P-Value and Expected Maximum Drawdown Calculator
26.03.2019, Miguel Figueres Moreno
In this entry you can find an Excel spreadsheet to calculate the values of:
- P-Value: The probability of a sports betting yield being a result of luck
- Expected Maximum Drawdown: The average drawdown of a series of bets
The p-value calculated in the PV_EMDD.xlsm spreadsheet answers the question:
Which is the probability of obtaining the yield ‘Y’ if the long-term expected yield is zero? A zero yield means that there is no skill when placing the bets.
The p-value depends on the Number of Bets, the actual Yield of the series of bets, and the Average Weighted Odds. The Average Bet Size does not affect the result of the p-value, so you can let a value of 1.
For example, if a series of 100 bets at average weighted odds of 1.83 (11/6) shows an actual yield of 5%, which is the probability of obtaining this yield just by chance? (assuming that the long term yield is 0% and the payout is close to 100%, as the tipster can choose the best odds from several bookmakers, or the bookmaker margin is small).
The results show that the probability is… 29.3% Almost one of every 3 tipsters could have obtained this yield just by chance.
How many bets does a tipster with that stats need to verify that the probability of obtaining this yield by chance is “small”? Small is, for example, 1%. If you try values of the number of bets, you will find that the tipster needs 1781 bets with a yield of 5% to verify that the probability of obtaining this yield by chance is 1%. However, this result is still not an irrefutable proof of being a profitable tipster, due to the fact that there are much more than hundreds of tipsters you can find around the internet, so I would be still more exigent and… requiere a higher yield, lower odds or a higher number of bets to be sure. A p-value of 0.1% or lower more reasonable from my point of view.
The following graph shows how much the p-value changes with the Number of bets, for different Average Weighted Odds of 1.5, 2 and 3; and a 5% Yield.
As you can see, with the same Yield, the higher Average Weighted Odds, the higher number of bets are required for the same p-value. You can find more graphs in the PV_EMDD.xlsm file.
In the following table, you can find the Number of Bets required to obtain 1% or 0.1% of p-value for different values of Yield and Average Weighted Odds.
Expected Maximum Drawdown
The Maximum Drawdown is the maximum difference between a previous value of the profit of a series of bets and a later profit. The Expected Maximum Drawdown is an estimation of the average drawdown that a tipster will suffer, taking into account the yield or average profit per bet, the average bet size, the average weighted odds and the number of bets.
The relationship between these variables and the Expected Maximum Drawdown (EMDD) is as follows. The EMDD:
- is linearly proportional to the average bet size
- is approximately proportional to the logarithm of the number of bets
- decreases when the yield or the average profit per bet increases
- increases when the average weighted odds are higher
By introducing the Number of bets, the Yield, the Average Weighted Odds, and the Average Bet Size in the PV_EMDD.xlsm table, you can obtain the average profit per bet, and then the Expected Maximum Drawdown.
Going back to the previous example, with a Yield of 5%, Average Weighted Odds of 1.83, and Average Bet Size of 1 unit, the EMDD after 100 bets is only 9.5 units. If we continue up to a value of 1781 bets (so that the probability of obtaing a yield of 5% is 1%), the EMDD is 26.067 units.
This means that, on average, the Maximum Drawdown is 26.067, but the tipster can be luckier or not, and have a lower or higher Drawdown during its unique series of bets. However, if the EMDD is very different from the actual Maximum Drawdown of a tipster, it can be suspicious of artificial or manipulated results. How much different, depends on the distribution of the Drawdown, and it can be obtained only with Monte Carlo simulations, that you can also perform with the “WinnerOdds Risk Management Tool“.
The following graph shows how much the Expected Maximum Drawdown increases with the Number of Bets, for different values of Average Weighted Odds, a 5% Yield, and Average Bet Size of 1 unit:
As you can see, the higher Odds and Number of Bets, the higher EMDD for the same value of Yield and Bet Size. You can also find more EMDD graphs in the PV_EMDD.xlsm file.
With this calculator, and taking into account that the EMDD is linearly proportional to the bet size, you can calculate the bet size to obtain a specific EMDD after a determined Number of Bets.
In the following table, the Bet size to obtain a EMDD of 25 units after 1000 and 5000 bets is shown for different values of Yield and Average Weighted Odds.
For example, if the yield is 5% and the Average Weighted Odds is 3, the Bet Size required to obtain a EMDD of 25 units after 1000 bets is 0.62, so that the profit would have been 31.09 units (1000 bets at 5% yield with a bet size of 0.62 units per bet).
This is very useful to determine the appropiate bet size when following a tipster.
A WinnerOdds average user places approximately 12 bets per day (you can check the “User History” results):
User history stats (28/03/2018)
In one year, the Number of Bets is around 4460. The average yield obtained is 5.27%. The Average Weighted Odds can be calculated as follows:
P = Won/Picks = 2788/4461 = 0.625
Y = 0.0527
C = (1+Y)/P = 1.68
The Average Bet Size is 6404/4461 = 1.435 units. The results with this figures are:
The probability of obtaining the 5.27% yield by luck is only 0.001%, meaning that it is (almost) sure that it is a profitable method in the long term. The EMDD of 38 units seems high, but when using proportional stake (by automatically updating the bankroll after every result), the Expected Maximum Drawdown, expressed as a percentage of the previous maximum bankroll, is lower than the value in units.
Moreover, an average profit of 337 units (almost 1 unit per day) is worth the risk of such an EMDD, as the reward / risk ratio is around 8.7 in a one year term.
I hope you like this info and find it useful. If you have any question or suggestion, please, leave a comment.