**More Information:**

- A Quantitative Introduction to the Kelly Criterion Part I -- Expected Value vs. Expected Growth
- Kelly Criterion Part II - Maximizing Expected Growth

**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 17-week 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 n-Kelly stake will not necessarily be n×full Kelly stake. (Mathematically speaking, the utility function for a Kelly multiplier of ?>0 is U(x;?)=(^{?}/_{(?-1)})*x^{1 - 1/?}for ??1, and U(x;?)=log_{e}(x) for ?=1. This implies^{dU}/_{dx}=x^{-1/?}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 n^{th}bet will be in US or decimal odds.**US/Decimal Odds:**

Odds for the n^{th}bet. For any given bet, decimal odds = edge + 1.**Win Probability/Edge Odds Selector:**

Indicates whether input for the n^{th}bet will be as win probability or edge.**Win Probability/Edge:**

Win probability/edge for the n^{th}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.

**Stakes:**

The text area displays the Kelly-optimal 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.

**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.

Set the event type selector to independent events, the # of events to "5", consecutive series to "17", starting bankroll to $15,000.00 (the two places after the decimal point will yield results to the nearest penny). The US Odds on all 5 bets would be -110 and the win probability would be 55%.

After clicking the "Calculate Kelly" button, you see that the optimal bet size for each of the 5 singles would be $657.93, for each of the 10 2-team parlays would be $38.29, for each of the 10 3-team parlays it would be $2.23, for each of the 5 4-team parlays it would be $0.13, and for the 5-team parlay it would be $0.01.

We see that this corresponds to expected profit of $207.39 and expected growth of $103.70. After repeating these bets each week over the course of a 17-week NFL season, you'd expect a bankroll of $18,943.85. Half the time your bankroll would be greater than $16,863.89, and half the time it would be less than $16,863.89.

If you decided you didn't want to bet any parlays of more than 2 teams, you'd set the stake sizes to zero for each parlay of 3 or more teams, and click the "Calculate Expectations" button. We see that this reduces expected profit by $3.66, expected growth by $0.04, expected bankroll after 17 weeks by $77.20, and median bankroll after 17 weeks by $0.74. It's left for each user to make hiw own determination as to whether he deems the reduction in number of bets placed worth the reduction in expectation.

**More Information:**

- A Quantitative Introduction to the Kelly Criterion Part I -- Expected Value vs. Expected Growth
- Kelly Criterion Part II - Maximizing Expected Growth

**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 17-week 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 n-Kelly stake will not necessarily be n×full Kelly stake. (Mathematically speaking, the utility function for a Kelly multiplier of ?>0 is U(x;?)=(^{?}/_{(?-1)})*x^{1 - 1/?}for ??1, and U(x;?)=log_{e}(x) for ?=1. This implies^{dU}/_{dx}=x^{-1/?}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 n^{th}bet will be in US or decimal odds.**US/Decimal Odds:**

Odds for the n^{th}bet. For any given bet, decimal odds = edge + 1.**Win Probability/Edge Odds Selector:**

Indicates whether input for the n^{th}bet will be as win probability or edge.**Win Probability/Edge:**

Win probability/edge for the n^{th}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.

**Stakes:**

The text area displays the Kelly-optimal 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.

**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.

Let's say you're confronted with 5 bets on football sides at -110, all of which you expect to win 55% of the time, and you wanted to calculate the growth maximizing optimal bet sizes and expectations. You also wanted to determined both your expected and most likely bankroll after 17 weeks betting similar opportunities. You start off with a bankroll of $15,000.

Set the event type selector to independent events, the # of events to "5", consecutive series to "17", starting bankroll to $15,000.00 (the two places after the decimal point will yield results to the nearest penny). The US Odds on all 5 bets would be -110 and the win probability would be 55%.

After clicking the "Calculate Kelly" button, you see that the optimal bet size for each of the 5 singles would be $657.93, for each of the 10 2-team parlays would be $38.29, for each of the 10 3-team parlays it would be $2.23, for each of the 5 4-team parlays it would be $0.13, and for the 5-team parlay it would be $0.01.

We see that this corresponds to expected profit of $207.39 and expected growth of $103.70. After repeating these bets each week over the course of a 17-week NFL season, you'd expect a bankroll of $18,943.85. Half the time your bankroll would be greater than $16,863.89, and half the time it would be less than $16,863.89.

If you decided you didn't want to bet any parlays of more than 2 teams, you'd set the stake sizes to zero for each parlay of 3 or more teams, and click the "Calculate Expectations" button. We see that this reduces expected profit by $3.66, expected growth by $0.04, expected bankroll after 17 weeks by $77.20, and median bankroll after 17 weeks by $0.74. It's left for each user to make hiw own determination as to whether he deems the reduction in number of bets placed worth the reduction in expectation.