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. Absorbing-boundary geometry can therefore generate state-dependent risk-averse-looking behavior without heterogeneous primitive preference parameters.
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