Expected value tells you the average outcome; variance tells you how badly a single draw can hurt. A chapter on EV might say +8¢ per share; variance asks whether that edge survives one resolution when you put 22% of bankroll on one binary. Module 01’s $1 payoff ladder is the simplest variance object; wide spreads raise effective risk; real arbitrage lowers variance when profit is locked—when legs are sound.
Variance and standard deviation
For random payoff X with mean μ = E[X], variance is Var(X) = E[(X − μ)²] and standard deviation σ = √Var(X). Variance is in squared dollars; σ is in dollars and is a rough “typical” deviation from the mean, not a hard bound on worst case.
Binary YES closed form
Buy YES at c with true probability p on a $1 face: YES pays (1 − c) with probability p, NO pays (−c) with probability (1 − p). Mean payoff μ = p − c—the EV from the prior chapter.
Remarkably, Var(X) = p(1 − p) depends only on p, not entry price c. So σ = √(p(1 − p)).
At p = 0.50, σ = 0.50 per share. At p = 0.20 or 0.80, σ = 0.40. At p = 0.90 or 0.10, σ = 0.30. Extremes feel “safer” in σ per share but you still lose often when wrong.
Same p = 52%, buy at 44¢ versus 60¢: EV per share is +0.08 versus −0.08, but Var per share is unchanged at 0.2496. Cheap entry moves μ, not σ.
One Kalshi bet
Buy 500 YES at 44¢ with p = 0.52 for illustration: μ per share = 0.08, portfolio EV = $40, Var per share = 0.52×0.48 = 0.2496, σ per share ≈ 0.50, portfolio σ ≈ 500×0.50 ≈ $250. You expect +$40 but one coin flip might swing roughly ±$250—resolution dominates.
On a $10,000 bankroll, risking $2,200 (500×44¢) is 22% at stake. Win pays 500×56¢ = $2,800; loss is −$2,200. Expected return on bankroll is ~0.4% while single-bet σ on bankroll might be ~2.5%. Positive EV with an uncomfortable swing is why fractional Kelly exists.
Many independent bets
Ten unrelated contracts each with +$4 EV and σ ≈ $50 per bet: total EV = $40, total variance (independent case) adds to 10×50² = 25,000, total σ ≈ $158. EV adds linearly; σ grows with √n—diversification compresses relative volatility only when independence is real.
Arbitrage and correlation
True arb with +7¢ locked has σ near fee and resolution noise only. Different rules, partial fills, or withdrawal delay reintroduce binary-scale risk. Ten Senate contracts one election night are not ten coins—correlation makes cluster loss likely.
Binary payoff is skewed: at c = 44¢, max win +$0.56, max loss −$0.44—not symmetric around μ. At 20¢ entry, upside is 80¢ and downside 20¢; at 80¢, upside 20¢ and downside 80¢. Favorite–longshot bias interacts with how that skew feels emotionally.
Brier scoring uses squared forecast error across many events—kin to variance of beliefs. Low σ on P&L one night does not mean calibrated; track both.
Compare two +EV trades
Trade A: p = 0.55, c = 0.50, EV +0.05, σ per share ≈ 0.4975. Trade B: p = 0.25, c = 0.20, same EV +0.05, σ ≈ 0.4330. Same EV; lower σ on B—but longshot psychology differs. Choose via Kelly and correlation, not σ alone.
Practical rules
Compute μ from EV. σ per share = √(p(1−p)) times share count. Divide σ by bankroll for single-bet stress. Cut size if above your cap. Count correlated exposures. Stress unhedged arb legs. Prefer many small independent +EV bets over one huge binary when possible.
Mistakes: “+EV so max bet”; assuming independence across the same headline; thinking 10¢ price means low risk (high payoff multiple); ignoring capital locked until resolution (opportunity cost, not Var, but real).
Ruin path on one resolution
Variance matters because prediction markets settle once. A +8¢ EV bet with 22% of bankroll at risk can still lose the stake with probability (1 − p). If p = 0.52, you lose 48% of the time on that single bet—almost a coin flip—while the EV line says you are “right on average.” Sizing is how you survive the losing paths long enough for EV to express itself across many trials.
Arb as variance collapse
When Module 03 structural trades lock +7¢, variance of the package is not p(1−p) on one leg—it is operational risk: one venue voids, one leg does not fill, withdrawal stalls. Stress the unhedged scenario: if only the buy fills, you are back to full binary variance on that contract.
Standard deviation and forecast error
Low P&L volatility one week does not prove calibration. Brier and log scores measure whether your f matched outcomes across many events. Use σ for bankroll stress and Brier for belief quality—different questions, both required for a forecasting career.
Longshot psychology versus σ
A 10¢ contract has σ ≈ 0.30 per share—lower than a coin flip’s 0.50—but the payoff multiple is huge and loss is still likely if you are wrong. Low σ does not mean “safe bet.” It means the per-share swing is moderate while the narrative feels like lottery ticket or doom, depending on side.
Expected value band
Over N similar independent bets with per-bet EV μ and variance σ², total EV is Nμ and total σ is σ√N. With μ = $4 and σ = $50, ten bets give $40 expected and ~$158 typical deviation—EV positive, comfort low. That inequality drives fractional Kelly and cluster caps in the next chapters.
Time and capital lock
Variance formulas describe payoff at resolution, not calendar time. A +EV position tied up through a six-month rulemaking docket carries opportunity cost—capital that cannot redeploy. That is not Var(X), but it belongs in the same sizing conversation as σ. Short-dated binaries concentrate resolution risk; long-dated ones concentrate boredom and headline clusters.
Reading variance from the book
Wide bid–ask on Module 02 books often coincides with high event uncertainty—not the same formula as p(1−p), but correlated in practice. When σ from belief is high and the book is wide, cut size twice: the model is noisy and the execution is expensive.
Favorite versus longshot variance
At p = 0.90, σ per share is 0.30—moderate—but you lose the stake 10% of the time on a big favorite ticket. At p = 0.10, the same σ formula applies on the YES side you bought; longshots lose often with small dollars at risk but loud narrative. Pair σ with dollars at risk, not only with σ alone.
Reporting to yourself
After resolution, compare realized P&L to the μ and rough σ band you expected. Surprises outside two σ should trigger model review, not immediate size-up. Calibration and variance together separate luck from process.
Half the story without EV
Variance without positive μ is pure gamble. EV without variance awareness is reckless. Module 04 pairs them deliberately: compute p − c, then ask whether one resolution can hurt at this size. Both answers fit on one journal line.
What comes next
Next: the Kelly criterion—turning EV and variance into growth-optimal (or disciplined fractional) position size.