In various operational problems, risk-sensitive decision makers often encounter the challenge of selecting an alternative with minimal tail risk from a collection of stochastic alternatives that generate random losses. Tail risk, in this context, refers to the potential for experiencing substantial losses, which will be formally defined shortly. Despite the significance of addressing this challenge, the majority of related studies still focus on identifying a subset of the alternatives with acceptable (or minimal) expected losses, rather than using tail risk as a ranking criterion. Our objective is to develop a tractable and effective solution to this problem in situations where decision makers aim to compare the alternatives based only on their tail risk. In practical scenarios, it would be ideal to apply our proposed solution to the aforementioned subset of the alternatives, which can be obtained via existing approaches, so that decision makers can ultimately find an alternative with both acceptable expected loss and minimal tail risk.