Priomha Capital was founded in 2009 by a group of senior finance and gaming executives who saw the opportunity to achieve superior returns through the application of portfolio and investment theory to sports betting. Investment decisions are made “through undertaking extensive statistical and trending analysis coupled with rigorous background research and market intelligence. Additionally, through the use of computer programmers, software developers and IT specialists Priomha have developed sport-specific algorithms that are used to identify value in all markets.”

The concept of a sports betting fund is appealing because it provides investors with an alternative to traditional investments such as bonds, property and shares. A sports fund enables investors to further diversify into an area that is independent of global market events.

Due to bookmaker margins, operating a sports betting fund is incredibly challenging. If you think of bookmaker margins as a transaction cost, the transaction costs involved with sports betting are fairly significant. Also, because most bookmakers accept ‘recreational gamblers’ only, a fund must make heavy or even exclusive use of betting exchanges such as Betfair. One challenge with an exchange is liquidity. A multi-million dollar fund may be limited to the more popular sporting events due to a lack of market depth in less popular sports. Typically however, the popular events are the most scrutinised ones, which could make finding value bets difficult.

There are punters who have made millions using Betfair, so the concept of a sports betting fund is certainly plausible. Due to the lure of an investment that is immune to global market events, it seems reasonable to expect this investment area to grow. According to a Priomha Capital investor report, “PricewaterhouseCoopers have estimated that on-line sports investing revenue will reach upward of US$144 billion this year while according to Merrill Lynch by 2015 this figure will be greater than US$500 billion.”

Time will tell how well these funds perform in the long term. If a fund can make a killing on a betting exchange perhaps this will lure additional participants whose presence will lead to more efficient odds markets and therefore less value betting opportunities. This could bringing returns back down to a level that is in line with the portfolio’s risk.

The Priomha CLONEY Fund’s return of 81.6% since January 2010 is impressive, but it would be interesting to know the amount of risk being taken. Typically with investing, the higher the expected return, the higher the risk. One risk measure is the standard deviation of returns. To get a better picture of a portfolio’s performance, you need to know both the expected return and standard deviation of the fund. If you’re getting double the returns of the stock market, are you taking on less than or more than twice the amount of risk to do so?

Until more data and funds history become available, it is hard to judge how well the sports betting hedge fund industry will do. Until then, watch this space!

This post continues Part 3 of a series on the Kelly criterion and its application to sports betting. Part 1 provides an introduction to the Kelly criterion and a worked example. Part 2 provides a simple derivation. This post discusses fractional Kelly betting, where a fraction of the amount recommended by the Kelly criterion is placed on each bet.

Why consider fractional Kelly betting?

There are two popular criticisms of the Kelly criterion. The first concerns Kelly betting in general, and the second concerns the application of the Kelly criterion to sports betting.

The first criticism is the roller-coaster ride that your account balance takes when you implement a Kelly strategy. The Kelly criterion can provide optimal bet amounts that exceed 50% of the account balance, which creates a high degree of volatility.

The second criticism is that, when applied to sports betting, the Kelly criterion does not account for the uncertainty in your perceived probability of occurrence. Kelly betting can be applied to Blackjack with a high degree of certainty in the calculated probabilities. With sports betting, however, you may feel the probability of an outcome is 50%, but you do not know that with certainty. The true probability may lie between 40% and 60%, or even 20% and 80%. Applying the Kelly criterion without acknowledging the uncertainty in the probability can lead to ruin.

These criticisms have led many practitioners to adopt fractional Kelly betting, also known as partial Kelly betting. It involves taking the bet amount recommended by the Kelly criterion and multiplying it by a certain fraction. This results in less volatile returns and a lower chance of the account balance hitting zero.

The Kelly criterion formula revisited

The Kelly criterion calculates the fraction, f, of the account balance that should be placed on a bet, given the available odds and your perceived probability of winning. The formula depends on how you express the betting odds, so two versions are presented below. Version A uses the decimal odds system that is popular in Australia. Decimal odds of 2.50 mean that if you win, a $10 bet would result in a $25 payout and a $15 profit. Version B uses the “b to 1″ odds system, where odds of “3 to 1″ mean that if you win, a $10 bet would result in a payout of $40 and a profit of $30.

Version:		A		B
Formula:
Variables:		f = fraction of your bankroll to bet d = decimal betting odds p = probability of bet winning q = 1 – p = probability of bet losing		f = fraction of your bankroll to bet b = “b to 1″ betting odds p = probability of bet winning q = 1 – p = probability of bet losing

Fractional Kelly betting

With fractional Kelly betting, you multiply f by a number between 0 and 1. Hence, if we use h as our multiplier, where 0 < h < 1, we now bet h*f of our account balance instead of f. The lower the value of h, the more conservative the strategy, because you are betting a smaller percentage of your account balance. For example, a quarter Kelly strategy, where h = 1/4, is more conservative than a half Kelly strategy, where h = 1/2. The graph below provides a stylized comparison of a full Kelly versus a half Kelly and a quarter Kelly. The graph has been drawn by simulating 100 bets with odds of 2.10 and a perceived probability of occurrence (which is modeled as being true) of 50%. Observe that the full Kelly criterion results in the most volatility, but is also expected to provide the highest long-term growth.

How to implement fractional Kelly betting

With full Kelly betting, the only ambiguous variable is your perceived probability, p, of an outcome occurring. The betting odds are known with certainty, and q is simply 1 – p. With fractional Kelly betting you now have the additional decision of how much to reduce f by. An aggressive strategy is to multiply f by a number close to 1, and a conservative strategy is to multiply f by a number close to zero.

So how do you choose a value for h?

A non-technical approach is to set an arbitrary value for h, say h = 0.25, and commence Kelly betting. If after a few hundred bets your strategy has enjoyed positive results, you could increase h to 0.30 for the next set of bets, and proceed in this manner until you find a value for h that you feel comfortable with.

A more technical approach is to record for a few hundred bets over time:

1. The betting odds
2. Your perceive probability of occurrence
3. The Kelly criterion value for f
4. The result of that bet (win/lose)

For an arbitrary value of h, simulate your portfolio performance had you bet h*f on these fixtures. Excel solver can be used to find the value of h that would have maximised your account balance growth over time. Alternatively, you could calculate the value of h that, in retrospect, would have provided a level of volatility that you feel comfortable with. This value of h could then be applied, and tweaked over time, as more betting data are recorded.

Variable fractional Kelly betting

While some practitioners use the same value of h for all bets, one alternative is to vary h depending on your confidence in the chosen value of p. Recall that the Kelly criterion calculates the optimal bet amount based on the disparity between p and the probability that is implied by the betting odds (1/d). By applying variable fractional Kelly betting, your bet amount will depend both on the disparity between p and 1/d, and your confidence in the chosen value for p.

Let h equal the maximum fraction you would ever consider to multiply f by. If you set h = 1, then in some circumstances you are willing to implement full Kelly betting. Alternatively you may only wish to bet at most 0.5*f, hence h = 0.5.

Now let the variable g represent your confidence in the chosen value of p, where 0 < g < 1. g = 0 equates to no confidence, and g = 1 represents full confidence. For each bet, calculate f and g, and bet the fraction f*h*g of your account balance. If you are absolutely certain in p, then g = 1, and you would bet your maximum fraction h*f.

The weakness in this approach is it provides no guidance on how to set g. How do you measure the level of certainty in your perceived probability? You may be more confident in your knowledge of some sports than others, but how to you quantify that? If you use a betting tracker spreadsheet you could compare your historical performance across different sports. Stronger performance indicates that you are better able to assess the probabilities in that sport. One approach is to set g based on your performance percentile for that sport compared to the rest. For example, if you regularly bet on eleven sporting markets, then you could set g as follows.

Sport	Return	Rank	g
Super Rugby	3%	7	0.4
Test Match Cricket	2%	8	0.3
Premier League Football	8%	3	0.8
Men’s Tennis	6%	4	0.7
NBA Basketball	5%	5	0.6
Women’s Tennis	-3%	10	0.1
Formula 1	-4%	11	0.0
NCAA Basketball	-1%	9	0.2
NRL Line Betting	12%	1	1.0
NRL Head to Head	11%	2	0.9
AFL Line Betting	4%	6	0.4

g has been calculated above as (11 – rank)/(11 – 1). Hence, for h = 0.5, you would bet 0.4*f on Premier League fixtures and 0.3*f on NBA fixtures. The value of h could be chosen using the methods outlined in the previous section.

This post continues Part 3 of a series on the Kelly criterion and its application to sports betting. Part 1 provides an introduction to the Kelly criterion and a worked example. Part 2 provides a simple derivation. This post extends the Kelly criterion to incorporate the possibility of a refund. It is recommended that you read Part 1 and Part 2 before proceeding.

Introduction

In the Kelly betting framework, recall from Part 2 the variable p (the probability that an outcome will occur) and the variable q = 1 – p (the probability that the outcome will not occur). If you are betting on this outcome, you believe you will win with probability p, and lose with probability q.

Note that q is defined as 1 – p to ensure that p + q = 1. This implies that no other outcome is possible: you either win or lose. There are, however, bets where refunds are a distinct possibility. In this case there are three possible outcomes and p + q < 1. This post will show that only a slight tweak to the Kelly criterion is required to incorporate the possibility of a refund.

When are refunds a possibility?

Refunds are a distinct possibility for sports such as:

• Football (soccer)
  bet365 refunds all losing pre-match Correct Score, Half-Time/Full-Time and Scorecast bets in the event of a 0-0 draw.
• Tennis
  Each bookmaker has it’s own policy, but Sportsbet, among others, refunds all bets if a player retires injured. Note that Betfair does not do this, which is something to keep in mind when comparing odds.
• Boxing
  For example, Sportsbet refunded losing bets on the Green v Briggs bout.
• Promotions
  For popular events bookmakers often run promotions where they refund certain losing bets in the event of a particular outcome. Some of these refunds are announced without warning. For the 2010 FIFA World Cup winner, Sportsbet refunded all losing bets on Australia because they felt the red cards Australia received were harsh.

Notation

This post continues with the notation used in Part 1 and Part 2. Let:

• W₀ = your account balance before you make a bet
• W_n = your account balance after making n bets
• f = the fraction of your bankroll (account balance) to bet on a particular outcome
• d = the decimal betting odds for that outcome (2.50 means a winning $10 back bet would have a payoff of $25 and a profit of $15)
• p = your perceived probability of that outcome occurring (where 0 < p < 1)
• q = your perceived probability of that outcome not occurring (where 0 < q < 1 and p + q < 1)

A new variable is introduced in this post. Let:

• j = your perceived probability of receiving a refund (where 0 < j < 1 and p + q + j = 1)

Incorporating the possibility of a refund

In the Kelly betting framework we now have three possible outcomes:
- win (d-1)f with probability p
- lose f with probability q
- have the bet refunded with probability j

The account balance after making one bet can be one of three possible values:
- W₁ = W₀(1 + f*(d-1)) with probability p
- W₁ = W₀(1 – f) with probability q
- W₁ = W₀ with probability j

Recall that in Part 2 the exponential rate of growth of the gambler’s capital was expressed as:

(1/n)*log(W_n/W₀) = log[(1 + f*(d-1))^k(1 – f)^(n-k)]^1/n

The blue text represents the gross return for a win and the red text represents the gross return for a loss. Note that in the event of a refund, the account balance is unchanged, so W₁ = W₀ and W₁/W₀ = 1. If, after n bets, you had k wins, m refunds, and n-k-m losses, then this expression can be modified to be:

(1/n)*log(W_n/W₀) = log[(1 + f*(d-1))^k(1 – f)^(n-k-m)(1)^m]^1/n

The number 1, raised to any power, is simply 1 so this expression simplifies to:

(1/n)*log(W_n/W₀) = log[(1 + f*(d-1))^k(1 – f)^(n-k-m)]^1/n

Note that for a sufficiently large n, (n-k-m)/n = q. If you plug the expression above into the workings in Part 2 and solve for f, you get:

Note that when q + p = 1 this expression simplifies to the standard Kelly criterion discussed in Part 1 and Part 2. Also note that the denominator term (p + q) could be rewritten (1 – j), where j is the probability of a refund.

The formula above uses the decimal odds system that is popular in Australia. For those who are more familiar with the “b to 1″ betting odds used in other literature, the corresponding formula is:

Interpretation

The impact of a potential refund depends on whether it reduces the possibility of winning, reduces the possibility of losing, or both. With football betting on bet365, the 0-0 draw refund has no impact on the possibility of winning, but it does reduce the possibility of losing, because the refund only applies to losing bets.

In tennis betting, if the bookmaker voids and refunds all bets in the event of injury, this reduces the probabilities of winning and losing because both players have the potential to retire injured. Having said that, it’s not uncommon for players to pick up a an injury during a tournament, but continue playing. You may feel that the injured player is more likely to withdraw hurt than their opponent. This would have an asymmetric impact on your perceived probabilities of winning and losing. Another point to consider is, for someone playing with an injury, there may be a higher chance they will retire hurt if they are losing the match, than if they are winning.

The Arsenal vs Chelsea fixture revisited

Recall the hypothetical Arsenal vs Chelsea fixture used in Part 1. To recap, the odds are as follows, with your perceived probabilities of occurrence shown in brackets:

Full-time result:

Arsenal: 2.60 (p = 20% chance of occurring)
Chelsea: 2.65 (p = 50% chance of occurring)
Draw: 3.25 (p = 30% chance of occurring)

Based on the standard Kelly criterion, you should bet
f = [0.5(2.65 - 1) - 0.50]/(2.65 – 1) = 0.197 = 19.7% of your account balance on Chelsea.

Now pretend the bookmaker refunds all losing full-time bets in the event of a 0-0 draw. Suppose you believe there is a 5% chance of a 0-0 draw, and you believe the outcome probabilities are now:

• Arsenal: 2.60
  p = 20% chance of winning, q = 75% chance of losing, j = 5% chance of a refund
• Chelsea: 2.65
  p = 50% chance of winning, q = 45% chance of losing, j = 5% chance of a refund
• Draw: 3.25
  p = 30% chance of winning, q = 70% chance of losing, j = 0% chance of a refund

Based on the Kelly criterion, after adjusting for the possibility of a refund, you should bet
f = [0.5(2.65 - 1) - 0.45]/[(2.65 – 1)(0.5 + 0.45)] = 0.239 = 23.9% of your account balance on Chelsea.

The optimal bet size has increased in this example because the perceived probability of losing has dropped.

Coming up in Part 3d

Part 3d will discuss fractional Kelly betting.

This post continues Part 3 of a series on the Kelly criterion and its application to sports betting. Part 1 provides an introduction to the Kelly criterion along with a worked example. Part 2 provides a simple derivation of the Kelly criterion. Part 3a extends the Kelly criterion to incorporate backing and laying bets using an exchange such as Betfair. This post extends the Kelly criterion to account for the time value of money. It is recommended that you read Part 1 and Part 2 before proceeding, if you have not done so already.

What is the time value of money?

The time value of money is the concept that a dollar today is worth more than a dollar in the future. If someone offered you a choice between receiving $1,000 now and receiving $1,000 in a year’s time, you would prefer to receive the money today. This is because even if you don’t need the money now, you can earn interest on the $1,000 between now and next year.

The time value of money depends on two variables: time and interest rate. The greater the time until you receive a payment, the less that payment is worth. This is because you have to forgo additional time that the money could be sitting in an interest bearing account. Also, the greater the interest rate, the less the payment in the future is worth. This is because money in a high interest account would have grown by a larger amount in the meantime had you received it now.

In essence, the time value of money relates to the opportunity cost of the funds. By receiving $1,000 in a year’s time instead of today, you have lost the opportunity to earn interest on that money in the meantime.

Visit Wikipedia to learn more about the time value of money.

The implication of time in sports betting

With sports betting, punters have access to a wide range of betting options for events around the globe. These bets can vary drastically in terms of the time it takes for settlement to occur. Settlement is the process of determining whether your bet has won or lost. Obviously, settlement can only occur after the conclusion of the sporting fixture. Typically, most bookmakers settle your bets within minutes of the end of the sporting event. For some events, settlement occurs within a few seconds of the conclusion of the fixture.

Below is a selected list of betting options and their associated settlement dates. Note that this post was written at 2pm on July 21, 2010.

For fixtures that settle within the next week or so, the time value of money can be safely ignored. However for tournaments that are held in upcoming years, the time value of money is an important consideration.

This post will illustrate that for fixtures that settle in the distant future, the optimal bet amount drops. The conclusion is that you should bias your betting towards fixtures that conclude sooner rather than later.

Notation

This post continues with the notation used in Part 1 and Part 2. Let:

• W₀ = your account balance before you make a bet
• W_n = your account balance after making n bets
• f = the fraction of your bankroll (account balance) to bet on a particular outcome of a sporting event
• d = decimal betting odds for that outcome (2.50 means a winning $10 back bet would have a payoff of $25 and a profit of $15)
• p = your perceived probability of an outcome occurring (where 0 < p < 1)
• q = 1 – p = probability of that outcome not occurring

We will introduce some new notation for this post. Let:

• r = the annual interest rate that you could get if you invest your funds elsewhere
• t = the time, measured in years, between now and the settlement of the bet

Calculating the time value of money

If you were given the choice of receiving $1,000 today or $x,xxx in three year’s time, what future figure would make you indifferent between your two options? Most people would turn down $1,001, and most would accept $3,000. If you could invest $1,000 at an interest rate of 5%, then you would have $1,000(1.05) = $1,050 in a year’s time. If you reinvested that figure for the second year you would have $1,050(1.05) = $1,102.50 in two years time. If you reinvested that figure for the third year you would have $1,102.50(1.05) = $1,157.63 in three years time. This means you would have to be offered at least $1,157.63 to consider taking the future amount rather than the $1,000 today.

We can recalculate this figure as $1,000(1.05)³ = $1,157.63. More generally, let PV (present value) denote a payment today, and let FV (future value) denote the future payment that is of equivalent value to the PV. We can calculate the FV as FV = PV(1+r)^t. If we knew the FV and wanted to know the PV, we could calculate PV = FV/(1+r)^t. Note that this is identical to the mathematical statement PV = FV(1+r)^-t.

For those who have a stronger mathematics background, you may prefer to use an exponential interest rate to be consistent with the Kelly criterion’s objective of maximizing the exponential growth rate of your sporting account balance. In this case FV = PV*e^rt and PV = FV*e^-rt. Note that the value for r in this formula will be slightly lower than that used in the previous paragraph, because it is the continuously compounded rate.

To make the workings understandable for a greater audience, PV = FV/(1+r) will be used henceforth. Simply substitute this expression with PV = FV*e^-rt if you wish instead to use a continuously compounded rate.

The Kelly criterion revisited

Recall that in Part 2 the exponential rate of growth of the gambler’s capital is expressed as:

(1/n)*log(W_n/W₀) = log[(1 + f*(d-1))^k(1 – f)^(n-k)]^1/n

The blue text represents the gross return for a win and the red text represents the gross return for a loss. Let $x denote the dollar value of a bet that you make. In terms of cash flow timings, a losing bet can be treated as occuring immediately. You make your bet of $x, which is taken from your account balance today and never returned. If you make a winning bet, you have $x taken from your account balance today and $x(d-1) returned to your account balance at some point in the future, where d are the decimal odds of your bet. You can think of this as an investment. You deposit $x now and receive $x(d-1) in the future. This payment in the future (FV) must be discounted to get our PV using the formula PV = FV/(1+r).

When we take into account the time value of money, the gross return for winning becomes (1 + f*(d-1)/(1+r)^t)^k, while the gross return when your bet loses remains unchanged. If you plug this expression into the workings in Part 2 and solve for f, you get:

The formula above is for the decimal odds system that is popular in Australia. For those who are more familiar with the “b to 1″ betting odds used in other literature, the corresponding formula is:

Observe that as r and t increase, the value of f decreases.

The Arsenal vs Chelsea fixture revisited

Recall the hypothetical Arsenal vs Chelsea fixture that was used in Part 1. To recap, the odds are as follows, with your perceived probabilities of occurrence shown in brackets:

Full-time result:

Arsenal: 2.60 (p_A = 20% chance of occurring)
Chelsea: 2.65 (p_C = 50% chance of occurring)
Draw: 3.25 (p_D = 30% chance of occurring)

Based on the Kelly criterion, you should bet
f = [0.5(2.65 - 1) - 0.50]/(2.65 – 1) = 0.197 = 19.7% of your account balance on Chelsea.

We will now recalculate the optimal bet amount assuming that the fixture will take place: in (A): 3 day’s time, (B): 30 day’s time, (C): 1 year’s time, and (D): 3 year’s time. Assume you can get 5% per annum in an interest bearing account. Using the formula above we get:

(A): t = 3/365 = 0.008219
(B): t = 30/365 = 0.08219
(C): t = 1
(D): t = 3

f_(A) = 0.5 – 0.5(1.05^0.008219)/1.65 = 0.197
f_(B) = 0.5 – 0.5(1.05^0.08219)/1.65 = 0.196
f_(C) = 0.5 – 0.5(1.05¹)/1.65 = 0.182
f_(D) = 0.5 – 0.5(1.05^0.008219)/1.65 = 0.149

Hence, for fixtures within the next month or so, you can pretty much ignore the time impact of money unless you are making substantially large bets. However, for events that conclude more than a year from now, the time value of money becomes an important consideration.

You may find that your optimal bet size drops below zero, meaning you shouldn’t make a bet at all. Consider the 2014 FIFA World Cup. You can get odds of 4.50 with bet365 for Brazil to win the tournament. Suppose you believe that Brazil has a 1 in 4 chance of winning. Ignoring the time value of money, you would calculate your optimal bet as:

f = [(0.25)(4.50 - 1) - 0.75]/(4.50 – 1) = 0.036

This means you would bet 3.6% of your account balance on Brazil to win the tournament.

However, if you consider the time value of money, where t = 3.98 and r = 5%, your optimal bet size becomes:

f = 0.25 – 0.75*1.05^3.98/3.5 = -0.01

Hence, if the time value of money is taken into account, you would chose not to place a bet on Brazil.

Coming up in Part 3c

The next post will provide an extension to the Kelly criterion when refunds are a distinct possibility. For example, bet365 refunds losing football (soccer) bets in the event of a 0-0 draw.

This post is Part 3 of a series on the Kelly criterion and its application to sports betting. Part 1 provides an introduction to the Kelly criterion along with a worked example. Part 2 provides a simple derivation of the Kelly criterion. Part 3 in this series provides some extensions to the Kelly criterion. This post extends the Kelly criterion to incorporate backing and laying bets using an exchange such as Betfair. It is recommended that you read Part 1 and Part 2 before proceeding, if you have not done so already.

What is backing and laying?

With standard bookmakers you can only bet for an outcome to occur. Because Betfair is an exchange, rather than a bookmaker, you can bet both for and against sporting outcomes.

If you ‘back’ a bet on an outcome, you are betting that the outcome will happen. For example you could back a bet on Federer to win Wimbledon. If the odds are 3.50, a $10 bet would result in a $25 profit if he wins, and a $10 loss if he doesn’t.

If you ‘lay’ a bet against an outcome, you are betting that the outcome will not happen. For example you could lay a bet on Federer to win Wimbledon. If the odds are 3.50, a $10 bet would result in a $25 loss if he wins, and a $10 profit if Federer doesn’t win. Essentially, when you lay a bet, you act as the bookmaker. You get to keep the other punter’s money if they lose, but have to pay them at the agreed odds if they win.

The Kelly criterion is designed for making ‘back’ bets. According to the formula, you would not make a bet if f < 0. This post adjusts the Kelly criterion to account for Betfair commissions. It also provides the corresponding formula for a lay bet.

Betfair commissions

Unlike standard bookmakers, who incorporate their fees directly into the odds themselves, Betfair charges a commission fee to the winner of each bet. The commission fee is calculated as a percentage of your net winnings (profit) as follows:

Commission = (market base rate) * (1 – discount rate)

The market base rate is typically 5%. The discount rate ranges between 0% and 60%, depending the level of your account activity. The more bets you make, the greater the discount. A discount rate of 60% corresponds to a commission of 2%, hence the commission fee charged by Betfair ranges between 2% and 5% of your net winnings.

Henceforth we will denote the commission fee (as a percentage of your net winnings) as c.

Notation

This post continues with the notation used in Part 1 and Part 2. Let:

• W₀ = your account balance before you make a bet
• W_n = your account balance after making n bets
• f = the fraction of your bankroll (account balance) to bet on a particular outcome of a sporting event
• d = decimal betting odds for that outcome (2.50 means a winning $10 back bet would have a payoff of $25 and a profit of $15)
• p = your perceived probability of an outcome occurring (where 0 < p < 1)
• q = 1 – p = probability of an outcome not occurring

This post will refer to the hypothetical Arsenal vs Chelsea fixture that was used in Part 1. To recap, the odds are as follows, with your perceived probabilities of occurrence shown in brackets:

Full-time result:

Arsenal: 2.60 (p_A = 20% chance of occurring)
Chelsea: 2.65 (p_C = 50% chance of occurring)
Draw: 3.25 (p_D = 30% chance of occurring)

For simplicity’s sake, it is assumed that due to the high liquidity of the betting market, the lay odds equal the back odds for each outcome. In reality, the lay odds are always slightly above the back odds. The more popular (and liquid) the market, the closer the back and lay odds are to each other.

Example odds in a popular market:

Example odds in an unpopular market:

The Kelly criterion for backing a bet

Recall that in Part 2 the exponential rate of growth of the gambler’s capital is expressed as:

(1/n)*log(W_n/W₀) = log[(1 + f*(d-1))^k(1 – f)^(n-k)]^1/n

The blue text represents the gross return for a win and the red text represents the gross return for a loss. In Betfair with a commission fee of c, the gross return for winning becomes (1 + f*(d-1)(1-c))^k, while the gross return when your bet loses remains unchanged (no commissions are charged to the loser). If you plug this expression into the workings in Part 2 and solve for f, you get:

The formula above is for the decimal odds system that is popular in Australia. For those who are more familiar with the “b to 1″ betting odds used in other literature, the corresponding formula is:

Note that for c = 0, the above formulas collapse to the original Kelly criterion used in Part 1 and Part 2 of this series. By taking the derivative of the expression for f with respect to c you will find that as c increases, f decreases. This is because the expected payoff for the bet drops as the commission rises, which makes the bet less worthwhile. Recall the example from Part 1:

Arsenal: f_A = [0.2(2.6 – 1) – 0.80]/(2.6 – 1) = –0.300
Chelsea: f_C = [0.5(2.65 – 1) – 0.50]/(2.65 – 1) = 0.197
Draw: f_D = [0.3(3.25 – 1) – 0.70]/(3.25 – 1) = –0.011

If these odds were offered by Betfair, we need to adjust them for the commission fee using the formula presented above. For a commission fee of 5% the figures become:

Arsenal: f_A = [0.2(2.6 – 1)(1 – 0.05) - 0.80]/((2.6 – 1)(1 – 0.05)) = –0.326
Chelsea: f_C = [0.5(2.65 – 1)(1 – 0.05) – 0.50]/((2.65 – 1)(1 – 0.05) = 0.181
Draw: f_D = [0.3(3.25 – 1)(1 – 0.05) – 0.70]/((3.25 – 1)(1 – 0.05) = –0.027

Now you would bet 18.1% of your account balance on Chelsea instead of 19.7%.

The Kelly criterion for laying a bet

When you lay a bet in Betfair you receive the other punter’s bet as a payout if the outcome does not occur, but you have to pay the punter at the agreed odds if the outcome does occur. As before, let p denote your perceived probability of the outcome occurring. In this case, you will lose the bet if the outcome does occur, and win the bet if it doesn’t.

Your account balance if you win will be W₀(1 + f*(1-c)), while your account balance if you lose will be W₀(1 – f*(d-1)).

Note that we must place the following restriction on f: 0 < f*(d-1) < 1, or equivalently, 0 < f < 1/(d-1). This prevents losing an amount that is greater than your account balance. You can think of this as your budget constraint. This contrasts with the budget constraint for backing an outcome: 0 < f < 1. Note that for a low odds bet of 1.01 and an account balance of $100, you could lay a bet amount up to $10,000, which is more than your account balance. This is because your total potential liability is ($10,000)(1.01 - 1) = $100. In contrast, for high odds of 101.00, the maximum you could lay for this bet is $1, which is far lower than your account balance of $100. We use < rather than < for the upper limit to avoid the possibility of your account dropping to zero.

After n bets, k of which result in the outcome occurring (a loss for you), and (n-k) of which result in the outcome not occurring (a win for you), your account balance will be:

W_n = W₀(1 – f*(d-1))^k(1 + f*(1-c))^(n-k)

We can express the exponential rate of growth of the gambler’s capital as:

G = (1/n)*log(W_n/W₀) = log[(1 - f*(d-1))^k(1 + f*(1-c))^(n-k)]^1/n

Setting the derivative of dG/df equal to zero and solving for f gives:

The formula above is for the decimal odds system. For those who are more familiar with the “b to 1″ betting odds used in other literature, the corresponding formula is:

If the above workings are difficult to follow, I recommend re-reading Part 2 of this series.

Referring back to the Arsenal vs Chelsea example, in the absence of commission fees you would calculate f as :

Arsenal: f_A = [0.8 – 0.2*(2.6 – 1)]/(2.6 – 1) = 0.300
Chelsea: f_C = [0.5 – 0.5*(2.65 – 1)]/(2.65 – 1) = –0.197
Draw: f_D = [0.7 – 0.3*(3.25 – 1)]/(3.25 – 1) = 0.011

Note that these values are the exact opposites in sign of the corresponding figures for back bets in the absence of commission fees. For a commission fee of 5%, you would calculate f as:

Arsenal: f_A = [0.8(1 – 0.05) 0.2*(2.6 – 1)]/[(2.6 – 1)(1 – 0.05)] = 0.289
Chelsea: f_C = [0.5(1 – 0.05) – 0.5*(2.65 – 1)]/[(2.65 – 1)(1 – 0.05)] = –0.223
Draw: f_D = [0.7(1 – 0.05) – 0.3*(3.25 – 1)]/[(3.25 – 1)(1 – 0.05)] = –0.005

Based on these figures you would choose to lay a bet against Arsenal using 28.9% of your account balance. It should be immediately noted that in doing so, it isn’t 28.9% of your balance that is at risk. If Arsenal fails to win, you would receive a profit equal to 28.9%(1 – 0.05) = 27.7455% of your account balance. If Arsenal does win, you will lose (2.60 – 1)(.289) = 46.24% of your account balance. Hence, when making a lay a bet, the fraction (d-1)f of your account balance is at risk and unavailable to bet elsewhere.

Should I back a bet or lay a bet?

In the example fixture between Arsenal and Chelsea, based on the Kelly criterion, you could back Chelsea using 18.1% of your account balance, or you could lay a bet against Arsenal using 28.9% of your account balance. So which action should you take? Should you do both?

Recall that the objective in Kelly betting is to maximise the exponential growth rate of your account balance. Let’s compare the expected exponential growth rate for:

A: Making a back bet on Chelsea
B: Making a lay bet against Arsenal
C: Making a back bet on Chelsea and making a lay bet against Arsenal

In formula form, your account balance for each strategy after each possible outcome is provided below.

Outcome	Arsenal Win	Chelsea Win	Draw
Probability	pA	pC	pD
Option A Payoff	W0(1 – fBC)	W0[1 + (1-c)(dC-1)fBC]	W0(1 – fBC)
Option B Payoff	W0[1 – (dA-1)fLA]	W0[1 + (1-c)fLA]	W0[1 + (1-c)fLA]
Option C Payoff	W0[1 – fBC – (dA-1)fLA]	W0[1 + (1-c)(dC-1)fBC + (1-c)fLA]	W0[1 – fBC + (1-c)fLA]

Plugging in the values for our Arsenal vs Chelsea example gives:

Outcome	Arsenal Win	Chelsea Win	Draw
Probability	0.2	0.5	0.3
Option A Payoff	W0(1 – 0.181)	W0[1 + (0.95)(1.65)(0.181)]	W0(1 – 0.181)
Option B Payoff	W0[1 – (1.6)(0.289)]	W0[1 + (0.95)(0.289)]	W0[1 + (0.95)(0.289)]
Option C Payoff	W0[1 – 0.181 – (1.6)(0.289)]	W0[1 + (0.95)(1.65)(0.181) + (0.95)(0.289)]	W0[1 – 0.181 + (0.95)(0.289)]

Note that the numbers below will differ slightly to the values calculated using the rounded figures above. Solving gives:

Outcome	Arsenal Win	Chelsea Win	Draw
Probability	0.2	0.5	0.3
Option A Payoff	W0(0.8190)	W0(1.2838)	W0(0.8190)
Option B Payoff	W0(0.5368)	W0(1.2750)	W0(1.2750)
Option C Payoff	W0(0.3558)	W0(1.5588)	W0(1.0940)

Let v_i denote the profit/loss for outcome i. A win will be associated with v_i > 0 and a loss will be associated with v_i < 0. We can calculate the expected exponential growth rate for each strategy using the formula:

W₁/W₀ = p_A*log(1 + v_A) + p_C*log(1 + v_C) + p_D*log(1 + v_D)

Option A: W₁/W₀ = 0.2*log(0.8190) + 0.5*log(1.2838) + 0.3*log(0.8190) = 0.011
Option B: W₁/W₀ = 0.2*log(0.5368) + 0.5*log(1.2750) + 0.3*log(1.2750) = 0.030
Option C: W₁/W₀ = 0.2*log(0.3558) + 0.5*log(1.5588) + 0.3*log(1.0940) = 0.018

In this example, option B of laying a bet against Arsenal provides the greatest expected exponential growth rate of the account balance. Hence based on the Kelly criterion and the above perceived probabilities of occurrence, you should make the single bet of laying 28.9% of your account balance against Arsenal to win.

Coming up in Part 3b

The next post will provide an extension to the Kelly criterion to account for the time value of money. Time is an important consideration when the bet settlement is in the distant future.

This post is Part 2 of a series on the Kelly criterion and its application to sports betting. Part 1 of this series provides an introduction to the Kelly criterion along with a worked example. This post provides a simple derivation of the Kelly criterion, which will hopefully provide additional insight into the model. Please read Part 1 before proceeding, if you have not done so already.

The proof below uses the decimal betting odds system. To use the “b to 1″ odds system, simply replace d with b + 1 in the workings below. Please note that this is not a rigorous proof. Readers should consult Kelly’s original paper for a formal derivation of the Kelly criterion.

Notation

We will continue with the notation used in Part 1. Let:

f = the fraction of your bankroll (account balance) to bet on a particular outcome of a sporting event (where 0 < f < 1)
d = decimal betting odds for that outcome (2.50 means a winning $10 bet would have a payoff of $25 and a profit of $15)
p = your perceived probability of the bet winning (where 0 < p < 1)
q = 1 – p = probability of the bet losing

Furthermore, let W₀ denote your account balance today, and let W_n denote your account balance after the settlement of n bets. Both W₀ and W_n are in monetary units.

Derivation

Suppose you have an account balance, W₀ and have the option of betting a on a sporting outcome that pays odds of d. You believe that the probability of this outcome occurring is p. What fraction, f, of your account balance should you place on this bet?

If you win, your bet will result in a payout of f*W₀*d. This corresponds of a profit of f*W₀*(d-1). If you lose, then your bet results in a payout and profit of –f*W₀, which is a loss.

At the end of your bet, your new bankroll will be:

W₁ = W₀ + f*W₀*(d-1) = W₀(1 + f*(d-1)) if you win, and:
W₁ = W₀ – f*W₀ = W₀(1 – f) if you lose

Now suppose you make the same bet two times. If you win both bets, you end up with:

W₂ = W₀(1 + f*(d-1))(1 + f*(d-1)) = W₀(1 + f*(d-1))²

This is because your initial bankroll after the first bet is W₁(1 + f*(d-1)) instead of W₀.

If you lose both bets, you end up with:

W₂ = W₀(1 – f)(1 – f) = W₀(1 – f)²

This is because your initial bankroll after the first best is W₁(1 – f) instead of W₀.

If you win the first bet and lose the second, you end up with:

W₂ = W₀(1 + f*(d-1))(1-f)

Note that this is the same value as if you had lost the first bet and won the second. This result holds true for any series of bets in this setup. If you win x times and lose y times, the order of the wins and losses is irrelevant to the final outcome.

Observe that (1 + f*(d-1)) is the multiplier whenever a win occurs, and (1-f) is the multiplier whenever a loss occurs. So, after 8 games, if you won 5 times and lost three times you would end up with:

W₈ = W_{0(1 + f*(d-1))⁵(1 – f)³}

If we generalise this to playing the game n times and winning k times, we have:

W_n = W₀(1 + f*(d-1))^k(1 – f)^(n-k)

When we divide both sides of the equation by W₀, we have an expression for the growth of our initial bankroll.

W_n/W₀ =(1 + f*(d-1))^k(1 – f)^(n-k)

This equation gives us the growth or our bankroll after n games. To calculate the growth in the bankroll per game, we need to raise both sides of the equation by a power of 1/n. To illustrate why we do this, suppose you invested $100 that returned $144 after two periods. If you wanted to know the growth rate per period, you would calculate it as follows:

144 = 100*(1+growth)²
144/100= (1+growth)²
1.44 = (1 + growth)²

We now raise both sides of the equation by 1/n, which is 1/2 in this case. This enables us to solve for the growth rate.

1.44^1/2 = (1 + growth)^2*1/2
1.2 = (1 + growth)
0.2 = growth
growth = 20%

Hence a growth rate of 20% results in a balance of $144 after two periods.

Returning to our example:

W_n/W₀ =(1 + f*(d-1))^k(1 – f)^(n-k)
(W_n/W₀)^1/n = [(1 + f*(d-1))^k(1 – f)^(n-k)]^1/n

The Kelly criterion seeks to maximise the exponential rate of growth per game, hence we seek to maximise the log of (W_n/W₀)^1/n. We achieve this by choosing the optimal fraction of our wealth to bet: f.

In Kelly’s original paper he expressed “the exponential rate of growth of the gambler’s capital” as (pg. 919):

Note that this is consistent with our workings because a known property of logs is log(W_n/W₀)^1/n = (1/n)*log(W_n/W₀).

(1/n)*log(W_n/W₀) = log[(1 + f*(d-1))^k(1 – f)^(n-k)]^1/n

Let G = (1/n)*log(W_n/W₀). Using known properties of logs we get:

G = (1/n)log[(1 + f*(d-1))^k(1 – f)^(n-k)]

G = (1/n)[k*log(1 + f*(d-1)) + (n-k)*log(1 – f)]

Anyone who remembers their calculus will recall that if we seek to maximise a function by choosing the optimal value for f, then we need to take the first derivative of the function with respect to f, and set it equal to zero.

Note that if our perceived probability p of success is true, then in the limit as n approaches infinity, we have p = k/n and q =(n-k)/n. Recall that k is the number of successes out of n games, and (n-k) is the number of failures out of n games.

Noting that p + q = 1 and solving for f gives:

Recall that this solution is for the decimal odds system that is popular in Australia. If you use the "b to 1" odds system then the corresponding formula is.

You can take my word for it that this value for f corresponds to a unique maximum for G(f) on [0,1). A formal proof of this can be found here.

Note that if f < 0, then you would anticipate a loss by placing a bet at the available odds. Hence in this circumstance you would bet nothing, and the optimal fraction, f, equals 0. However if you had a Betfair account you may choose to lay a bet against the outcome, although you would have to recalculate f using a model that pays odds of 2.00 if the outcome does not occur, and costs you the lay odds if the outcome does occur. This extension of the Kelly criterion will be discussed in Part 3 of this series.

Coming up in Part 3

Part 3 will provide some extensions to the Kelly criterion. At this stage I may actually break it down into a number of posts. The extensions will be:
- 3a – the Kelly criterion for lay bets in Betfair
- 3b – the Kelly criterion that accounts for the time value of money
- 3c – the Kelly criterion when refunds are a distinct possibility
- 3d – fractional Kelly betting

Sources

• Kelly Criterion Explained
• Original paper by Kelly: A New Interpretation of Information Rate
• The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market
• Wikipedia – Kelly criterion

This post will serve as the first of a series to discuss the Kelly criterion. They Kelly criterion is a formula used to determine how much of your money to place on a particular gamble. The formula was derived by J.L. Kelly, Jr in 1956. The formula has a number of applications, one of which is sports betting.

This post provides an introduction to the Kelly criterion. At this stage, my intention for the rest of this series is as follows:
- Part 2 will provide a simple derivation of the formula
- Part 3 will discuss some extensions to the model

The formula

The formula calculates the fraction, f, of your account balance that you should place on a bet, given the available odds and your perceived probability of winning. The formula depends on how you express the betting odds, so I have provided two versions below. Version A uses the decimal odds system that is popular in Australia. Decimal odds of 2.50 mean that if you win, a $10 bet would result in a $25 payout and a $15 profit. Version B uses the “b to 1″ odds system, where odds of “3 to 1″ mean that if you win, a $10 bet would result in a payout of $40 and a profit of $30.

Version:		A		B
Formula:
Variables:		f = fraction of your bankroll to bet d = decimal betting odds p = probability of bet winning q = 1 – p = probability of bet losing		f = fraction of your bankroll to bet b = “b to 1″ betting odds p = probability of bet winning q = 1 – p = probability of bet losing

The betting odds are observable and easy to obtain. The key variable in this formula is the probability, p, of the bet winning. Observe that q is simply 1 – p. So if the probability of winning is 60%, then the probability of losing must be 40%. Note that this ignores the chance of a refunded bet. If you play around with the formula you will find that the greater your perceived odds of winning, relative to those implied by the bookmaker, the larger the percentage of your account you should bet. The probability implied by the bookmaker for a particular outcome is the inverse of the decimal odds. For example, 2.50 odds imply a 1/2.5 = 40% chance of winning.

A worked example

The example below uses the decimal odds system. To obtain b, if that’s what your most comfortable using, simply subtract 1 from d.

Suppose you observe the following odds for the football game between Arsenal and Chelsea.

Full-time result:
Arsenal: 2.60
Chelsea: 2.65
Draw: 3.25

You take a look at the current form of the two sides, and believe the probabilities of the respective outcomes are as follows:

Hence, in your opinion, there is a 20% chance of Arsenal winning, a 50% chance of Chelsea winning, and a 30% chance of a draw.

Note that the sum of p an q for each outcome is 100%. Also note that the sum of the values for the p’s also equals 100%. Using the Kelly criterion, you calculate the optimal fraction of your wealth to bet on each outcome as follows:

Arsenal: f_A = [0.2(2.6 - 1) - 0.80]/(2.6 – 1) = -0.300
Chelsea: f_C = [0.5(2.65 - 1) - 0.50]/(2.65 – 1) = 0.197
Draw: f_D = [0.3(3.25 - 1) - 0.70]/(3.25 – 1) = -0.011

Because the fractions f_A for Arsenal and f_D for the Draw are below zero, you would not bet on these outcomes. The fraction, f_C, for Chelsea is a positive value of 0.197. This means you should bet 19.7% of your betting account balance on Chelsea to win. If your account balance is currently $200, then you should place a $39.40 bet on Chelsea.

Note that if your probabilities for Arsenal, Chelsea and a draw were 35%, 35% and 30%, respectively. Then you would not make any bet on this event. Plug the corresponding probabilities into the formula to verify this result for yourself.

Coming up in Part 2

This post simply provides the formula and illustrates how to use it. The derivation of the formula will be discussed in Part 2. To give you some idea of the methodology, the Kelly criterion aims to maximise the log of the expected growth rate of your account balance.

Sources:

• Wikipedia – Kelly Criterion
• Original paper by Kelly: A New Interpretation of Information Rate

Like most people though, I never predicted Schiavone would win. Her backer’s would have made a fortune, because even halfway through the tournament you could have got 90-1 on her. I had bet heavily on Stosur to win the tournament using Betfair and Luxbet, so in the tournament final I hedged by bets by backing Schiavone at 5.10 odds. All up I made $150.13 on $76.79 of bets, making a $73.34 profit.

To elaborate on “greening up”, you can do this by backing an outcome at X odds, and then betting against the outcome for Y odds later, where Y < X. During the tournament I bet on Venus Williams, Stosur, Hantuchova, and Rezai. I was able to make counter bets against everyone except Rezai later on in the tournament to hedge my positions. Below is a screen shot of my position on May 30th. Basically you try to green up by picking people who you think are undervalued and who will progress further in the tournament. For example, I was able to back Hantuchova for 540-1 odds on May 26th, and by May 30th, I could bet against her at 180-1 odds. I plan to see if I can do the same for the 2010 FIFA World Cup. I have enjoyed sustained success with tennis, but have yet to try it with football.

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The main result from this article is that the further away the payout for your winning bet, the higher the odds need to be to compensate you for the delay in receiving your funds. Bookmakers typically don’t provide any compensation for this, which means you should bias your betting towards bets with more immediate payouts.

Example without time value of money

Consider a betting decision in the absence of time. Here you compare the inverse of the betting odds to your perceived probability of occurrence. Suppose the odds for the NHL fixture ‘Los Angeles Kings at Vancouver Canucks’ are as follows.
     LA Kings 2.60
     Vancouver Canucks 1.52

The bookmaker’s odds translate to an implied probability of 1/1.52 = 65.8% for Vancouver to win, and 1/2.60 = 38.5% for LA to win. Note that the probability of LA winning equates to a 1 – 38.5% = 61.5% probability of Vancouver winning. At this point, or preferably before you’ve looked at the odds, you determine your expected probability of Vancouver winning, which I will denote P. Your probability can fall within one of three ranges:
     P < 61.5%
     61.5% < P < 65.8%
     P > 65.8%

If you believe the probability of Vancouver winning is less than 61.5%, you should bet on Los Angeles. If you personally believe the probability is greater than 65.8%, you should bet on Vancouver. Finally, if you believe the true probability lies between 61.5% and 65.8%, then you should not make a bet on this fixture.

Example with time value of money

In the previous example, the outcome will be determined within a few days of placing a bet, so timing considerations aren’t important. The funds you place on the bet will unavailable for use for only a short period of time. But what if you make a bet on, say, the 2013 Rugby League World Cup? Yes, I know I’ve chosen an extreme example here, but it will illustrate the concept well. Sportsbet’s odds for the three favourites to win the tournament are as follows:
     Australia 1.20
     New Zealand 6.00
     England 8.00

These odd correspond to the implied probabilities below:
     Australia 83.3%
     New Zealand 16.7%
     England 12.5%

You should bet on Australia if your perceived odds of them winning are more than 83.3%, but this ignores the fact that you have made your money unavailable for other uses until after the tournament final in 2013. We clearly need to take this into account before making a betting decision.

How to determine the time value of money

The time value of money equates to the opportunity cost of your betting funds. It is the amount you could earn elsewhere had you not made your bet. This takes into account two things: time and interest rate.

The interest rate is what could get for your money had you chosen not to bet it. I looked at ING Direct this morning, and Australian residents can currently get 4.65% p.a. The time is the date of the expected payout (if your bet wins) minus the date on which you place your bet. Today is the 24th of April, 2010, and let’s assume the 2013 final will take place on November 24th, 2013, so that the payout will be in three years and 7 months time. Mathematically, this is 3 + 7/12 = 3.583 years from now.

We now have the two components with which to calculate the time value of money:
Interest rate = r = 0.0465
Time = t = 3.583

Using the equation we can calculate how much to discount any future payout to arrive at the equivalent value had we won the bet today. Note, to keep things simple I have only compounded the interest once per year. You can learn more about compound interest here.

In our case we get 1/(1.0465)^3.583 = 0.8497 when applying the above formula.

Now that we have determined our discount for the time value of money, we need to multiply the bookmaker odds by this value, and then recompute the implied probabilities:

     Australia 1.20 x 0.8497 = 1.0196
     New Zealand 6.00 x 0.8497 = 5.0982
     England 8.00 x 0.8497 = 6.7976

The new implied probabilities are:

     Australia 98.1%
     New Zealand 19.6%
     England 14.7%

This means that instead of betting on Australia if you believe the odds are over 83.3%, you should only consider betting if your perceived odds are over 98.1%. These required probabilities are drastically higher because if you put your money into a savings account you could get the equivalent of 1.0465^3.583 = 1.1769 odds without taking any risks! Note that 1/0.8497 = 1.1719.

General formula

Below is a general formula for comparing adjusted betting odds to your perceived probability of occurrence. Note that I have used the property (1/x)/(1/y) = y/x. Play around with the formulas to verify the result for yourself.

P_X = your perceived probability of outcome X occurring.
B_X = the bookmaker odds for outcome X
B_X^C = bookmaker odds for outcome X to not occur (where available)
L denotes “low”
H denotes “high”
NB denotes “no bet”

When you can bet either for or against outcome X, like in a hockey game, your decision is based on the comparisons below:

If P_X < P_L then you should bet AGAINST outcome X
If P_X > P_H then you should bet on outcome X
If P_X = P_NB where P_L < P_NB < P_H then you should not bet on the event

When you can’t easily bet against outcome X, like for the Rugby League World Cup winner, your decision is based on the equation below:

If P_X > P_H then you should bet on outcome X
If P_X = P_NB where P_NB < P_H then you should not bet on the event.

Closing remarks

While the Rugby League example I have used is extreme, I honestly believe that you should take into account the time value of money when making bets that are to take place more than three months away. For three months your time is 3/12 = 0.25, so you should multiply the available odds by 1/1.0465^0.25 = 0.9887 before comparing them to your perceived probability of occurrence. Looking again at our LA Kings vs Vancouver Canucks example, if the fixture was to take place three months from now, the range for which you would not bet would expand from 61.5% < P < 65.8% to 61.1% < P < 66.5%. Note that the range of perceived probabilities for which you would not bet always expands when you factor in the time value of money.

At this stage I must point out some caveats:

First, the above analysis assumes that you are 100% confident that your estimated probability is correct. In reality, there is uncertainty in your probability estimate. For example, you may guess that probability of an outcome is 60%, but believe that the true probability could lie anywhere between 50% and 70%. It turns out that the decision of whether to bet is independent of your confidence in the probability estimate. The decision of how much to bet will take into account this uncertainty. I will discuss the issue of how much to bet, which directly relates to your confidence in the estimated probability, on a later date.

Second, most sports punters are pure gamblers. Few consider withdrawing money from a sports betting account and putting into a bank account. This makes the use of a savings account interest rate as an opportunity cost irrelevant. I suppose I'm reaching out to a more rational sports betting audience here. Most likely, any decision not to bet simply frees up money to bet elsewhere. In this case your opportunity cost is actually a comparison between two possible bets. If you are constrained for funds, you could bet on event E₁ that takes place t₁ days time, or you could bet on event E₂ that takes place in t₂ days time. You would need to discount one using the expected payoff from the other to determine whether to bet on it. Another option is to look at your historical betting performance, and determine an expected return on your betting account each year. If it is 20%, then a bet on an event that pays out one year from now should have the odds discounted by 1/1.2 first. Of course, the downfall of this method is that many punters have negative earnings history, which would lead to erroneous results.

The analysis here is intended to provide food for thought. The main concept is that the further away the payout for your winning bet, the higher the odds need to be to compensate you for the delay in receiving your funds. Bookmakers typically don’t provide any compensation for this, which means you should bias your betting towards bets with more immediate payouts. This result draws from the fact that the P_NB region expands as t increases, making you less inclined to make a wager on later fixtures.