Sharpe is scale-invariant. Stop trying to make it not.

A sibling note observed that long-only-buys-asymmetric-exposure — the long-only arm of the 15-arm harness had 2.5× the per-seed Sharpe dispersion of the long-short baseline. The discipline rule named there: “compare stdev across seeds, not just Sharpe.”

A natural follow-up: scale the long-only’s gross_leverage down to ~0.71 so its per-period vol matches the long-short arm at 2× gross. Then Sharpe is comparable, right?

No. Sharpe is scale-invariant under gross_leverage. The arm verifies it directly:

long_only (gross=1.0):        sharpe +1.881, final $+147K, mdd $-36K
equal_risk_long_only (0.71):  sharpe +1.881, final $+104K, mdd $-26K

Same Sharpe. Different dollars.

Why scaling doesn’t change Sharpe

Sharpe = mean(returns) / stdev(returns), annualised. When you scale the strategy’s gross exposure by a constant c:

• Daily PnL scales by c → mean scales by c.
• Daily PnL variance scales by c² → stdev scales by c.
• Sharpe = (c·mean) / (c·stdev) = mean/stdev.

The c cancels. The strategy’s Sharpe is invariant under any scalar rescaling of its position sizing — leverage, gross, notional, doesn’t matter.

This is high-school algebra dressed up as a finance result. Practitioners know it but Sharpe-vs-Sharpe comparisons sneak past the algebra all the time because the comparison feels natural when the strategies LOOK like they’re doing different things.

Where this bites in practice

Three concrete patterns:

1. “Strategy A has 2× the Sharpe of strategy B at half the leverage.” No, strategy A has 2× the Sharpe of strategy B, period. The leverage matters for dollar PnL and dollar risk budget. It doesn’t matter for Sharpe.

2. “Equal-risk Sharpe comparison.” When operators rescale strategy A to match strategy B’s volatility before comparing Sharpes, they’re not comparing anything new. They’ve just scaled the dollar outcomes — Sharpes were equal before the rescale.

3. “Better risk-adjusted returns at lower leverage.” This is nearly always wrong about Sharpe (still scale-invariant) and sometimes right about other risk-adjusted measures (Sortino, Calmar, MAR) that aren’t homogeneous in scale. If the claim is real, the operator needs to name which risk-adjusted measure, and verify it’s not just Sharpe in disguise.

What does change under scaling

• Dollar PnL: linear in scale.
• Dollar max drawdown: linear in scale.
• Calmar ratio (annual return / max drawdown): scale-invariant (same reasons as Sharpe).
• Cumulative excess return as a fraction of risk-free: scale-invariant.
• Per-period volatility in $: linear in scale.
• Probability of ruin given a fixed initial bankroll: NOT scale-invariant. Higher leverage → higher probability of drawdown breaching the bankroll. This is the right metric for the “lower leverage is safer” claim.

The harness as a teaching surface

The 15-arm A/B harness’s two long-only arms exist specifically to make this concrete. An operator who reads the output sees:

arm                    mean    stdev    min      max
long_only             +0.805   1.487   -0.930   +2.446
equal_risk_long_only  +0.805   1.487   -0.930   +2.446

Identical numbers, character-for-character. The single-seed output shows them with identical Sharpes and different dollar outcomes. That visual is more persuasive than any one-page derivation; the takeaway is built into the data.

The discipline rule

Don’t claim a Sharpe advantage from rescaling gross_leverage. If you mean “lower dollar risk for the same Sharpe,” say that. If you mean “lower probability of ruin,” compute that. Sharpe is mean over stdev; both halves of the ratio scale together.

A corollary: when comparing strategies at the same Sharpe but different leverage, the operator’s risk budget picks between them. Sharpe doesn’t.