Risk bounds under right-tail uncertainty

We investigate upper and lower bounds for spectral risk measures, when there exists uncertainty regarding the probability distribution of large losses. Initially, we focus on scenarios in which information is only available on the left-tail of the relevant random variable. Subsequently, we progressively incorporate knowledge of the first two moments of the distribution, culminating in uncertainty sets for both the mean and the variance. Throughout our analysis, we provide closed-form bounds and discuss their sharpness. A pivotal aspect of our study is to show that while the sole knowledge of the left-tail leaves a spectral risk measure unbounded, such partial information combined with additional assumptions on the moments of the distribution can notably improve the worst-case scenario, with respect to the conventional case explored in Li (2018), in which only the mean and variance are fixed. Furthermore, we offer a numerical analysis of our findings.