Geometric compounding & volatility drag

Definition

A portfolio lives one path, not an average of paths — and over that path, wealth compounds GEOMETRICALLY: what matters is the product of return relatives, not their sum. The consequences are arithmetic, not opinion: a −50% year requires +100% to recover; volatility itself taxes growth (geometric mean ≈ arithmetic mean − σ²/2, the VOLATILITY DRAG); and the sizing that maximizes long-run growth (the Kelly fraction — risk-kelly-criterion) is finite, above which MORE risk produces LESS terminal wealth. This entry is the philosophy underneath every sizing rule in pillar 7 — the reason position sizing is not a detail.

How it works / structure

• The core identities: terminal wealth = Π(1+rᵢ) — order-independent but ruin-sensitive (one −100% zeroes the product forever, the formal ground of every never-risk-ruin rule); the drag approximation g ≈ μ − σ²/2 (two assets with equal average returns and different volatility compound differently — the quiet one wins; documented in the leveraged-ETF reset record, instrument-leveraged-inverse-etf, where the drag is the product’s documented decay).
• The Kelly consequence (MacLean-Thorp-Ziemba): growth as a function of sizing is a hill — maximum at the Kelly fraction, ZERO again at roughly twice it; oversizing is not aggressive, it is mathematically self-defeating; estimation error makes practical sizing fractional-Kelly (the documented industry convention, and pillar 7’s justification for conservative defaults).
• The rebalancing/diversification connection: combining imperfectly correlated assets lowers σ² at similar μ — diversification is a GROWTH strategy, not just comfort (port-diversification-math’s deepest argument); volatility-managed exposure (risk-volatility-targeting) is drag management restated.
• The time-horizon honesty (Markowitz’s analysis): the geometric-mean criterion dominates for long horizons but is not a universal utility — short horizons and consumption needs change the objective; the KB uses growth-optimality as the long-run frame, with drawdown budgets (risk-max-drawdown-budget) as the binding short-run constraint.

When it applies

Every sizing decision (the philosophy IS pillar 7’s foundation); leverage evaluation (drag grows with the square of exposure — the arithmetic that retires most leverage proposals); strategy comparison (Sharpe-equal strategies with different vol compound differently); multi-year simulation design (the engine simulates paths, not averages, for exactly this reason).

Risk profile & failure modes

• Arithmetic-mean seduction (the signature error): evaluating strategies by average return ignores the σ²/2 toll — high-vol strategies look better in expectation tables than in compounded ledgers; the platform reports geometric outcomes.
• Oversizing via optimism: Kelly computed on estimated (optimistic) edges lands past the peak — the documented LTCM-family failure (episode-ltcm-1998); fractional sizing prices the estimation error.
• Ruin-blindness in fat tails: drag formulas assume no absorbing barrier — gap risk and leverage create one; the geometric frame makes ruin infinitely costly, which is the formal case for scenario floors (risk-scenario-analysis).
• Misapplied universality: short-horizon, liability-driven, or option-shaped objectives are not growth-optimal problems — the frame is stated per mandate.

Evidence & limits

The identities are mathematics; MacLean-Thorp-Ziemba compiles the growth-criterion literature including its critiques (Samuelson’s utility objections — labeled); Markowitz (1976) carries the long-run case. Nothing here is empirical claim — it is the arithmetic frame the platform’s risk entries instantiate.

Falsifiable-thesis examples

Illustrations only, not signals:

• “Across the strategy library, realized geometric return tracks μ − σ²/2 within measurement error (drag-identity audit)” — falsified by the ledger regression.
• “Half-Kelly sizing outperforms full-Kelly on terminal wealth in replays where edges are estimated with realistic error (estimation-robustness check)” — falsified by the paired simulation.

Cross-references

• The sizing machinery it grounds: risk-kelly-criterion, risk-fixed-fractional, risk-volatility-targeting
• The constraint partner: risk-max-drawdown-budget
• The growth case for diversification: port-diversification-math
• The cautionary record: episode-ltcm-1998, instrument-leveraged-inverse-etf