Risk of Ruin: The Bankroll Formula Every Counter Should Know

Risk of ruin is the single most important bankroll equation in advantage play — and it's an exponential, which means small bet-size changes have outsized blowup consequences. The Schlesinger/Sileo formula, the Kelly relationship, and the working numbers for a 1% counter.

A working Hi-Lo counter has built up a $20,000 bankroll over two years of weekend trips. He's earning a published 0.8% edge with a 1-12 spread at his local 6D H17 DAS LS 3:2 game. He's been betting $25 units, generating about $10/hr, and the bankroll has grown slowly. He decides he's ready to step up — he doubles his unit from $25 to $50. The hourly EV doubles to $20/hr. So does the variance per hand. But the risk-of-ruin formula says his probability of busting the bankroll before doubling it has jumped from approximately 1.8% to approximately 13%. The bet-size doubling has made the blowup risk seven times worse, not two times worse, because risk of ruin is exponential in bet size — not linear.

This is the equation that separates a working advantage player from a gambler with a positive edge. Edge tells you what you'll average; standard deviation tells you the spread of any single session; risk of ruin (RoR) tells you whether you'll be around long enough for the edge to matter. The Schlesinger/Sileo RoR formula — published in Don Schlesinger's Blackjack Attack and refined by John Sileo for variable-bet counting — is the working tool every serious counter uses to size bets. This post is the derivation, the working numbers for typical counter scenarios, the Kelly relationship, and the four conditions where the formula understates real-world ruin risk.

The formula

For a player with a positive edge, the standard exponential approximation is:

RoR ≈ e^(-2 × bankroll × edge / (variance × bet))

Where: bankroll is in dollars, edge is the player advantage as a decimal fraction (0.01 for 1%), variance is the per-unit-bet variance of the game, and bet is the unit bet size in dollars.

For blackjack with optimal counting, the per-unit standard deviation is approximately 1.14 — so variance (which is SD squared) is approximately 1.30. The simulator in lib/strategy/risk-of-ruin.ts uses 1.14 as the STD_DEV constant for blackjack.

Substituting: RoR ≈ e^(-2 × bankroll × edge / (1.30 × bet)) ≈ e^(-1.54 × bankroll × edge / bet)

The exponential shape is the whole point

Notice that bet appears in the denominator of the exponent. Doubling bet halves the magnitude of the (negative) exponent. Halving the magnitude of a negative exponent doesn't halve the result of e^x — it SQUARES it (or roughly so, depending on the original value). That's why doubling the unit bet at fixed bankroll roughly squares the RoR: a 1.8% RoR becomes about 13%, not 3.6%.

The same shape works the other way: halving the bet at fixed bankroll roughly square-roots the RoR. A 13% RoR becomes about 3.6%. Bankroll changes are linear in the exponent and therefore have less dramatic effect: doubling the bankroll squares the (very small) survival probability — which for a counter at typical sizes just drops RoR from 1.8% to 0.03%.

Most beginner counters underestimate this shape and oversize their bets relative to bankroll. A 1% edge with a $5,000 bankroll and $25 units gives an RoR around 5% (acceptable for casual trip play). The same edge with $5,000 and $50 units gives an RoR around 22% — more than four times worse, not two times. The intuition that 'I have a positive edge, so I'll come out ahead' breaks because the timing of bust is what matters, not the long-run average. Bet small enough that you survive the variance long enough for the edge to compound.

Working numbers for the typical counter

Using the Schlesinger formula with 0.8% edge (typical 1-12 spread Hi-Lo at 6D H17 DAS LS 3:2) and per-unit SD 1.14:

$5,000 BR, $25 unit (200× bet): RoR ≈ 0.4%
$5,000 BR, $50 unit (100× bet): RoR ≈ 6%
$5,000 BR, $100 unit (50× bet): RoR ≈ 25%
$10,000 BR, $50 unit (200× bet): RoR ≈ 0.4%
$10,000 BR, $100 unit (100× bet): RoR ≈ 6%
$20,000 BR, $100 unit (200× bet): RoR ≈ 0.4%
$20,000 BR, $50 unit (400× bet): RoR ≈ 0.002% — overkill

The pattern: a 200× bet-to-bankroll ratio gives a sub-1% RoR across all reasonable scales. That's the working rule professional APs use — and it's why the published Schlesinger guideline is ≥50× max bet for any sustained play and ≥200× unit bet for trip-safe play.

TableSharp's risk-of-ruin calculator at /calc/risk-of-ruin runs these numbers for any combination of edge, variance, and bet size; the assessBankroll function in lib/strategy/risk-of-ruin.ts flags a $100 BR / $5 unit (20×) configuration as RECKLESS regardless of edge — under 20× single-bet, the RoR spike is too steep for any positive edge to compensate.

The Kelly relationship

Kelly criterion gives the bet size that maximizes long-run growth rate. The full-Kelly fraction for a favorable game is:

f* = edge / variance

For a 0.8% edge with variance 1.30, full Kelly is 0.8% / 1.30 ≈ 0.62% of bankroll per hand-unit. On a $10,000 bankroll, that's $62 per unit — meaning a 1-12 spread would peak around $750 at high counts.

Full Kelly bet sizing produces a risk of ruin around 13.5% — by construction. Kelly is the bet size that maximizes growth rate while accepting a 13.5% blowup risk; that's the price of the maximum-growth strategy. Most working counters bet at half-Kelly (cuts RoR to about 1.85%, captures about 75% of the growth) or quarter-Kelly (cuts RoR to about 0.03%, captures about half the growth).

The kellyBet function in lib/strategy/risk-of-ruin.ts defaults to half-Kelly fraction = 0.5 for this reason. Anything above full Kelly is gambling against your own math; anything below quarter-Kelly is over-conservative for a player with a known sustainable edge.

Why half-Kelly is the working sweet spot

Half-Kelly trades a small amount of long-run growth for a large reduction in blowup risk. The math: cutting bet size in half (from full Kelly to half Kelly) cuts growth rate by 25% (long-run growth is quadratic in bet size near optimum) but cuts RoR from 13.5% to 1.85% — a 7x risk reduction for a 25% return reduction. That's why almost every professional bankroll-managed strategy starts at half-Kelly and slides down toward quarter-Kelly when the edge is uncertain or the variance estimate is shaky.

Trip bankroll vs lifetime bankroll

The Schlesinger formula assumes an infinite-horizon counter who replays whenever the bankroll grows. For trip bankrolls — a fixed amount brought to a single trip with no replenishment — the relevant calculation is different. Trip RoR doesn't asymptote to zero with growth because there's no replenishment; instead, the relevant question is the probability of busting during a fixed number of hours.

Working approximation for trip play: the trip bankroll should cover the 95th percentile of expected loss across the trip's planned hours. For a 4-hour session at $50/unit with the typical 0.8% edge: expected loss is positive (player wins on average) but the 95th-percentile drawdown over 4 hours is approximately 1.645 × session SD. Session SD for 4 hours at 80 hands/hour with $50 units is roughly 1.645 × sqrt(320) × 1.14 × $50 ≈ $1,675. A $2,500 trip bankroll comfortably covers that drawdown; a $1,000 trip bankroll busts in about 1 in 4 such trips.

The expectedSessionLoss and sessionStdDev functions in lib/strategy/risk-of-ruin.ts compute these; the assessBankroll function rolls them into a SAFE / AGGRESSIVE / RECKLESS rating that's faster to apply at the felt than running the formula in your head.

Four things the formula understates

The Schlesinger exponential RoR is a tight approximation but it depends on assumptions that real play violates. Four common slippages:

Variable edge across the bet ramp. Counters bet larger at high counts (positive edge) and smaller at low counts (negative edge). The formula uses an effective average edge — but the variance at high counts is larger because the bigger bets contribute more variance per round. Practical effect: the formula slightly underestimates RoR for high-spread players. Add about 20% margin to the bankroll the formula recommends for a 1-12+ spread.
Heat and backoffs. Real RoR includes the probability that you're physically prevented from betting your computed Kelly fraction because the casino restricts your bet, asks you to flat-bet, or backs you off entirely. None of this is in the math. Practical effect: add another 10-20% margin if you're playing properties that watch counters closely.
Camouflage costs. Cover plays (occasionally taking insurance at low count, occasionally NOT splitting a pair when basic strategy says split, etc.) cost EV. The formula doesn't subtract camouflage cost from the edge input. If you spend 0.1% of edge on cover, your effective edge drops from 0.8% to 0.7% and your RoR shifts upward accordingly.
Variance estimation error. The variance of blackjack at a given spread is published but not exactly knowable for your specific play style. A counter who's actually playing closer to 1.20 per-unit SD (instead of the assumed 1.14) has materially higher RoR than the formula computes — variance enters the exponent quadratically, so a 5% variance underestimate roughly doubles the computed RoR. Pad your variance input upward or pad your bankroll input upward to compensate.

Working APs apply a 25-30% margin to whatever the bare formula prescribes. The formula gives the floor; the margin gives the survival.

The bet-sizing protocol that follows from RoR

Most working counters follow some version of this protocol:

Pick a target RoR. 1% is the professional standard for sustained play. 5% is acceptable for trip-only play.
Estimate edge (from your spread and rules) and per-unit SD (use 1.14 for typical blackjack).
Solve the RoR equation for unit bet size given your current bankroll. Add a 25% safety margin.
Bet that size until your bankroll grows by at least 50% or shrinks by 25%. Then recompute.
Tie the bet ramp (1, 2, 4, 8, 12 units across TC +1 through +5) to that unit, not to a fixed dollar amount.

This produces dynamic bet sizing — units grow as the bankroll grows, shrink as it shrinks, and the relative spread within a shoe stays constant. The bankrollSizing calculator at /calc/bankroll-sizing automates this for any (bankroll, edge, target RoR) triple; the bet-spread chart at /advanced/bet-spread shows the ramp that hangs off the unit.

What RoR doesn't tell you

RoR is the probability of going broke before reaching a goal. It does NOT tell you how long the journey takes, how variance behaves session-to-session, or what your win rate looks like over any specific horizon. For those questions, the relevant numbers are session standard deviation (which the variance calculator at /calc/variance computes) and N0 (the number of rounds it takes for cumulative EV to equal cumulative SD — i.e., the point at which you should be ahead by more than you might be unlucky behind; available at /calc/n0).

RoR alone tells you whether your bankroll size is appropriate for your bet size. N0 tells you how long it takes for your edge to manifest in dollars. Together, they're the two equations that turn 'I have an edge' into 'I have a bankroll-EV strategy.'

Run your current bankroll, bet size, edge, and variance through the calculator at /calc/risk-of-ruin to see your actual RoR — and through /calc/bankroll-sizing to back-solve the recommended unit for a 1% target. Pair both with the bet-spread analysis at /blog/bet-spread-card-counting-1-to-12 (the sibling post in this batch) to translate the per-hand bet size into the ramp you'll actually use at the table, and with /blog/wonging-vs-flat-betting to see how back-counting changes the effective edge that flows into the RoR formula. The Kelly calculator at /calc/kelly closes the loop with the bet-size-for-maximum-growth side of the same math.