Ask ten quants which portfolio is "optimal" and you'll get two camps staring at each other across a very expensive spreadsheet. Both camps are technically correct within their own frameworks. The trouble is those frameworks are answering subtly different questions — and the practical difference in outcomes is far from subtle.
The Sharpe ratio maximises risk-adjusted return in a single-period, arithmetic sense. Geometric mean maximisation — rooted in the Kelly Criterion — asks a different question entirely: what allocation maximises the long-run compound growth rate of wealth? These sound similar. They are not. The villain separating them has a name: variance drag.
Variance drag is the wedge mathematics drives between what you earn on average and what you actually keep. The geometric mean of a return series approximates the arithmetic mean minus half the variance: g ≈ μ − σ²/2. Double the volatility and you quarter the drag term — but it compounds against you every single period. A strategy returning 20% arithmetic with 40% volatility has a geometric mean around 12%. A boring 12% arithmetic strategy with 10% volatility clocks in at about 11.5%. The Sharpe ratios might be identical. The wealth outcomes after twenty years are not.
This is why Bernstein and Wilkinson's work — and the broader Kelly Criterion literature — consistently lands on more conservative allocations than mean-variance optimisation suggests. Kelly-derived sizing penalises variance explicitly. A Sharpe-maximising framework treats two portfolios with the same ratio as equals. A geometric framework sees the higher-volatility version as quietly haemorrhaging compound return every period. For traders with multi-year horizons, that distinction is the whole ballgame. A deeper grounding in the mathematics of the Kelly Criterion, the formal definition of geometric mean returns, and the mechanics of Sharpe ratio construction reveals exactly where the two frameworks diverge — and why neither is universally superior.
The practical takeaway is straightforward: if your horizon is long and your returns compound, calculate both metrics before sizing a position. When they disagree, your geometric mean is telling you the Sharpe is flattering a volatility problem you haven't priced in yet.
Sharpe tells you how good the ride looks on paper. Geometric mean tells you how much money you actually have at the end.
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