Written by Adrian Fabich Balk-Møller - MSc in Statistics from the University of Copenhagen
This article analyses how implied edges and number of bets affects different aspects of a betting history given the fractional Kelly staking strategy - in particular the expected fund growth and profit align with the probabilities of going bankrupt and of obtaining no profit.
The analysis is carried out by use of simulations in R, where the goal was to imitate real betting histories as close as possible.
First of all, we will make a distinction between observed and true edges. The observed edges are the ones computed by Trademate Sports and that are shown in the Tradefeed.
The true edges are the edges computed from the closing lines, i.e. when the game starts. Sometimes positive observed edges will shift into negative edges due to line movements.
To account for this, the observed edges eobs are simulated from gamma distributions with means equal to the expected edge e0 of interest:
This ensures that the range of observed edges we will consider are always positive and follow a right-skewed distribution, which was found empirically to be the case by the author.
The true edges, e_true are computed from the observed edges by including some random noise from adding a normal random variable c:
The noise is capped at 10%. By linearity of expectation, the true edges still have the desired expected edge, but with slightly larger variability and with the possibility of being negative.
The densities from which the observed edges are simulated for e_obs=1,2,3,4 are shown below in:
The odds are simulated from a uniform distribution bounded by 1.2 and 2.8, such that the average odds is 2:
The odds density is shown in figure 2:
The staking strategy used in the simulations is fractional Kelly staking with = 0.3 and a starting bankroll of $3,000 with a maximum staking size of $60 (2% of the starting bankroll).
As an illustration, a single simulation of betting history is shown in with expected edges of 2.5% (the average ROI of Trademate customers is 2.5%). The blue line is the bankroll and the red dashed line is the starting bankroll.
While this particular simulation resulted in a profit over 500 bets (final bankroll ~ $3400), this will of course vary for each simulation. The number 500 was chosen since it corresponds roughly to one month of betting.
The idea is to simulate a history as seen in Figure 4 a lot of times, and for each simulation compute the quantities of interest. Figure 4 below shows 1,000 simulated histories with 500 bets each overlayed in the same plot.
The figure reveals a lot of variation in the final bankrolls, and the histogram of final bankrolls summarises this variability as seen in Figure 6.
Next, let’s have a look at what the results would look like for 1000 simulations of betting histories with 5000 bets. This is the equivalent of 1 year of casual betting using Trademate, averaging ca. 100 bets per week.
Next we will compare the histograms of the final bankrolls for 500 bets vs 5000 bets. The histograms show the frequency of how much money one would have after doing e.g. 1000 simulations of 500 bets.
The mean of the final bankrolls (dashed black line) will be a good estimate of the true expected final bankroll given the distribution of edges and odds, the staking strategy and number of bets - if the number of simulations is sufficiently large.
What we can see in Figure 6, is that with a starting bankroll of $3000, one would in average end up with an ending bankroll of $3585 over 1000 simulations. Within +1 standard deviation from the mean one would end up with $4398 a profit of $1398. Within - 1 standard deviation from the mean, one would end up with $2772, a $228 loss.
Compare this to the simulation over 5000 bets in Figure 7 and the average ending bankroll would be $12 170. Within +1 standard deviation from the mean one would end up with $18 241, a profit of $15 241. Within - 1 standard deviation from the mean, one would end up with $6098, a $3098 profit.
One factor that the simulation does not take into consideration is that one will eventually face betting limits when winning over time with the bookies. Still, in summary there is a good chance that one will make a big profit, but also a chance that one will make a small loss.
We will estimate the quantities of interest: expected fund growth, profit, the probabilities of going bankrupt and of obtaining no profit. We’ll also estimate the standard deviation for the former two, for a range of different edges and number of bets.
We define going bankrupt as going below one half of the starting bankroll at any time in the history, as the majority of bettors lose faith in the strategy at this stage.
For each combination of the sequences of mean edges (1,...,4) and number of trades (100,...,5000), each of length 50 with equidistant increments of 100, the different quantities were estimated from n=5000 simulated betting histories.
As such, each “pixel” in figures 8-13 below contain the estimated quantity of interest. The images are colour graded according to the values, which can be read from the scales on the right side of each image.
The expected fund growth was estimated as:
Where K is the starting bankroll. In finance expected fund growth would be referred to as Return on Capital (ROC). The expected final bankrolls are shown in Figure 8 and 9 below.
The expected final bankroll is the final amount of capital one would have after placing the 500 and 5000 bets respectively. Next you can see the expected bankroll for different edge averages.
The probability of no profit was estimated as:
i.e. the fraction of instances where the final bankroll succeeded the starting bankroll K to the total amount of simulated histories n. Next we see the probability of not making a profit up to 5000 bets with varying edges. Note that as the number of bets and avg. edge increases, the probability of losing money decreases.
The probability of going bankrupt was estimated as:
i.e. the fraction of instances where the bankroll succeeds half the starting bankroll K to the total amount of simulated histories n. The Figure below might look a bit counter-intuitive, but this is explained in the conclusion section. Note that in general there is a very low probability of losing half of your starting bankroll (0-2%) after 2000 bets.
Finally, to analyse the long-term properties of a losing betting strategy, we simulated the same strategy as before but with an expected edge of -2.5% for both 500 and 5000 bets. Figure 14 show the results for 500 bets, and Figure 15 for 5000 bets.
We see that for the losing strategy, the expected final bankroll falls well below the starting bankroll - but there is still a significant portion of final bankrolls that turns up with profit. The same procedure for 5,000 bets is shown in figures x and y.
It is evident that a larger amount of bets results in a higher proportion of final bankrolls that yields a loss for the losing strategy. This trend is illustrated in Figure 18, where the percentage of final bankrolls with profits for the two strategies are shown as function of the number of bets.
(Written jointly by Adrian Balk-Møller and Marius Norheim)
Not surprisingly our findings from the simulation study suggest that both the expected edge and the number of bets play a significant role in obtaining success at sports betting.
Looking at the figures we see that in the best case (maximising both expected edge and number of bets), one can expect to multiply their starting bankroll by around 88 - or equivalently, increasing the bankroll from 3,000 to 25,000. Also note that even at −1−1 standard deviation away from the mean one would still end up with a very decent bankroll (~15,000). This is very much achievable in practice using Trademate, if one has access to a good amount of bookmakers.
Note that the standard deviation of the final bankroll is increasing as well with both the edges and number of bets, which is expected since longer histories should be expected to vary more, and larger edges will result in larger stake sizes.
The probabilities of obtaining no profits is decreasing with both edges and number of trades - however, note that even at 1,000 bets and with an expected edge of 44 there is still around 5-10% probability of obtaining no profits.
Sports betting is indeed a long term game and nothing is guaranteed even when maximising your parameters.
At first it might seem non-intuitive that the probability of going bankrupt increases with both edges and number of trades. This can be explained by the increasing standard deviation of the final bankrolls - more variability increases the risk of going below half of the starting bankroll.
However, this does not mean that these particular simulated betting histories themselves yielded a final bankroll with no profit, just that at some point in time, the bankroll went under half the initial bankroll.
The chance of this happening is not negligible at around 1-2%, but the best advice based on this study is to continue even after streaks of bad luck in order to maximise the expected profit along with minimising the probability of obtaining no profit.
Our study further shows the importance of the number of bets needed in order to identify a losing strategy, in addition to separating a winning strategy from a losing one. It is also worth noting that when the sample size is small, one can make a loss despite following a long-term winning strategy. This again highlights the importance of keep getting bets in. You can read more about this in the article about The Law of Large Numbers.
If you want to run Monte Carlo simulations of your own betting history, you can check out the betting simulator at sportsbettingcalcs. It’s a more simplistic simulation, but still good cause it enables you to test our different scenarios and see how factors such as staking or edge affects your results.
Finally, one thing is simulating a betting history. Another thing is placing bets in practice. This article covers the results of the Trademate users. What one should notice is that it very much reflects the results shown in this article. The more bets customers place, the better results they are able to achieve. However, not everyone will end up making a profit in the end, and that is simply a result of the inherent variance that exists within value betting. Even when following a profitable strategy. In summary there is a chance that one will make a small loss, but also a good chance that one will make a big profit, returning the capital invested several times over.
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