Objective and subjective entropy measures of portfolio suboptimality

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.