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We investigate a discrete-time mean-risk portfolio selection problem, where risk is measured by the conditional value-at-risk (CVaR). A substantial challenge is the combination of a time-inconsistent objective with an incomplete and dynamic model for the financial market. We are able to solve this problem analytically by embedding the original, time-inconsistent problem into a family of time-consistent expected utility maximization problems with a piecewise linear utility function. The optimal investment strategy is a fully adaptive feedback policy and the cumulated amount invested in the risky assets is of a characteristic V-shaped pattern as a function of the current wealth. For the incomplete, discrete-time market considered herein, the mean-CVaR efficient frontier is a straight line in the mean-CVaR plane and thus economically meaningful. This contrasts the complete, continuous-time setting where the mean-CVaR …