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How to optimize bankroll in mathematics

1) First - the goal, then the formulas

Bankroll optimization begins with a clear goal:

Playtime and stability - maximum playing time in controlled drawdowns.
Winning the bonus - "live to the end" with a minimum RoR (risk of ruin).
Capital growth at EV> 0 - carefully scale the advantage.
The target determines the rate in% of BR (bankroll), the length of the session and the risk thresholds.

2) Base: how much your game "costs"

edge = 1 − RTP (in fractions).

Turnover = bet × number of games/spins.

Expected total: 'Total ≈ − edge × Turnover'.

Hence the three optimization levers: reduce edge, reduce turnover, reduce the stake share.

3) Rate as percentage of current BR

Quick benchmarks for volatility:

High volatility: 0. 25–0. 75% BR per attempt.
Average: ~ 1% BR.
Low/rates 1:1: 1-2% BR.
The logic is simple: the wider the spread, the smaller the share. This reduces the depth of typical drawdowns and the chance to "break" to the target.

4) How to schedule a session: units, stop levels, time

1. Define unit: rate = X% of current BR.

2. Stop loss/break profit:

High-Vol: − 30... − 40 %/+ 60... + 150% (of the session budget).
Medium: −20% / +30%.
3. Time/spin limit - controls the turnover (and therefore the expected "price" of the session).
4. Game speed: slower = less turnover/hour = lower expected minus/hour.

5) Risk of ruin (RoR) - practical

RoR rises with volatility and rate share. The one-size-fits-all rule: It's best to survive a long unfavorable streak on your bet.

Mini-checks:

Is there a margin of 100-300 empty backs in high-vol slots?
BR/rate ≥ 200-400 for quiet playtime (landmark).
If the goal is to live up to the bonus with an average spin distance B, the requirement is' BR/bet ≥ 2-3 × B '.

6) Bonuses and vager: optimization for survival

Cost: 'Cost ≈ Bonus × Wager × edge (allowed games)'.

To reduce RoR: take the allowed games with higher RTP and lower variance, keep the bet 0. 25–0. 5% BR, we comply with the rate/exclusion limit.

7) Fractional Kelly - only when EV> 0

If you have a confirmed advantage (sports/exchange/rare payment tables):

[
f^=\frac{k,p-1}{k-1}
]

where (k) is the decimal factor, (p) is your probability. In practice, use ¼ - Kelly ½ to reduce drawdowns and sensitivity to errors in p.

💡 When EV≤0 (classic casino games), Kelly is not used - the optimal share is 0%.

8) Rebalance and rolling interest

Optimization = rate from current BR, not fixed amount. This is automatic:

reduces the risk of drawdown (the bet falls), scales the game with the growth of BR (the bet grows).
Rebalance in time: recalculation of the rate every N spins/distributions or when BR changes by ± 10-20%.

9) Diversification by game mode

Break the session into "baskets":

Core basket (70-90%) - games/bets with moderate variance and high RTP; rate ~ 1% BR.
High-Vol basket (10-30%) - hunting for big hits; bet 0. 25–0. 5% BR, fewer auto-spins.
This way you keep the playtime and upside chance without blowing up the overall risk.

10) Withdrawal policy and "insurance fund"

Conclusion rule: fix, for example, the conclusion of 30-50% of the BR gain after a successful series. This "closes profits" and stabilizes psychology.

Insurance cushion: keep 10-20% of BR out of the game (not in the casino account). In case of failure, this is a fresh start without dogons.

11) Anti-crisis protocol (when everything goes wrong)

Caught − 20% BR faster than planned? Pause and decrease in the rate share by 25-50%.

Are empty spin series longer than expected? Slow down/auto-backs and go back to the core basket.

Growth of emotions/dogons? We stop the session - progressions only accelerate the turnover and mathematical minus.

12) Quick presets (copy)

A. Calm playtime (average volatility)

BR 200; 1% rate (2 cu); 800-1000 spins; SL −20%, TP +30%.

Turnover ~ 1600-2000; at edge 4% session price 64-80 cu.

B. High-Vol hit hunt

BR 200; bet 0. 25–0. 5% (0. 5-1 cu.); SL −30–40%, TP +60–150%; fewer auto-spins.

C. Wager × 35

BR 300; bet 0. 25–0. 5% (0. 75–1. 5 cu.); allowed games with frequent hits; rate limit control.

D. Plus sport (there is a model)

Fractional Kelly ¼ - ½ from (f ^); daily risk ≤5 -8% BR in total; avoid correlated express trains.

13) Frequent mistakes and how to fix them

Fixed amount instead of% BR. Correction: the rate is always from the current BR.

Ignore volatility. Lower% on high-vol, raise carefully on low-vol.

Progressions/dogons. Accelerate turnover and RoR - refuse.

No stop levels. Enter SL/TP and time limit.

RTP version is invalid. Look for the actual RTP of a specific game in your casino.

Hasty conclusions on the short distance. Keep a log of sessions: turnover, total, drawdown,% rate - adjust the plan for the data.

14) Optimization checklist (in 60 seconds)

1. Target: playtime/vager/EV> 0?

2. edge and RTP are known for the selected game?

3. Rate in% of current BR including volatility?

4. SL/TP and time limit set?

5. For bonus calculated 'Bonus × Vager × edge'?

6. BR rebalance ± 10-20% enabled?

7. Withdrawal policy and "insurance fund" defined?

Bankroll optimization is a system: goal → interest rate for volatility → sales control and SL/TP → rebalance and discipline. At EV <0, it makes the game predictable in risk and cost; at EV> 0 allows to scale the preponderance without self-destructive drawdowns. Math sets the framework, and you set the rules by which your bankroll lives and grows.