/* File : portfolio_6_factor.java Copyright : Copyright (c) MOSEK ApS, Denmark. All rights reserved. Purpose : Implements a portfolio optimization model using factor model. */ package com.mosek.fusion.examples; import mosek.fusion.*; import mosek.LinAlg; import java.io.FileReader; import java.io.BufferedReader; import java.util.Arrays; public class portfolio_6_factor { public static double sum(double[] x) { double r = 0.0; for (int i = 0; i < x.length; ++i) r += x[i]; return r; } public static double dot(double[] x, double[] y) { double r = 0.0; for (int i = 0; i < x.length; ++i) r += x[i] * y[i]; return r; } public static double[][] transpose(double[][] m) { int ni = m.length; int nj = m[0].length; double[][] mt = new double[nj][ni]; for (int i = 0; i < ni; ++i) { for (int j = 0; j < nj; ++j) { mt[j][i] = m[i][j]; } } return mt; } public static double[] vector_sqrt(double[] m) { int ni = m.length; double[] sqrtm = new double[ni]; for (int i = 0; i < ni; ++i) { sqrtm[i] = Math.sqrt(m[i]); } return sqrtm; } // Vectorize matrix (column-major order) public static double[] mat_to_vec_c(double[][] m) { int ni = m.length; int nj = m[0].length; double[] c = new double[nj * ni]; for (int j = 0; j < nj; ++j) { for (int i = 0; i < ni; ++i) { c[j * ni + i] = m[i][j]; } } return c; } // Reshape vector to matrix (column-major order) public static double[][] vec_to_mat_c(double[] c, int ni, int nj) { double[][] m = new double[ni][nj]; for (int j = 0; j < nj; ++j) { for (int i = 0; i < ni; ++i) { m[i][j] = c[j * ni + i]; } } return m; } public static double[][] cholesky(double[][] m) { int n = m.length; double[] vecs = mat_to_vec_c(m); LinAlg.potrf(mosek.uplo.lo, n, vecs); double[][] s = vec_to_mat_c(vecs, n, n); // Zero out upper triangular part (LinAlg.potrf does not use it, original matrix values remain there) for (int i = 0; i < n; ++i) { for (int j = i+1; j < n; ++j) { s[i][j] = 0.0; } } return s; } // Matrix multiplication public static double[][] matrix_mul(double[][] a, double[][] b) { int na = a.length; int nb = b[0].length; int k = b.length; double[] vecm = new double[na * nb]; Arrays.fill(vecm, 0.0); LinAlg.gemm(mosek.transpose.no, mosek.transpose.no, na, nb, k, 1.0, mat_to_vec_c(a), mat_to_vec_c(b), 1.0, vecm); double[][] m = vec_to_mat_c(vecm, na, nb); return m; } /* Purpose: Computes the optimal portfolio for a given risk using factor model approximation of the covariance. Input: n: Number of assets mu: An n dimmensional vector of expected returns G_factor_T: The factor (dense) part of the factorized risk theta: specific risk vector x0: Initial holdings w: Initial cash holding gamma: Maximum risk (=std. dev) accepted Output: Optimal expected return and the optimal portfolio */ public static double FactorMarkowitz ( int n, double[] mu, double[][] G_factor_T, double[] theta, double[] x0, double w, double gamma) throws mosek.fusion.SolutionError { Model M = new Model("Factor model Markowitz"); try { // Redirect log output from the solver to stdout for debugging. // if uncommented. //M.setLogHandler(new java.io.PrintWriter(System.out)); // Defines the variables (holdings). Shortselling is not allowed. Variable x = M.variable("x", n, Domain.greaterThan(0.0)); // Maximize expected return M.objective("obj", ObjectiveSense.Maximize, Expr.dot(mu, x)); // The amount invested must be identical to intial wealth M.constraint("budget", Expr.sum(x), Domain.equalsTo(w + sum(x0))); // Imposes a bound on the risk M.constraint("risk", Expr.vstack(new Expression[]{Expr.constTerm(gamma), Expr.mul(G_factor_T, x), Expr.mulElm(vector_sqrt(theta), x)}), Domain.inQCone()); // Solves the model. M.solve(); // Check if the solution is an optimal point SolutionStatus solsta = M.getPrimalSolutionStatus(); if (solsta != SolutionStatus.Optimal) { // See https://docs.mosek.com/latest/javafusion/accessing-solution.html about handling solution statuses. throw new SolutionError(String.format("Unexpected solution status: %s", solsta.toString())); } return dot(mu, x.level()); } finally { M.dispose(); } } /* The example. Reads in data and solves the portfolio models. */ public static void main(String[] argv) throws java.io.IOException, java.io.FileNotFoundException, mosek.fusion.SolutionError { int n = 8; double w = 1.0; double[] mu = {0.07197, 0.15518, 0.17535, 0.08981, 0.42896, 0.39292, 0.32171, 0.18379}; double[] x0 = {0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0}; // Factor exposure matrix double[][] B = { {0.4256, 0.1869}, {0.2413, 0.3877}, {0.2235, 0.3697}, {0.1503, 0.4612}, {1.5325, -0.2633}, {1.2741, -0.2613}, {0.6939, 0.2372}, {0.5425, 0.2116} }; // Factor covariance matrix double[][] S_F = { {0.0620, 0.0577}, {0.0577, 0.0908} }; // Specific risk components double[] theta = {0.0720, 0.0508, 0.0377, 0.0394, 0.0663, 0.0224, 0.0417, 0.0459}; double[][] P = cholesky(S_F); double[][] G_factor = matrix_mul(B, P); double[][] G_factor_T = transpose(G_factor); double[] gammas = {0.24, 0.28, 0.32, 0.36, 0.4, 0.44, 0.48}; System.out.println("\n-----------------------------------------------------------------------------------"); System.out.println("Markowitz portfolio optimization based on a factor model."); System.out.println("-----------------------------------------------------------------------------------\n"); for ( int i = 0; i < gammas.length; ++i) { double expret = FactorMarkowitz(n, mu, G_factor_T, theta, x0, w, gammas[i]); System.out.format( "Expected return: %.4e Std. deviation: %.4e\n", expret, gammas[i] ); } } }