Bankroll management: odds, advantage and variance

Betting bankroll management and variance awareness are essential skills for players. What is the relationship between odds, advantage and variance? What are the bankroll implications of different odds? Read on to find out.

By understanding what to expect from a series of bets, prudent bankroll management can help a player avoid certain behavioral biases such as overconfidence, self-esteem, and the illusion of mastery that can undermine expected profitability in the long run. This article explores how odds, advantage and variance interact and can help players manage their bankroll wisely.

Bankroll Management

Bankroll management and understanding variance are essential skills for any player. From poker players to sports players, traits that all successful players will possess include their ability to understand and quantify their advantage, and attribute variance to either luck or failure.

Consider a bet with odds of 2.0, which implies a probability (no margin) of 50%. If the player can accurately determine that the true probability is 52% (true price 1.92), then the expected return for every bet placed at 2.0 would be 4% (2.0 / 1.92 – 1). This can be called the “advantage” of the player.

Now, suppose the player starts with a bankroll of 100 and bets a fixed one. After 100 such bets, the player’s bankroll could be from 0 to 200 units, but it is expected to be 104 units – a profit of 4%.

By simulating this scenario 10,000 times, we can see the effect of variance on a player’s bankroll on Chart 1 below:

Understanding variance

While the average score was just under four units in terms of bankroll growth, the difference between the best (+38 units) and worst (-30 units) results is significant. It is important for the player to understand the variance and remember that a 4% advantage does not guarantee 4% profit.

With such a simulation of 100 bets in 90% of cases, the player can expect a return from -12 units to +20 units. A drawdown of 10 units (from your starting bankroll) can be expected about 20% of the time, but only 2% of the time the player will experience a drawdown of 20 units.

Interestingly, in 32% of cases, a player can expect to be reduced after 100 bets, despite the 4% advantage on each bet.

If we increase the player’s advantage to 10% (true 55% probability for a bet at 2.0), then a loss will occur 13% of the time after 100 bets.

The probability of a drawdown of 20 units or more was only 0.4%. Of course, as the edge increases, the probability of a bad bet decreases, but what happens when the number of bets increases, say, to 5,000? The Chart 2 below shows the first scenario above (52% true probability, odds of 2.0) simulated 10,000 times:

While the worst result was at -72 units, only 28 (0.28%) out of 10,000 simulations lost after 5000 bets. In 90% of simulations, recoil was obtained from +82 units to +314 units. This reflects a Return on Investment (ROI) ranging from 1.64% to 6.28%.

How will the scenario change if, instead of a 2.0 bet, the odds are 4.0 (an implied probability of 25%)? The process is reflected in Graph 3 below. If we set the true probability to be 26% (true price 3.846), then the expected return for each bet will remain the same at + 4% (4.0 / 3.846-1), but what happens to the variance?

How are the graphs compared?

Comparing these two graphs, we can see that the variance has increased significantly, despite the same bet size, number of bets and expected returns. The standard deviation of returns increased from 1.4% to 2.4%. The range of simulated outcomes is 64% larger in the 4.0 rate scenario, and the 90% confidence range is 72% wider, representing an ROI between 0% and 8%.

In the first scenario, the player lost his entire bankroll of 100 units in only two out of 10,000 simulations (0.02%). In the latter case, the entire 100 unit bankroll was lost in 6.3% of simulations. A 50 unit drawdown was significantly more likely (25.7%) supported by outsider 4.0 compared to a 2.0 rate (2.0%).

In the worst case, betting at 4.0 would have lost almost three whole bankrolls (-276 units). This example shows that with a constant bet size, number of bets and expected return, the variance increases as the odds increase.

Thus, a player who mainly supports outsiders can expect to have much more fluctuation in bankroll size than a player who supports favorites, even if their advantage is the same.

Given that it can take months or even years for a sports bettor to place 5,000 bets, it is probably more important to understand the implications of a bankroll by placing significantly fewer bets.

Assuming a player can find a 4% advantage at 2.0 and bet a fixed one, the Chart 4 below shows the likelihood that there will be a one-point drawdown of your starting bankroll during series from 100 to 1000 bets based on 10,000 simulations.

By placing 1,000 bets at 2.0 odds and 4% margin, the chance of experiencing a certain drawdown seems to be approaching its upper limit, especially for small drawdowns. As the player’s advantage increases, the probability of a certain drawdown decreases. The Chart 5 below shows this probability for a series of 1,000 bets at odds of 2.0 based on 10,000 simulations.

For example, with a 4% margin, the chance of experiencing a 20 unit drawdown over 1000 bets at 2.0 was 17.4%. However, the probability of a decline of 20 units or more after 1000 such rates was only 2.8%. Understanding this difference ensures that the player can look at short-term variance in terms of long-term advantage.

Various bankroll implications

What are the bankroll implications if we keep the bet size and edge constant, but change the odds? Chart 6 below shows the likelihood of different drawdowns (starting from the starting bankroll) when a player makes 1000 single bets at different odds, with a margin of 4%. Each series of 1000 bets was simulated 10,000 times.

Recall that with bets with odds of 2.0, the probability that at some stage in a series of 1000 bets will fall by 20 units was 17.4%. With a factor of 5.0, the probability of a drawdown of 20 units increases to just under 60%. With the same bet, advantage and expected profit from a series of bets, preferential support of favorites or, conversely, underdogs, has completely different consequences for the bankroll in terms of variance.

Understanding which type of player you are is critical to dealing with the inevitable fluctuations you will experience.

To quantify this variance, consider a 1000-bet streak again. By varying the odds (implied probability 10% to 90%) and the advantage, Chart 7 below shows the standard deviation of returns:

We can clearly see that the variance increases as the odds increase (or as the implied probability decreases), according to the analysis above. From Chart 7 above, you can see that with 1,000 bets per 1 unit with a 10% advantage, the standard deviation is 6.5% if all bets are set at 5.0 versus 2.5% bets at 1.67. In both cases, the expected return is +100 units (+ 10%).

An interesting finding is that for odds less than 2.0, the advantage (and therefore the expected return) increases, while the standard deviation actually decreases. Finding an increasing advantage in odds less than 2.0 is rewarded not only by an increase in the expected return, but also by a decrease in variance.

Summary based on this data

This article explored the relationship between odds, edge and variance by simulating a series of positive edge bets.

While more advantage and more bets increases the likelihood of getting ahead of bad luck, it is important for bettors to understand what type of player they are and be able to quantify their advantage.

This will make it easier for them to avoid disappointment during a downturn or to be overconfident when the results are working in their favor.

While a player may not know their exact edge at the time of placing each bet, some previous articles from Pinnacle have discussed the reasons for using Pinnacle’s closing price as a measure of fair value.

If the closing price can be consistently broken, Pinnacle’s low margin means that the player is likely to receive positive long-term profits.

However, if a player is able to generate long-term profits by betting on Pinnacle’s closing prices, they may have discovered inefficiencies that the market cannot account for. Pinnacle’s winning bidding policy ensures that the edge remains available to any player as long as it exists.

Bankroll management: odds, advantage and variance

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