We use distributionally robust optimization (DRO) to model a general class of newsvendor problems with unknown demand distribution. The goal is to find an order quantity that minimizes the worst-case expected cost among an ambiguity set of distributions. The ambiguity set consists of those distributions that are not far-in the sense of the total variation distance-from a nominal distribution. The maximum distance allowed in the ambiguity set (called level of robustness) places the DRO between the risk-neutral stochastic programming and robust optimization models. An important problem a decision maker faces is how to determine the level of robustness-or, equivalently, how to find an appropriate level of risk-aversion. We answer this question in two ways. Our first approach relates the level of robustness and risk to the regions of demand that are critical (in a precise sense we introduce) to the optimal cost. Our second approach establishes new quantitative relationships between the DRO model and the corresponding risk-neutral and classical robust optimization models. To achieve these goals, we first focus on a single-product setting and derive explicit formulas and properties of the optimal solution as a function of the level of robustness. Then, we demonstrate the practical and managerial relevance of our results by applying our findings to a healthcare problem to reserve operating room time for cardiovascular surgeries. Finally, we extend some of our results to the multi-product setting and illustrate them numerically. (C) 2019 Elsevier B.V. All rights reserved.