



In probability theory, the Kelly criterion is a formula used to determine the optimal size of a series of bets. We use the Kelly formula as a base for all of our bet sizing. It is mathematically proven to provide the greatest rate of return while minimizing risk of ruin.
Inputs
 Event Type Selector:
Either simultaneous independent events (as in several distinct games) or mutually exclusive outcomes (as in a single event that can have one of several winners, e.g., a horse race or the American Idol competition).  # Events/Outcomes:
The number of either simultaneous independent events or mutually exclusive outcomes.  Consecutive Series:
The number of times that this set of bets is to be sequentially repeated. This is included in order to determine expected and median bankrolls over multiple trials. (For example, if the user expects that every Sunday he'll have 5 betting opportunities and wanted to determine bankroll expectations over the course of a 17week NFL season, he'd set "# Independent Events" to 5 and "Consecutive Series" to 17).  Kelly Multiplier:
The true Kelly multiplier such that 1 implies full Kelly, 0.5 implies half Kelly, 0 implies total risk aversion, and 8 implies total risk neutrality. Note that because this is the "true" Kelly multiplier, the nKelly stake will not necessarily be n×full Kelly stake. (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=x1/? for all ? > 0). Currently, this is only implemented for independent events, for mutually exclusive events, the Kelly multiplier is hard coded to a value of 1.  Starting Bankroll:
The starting bankroll prior to the first bet being made. A value of "1" or "100%" will yield outcome stakes and expected profit/growth as percentages, while any other input will be treated as a dollar figure yielding outcomes and single period expectations in dollar terms. If dollar input is used then the number of decimal places in the input determine the number of decimal places in teh output (so a starting bankroll of "$10,000" would yield whole dollar outputs, while a starting bankroll of "$10,000.000" would yield output precision to a tenth of a penny).  US/Decimal Odds Selector:
Indicates whether input for the nth bet will be in US or decimal odds.  US/Decimal Odds:
Odds for the nth bet. For any given bet, decimal odds = edge + 1.  Win Probability/Edge Odds Selector:
Indicates whether input for the nth bet will be as win probability or edge.  Win Probability/Edge:
Win probability/edge for the nth bet. Inputs are in percentage terms, so a value of "5" would correspond to 5%. The sum of win probabilities for all mutually exclusive events may not total to more than 100%. For any given bet, edge = decimal odds  1.  Calculate Kelly:
Calculates Kelly stakes and expectations. Note that every additional variable increases calculation time by a factor of 4, so processing times for a large number of variables can be quite long.
Outputs/Inputs
 Stakes:
The text area displays the Kellyoptimal stakes for singles and parlays of constituent bets (for mutually exclusive events optimal parlays sizes will always be zero and so are not displayed). Stake output format will be as specified in "Starting Bankroll" text box. Stake sizes may be edited (in all tabs except the "All" tab) so expectations may be recalculated for a user specified set of (suboptimal) stakes.
Outputs
 Expected Profit/Growth:
These boxes display expected profit and growth (format as specified in the "Starting Bankroll" text box) from a single set of bets. (To be precise, expected profit corresponds to the arithmetic average profit per set of bets were the set repeated an infinite number of times with bet size held to a constant dollar amount, and expected growth corresponds to the geometric average growth per set of bets were the set repeated an infinite number of times with bet size held to a constant percent of bankroll).  Expected Bankroll/Median Bankroll:
These boxes display the expected mean and median bankrolls after a number of series of bets equivalent to the value in the "Consecutive Series" text box.  Calculate Expectations:
This button calculates expectations after the user has made modifications to stake sizes in the above text area.
Example:
Let's say you're confronted with 8 bets on football sides at 110, all of which you expect to win 57% of the time, and you wanted to calculate the growth maximizing optimal bet sizes and expectations. You also wanted to determine both your expected and most likely bankroll after 13 weeks of similar betting opportunities. You start off with a bankroll of $1,000.
Set the event type selector to independent events, the # of events to "8", consecutive series to "13", starting bankroll to $1,000.00 (the two places after the decimal point will yield results to the nearest penny). The US Odds on all 8 bets would be 110 and the win probability would be 57%.
After clicking the "Calculate Kelly" button, you see that the optimal bet size for each of the 8 singles would be $47.49, for each of the 2team parlays would be $5.10, for each of the 3team parlays it would be $0.55, and for each of the 4team parlays it would be $0.06.
At RLD Investments we never advise our clients to bet using parlays. We included them in the calculator as examples of how the Kelly criterion works only.
We see that this corresponds to expected profit of $70.51 and expected growth of $34.96. After repeating these bets each week over the course of our 13week NFL betting season, you'd expect a bankroll of $2,424.91. Half the time your bankroll would be greater than $1,563.22, and half the time it would be less than $1,563.22.