We analyze systems of agents sharing light-tailed risky claims issued by different financial objects. Assuming exponentially distributed claims, we obtain that both agents' and system's losses follow generalized exponential mixture distributions. We show that this leads to qualitatively different results on individual and system risks compared to heavy-tailed claims previously studied in the literature. By deducing conditional loss distributions we investigate the impact of stress situations on agents' and system's losses. Moreover, we present a criterion for agents to decide whether holding few objects or portfolio diversification minimizes their risks in system crisis situations. ; Comment: 29 pages, 3 figures