We study the design of risk-sensitive online algorithms, in which risk measures are used in the competitive analysis of randomized online algorithms. We introduce the CVaR$_δ$-competitive ratio ($δ$-CR) using the conditional value-at-risk of an algorithm's cost, which measures the expectation of the $(1-δ)$-fraction of worst outcomes against the offline optimal cost, and use this measure to study three online optimization problems: continuous-time ski rental, discrete-time ski rental, and one-max search. The structure of the optimal $δ$-CR and algorithm varies significantly between problems: we prove that the optimal $δ$-CR for continuous-time ski rental is $2-2^{-Θ(\frac{1}{1-δ})}$, obtained by an algorithm described by a delay differential equation. In contrast, in discrete-time ski rental with buying cost $B$, there is an abrupt phase transition at $δ= 1 - Θ(\frac{1}{\log B})$, after which the classic deterministic strategy is optimal. Similarly, one-max search exhibits a phase transition at $δ= \frac{1}{2}$, after which the classic deterministic strategy is optimal; we also obtain an algorithm that is asymptotically optimal as $δ\downarrow 0$ that arises as the solution to a delay differential equation.