<p>This study assumes homothetic robust Epstein-Zin (HREZ) utility and analyzes the consumption–investment problem under a quadratic security market model. In HREZ utility, characterized by relative risk aversion and relative ambiguity aversion, if the sum of these equals for two different utilities, then they are observationally indistinguishable. We show that under the worst-case probability, the market price of risk is replaced by the “investor price of uncertainty.” We introduce the notions of “worst-case premiums” and “worst-case long-term premiums” on securities, and derive analytical expressions of optimal robust control, investor price of uncertainty, worst-case premiums, and worst-case long-term premiums. Our numerical analysis suggests that we can identify the two different HREZ utilities based on the information related to the worst-case long-term premiums.</p>