceive us in the short run.

To your point though, I know it sucks missing a ton of free bets. I was probably like 0/20 no joke on free bets since legal books launched in my state. Then I hit one that more than paid for the rest and decided I was a free bet master so I made this post haha.

But in all seriousness, what I think is so interesting about picking optimal free bet odds with relation to your bankroll is it should be formulaic, especially if we only consider neutral EV bets. I may take a crack at it if I have time just because I am so curious. @matt_courchene may have to check my work when I am done.

It comes down to the rate of bankroll growth

Let B = bankroll size
f = fraction of bankroll that the freebet represents
O = International odds youāre receiving (11.00 for a +1000 freebet)
p = probability of winning freebet

Then you want to maximize:
B*(1 + f*(O-1))^p

In practice you might have a \$500 freebet, \$1000 bankroll, and the following wagers:

100-1 odds with 1.5% probability
50-1 odds with 2.5% probability
20-1 odds with 5.5% probability
10-1 odds with 10% probability
5-1 odds with 18% probability

Here f = 0.5

If you plug in the numbers you find that the 5-1 wager results in the fastest possible bankroll growth

Fiddle around with a lot of numbers and youāll find that the sweet spot is about the 5-1 range. So in general you want to look at the best odds on a middling golfer to score a top 20 (or a good golfer to score a top 10), based on what Data Golf has as the top pick

Or you can just pick someone like Rahm to win in a good spot (Cantlay this week certainly qualifies too).

Your conclusion only holds when your freebet represents a very large portion of your bankroll (such as the 50% you used), which in practice is highly unlikely unless youāre redeeming some sort of welcome offer or have a very small bankroll. Run the numbers with something like a \$10-25 free bet on a \$1000 bankroll and the longer odds free bets have a superior EV.

The formula is there so you can use them in real-life situations and see what comes up

So I used the formula
It put me on Corey Shaun at +6000 to win on the Korn Ferry Tour
Without the formula I definitely would have picked something else

Trust the math, I guess (and Iāll probably hit one of these every 5 years)

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I just want to follow up on this with a few comments:

I am not completely convinced that the rate of bankroll growth is necessarily the right thing for everyone to consider in the context of free bets. The Kelly criterion for bet sizing was derived in the following context:

• A bettor is presented with a positive expectation wager, and a bankroll of fixed.
-The bettor can risk as much or as little of their bankroll on each wager
-The bettor can play for as long as they wish (the number of plus-EV wagers available in time is not finite)

The formula you provide is a tool for deciding what fraction of a bankroll to wager on each bet in the above situation. Keep in mind that in the context that this formula is derived, we are trying to maximize the rate of bankroll growth-- and (loosely speaking) the reason we are trying to do that is so that we may risk more money on the next bet.

Note that from the standpoint of many bettors (non professional who are trying to extract value from their free bets, rather than professionals trying to grow their bankroll), the situation involving free bets is different in the following ways:

• The size of the bet is predetermined, and it independent of the bankroll
• The number of free bets is finite, and typically small

If we are restricting our attention to just the free bets, and not thinking about them in the context of an overall strategy that allows us to dynamically size bets, a bettor still has to make a determination about the EV/variance tradeoff that suits them best.

Another way of thinking about this tradeoff would be to consider the Sharpe ratio of a single free bet. A back of the envelope calculation shows that even for a single bet, the Sharpe ratio is increasing as a pā> 0

|Probability of Win |EV|Variance| Sharpe|
|0.5 |500|250000| 1|
|0.1 |900|90000| 300|3|
|0.01 |990|9900|99.4|9.94|
|0.005 |995|4975|70.5|14.1|

Of course, as the number of free bets increases this will only favor the longer odds wagers even more. At this point I am inclined to say that a free bet represents a rare moment when the math says you really should just send it on your favorite longshot.

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Just trying to digest this, minor question: what are the final 3 rows of that table showing? Variance should be increasing as p ā 0, e.g. variance of a \$1000 bet when p = 0.005 and fair odds of 1/0.005 is around \$200M.

edit: I realize now that the 2nd column from the right is just the standard deviation in the last 3 rows. But I think using the correct variance/sd, sharpe ratio for the 4 bets will be 1, 0.3, 0.09, 0.07, which appear to be roughly the inverses of what you have.

To calculate the variance, I am using p*(1-p)*bet_size^2 ā¦ basically a Bernoulli random variable times the size of the bet---- is that wrong?

Ah, yes I think this is a rabbit hole Iāve been down before. Itās not Bernoulli, because Bernouilli is equal to 1 with probability p. But here, as p ā 0, the value it takes on when you win is increasing (equal to 1/p).

I just calculated variance without simplifying the formula at all, e.g. for 10%:

0.1 * (1000/.1 - 1000 - 900)^2 + 0.9 * (0-900)^2 ~ 7,290,000.

And actually I think got my variances slightly wrong above in that edit, but the ratios still decline as p ā 0, assuming this formula is correct here.

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Ok that makes complete sense, you are rightā it should be p(1-p)*payout^2 to calculate it correctly, which would match what you have.

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Now that I have spent more time thinking about this than I probably should have, I also donāt really know that the Sharpe ratio (that I suggested above) is a very useful tool for making this decision either, given that the āvarianceā in the denominator is supposed to penalize wagers (or investments, trades, etc) that carry a risk of loss. The source of the large variance in these long shot free bets is almost all on the upside (the good kind of variance).

I guess what I am trying to get at is that when it comes to free bets you can kind think about them in three different-- and in my view, equally defensible-- ways:

1. As a bankroll growth maximization problem, if you are pro who is taking advantage of a never-ending stream of +EV bets
2. As a problem of deciding how much to pay in expectation to convert something worth almost 1000 dollars to actual dollars
3. As a mathematically defensible (if not via the Sharpe ratio!!) opportunity to gamble on whatever 0EV longshot bet you would not otherwise take.

I was trying to provide an answer to question 2, and here is one more attempt: another way of thinking about the tradeoff between expectation and probability of winning is to think about the question this way: when you pay money in expectation in the form of a lower odds wager, what do you get in return?

Trivially, for a +100 bet versus a +10000 bet, you are paying 500 dollars in expectation to probably make more money than you would have otherwise. Put another way, the median outcome under bet 1 is much better than the median under bet 2. But how does this scale as the number of bets increases?

Below I have the the probability of a +100 bet returningas much or more money to the bettor under N independent free bets for win probabilities listed on the top row:

0.200 0.100 0.050 0.010 0.005
1 0.79985 0.90092 0.94970 0.98955 0.99463
2 0.64198 0.81002 0.90240 0.98023 0.99023
5 0.53856 0.59897 0.77278 0.94961 0.97604
10 0.56113 0.58874 0.59988 0.90337 0.95088
100 0.51601 0.52642 0.53503 0.56527 0.60533

And here is the same table but the estimated probabilities that you actually make more money with a +100 bet:

0.200 0.100 0.050 0.010 0.005
1 0.39754 0.45079 0.47567 0.49258 0.49763
2 0.47798 0.60855 0.67743 0.73509 0.74395
5 0.52165 0.57283 0.74893 0.92212 0.94555
10 0.48647 0.49526 0.59774 0.90375 0.94878
100 0.49543 0.50404 0.51698 0.53321 0.60560

(Notice the first row looks much worse since you get zero half the time on a +100 RFB)

So even after 5 bets, 4k starts to look like a big price to pay versus a +1000 bet, for example, that will return as much more more money almost half the time.

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