Even when you know which investment has the highest expected return, betting everything on it is a losing strategy. The reason has less to do with finance than with biology.
This paper extends the Brennan–Lo evolutionary framework to continuous-time asset allocation. The move that does the work is small but consequential: we shift the unit of natural selection from investors to individual dollars. Once you do that, the growth-optimal strategy stops looking like a single best bet and starts looking like a deliberate spread — a Kelly–Merton deterministic core, plus a stochastic term whose size is governed by how well the investor's beliefs line up with the market's actual randomness.
The headline result is asymmetric. Misaligned beliefs destroy roughly three times what aligned beliefs create. And the spread around Kelly is always positive over any finite horizon — meaning some randomization is always optimal, even when you think you know the answer.
The puzzle
Optimization tells you what to hold. It does not tell you why a market populated by selection pressure should produce investors who hold it. The Kelly criterion and Merton's portfolio rules both fall out of clean maximization problems, but the maximization is assumed, not explained. They describe the behavior of an investor who already has the right objective. They are silent on why that objective should survive.
Brennan and Lo asked the harder question. Run a population through natural selection and watch what comes out. Their answer has a sharp edge: when the shock is systematic — when everyone is in the same boat at the same time — a fixed, deterministic rule does not survive. A bad enough draw eventually ruins anyone holding it. What survives instead is randomization. Behaviors that look irrational to an individual turn out to be optimal for the population.
A single risky asset is the purest case of a systematic shock: every investor is exposed to the same Brownian increment at the same instant. Taken at face value, this seems to rule out any deterministic portfolio rule at all — which would leave Kelly and Merton with no evolutionary footing. That is the gap. The resolution is not to escape the systematic risk. It is to find the right population.
Shifting the unit of selection
In the Brennan–Lo model, a population of organisms faces a binary choice, and each choice yields a random number of offspring. Selection maximizes the geometric growth rate of the population. The crisp result is a dichotomy: idiosyncratic shocks favor deterministic, individually-optimal behavior; systematic shocks favor randomization. The natural reading maps organisms to investors. Under that reading, the single-asset problem stalls, because the shock is fully systematic.
The pivot is to read the same logic one level down. The dollars inside a portfolio are themselves a population, and selection acts on them exactly as it acts on organisms. Dollars are to the investor what investors are to the market. Each dollar faces the same systematic increment, so the Brennan–Lo conclusion for systematic environments applies without modification: each dollar should randomize. Every dollar \(i\) draws its allocation independently from a Gaussian — a biased coin flip:
The mean is the Kelly weight; the variance is set by the horizon \(\mathrm{d}t\). Averaging over independent draws, the law of large numbers returns the familiar portfolio weight — not as an assumption, but as the aggregate of optimal dollar-level randomization.
The reframing is what changes the answer. At the investor level the systematic shock is a dead end. At the dollar level it is the precondition for the result. The same selection logic, pointed at the right population, turns an impossibility into a derivation.
What the math says
Maximizing instantaneous log-portfolio growth yields an optimal allocation in two parts:
The first term is the Kelly weight: excess return \(\mu_{e}\) divided by variance \(\sigma^{2}\). In the continuous-time limit this is also Merton's fraction under log utility. The second term is stochastic — it tracks the Brownian innovation at every instant. The optimal allocation is therefore not a number but a process: a fixed deterministic core with a randomizing term layered on top.
That stochastic term is where the investor's beliefs enter. Each dollar's allocation is a draw from a Gaussian centered exactly at the Kelly weight, with a variance set by the investment horizon. The center never moves. What changes with the investor's information is how that randomization is oriented relative to the market's true innovation — and what changes with the horizon is how wide the draw is. The practical instruction is not "find the single best leverage and hold it." It is: hold Kelly on average, and randomize around it by an amount the horizon dictates.
The 3× asymmetry
Intelligence, in this framework, has a precise definition: the correlation \(\rho\) between the investor's belief about market randomness and the market's actual innovation. An investor with \(\rho = 1\) reads the market perfectly. An investor with \(\rho = -1\) is perfectly, exactly wrong. Substituting the optimal allocation back into the growth dynamics produces an intelligence correction \(\rho - \tfrac{1}{2}\):
That correction runs from \(-\tfrac{3}{2}\) at \(\rho = -1\) to \(+\tfrac{1}{2}\) at \(\rho = 1\). The interval is not symmetric around zero. Perfect alignment buys a growth premium of \(\tfrac{1}{2}\). Perfect misalignment costs \(\tfrac{3}{2}\). Being wrong is three times as expensive as being right is valuable. This is the continuous-time, quantitative version of the Brennan–Lo observation that subpopulations with negative intelligence simply do not survive: the perfectly misaligned investor leans against every market move and compounds the loss at every instant.
The asymmetry has a structural reason. Building positive ρ takes information, processing, and execution — all rate-limited, all costly. Achieving negative ρ requires none of that; you can be exactly backward instantly and for free. The upside of intelligence is bounded and hard-won. The downside is unbounded and cheap. For anyone weighing their own confidence, that is the asymmetry to sit with: the penalty for conviction in the wrong direction dwarfs the reward for conviction in the right one.
Implications for real portfolios
The framework also dissolves a long-standing puzzle. Practitioners have long noted that Kelly dominates in the long run but can underperform painfully in the short run, and have mostly treated this as an unavoidable fact. The resolution here is that there are two distinct objects, not one:
\(\ell^{*}\) maximizes instantaneous growth and keeps the randomization term. \(\ell^{**}\) is the pure Kelly weight — only the limit of \(\ell^{*}\) as the horizon goes to infinity, a horizon no investor reaches. The profession has been applying the long-run rule where the short-run rule belongs.
That randomization term is a hedge, and its size is governed entirely by the horizon through \(\operatorname{Var}[\ell^{*}_{i}] = 1/(\mathrm{d}t\cdot\sigma^{2})\): shorter horizon, wider spread, larger hedge; longer horizon, tighter spread. The center stays at Kelly regardless. This produces a clean separation between preference and mechanics. The investor chooses the horizon — that choice belongs to goals, mandate, and life stage. Once the horizon is fixed, the optimal hedge follows mechanically from the horizon and the volatility. Nothing else is discretionary.
The framework also reads the coexistence of indexers, active managers, and quantitative strategies as an equilibrium rather than a mistake. Each sits at a different point on the cost-benefit frontier of intelligence and a different horizon-specific edge. A higher-cost strategy is suboptimal only relative to a cheaper cost structure that may not be available to the investor pursuing it. No single strategy is positioned to drive the others extinct. The practical posture for an investor or a risk team is therefore conservative by construction: size the hedge to the horizon, treat conviction asymmetrically, and do not mistake the long-run weight for what to hold today.
Read the full paper
The working paper develops the framework in full, including the continuous-time derivation, the closed-form solution for the optimal randomization term, and the comparative statics around belief alignment.