# Design high-order portfolio based on weighted linear combination of first four moments Design high-order portfolio based on weighted linear combination of first four moments (i.e., mean, variance, skewness, and kurtosis): ``` minimize - lmd1*(w'*mu) + lmd2*(w'*Sigma*w) - lmd3*(w'*Phi*w*w) + lmd4*(w'*Psi*w*w*w) subject to ||w||_1 <= leverage, sum(w) == 1. ``` ```r design_MVSK_portfolio_via_sample_moments( lmd = rep(1, 4), X_moments, w_init = rep(1/length(X_moments$mu), length(X_moments$mu)), leverage = 1, method = c("Q-MVSK", "MM", "DC"), tau_w = 0, gamma = 1, zeta = 1e-08, maxiter = 100, ftol = 1e-05, wtol = 1e-04, stopval = -Inf ) ``` ## Arguments - `lmd`: Numerical vector of length 4 indicating the weights of first four moments. - `X_moments`: List of moment parameters, see `estimate_sample_moments()`. - `w_init`: Numerical vector indicating the initial value of portfolio weights. - `leverage`: Number (>= 1) indicating the leverage of portfolio. - `method`: String indicating the algorithm method, must be one of: "Q-MVSK", "MM", "DC". - `tau_w`: Number (>= 0) guaranteeing the strong convexity of approximating function. - `gamma`: Number (0 < gamma <= 1) indicating the initial value of gamma. - `zeta`: Number (0 < zeta < 1) indicating the diminishing paramater of gamma. - `maxiter`: Positive integer setting the maximum iteration. - `ftol`: Positive number setting the convergence criterion of function objective. - `wtol`: Positive number setting the convergence criterion of portfolio weights. - `stopval`: Number setting the stop value of objective. ## Returns A list containing the following elements: - **`w`**: Optimal portfolio vector. - **`cpu_time_vs_iterations`**: Time usage over iterations. - **`objfun_vs_iterations`**: Objective function over iterations. - **`iterations`**: Iterations index. - **`convergence`**: Boolean flag to indicate whether or not the optimization converged. - **`moments`**: Moments of portfolio return at optimal portfolio weights. ## Examples ```r library(highOrderPortfolios) data(X50) # estimate moments X_moments <- estimate_sample_moments(X50[, 1:10]) # decide moment weights xi <- 10 lmd <- c(1, xi/2, xi*(xi+1)/6, xi*(xi+1)*(xi+2)/24) # portfolio optimization sol <- design_MVSK_portfolio_via_sample_moments(lmd, X_moments) ``` ## References R. Zhou and D. P. Palomar, "Solving High-Order Portfolios via Successive Convex Approximation Algorithms," in **IEEE Transactions on Signal Processing**, vol. 69, pp. 892-904, 2021. . X. Wang, R. Zhou, J. Ying, and D. P. Palomar, "Efficient and Scalable High-Order Portfolios Design via Parametric Skew-t Distribution," Available in arXiv, 2022. . ## Author(s) Rui Zhou and Daniel P. Palomar

design_MVSK_portfolio_via_sample_moments function