Payout ratio

In the case of lotteries , sports betting and similar games, the payout quota indicates the proportion of the stakes that is distributed back to the players in the form of winnings.

Games based on the totalizator principle

In games after the totalizator principle such as the Lotto 6 out of 49 and 6 out of 45, the Toto , the bets on the totalizator in horse racing or the Calcutta Auction , the payout ratio is set fix: summed The stakes of this pool is a A fixed percentage is deducted and the rest is distributed to the players or betting participants.

Fixed odds lotteries

For lotteries with fixed odds, e.g. For example, in the class lottery , the expected or average distribution rate is determined by the following thought experiment : The sum of all possible winnings ( total winnings ) is calculated and this is related to the costs of buying all the tickets. This quotient indicates the expected value or the fair value of a lot - which is of course smaller than the lot price.

If the lottery company succeeds in selling all of the tickets, the distribution rate calculated in this way corresponds to the actual distribution rate. However, if the organizer cannot sell all of the tickets, the actual payout quota can vary considerably from the expected payout rate. The - admittedly unrealistic - extreme case would exist if the lottery company can only sell a single ticket and this is the main prize.

Sports betting

In principle, the payout rate for a bookmaker bet, ie a sports bet with fixed odds , can be defined as the expected value as follows: If p is the probability of winning and Q is the (gross) profit rate, the expected payout rate is the product

{\ displaystyle \ mathbb {E} = p \ cdot Q \ cdot 100 \, \%}

However, this definition is not very satisfactory for practical purposes since the probability of winning - e.g. B. What is the probability of Fernando Alonso winning the Monaco Grand Prix ? - cannot be precisely determined and can hardly be estimated with sufficient accuracy.

The payout rate can, however, be approximately determined with the help of a thought experiment , this is explained using the following example. Assuming a bookmaker offers the odds of 1.25, 5.00 and 10.00 for the three possible outcomes 1 , X and 2 on the occasion of a soccer game, we mentally bet enough on each of the three outcomes that we get 100 € back if we win; so

100 / 1.25 = 80 € on 1 ,
100/5 = € 20 on X and
100/10 = 10 € to 2 .

The total stake is 80 + 20 + 10 = 110 €, we now receive - regardless of the outcome of the game - 100 € back, ie 90.91% of our stake.

The formula for calculating the payout ratio is: ${\ displaystyle \ mathbb {E}}$

{\ displaystyle \ mathbb {E} = {\ frac {1} {{\ frac {1} {Q (1)}} + {\ frac {1} {Q (X)}} + {\ frac {1} {Q (2)}}}} \ cdot 100 \, \%}

,

in the example above:

{\ displaystyle \ mathbb {E} = {\ frac {1} {{\ frac {1} {1 {,} 25}} + {\ frac {1} {5 {,} 00}} + {\ frac { 1} {10 {,} 00}}}} \ cdot 100 \, \% = 90 {,} 91 \, \%}

This process is known as hedging in the financial sector , so the payout ratio calculated in this way is also known as the hedge ratio . If in the example above all three bets had the same expected value, this would be the same as the hedge rate. In fact, the expected value for a bet on the favorite, in the above example the bet on 1 , is slightly higher and the expected value for the bet on the underdog, in the above example the bet on 2 , is slightly lower. The hedge rate thus represents a weighted mean of the expected values.

The hedge odds for bets with two possible outcomes (e.g. tennis ) are mostly around 90%, for bets with three possible outcomes such as football, they are often only around 85%, and for events with more than three possible outcomes in many cases even significantly lower, for comparison: the payout rate for the multiple chances of roulette is 97.30%, for the simple ones even 98.65%.

If one sets in an event on all possible outcomes with different bookmakers at the best available odds in each case, the amount to always be paid regardless of the outcome of the event. For example, to receive € 100, you can occasionally achieve an arbitrage profit, namely exactly when the following applies - however, such bundles of bets called Sure Bet are rarely possible. ${\ displaystyle \ mathbb {E}> 1}$

Relationship between payout ratio and bank advantage

While it is common in lotteries and sports betting to indicate the payout percentages, in other games of chance such as roulette, baccarat or blackjack, it is common to calculate the bank advantage . These two terms are very closely related: the payout ratio plus bank advantage always results in 100%; if the payout ratio is 98%, the bank advantage is 2%.

In a fair game (cf. Martingale ), i. H. In a game in which neither participant has a mathematical advantage over the other side, the expected payout rate is 100%. The payout quota of the games of chance offered by lottery companies, betting operators and casinos is of course below 100% in order to enable the operator to make a profit in the long term.

Examples of payout ratios in popular games of chance

South German class lottery : 51.36%
North German class lottery : 48.64%
German Lotto "6 out of 49": 50%
Totals (6 out of 49): 50%
Eurojackpot lottery: 50%
Spanish Christmas Lottery : 70%
French Roulette (one zero, no lock at Zéro): 97.3% (i.e. bank advantage 2.7%)

Individual evidence

[1] ttps://gluecksspiel.uni-hohenheim.de/fileadmin/einrichtungen/gluecksspiel/Oekonomie/Ausschuettungsquoten.pdf

[2] Report of March 21, 2016 at www.lottodeals.org