This paper studies mean-risk portfolio optimization models using the conditional value-at-risk (CVaR) as a risk measure. We also employ a cardinality constraint for limiting the number of invested assets. Solving such a cardinality-constrained mean-CVaR model is computationally challenging for two main reasons. First, this model is formulated as a mixed-integer optimization (MIO) problem because of the cardinality constraint, so solving it exactly is very hard when the number of investable assets is large. Second, the problem size depends on the number of asset return scenarios, and the computational efficiency decreases when the number of scenarios is large. To overcome these challenges, we propose a high-performance algorithm named the bilevel cutting-plane algorithm for exactly solving the cardinality-constrained mean-CVaR portfolio optimization problem. We begin by reformulating the problem as a bilevel optimization problem and then develop a cutting-plane algorithm for solving the upper-level problem. To speed up computations for cut generation, we apply to the lower-level problem another cutting-plane algorithm for efficiently minimizing CVaR with a large number of scenarios. Moreover, we prove the convergence properties of our bilevel cutting-plane algorithm. Numerical experiments demonstrate that, compared with other MIO approaches, our algorithm can provide optimal solutions to large problem instances faster.