# Kelly Calculator

### The Kelly calculator will determine stakes for bets on up to 15 events occurring at the same time or 15 different outcomes on a single event.

More information can be found at ‘A Quantitative Introduction to the Kelly Criterion Part 1 – Expected Value vs Expected Growth’ and ‘Kelly Criterion Part II – Maximizing Expected Growth’.

### Inputs

The inputs for the Kelly calculator are:

• An event type selector which will look at events which are either events which are independent of each other and which are occurring at the same time (such as several games at the same time) or an event which can have outcomes which are mutually exclusive (such as a race or a TV talent show).
• The events and outcomes which are the number of events or the number of outcomes.
• The consecutive series, which is the total number of times that the bet is to be carried out in sequence. This will take into consideration the expected and average bankrolls which occur over a number of trials. An example of this is if the user believes that he will have 5 opportunities for betting each Sunday, to determine the bankroll expectations over the course of a season (17 weeks NFL) he would enter 5 in the ‘Independent Events’ section and 17 in the ‘Consecutive Series’ section.
• The Kelly multiplier means that 1 is equal to a full Kelly, 0.5 is equal to a half Kelly and 0 means that there is total risk aversion. Entering a figure of 8 would imply that there is total risk neutrality. It should be noted that as this is the ‘true’ Kelly multiplier, the stake n-Kelly will not always be nxfull Kelly. (Mathematically speaking, the utility function for a Kelly multiplier of ?>0 is U(x;?)=(?/(?-1))*x1 – 1/? for ??1, and U(x;?)=loge(x) for ?=1. This implies dU/dx=x-1/? for all ? > 0). At the moment this is only used for events which are independent and for events which are mutually exclusive the multiplier needs to be hard coded to 1.
• The starting bankroll which is how it stands before any bet is made. Putting in a value of 1 or 100% will show the outcome stakes and the expected profit or growth as a percentage. Other inputs will be shown as a dollar figure for outcomes and expectations. When the dollar input is implemented then the number of decimal places which are shown will be indicative of the number of decimal places shown in the output. (If entering \$10,000 there will be figures in the output of whole dollars. If the figure entered is \$10,000.000 then the output figure will be shown to a 10th of a penny).
• The US/Decimal Odds Selector which will show if the input for the bet will be shown as US or decimal odds.
• US/Decimal Odds are the odds for the nth bet. For all bets, the decimal odds are calculated as edge + 1.
• The win probability or edge odds selector will show if the input for the bet will be shown as edge or win probability.
• The win probability or edge is shown for the nth bet. These inputs are done as percentages, so adding in a figure of 5 would equal 5%. The sum of the probabilities calculated for all the mutually exclusive events should not come to more than 100%. For all bets the edge is equal to the decimal odds minus 1.
• The calculate Kelly stakes and expectations needs to be inputted and it should be noted that every additional variable will increase the calculation time by 4, so the processing times for many of the variables can take a while.

### Inputs and Outputs

For the outputs and inputs there are:

• Stakes: The text section shows the Kelly-optimal stakes for both the singles and the parlays of the constituent bets. It should be noted that for the mutually exclusive events the sizes of the optimal parlays should always be at zero and these will not be shown. The format of the stake output will as stated in the ‘Starting Bankroll’ box. The stake sizes can be altered so the expectations can be recalculated for the set of stakes which are specified by the user.

### Outputs

For the outputs there are:

• The expected growth and profit. These are the boxes which show the growth and profit which is expected. The format should be as specified in the ‘Starting Bankroll’ text box and this is from a single set of bets. The expected profit will correspond to the average profit for each set of bets, but only where the set has been repeated a number of times and the bet remained constant in terms of dollar amount. The expected growth will correspond to an average growth for each set of bets where the set is also repeated and the bet remains constant in terms of the percentage of the bankroll.
• The expected or median bankroll. These are the boxes which show the mean and median bankrolls after a number of bets have been added which are the same as the value shown in the ‘Consecutive Series’ box.
• The calculate expectations button will show the values after the user has made any alterations to the sizes of the stakes in the text area.

As an example, the user could have 5 bets on different football games at -110 and all of these could be expected to win around 55% of games. The user may want to calculate the growth maximizing optimal bet sizes and expectations. The user may also want to determine the expected and likely bankroll after 17 weeks of betting on similar situations. The user starts off with a bankroll totaling \$15,000.

The user should set the event type selector to show ‘independent events’, set the number of events to 5 and the bankroll to \$15,000.00, clearly showing the two decimal places to get the calculation to the nearest penny. For this the US odds will be -110 on all the bets and the win probability is entered at 55%.

The user should click on the ‘Calculate Kelly’ button, which will then show that the optimal bet size for each of the single games is calculated at \$657.93, though for each of the 10 2-team parlays this is \$38.29. The optimal bet size for each of the 10 3-team parlays comes to \$2.23 and this is reduced to \$0.13 for the 5 4-team parlays. For a 5-team parlay it is a figure of \$0.01.

This will correspond then to an expected profit total of \$207.39 and will show an expected growth of \$103.70. As these bets are repeated each week over the NFL season of 17 weeks the bankroll is expected to be \$18,943.85. For half of the time the bankroll will show as more than \$16,863.89 and the other half it would be less than this.

If the user made the decision not to bet any parlays which were for more than 2 teams then the stake sizes should be zero for any parlays of more than this. Again, click on the ‘calculate expectations’ button. This will reduce the figure for the expected profit by \$3.66, the expected growth would drop by \$0.04 and the expected bankroll after the 17 week period would drop by \$77.20. The median bankroll over the same period of time would drop by \$0.74. The user should then decide if these reductions are worth the bet.