Value bets and the Kelly criterion: the honest maths

A value bet is a bet with positive expected value: the probability multiplied by the odds comes to more than one. If an outcome has a 50% chance and the odds are 2.10, the expected value is 0.50 x 2.10 - 1 = +0.05, so +5 cents for every euro staked. The formula fits on one line. The hard part is somewhere else - knowing the probability better than the bookmaker does. This lesson is part of our betting academy, and every bet type is collected on our sportsbooks page.

What is a value bet

A value bet is one where your estimate of the probability is higher than the probability built into the odds.

Odds of 2.10 imply a probability of 1 / 2.10 = 47.6%. That is the break-even point: if the outcome really happens more often than 47.6%, the bet has positive expected value; if less often, negative. At 50% it is +5%. At 45%, the same odds give 0.45 x 2.10 - 1 = -5.5%.

Every number in this article is an example. A true 50/50 outcome priced at 2.10 is rare in real life - it was picked for round numbers.

Probability is the hard part

The right formula gives you nothing if the probability you feed it is wrong.

Here is how the expected value per euro changes at odds of 2.10 when only your estimate changes:

• 44%: -7.6%
• 46%: -3.4%
• 47.6%: 0%
• 48%: +0.8%
• 50%: +5.0%
• 52%: +9.2%

Misjudge by four percentage points and +5% becomes -3.4%. That is how thin the line is between "I found value" and "I paid for my confidence".

A bookmaker's price is not one person's guess. It is a model corrected by the money of the whole market, with the bookmaker's margin built in. Your edge can only be the gap between your estimate and the fair price with the margin removed. How to take the margin out of the odds is the subject of the previous lesson on bookmaker margin. Football is where "value" talk lives, because there are so many markets, but the major leagues also carry some of the sharpest prices - more on our football betting page.

How to check whether you have an edge

After a few dozen bets you cannot tell an edge from luck - only after many hundreds.

Say you really do have a 5% edge at odds of 2.10 and you stake one euro every time. Each bet ends at +1.10 or -1 euro, so the standard deviation per bet is 1.05 euros - far more than the expected 5 cents.

• After 100 bets: expected result +5 euros, standard deviation 10.5 euros. About one bettor in three with this edge is still down (32%).
• After 1,000 bets: expected result +50 euros, standard deviation 33.2 euros. About one in fifteen is still down (7%).

Calculated with the normal approximation. The conclusion is simple: fifty good weeks prove nothing, and neither do fifty bad ones.

A quicker check is how the odds move. If, after your bet, the odds on the same outcome shortened before kick-off (say from 2.10 to 1.95), the market reached the same conclusion you did. If it regularly goes the other way, the "value" existed only in your head. That last price before the start is called the closing odds.

The Kelly criterion: the formula and its limit

The Kelly criterion tells you what share of your bankroll to stake if your probability is right.

The formula: f = (b x p - q) / b, where b is the odds minus one, p is your probability and q = 1 - p. With odds of 2.10 and a 50% probability: b = 1.10, q = 0.50, so f = (1.10 x 0.50 - 0.50) / 1.10 = 0.05 / 1.10 = 4.55% of the bankroll.

The formula was published in 1956 by the engineer John L. Kelly in a paper on information rate (Bell System Technical Journal, volume 35). It maximises long-run bankroll growth - but only if p is accurate.

That is why many people who use it stake half or a quarter of the Kelly fraction. Kelly is a ceiling, not a target.

When your estimate is wrong

With a wrong probability, Kelly puts a large stake on a losing proposition.

Take the same 4.55% stake, but the true probability is 46%, not 50%. The bet's expected value is now -3.4%, and the stake is more than twice the 1-2% flat stake we suggest in our beginners' guide. The mistake costs twice: a losing bet and a big one at the same time.

There is a second rule. Stake twice the Kelly fraction and long-run bankroll growth is roughly zero, even with a genuine edge. Stake more than that and the bankroll shrinks, although every single bet is positive. Overconfidence costs more than ignorance here.

Why bookmakers limit winning customers

Bookmakers set maximum stakes and can also change an individual customer's limits.

Optibet's terms define the stake limit as a limit the company sets for each event, and the maximum stake can range from 1.00 to 20,000.00 EUR depending on the event type (4.1.3). In clause 2.4.5.3 the company reserves the right to change customer limits without having to give a reason. So a bettor who keeps beating the market price can end up with a lower limit - and the "value" stays on paper.

What follows from this

A small flat stake is a more sensible route than a formula whose inputs you cannot check.

If you bet regularly, a flat stake of 1-2% of your bankroll, as in our beginners' guide, survives estimation errors far more easily. Use Kelly only as an upper bound, never as the stake itself. If you start hunting for an "edge" to win back what you lost, that is the signal to stop - our article on responsible gambling tools lists the limits and breaks Latvian sportsbooks offer.

Previous lesson: bookmaker margin and how to calculate it.

Play within your means

Sports betting is not a source of income, and no formula changes that. Set a budget before you start and stop when it is spent. Our responsible gambling page collects the tools that help, and you can add yourself to the State Revenue Service's self-exclusion register at any time. The Centre for Disease Prevention and Control runs an anonymous addiction helpline on 67037333. Betting is for people aged 18 and over.