How does international trade affect the risk exposure of firms and countries? Trade induces specialization, thus increasing economies' exposure to idiosyncratic supply shocks. But greater geographic diversification in trade destinations offers natural hedging properties against demand shocks. In this paper, we offer an integrated economic and econometric view of the impact of trade on firms and countries volatility. Exporters' volatility is shown to directly depend on the (lack of) diversification in their portfolio of clients. Indeed, most exporters, including the largest, have one or two main clients that dwarf the others. This structure of trade networks implies that individual exporters are strongly exposed to microeconomic demand shocks. The concentration of trade flows further implies that such risk does not wash out across firms, thus contributing to aggregate fluctuations.