Merge pull request #128 from Ayush-iitkgp/pull-request/d6caa548

Added solution Unconstrained Markowitz Efficient Frontier example

examples/portfolio_optimization3_unconstrained_markowitz_efficient_frontier.ipynb

@@ -0,0 +1,97 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#  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."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "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\");"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<img src=\"efficient_frontier.png\">"
+   ]
+  }
+ ],
+ "metadata": {
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+   "display_name": "Julia 0.5.0-dev",
+   "language": "julia",
+   "name": "julia-0.5"
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+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "0.5.0"
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+ "nbformat": 4,
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+}