How to use Kelly's formula to manage rates

1) Kelly criterion intuition

Kelly chooses the bankroll share of the bet so as to maximize the average logarithm of capital growth (long-term growth rate).

The idea is simple: if the EV rate has> 0 (real advantage), too small a share is slow growth, too large is a high chance of deep drawdown and bankroll "transplantation"; Kelly is looking for balance.

💡 Important: Kelly does not create an advantage. If there is no advantage (EV≤0), the optimal share is 0%. In classic casino games without special conditions, Kelly is not used.

2) Binary bet (one win/lose outcome)

Let the decimal factor 'k', the net payout 'b = k − 1', your estimate of the probability of winning 'p', losing 'q = 1 − p'.

Full Kelly:

[
f^;=; \frac{b p - q}{b};=; \frac{k p - 1}{k - 1}
]

where (f ^) is the bankroll share of the bet.

If (k p\le 1) ⇒ (f ^\le 0) ⇒ we skip the bet.

If (k p> 1) ⇒ (f ^> 0), there is a positive expectation.

Example: k = 2. 10, p = 0. 52.

(k p - 1 = 2. 10×0. 52 − 1 = 0. 092).

(f^ = 0. 092 / (2. 10−1) = 0. 092/1. 10 ≈ 0. 0836 = 8. 36%) bankroll.

In practice, they play fractional Kelly: ½ → ~ 4. 2%, ¼ → ~2. 1%.

3) Why fractional Kelly is commonly used

The full Kelly is optimal with perfectly accurate probabilities and unlimited betting. In reality:

A p-score error (even by a couple of pp) can turn a plus into a minus.
Yield volatility at full Kelly is high; drawdowns are psychologically difficult.
Bookmaker/exchange limits, commissions and taxes reduce the actual edge.

Practice: ½ Kelly or ¼ Kelly give a better "recyclability" advantage with less drawdowns.

4) Alternative forms and rapid tests

EV test: bet makes sense if (k p> 1).

The shape through the "overlay" (edge): (e = k p - 1). Then (f ^ = e/( k-1)).

American coefficients: convert to decimals, then apply the formula.

Fractional coefficients a/b: (k = 1 + a/b).

5) Multiple events and correlation

If you have several simultaneous bets, the correct Kelly is a portfolio optimization problem (vector version), where the covariances of the results are taken into account. Euristics:

With independent rates, you can distribute the bankroll proportionally to each (f_i^) and make sure that the sum of the shares does not exceed 1 (conservatively).
With correlated (for example, bets in one match), scale the shares down (for example, ½ -Kelly per portfolio) or take into account the relationship of events (one goal affects the total and outcome).

6) Practical scale for positive markets

Weak edge (1-3%): Kelly's ¼ or less.

Average edge (3-7%): ¼ - ½ Kelly.

Strong edge (> 7%): Kelly ½ maximum; complete - rarely and with high confidence in the model.

High variance of the outcome (for example, "outlets," express): reduce the proportion even more.

7) Risk, drawdowns and "geometric" growth

Kelly maximizes the geometric average of growth. This is not the same as maximizing the chance to "earn tomorrow."

Typical observations:

Full Kelly gives deep, but less often drawdowns (for example, − 30... − 50% are possible).
Kelly's ½ reduces drawdowns by about 1. 5-2 times with moderate loss of growth rate.
If your risk profile is conservative, start with ¼ Kelly.

8) Limitations and hygiene assessments

1. Data → Probability → Model. p - not an opinion, but the result of the calculation (statistics, regressions, beyes, market spreads, news injection, etc.).

2. Conservatism: "cut" p in favor of the market (regularization).

3. Sensitivity test: check (f ^) at p ± 2-3 pp If the sign changes, the rate is fragile.

4. Consider the costs: fees, currency conversions, taxes reduce (e = k p - 1) and (f ^).

5. Operator limits: if the maximum allowed rate is less (f ^\cdot BR), use the available one, do not forcibly average by "catching up."

9) Examples "from and to"

Example A: light value per total

Score p = 0. 54 (54%), k = 1. 95.

(e = 1. 95×0. 54 − 1 = 0. 053) (5. 3%).

(f^ = 0. 053/(1. 95−1) = 0. 0558 \approx 5. 6%).

We play ¼ Kelly ≈ 1. 4% BR.

Example B: Strong overlay

p = 0. 60, k = 2. 05.

(e = 2. 05×0. 60 − 1 = 0. 23) (23%).

(f^ = 0. 23/(1. 05) ≈ 21. 9%).

It is realistic to take Kelly's ½ ~ 11%, given the risk and possible correlation with other rates.

Example C: Casino Game (EV <0)

European roulette: k = 2. 00 to "red," p=18/37≈0. 4865.

(k p − 1 = 2×0. 4865 − 1 = −0. 027) ⇒ (f^<0).

Kelly says: do not bet.

10) Kelly and the Express (Multistarts)

Express = coefficient product; margin and variance grow, and real p is often overestimated by the player.

Recommendation:

Either decompose the express into single bets and apply Kelly to each, or apply a strongly fractional Kelly to the express (⅛ and less) if confident in the joint probability of outcomes.

11) Operational implementation algorithm

1. Collect the data and construct a probability model p (including regularization).

2. Clear fees/taxes; get an effective k.

3. Filter value: take only markets with (k p> 1).

4. Fraction calculation: (f ^ = (k p − 1 )/( k − 1)).

5. Fractional Kelly: multiply (f ^) by ¼ - ½.

6. Limits: daily risk ceiling (for example, total ≤5 -8% BR), limit per bet, anti-correlation rules.

7. Log: fix p, k, f, result; calibrate the model regularly.

8. Pause during the series: if you observe atypical drawdowns, check the p calibration and costs, temporarily reduce the fractions.

12) Frequent errors

Kelly at the EV≤0. This is an accelerated road to drawdown.

Revaluation of p. Optimism in probabilities is the main reason for the "paper" plus and real minus.

Ignoring correlation. Multiple bets on a single event add up the risk.

Complete Kelly with no experience. Psychologically difficult and requires a large sample.

Violation of operator limits. Catching up for the sake of an "even share" breaks discipline.

13) Mini cheat sheet

Condition value: (k p> 1).

Full Kelly: (f ^ = (k p − 1 )/( k − 1)).

Working share: ¼ - ½ Kelly.

Total daily risk: ≤5 -8% BR (benchmark).

When in doubt about p: cut f twice.

The Kelly criterion is a tool for scaling the preponderance, not a way to create it. He answers the question "how much to bet" when you have already proved to yourself that the bet is positive. In real work, fractional Kelly wins plus discipline: neat fractions, accounting for costs and correlations, risk limits, and constant recalibration of probabilities. So Kelly turns from a beautiful formula into a practical bankroll management system.