“I will pay you a penny for the first day you work and I will double it every day for a month” Have you ever heard this? Most people asked if they would do manual labor under those terms say no. They obvious haven’t counted out that the penny doubled every day for 30 day is $10,737,418.23! (This is no lie; do the math!)

The same scenario is common with winning sports bettors. While I’m not sure if he’s a winner, take for example this 2+2 thread where a poster was disappointed to learn someone with a $500,000 bankroll who bets 1% of it ($5,000) per wager, on 30 bets per month with a 2% edge averages only $3,000 profit per month. To him $3,000 a month is a living.

But, *“ wow a $500,000 bankroll is needed to earn that”* (when using what is in his mind “proper bet sizing”). After telling this poster exactly what he needed to know, his last response was

*“*!

**Can someone here offer any real insight?**”Many other sports bettors do the exact same thing. They have a chance at a penny doubled per day type scenario but never learn optimal bet sizing. Most continue to flat bet because they need to cash out to pay their living expenses or whatever else. But get this! If you started with only $500 and every week increased it 5%, it would take only 3 years to turn $500 into $1,000,000.

Is the above example realistic? Perhaps not and perhaps so, but; how close to $1,000,000 do you suppose you would get if you spent 3 years flat betting 1% ($5) per game, never increasing your stake. Do you see the point? This is why the topic I’m going to introduce is valuable. First let’s make sure we all understand expected value.

### What is Expected Value (+EV / -EV)

Expected value **(EV) is how much a player expects to profit per wager** on average.

If we were to make an infant number of even money (risk 1 to win 1) bets on the result of a coin flip, this is neutral expected value. This is because we’re getting 50% odds and the chances of winning are 50%.

**Example**

Let’s instead say that on a coin flip we’re getting +150 odds (risk 1.5 to win 1). This an extra 0.5 per win, and we win half the time. In this case we have positive expected value (+EV) of +0.25. If we had odds -150 (risk 1.5 to win 1) we’d have negative expected value (-EV) of -0.25. Note that the EV equation is: (win probability * what you’ll be paid if you win) – (loss probability * amount staked)=EV

This article is written only for the benefit of those who make primarily +EV bets and **never knowingly make –EV bets**. In other words, this article has **no benefit to the recreational gambler**. This is because as an investment strategy the optimal stake on –EV bets is zero. There is nothing wrong with gambling for entertainment and that outside chance of striking it big due to luck. However, in that scenario asking how much you should bet is like asking how much you should budget for alcohol or Facebook games. That’s not a question I can answer.

For those new to betting for profit understand that there are many ways to find +EV bets. It is a lot of work, often is not fun, but the rewards can be well worth it. For where to get started see my info on best sports betting books. For those whom already know how to make +EV bets, this is where understanding expected growth is important.

### Understanding Expected Growth

Let’s say that we find a 12% advantage on a line that pays American odds +49,800,000,000 (risk $1 to win $498 million). The expected value on $10,000 bet is still $1,200. However, if making this bet once per day, the average American male would need to die and be reborn over 17,952 times to be expected to have averaged just a single win. So 12% +EV but you are not going to win (period). The expected value is +$1200 but the expected bankroll growth is -$1200. This extreme example was just to prove the point.

Let’s now look at a realistic example relative to sports betting. You bet small underdogs at odds +110 that you’ve quantified have a 50% chance of winning. Your current bankroll is $30,000 and you decide to stake 1% of your bankroll on these. Where do you expect to be after just two bets?

**Understand there are only 4 possible outcomes.**

- Win both bets: Our bankroll was $30,000 we bet 1% ($300 to win $330) and won. This brought our bankroll to $30,330. We then bet 1% again ($303.3 to win $ 333.63) and won bringing our bankroll to $30,663.63
- Lose First / Win Second: Our bankroll was $30,000 we bet 1% ($300 to win $330) and lost reducing our bankroll to $29,700. We then bet 1% again ($297 to win $326.7) and won bringing our bankroll to $30,026.7
- Win First / Lose Second: Our bankroll was $30,000 we bet 1% ($300 to win $330) and won. This brought our bankroll to $30,330. We then bet 1% again ($303.3 to win $ 333.63) and lost bringing our bankroll to $30,026.7
- Lose Both Bets

:: Our bankroll was $30,000 we bet 1% ($300 to win $330) and lost reducing our bankroll to $29,700. We then bet 1% ($297 to win $326.7) and lost again bringing our bankroll to $29,403

The math is such that each of the potential outcomes above have an equal probability of occurring. In other words, with four outcomes each had a 25% chance of happening. To calculate our expected value we take 25% of the ending bankrolls from each. In this case:

($30,663.63*0.25)+(30,026.70*0.25)+(30,026.70*0.25)+($29,403*0.25)=$30,030.01

As $30,000 was our starting bankroll our expected value is the $30.01 over that. As 30.01/30,000=0.00100033 in a percentage that’s about exactly 1% EV.

But understand this. Making 2 bets that have a 50% probability we expect on average to win one and lose the other. That would be results #2 and result #3 on the above list. Notice both of those have us wining $26.70 more than our starting bankroll, which is an 0.89% increase. That $26.70 win (0.89%) is our expected bankroll growth.

Expected growth is always less than expected value. In this case it is only slightly less because we’re dealing with a small favorite. In the $489 million for $1 example showed earlier even with a 12% edge the gap is near the full amount. In short the bigger the underdog the larger the gap between expected value and expected growth. Vice versa also applies where the bigger the favorite is the less difference there is between expected value and expected growth. I’ll come back to this point.

## Introducing Kelly Criterion

The goal a winning sports bettor should have is to grow their bankroll as fast possible without ever going broke. The best way to do this is to bet size based on Kelly Criterion (fraction Kelly is probably ideal).

I provided that link for the geeks who are into all the details, but in short this is a well-known and recommended investment strategy. It is derived using calculus. It’s highly probable that Warren Buffet uses it for investing and Billy Walters uses it for sports. What it does is exactly what I suggested we need. It calculates the proper stake size based on odds and edge to maximize expected bankroll growth.

The formula for calculating Kelly Stake as a percentage is simple. The formula is:

**(Prob * Decimal Odds)–1 / (Decimal Odds–1) = Kelly Stake**

As many reading this are used to dealing with American odds, you may wish to use our odds converter to get the decimal stake. To give an example: Let’s says American odds are -200 (which is 1.50 decimal odds) and you’ve calculated it has a 70% chance of winning. This plugs in as:

(0.70*1.5)-1 / 1.5-1)= 0.10 and therefore the proper Kelly stake is 10% of your bankroll.

I should warn that Kelly Criterion is very aggressive. If you often over estimate your edge you can go broke quickly. The risk of ruin is also a bit too high for many people’s comfort level. For that reason a lot of Kelly bettors bet fraction Kelly. If full Kelly calculates to 10% then half Kelly is 5%, and quarter Kelly is 2.5%. For new bettors it is probably best starting at around 1/3 Kelly until you learn much more about the topic.

### Kelly Calculators

There are many Kelly Calculators on the web, which can be found with simple Google search. This one here is the best – that link will take you to the one coded by now Heritage Sports employee and former SBR moderator Ganchrow. If you start reading more about Kelly betting, his posts over at SBR are for sure the ones you want to pay the most attention to. With these calculators there is no tedious work involved to calculate the proper Kelly Stake.

### Kelly Stake Observations

In order to make another point about Kelly Criterion let’s look at a few calculations I did.

- If you have a 5% edge on a -200 line the proper Kelly stake is 10% of bankroll
- If you have a 5% edge on a +100 line the proper Kelly stake is 5% of bankroll
- If you have a 5% edge on a +200 line the proper Kelly stake is 2.5% of bankroll
- If you have a 5% edge on a +300 line the proper Kelly stake is 1.67% of bankroll
- If you have a 5% edge on a +400 line the proper Kelly stake is 1.25% of bankroll

Did you notice the trend? When dealing with favorites you stake a lot more of your bankroll. This is how most that do flat betting operate anyhow in that they risk however many units to win 1 unit. With underdogs most are making a mistake. Here it would be much smarter to risk however many units to win 1 unit, as opposed to the ridiculous overbets people make now with risk 1 unit to win multiple units. Why is this a mistake?

As shown above a 5% edge on -200 has a 10% Kelly stake. If a player makes this same bet 100 times he has expected bankroll growth of +64.7%, a 36.7% chance of making no profit, and a 3.4% chance of losing more than 2/3 his bankroll. This is all good and averages out in the long run. If a player were however to bet the same 10% of bankroll on same 5% edge over the same 100 bet sample on +400 odds he’d have the same 64.7% expectation. But (and this is HUGE!), he’d have a 73.5% chance of making no profit, and 55.8% of the time he’d lose more than 2/3 of his bankroll.

So two points. If you must flat bet, do it “to win that flat amount” as opposed to risking that flat amount. Secondly, back to an earlier topic, the goal is to compound ones bankroll. Flat bettors are often guilty of moving money out of their bankroll more often than they should. Yes we all have rent or mortgage, women that want bags and smart phones, need to eat have fun etc. Just be conscious of the compounding factor. Again, start with $500 and grow it 5% per week, you have a $1 million in 3 years.

### Final Kelly Betting Considerations

Kelly betting is actually a much more advanced topic than what has been covered in this article. I decided to write this simply because linking people to Ganchrow’s article is often brushed off and the same questions come up over and over again. His articles are however the ones you want to read to take this further, and hopefully mine here helps draw attention to that fact. Keeping it simple with final points:

Again Kelly Betting is about maximizing expected bankroll growth. Professionals often decide against tying their money up long term even when they’ve spotted a +EV bet. For example regular season win totals, or futures on who will win the super bowl. Even though large +EV bets can be found, the ability to have ones bankroll free so it can be turned over again and again thus compounding while those bets would have been pending is sometimes more ideal.

Also note that if given the choice a professional would much rather bet a 5% edge at -200 then a 5% edge at +400. In fact, when dealing with off market lines, many will focus on looking much closer at favorites than they do for underdogs. This is not set in stone but it is a consideration.

Finally, be very careful of correlated bets. If you are betting a prop that a quarterback will have under X number of passing yards in a game, but also bet that team to cover a large spread, these bets are correlated. The QB winning in a blowout does very little passing late in the game, making it more likely their yards go under. If you bet only the independently calculated stakes on each you are over betting your bankroll. For now just be conscious of this and reduce stake when correlated wagers are involved. As you search and read more about Kelly criterion you’ll learn how to deal with the correlation.

*Author: Jim Griffin*