examples/portfolio_optimization3_unconstrained_markowitz_efficient_frontier.ipynb

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    "#  Portfolio Optimization - Unconstrained Markowitz Efficient Frontier \n",
    "\n",
    "In this problem, we will find the unconstrained portfolio allocation where we introduce the weighting parameter $\\lambda(0 \\leq \\lambda \\leq$ 1)and minimize the $\\lambda * risk - (1-\\lambda)* return$. By varying the values of $\\lambda$, we trace out the efficient frontier.  \n",
    "\n",
    "Suppose that we know the mean returns $R \\in \\mathbf{R}^n$ of each asset and the covariance $Q \\in \\mathbf{R}^{n \\times n}$ between the assets. Our objective is to find a portfolio allocation that minimizes the *risk* (which we measure as the variance $w^T Q w$) and maximizes the *return* ($w^T R$) of the portfolio of the simulataneously. We suppose further that our portfolio allocation must comply with some lower and upper bounds on the allocation, $w_\\mbox{lower} \\leq w \\leq w_\\mbox{upper}$ and also $w \\in \\mathbf{R}^n$ $\\sum_i w_i = 1$.\n",
    "\n",
    "This problem can be written as\n",
    "\n",
    "\\begin{array}{ll}\n",
    "    \\mbox{minimize}   & \\lambda*w^T Q w - (1-\\lambda)*w^T R \\\\\n",
    "    \\mbox{subject to} & \\sum_i w_i = 1 \\\\\n",
    "                      & w_\\mbox{lower} \\leq w \\leq w_\\mbox{upper}\n",
    "\\end{array}\n",
    "\n",
    "where $w \\in \\mathbf{R}^n$ is the vector containing weights allocated to each asset in the efficient frontier.\n",
    "\n",
    "We can solve this problem as follows."
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    "using Convex, ECOS    #We are using ECOS solver. Install using Pkg.add(\"ECOS\")\n",
    "\n",
    "# generate problem data\n",
    "srand(0);     #Set the seed\n",
    "n = 5;        # Assume that we have portfolio of 5 assets.\n",
    "R = 5 * randn(n);\n",
    "A = randn(n, 5);\n",
    "Q = A * A' + diagm(rand(n));\n",
    "w_lower = 0;\n",
    "w_upper = 1;\n",
    "\n",
    "\n",
    "risk = zeros(2000);   # Initialized the risk and the return vectors.   \n",
    "ret = zeros(2000);   # lambda varies in the interval(0,1) in the steps of 1/2000.\n",
    "\n",
    "w = Variable(length(R));\n",
    "\n",
    "#Defining constraints\n",
    "c1 = sum(w) == 1;\n",
    "c2 = w_lower <= x; \n",
    "c3 = w <= w_upper;\n",
    "for i in 1:2000\n",
    "    λ = i/2000;\n",
    "\n",
    "    #Defining Objective function\n",
    "    objective = λ * quadform(w,Q) - (1-λ)* w' *R;\n",
    "    p = minimize(objective, c1,c2,c3);\n",
    "    solve!(p, ECOSSolver(verbose = false));\n",
    "    risk[i] = (w.value' * Q * w.value)[1];\n",
    "    ret[i] = (w.value'R)[1];\n",
    "    #println(\"$i \",\"$(λ*risk[i] - (1-λ)*ret[i]) \",\"$p.optval\");\n",
    "    end\n",
    "\n",
    "using PyPlot            #Install PyPlot if you don't have it installed. Pkg.add(\"PyPlot\")\n",
    "plot(risk,ret)\n",
    "title(\"Markowitz Efficient Frontier\");\n",
    "xlabel(\"Expected Risk-Variance\");\n",
    "ylabel(\"Expected Return\");"
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   "source": [
    "<img src=\"efficient_frontier.png\">"
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