By Tom Pohlschmidt, Shihao Zhu
Portfolio decisions are often made under uncertainty about expected returns, and investors revise their beliefs as new information arrives. Although richer information generally cannot reduce optimized value ex ante, the trading strategy it induces need not improve on the strategy induced by a less informative information structure in every fixed-drift scenario. We introduce scenario-conditional expected utility and value of information to evaluate strategies selected under the investor's current belief. In a finite-horizon Gaussian portfolio choice model, we compare a fixed-belief investor, a price observer, and an attentive investor who also processes an auxiliary signal under an attention capacity constraint. Ex ante, price learning and auxiliary attention are valuable, and the total value of attention increases with capacity. Scenario-conditionally, learning creates a correction benefit by reducing the initial trading-position gap relative to the full-information strategy, but also a fluctuation cost through belief revisions, so the ex ante ranking can reverse. We derive the current-belief distribution of scenario-conditional value and the associated underperformance probability, defined as the probability of negative scenario-conditional value relative to fixed-belief choice. Over sufficiently short horizons, the scenario-conditional value of learning is negative when belief error is small relative to posterior uncertainty. For a given horizon, sufficiently high attention capacity yields a higher scenario-conditional value than any fixed finite-capacity benchmark across all drift scenarios. We also derive sharp convergence rates to full-information performance under Shannon and Tsallis technologies.
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