Boundary-Induced Apparent Risk Aversion in Nonergodic Multiplicative Growth

Observed risk-taking behavior is often rationalized through expected-utility curvature, yet the curvature required to fit choices in one context can differ sharply from the curvature required in another, a tension highlighted by calibration critiques of expected-utility theory. Finite multiplicative systems often cease to evolve when a lower continuation threshold is reached, whereas standard growth-optimal benchmarks assume uninterrupted continuation. We study a finite-horizon binary multiplicative process in which a fixed exposure is chosen ex ante and paths crossing an absorbing boundary are assigned a residual value. Exact lattice propagation yields the optimal exposure as a function of initial log distance to the boundary, horizon, and residual ratio. Costly absorption compresses exposure below the no-boundary Kelly fraction near the boundary. When interpreted through an unconstrained constant-relative-risk-aversion benchmark, this compression appears as elevated risk aversion. As the residual value approaches the boundary, a local above-Kelly reversal can occur. In this minimal finite-horizon setting, absorbing-boundary geometry is therefore sufficient to generate state-dependent risk-averse-looking behavior without heterogeneous primitive preference parameters.