In risk theory, financial asset returns often follow heavy-tailed distributions. Investors and risk managers used to compare risk measures as the value at risk or tail value at risk in order over the whole confidence levels to avoid the exposure to to large risks. In this paper we analyze the comparison between tail values at risk from a confidence level and beyond which is a reasonable criterion when we are focused on large losses or simply we cannot give a complete ordering over all the confidence levels. A family of stochastic orders indexed by $p_0\in(0,1)$ is proposed. We study their properties and connections with other classical criteria as the increasing convex and tail convex orders and we rank some parametrical families of distributions. Finally, two applications with real datasets are given as well.