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When placing a sports or race betting wager you are always confronted with two decisions:
(1) Which selection(s), if any, to back; and
(2) How much to wager on the selection(s) (i.e. how much to stake)
While the first decision plays a massive role in determining your long-term performance, the second decision is almost as important because a disciplined staking plan will help you to avoid losing money rapidly if you don’t have an edge over the bookmaker.
A staking plan, also referred to as bankroll management, is a formal methodology for choosing your stake amounts when placing bets. If you have $100 in your account and you’re dead certain a 1.50 odds selection will win, should you wager $1, $5, $20, $100?
The Introduction to Staking Plans outlined strategies such as Level Stakes, Percentage Stakes, Square Root Stakes, the Martingale and Reverse Martingale Strategies. Staking Plans with Simultaneous Bets then outlined how to apply them when placing simultaneous wagers.
Limitations of Staking Plans Such as Percentage Stakes and Square Root Stakes
The problem with simple staking strategies is they don’t take into account whether you believe one wager is better value than another. They also don’t take into account bookmaker odds. The would recommend the same stake regardless of whether the odds are 1.01 or 101.00.
Due to these limitations we recommend you use the stake calculations to determine the maximum stake you would consider wagering rather than the actual stake. You could then discount the stake you actually apply based on additional information.
Modified Percentage Stakes
We will illustrate how to modify Percentage Stakes for bookmaker odds and/or bet ratings. A similar approach can be applied to other betting systems.
This plan simply involves setting a percentage of your account balance to wager on any given selection. For example, if your balance is $100 and you set a stake percentage of 5%, your next stake will be 0.05 * $100 = $5.00.
Bookmaker Odds
To adjust the stake amount for the level of risk, you could simply divide the maximum stake amount by the decimal odds of the wager. Suppose you have a $100 balance and are using Percentage Stakes with a 5% maximum stake. If you place a bet at 2.50 odds you would wager $100 * 0.05 * (1/2.50) = $2.00.
Now suppose you have a maximum risk limit of 20% for all wagers. If you placed five simultaneous bets with the following array of odds: [2.50, 1.50, 1.02, 8.00, 35.00], you would calculate your wager amounts using the following approach:
- Calculate the actual wagers based on the bookmaker odds [$5/2.50, $5/1.50, $5/1.02, $5/8, $5/35] = [$2.00, $3.33, $4.90, $0.64, $0.14]
- Compare the sum of these stakes to the maximum total stake limit of $100 * 0.20 = $20.00.
The total is $2.00 + $3.33 + $4.90 + $0.64 + $0.14 = $11.00, which is below the limit, so there’s no problem. If the total exceeded the limited you could instead take the following approach:- Calculate the maximum limit per wager using your total risk limit: $100 * 0.20 / 5 = $4. Because this is less than the $5 maximum for a single wager the lower limit of $4 applies.
- Calculate the actual wagers based on the bookmaker odds and the lower stake limit:
[$4/2.50, $4/1.50, $4/1.02, $4/8.00, $4/35.00] = [$1.60, $2.67, $3.92, $0.50, $0.11]
Value Rating
Likewise, you could adjust each bet by multiplying the maximum stake amount by the bet rating divided by the maximum rating attainable. Note that deriving a ratings system is beyond the scope of this article, although it would work as a crude alternative to the Kelly Criterion.
To illustrate, suppose you have a system where you rate your potential bets based on your perceived value of the odds. You set a maximum rating of 10 to denote fantastic value and a minimum rating of 1 to denote just enough value to be worthwhile betting. Suppose you have a $100 balance and are using Percentage Stakes with a 5% maximum stake. If your bet has a rating of 6 you would wager $100 * 0.05 * (6 / 10) = $3.00. Note that for a maximum rating of 10 you would wager the full $100 * 0.05 * (10 / 10) = $5.00.
Now suppose you have a maximum risk limit of 20% for all wagers. If you placed five simultaneous bets with the following array of ratings:
[2, 4, 6, 8, 10], you would calculate your wager amounts using the following approach:
- Calculate the actual wagers based on your bet ratings [$5*2/10, $5*6/10, $5*6/10, $5*8/10, $5*10/10] = [$1.00, $2.00, $3.00, $4.00, $5.00]
- Compare the sum of these stakes to the maximum total stake limit of $100 * 0.20 = $20.00.
The total is $1.00 + $2.00 + $3.00 + $4.00 + $5.00 = $15.00, which is below the limit, so there’s no problem. If the total exceeded the limited you could instead take the following approach:- Calculate the maximum limit per wager using your total risk limit: $100 * 0.20 / 5 = $4. Because this is less than the $5 maximum for a single wager the lower limit of $4 applies.
- Calculate the actual wagers based on your bet ratings and the lower stake limit:
[$4*2/10, $4*6/10, $4*6/10, $4*8/10, $4*10/10] = [$0.80, $1.60, $2.40, $3.20, $4.00]
Bookmaker Odds and Value Rating Combined
There’s no reason why you can’t combine bookmaker odds with the value rating to discount your maximum theoretical wager. Suppose you have a $100 balance with a 5% maximum stake. For a wager with 1.90 odds and a value rating of 8 you would wager $100 * 0.05 * (8/10) * (1/1.90) = $2.11.
Now suppose you have a maximum risk limit of 20% for all wagers. If you placed five simultaneous bets with the following array of odds:
[2.50, 1.50, 1.02, 8.00, 35.00] and the following array of ratings: [2, 4, 6, 8, 10], you would calculate your wager amounts as follows:
- Calculate the actual wagers based on the bookmaker odds and value ratings:
$5.00 * (2/10) * (1/2.50) = $0.40
$5.00 * (4/10) * (1/1.50) = $1.33
$5.00 * (6/10) * (1/1.02) = $2.94
$5.00 * (8/10) * (1/8.00) = $0.50
$5.00 * (10/10) * (1/35.00) = $0.14 - Compare the sum of these stakes to the maximum total stake limit of $100 * 0.20 = $20.00. The total is $0.40 + $1.33 + $2.94 + $0.50 + $0.14 = $5.31, which is below the limit, so there’s no problem. If the total exceeded the limited you could instead take the following approach:
- Calculate the maximum limit per wager = $100 * 0.20 / 5 = $4. Because this is less than the $5 maximum for a single wager the lower limit of $4 applies.
- Calculate the actual wagers based on the lower stake limit:
$4.00 * (2/10) * (1/2.50) = $0.32
$4.00 * (4/10) * (1/1.50) = $1.07
$4.00 * (6/10) * (1/1.02) = $2.35
$4.00 * (8/10) * (1/8.00) = $0.40
$4.00 * (10/10) * (1/35.00) = $0.11
The Kelly Criterion
In their basic form, staking plans such as Percentage Stakes and Square Root Stakes do not account for your perceived value of each wager. They wager the same amount regardless of whether you think a selection with 2.00 odds has a 55% chance or a 70% chance of winning. We have introduced simple odds-adjusted and rating-adjusted versions of these plans, but for those who want something with firmer academic grounding, the Kelly Criterion, discussed in the next section, moderates your wager amount based on the disparity between the bookmaker odds and your perceived probability of the selection winning.
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