This work introduces a comprehensive approach to the sensitivity analysis (SA) of risk-coherent inventory models. We address the issues posed by i) the piecewise-defined nature of risk-coherent objective functions and ii) by the need of multiple model evaluations. The solutions of these issues is found by introducing the extended finite change sensitivity indices (FCSI's). We obtain properties and invariance conditions for the sensitivity of risk-coherent optimization problems. An inventory management case study involving risk-neutral and conditional value-at-risk (CVaR) objective function illustrates our methodology. Three SA settings are formulated to obtain managerial insights. Numerical findings show that risk-neutral decision-makers are more exposed to variations in exogenous variables than CVaR decision-makers.