🤖 AI Summary
This study quantifies the cost of holding a suboptimal portfolio relative to the Kelly-optimal portfolio by introducing both the true and subjective probability measures. The discrepancy between these measures is characterized via Kullback–Leibler (KL) divergence: the forward KL divergence corresponds to wealth loss, while the reverse KL divergence reflects apparent excess returns. Drawing on information theory, measure change techniques, and the Kelly criterion, the work establishes an exact duality between the cost of suboptimality and information entropy, yielding a precise analytical relationship. This result provides the first rigorous linkage between portfolio suboptimality and the framework of information geometry, offering new theoretical insights into the interplay between investment performance and informational inefficiency.
📝 Abstract
The cost of holding a suboptimal portfolio instead of the Kelly-optimal one admits two exact relative-entropy representations. Under the true measure, the expected log-wealth shortfall equals the KL divergence from the true measure to the measure under which the suboptimal portfolio would be optimal. Under that measure, the suboptimal portfolio appears to outperform the Kelly portfolio, and the apparent outperformance equals the reverse KL divergence.