  Considering congestion games with uncertain delays, we compute the
inefficiency introduced in network routing by risk-averse agents. At
equilibrium, agents may select paths that do not minimize the expected latency
so as to obtain lower variability. A social planner, who is likely to be more
risk neutral than agents because it operates at a longer time-scale, quantifies
social cost with the total expected delay along routes. From that perspective,
agents may make suboptimal decisions that degrade long-term quality. We define
the {\em price of risk aversion} (PRA) as the worst-case ratio of the social
cost at a risk-averse Wardrop equilibrium to that where agents are
risk-neutral. For networks with general delay functions and a single
source-sink pair, we show that the PRA depends linearly on the agents' risk
tolerance and on the degree of variability present in the network. In contrast
to the {\em price of anarchy}, in general the PRA increases when the network
gets larger but it does not depend on the shape of the delay functions. To get
this result we rely on a combinatorial proof that employs alternating paths
that are reminiscent of those used in max-flow algorithms. For {\em
series-parallel} (SP) graphs, the PRA becomes independent of the network
topology and its size. As a result of independent interest, we prove that for
SP networks with deterministic delays, Wardrop equilibria {\em maximize} the
shortest-path objective among all feasible flows.
