Significant sex differences exist in the activity of the hypothalamic-pituitary-adrenal (HPA) axis. These differences are thought to contribute to the disparity in the prevalence of various autoimmune and infectious diseases between males and females. We use a mathematical model of the HPA axis to evaluate the hypothesis that differential sensitivity and negative feedback of the HPA axis network are causal factors for the observed sex differences in its activity. In doing so, we implicitly account for the differential influence of gonadal hormones on the HPA axis. Furthermore, we determine whether the putative mechanisms responsible for differences in basal HPA axis activity might also contribute to the observed differences in its stimulus-driven response. Model simulations predict the female HPA axis has greater adrenal sensitivity and weaker negative feedback. We identify two distinct sex-specific parameter spaces that generate corticosterone profiles in qualitative agreement with experimental results. We propose that these parameter sub-spaces are indicative of the inter-individual variability in the regulatory mechanisms of the HPA axis. Furthermore, the model predicts that the maintenance of homeostatic rhythms in response to chronic stress requires specific regulatory adaptations resulting in a phenotype of allostatically-driven chronic stress-sensitization. We propose these adaptations are indicative of a physiological cost of adaptation to chronic stress. Model simulations suggest that individuals with high adrenal sensitivity are more vulnerable to chronic stress sensitization and might be more susceptible to the development of neuropsychiatric disorders. These results contribute to the study of sex differences in physiological feedback systems within a quantitative framework.