The paper deals with three bilevel optimization problems applied to portfolio selection. First, having a given mean-CVaR inefficient portfolio as the benchmark, problem identifying of the closest mean-CVaR efficient portfolio to the benchmark is addressed. For a given CVaR level \(\alpha \) , the lower level guarantees that only mean-CVaR efficient portfolios are feasible, and the upper level chooses the portfolio that minimizes proportional transaction costs when changing from the benchmark. That is, the problem searches for a mean-CVaR efficient portfolio closest to the benchmark with respect to the norm \(L_1\) . The second problem combines in-sample mean-CVaR efficiency with the out-of-sample gross return maximization. The lower level maximizes the mean-CVaR criterion for in-sample data, but the upper level now maximizes the out-of-sample gross return. Contrary to the previous model, the mean-CVaR efficiency is considered with respect to both the risk aversion parameter and the CVaR confidence level. This means that the lower level produces a three-dimensional efficient frontier as the feasibility set for the upper level. The last problem converts the second one to the moving window framework, where every window corresponds to one lower-level mean-CVaR criterion and the upper level maximizes the total gross return over all out-of-sample periods. Two versions of this bilevel problem are considered: (1) risk aversion parameters and CVaR levels are the same in all windows; (2) risk aversion parameters and CVaR levels could change from window to window. Both problems are solved and evaluated based on the out-of-time performance.
