Markowitz portfolio optimization using MOSEK. MOSEK Technical report: TR-2009-2

In this tutorial paper we introduce different approaches to Markowitz portfolio optimization, and we show how to solve such problems in MATLAB, R and Python using the MOSEK optimization toolbox for MATLAB, the Rmosek package, and the MOSEK Python API, respectively. We first consider conic formulations of the basic portfolio selection problem, and we then discuss more advanced models for transaction costs. 1 Markowitz portfolio selection We start by reviewing basic Markowitz portfolio optimization, while introducing the necessary notation used in the remaining tutorial. In Markowitz portfolio selection we optimize a portfolio of assets based on a simple probabilistic model of the stock market. Traditionally we assume that the return of n different assets over time can be modeled as a multivariate random variable r ∈ R with known mean E r = μr and covariance matrix E(r − μr)(r − μr) = Σr, where the mean is a vector, and Σr is the covariance matrix (i.e., μr ∈ R and Σ ∈ R). We have an initial holding of w j dollars of asset j, j = 1, . . . , n and we invest xj dollars in asset j, i.e., after the investment period our portfolio is w + x. The return of our investment is also a random variable, y = r (w + x), with mean value (or expected return) E y = μr (w 0 + x) (1) and variance (or risk) (y −E y) = (w + x)Σr(w + x). (2) In Markowitz portfolio selection, we seek to optimize a trade-off between expected return and risk. ∗MOSEK ApS, Fruebjergvej 3, Box 16, 2100 Copenhagen, Denmark. Email: [email protected]