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    "**********************\n",
    "Financial Applications\n",
    "**********************"
   ]
  },
  {
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   "source": [
    "Exercise 13.1\n",
    "=============\n",
    "\n",
    "The parametric simplex method should be used when the number of investments to\n",
    "analyze is large."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "from pymprog import model\n",
    "\n",
    "def find_optimal_portfolio(R, risk):\n",
    "    mu = max(0, risk)\n",
    "\n",
    "    #monthly returns per dollar for each of the n investments over T months\n",
    "    T, n = R.shape\n",
    "\n",
    "    #R = numpy.cumsum(M, axis=0) / numpy.arange(1, T + 1)[:, None]\n",
    "\n",
    "    #approximate expected return using historical data\n",
    "    #r = R[-1, :]\n",
    "    r = numpy.mean(R, axis=0)\n",
    "\n",
    "    lp = model('Portfolio Selection')\n",
    "    #lp.verbose(True)\n",
    "    x = lp.var('x', n, bounds=(0, None))\n",
    "    y = lp.var('y', T, bounds=(0, None))\n",
    "\n",
    "    lp.maximize(mu * sum(x_j * r_j for x_j, r_j in zip(x, r)) - sum(y) / T)\n",
    "    sum(x) == 1\n",
    "    for t in range(T):\n",
    "        -y[t] <= sum(x[j] * (R[t, j] - r[j]) for j in range(n)) <= y[t]\n",
    "\n",
    "    lp.solve()\n",
    "    #lp.sensitivity()\n",
    "\n",
    "    lp.end()\n",
    "\n",
    "    return numpy.asfarray([x[j].primal for j in range(n)]), lp.vobj()\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy\n",
    "\n",
    "M = numpy.asfarray([[1.000, 1.044, 1.068,1.016],\n",
    "                    [1.003, 1.015, 1.051, 1.039],\n",
    "                    [1.005, 1.024, 1.062, 0.994],\n",
    "                    [1.007, 1.027, 0.980, 0.971],\n",
    "                    [1.002, 1.040, 0.991, 1.009],\n",
    "                    [1.001, 0.995, 0.969, 1.030]])\n",
    "\n",
    "risks = numpy.concatenate((numpy.linspace(0, 3, 48),\n",
    "                           numpy.linspace(3, 5, 6)[1:]))\n",
    "X = []\n",
    "for _ in risks:\n",
    "    _ = find_optimal_portfolio(M, _)\n",
    "    X.append(_[0])\n",
    "\n",
    "plt.plot(risks, X)\n",
    "plt.xlabel(r'Risk $\\mu$')\n",
    "plt.ylabel('Fraction of Portfolio')\n",
    "plt.title(r'Optimal Portfolios')\n",
    "plt.legend(['SHY', 'XLB', 'XLE', 'XLF'], loc='lower right')\n",
    "plt.show()"
   ]
  },
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   "source": [
    "Exercise 13.2\n",
    "=============\n",
    "\n",
    "Planet Claire's efficient frontier is a plot of every portfolio in terms of\n",
    "risk–reward."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "M = numpy.asfarray([[1, 2, 1],\n",
    "                    [2, 2, 1],\n",
    "                    [2, 0.5, 1],\n",
    "                    [0.5, 2, 1]])\n",
    "\n",
    "risks = numpy.linspace(0, 5, 32)\n",
    "X = []\n",
    "rewards = []\n",
    "for _ in risks:\n",
    "    _ = find_optimal_portfolio(M, _)\n",
    "    X.append(_[0])\n",
    "    rewards.append(_[1])\n",
    "\n",
    "plt.plot(risks, X)\n",
    "plt.xlabel(r'Risk $\\mu$')\n",
    "plt.ylabel('Fraction of Portfolio')\n",
    "plt.title(r'Optimal Portfolios')\n",
    "plt.legend(['Hair Products', 'Cosmetics', 'Cash'],\n",
    "           loc='center right')\n",
    "plt.show()"
   ]
  },
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   "source": [
    "Exercise 13.3\n",
    "=============\n",
    "\n",
    "Converting (13.5) to :doc:`standard form </nb/linear-programming-vanderbei/chapter-05>`\n",
    "gives\n",
    "\n",
    ".. math::\n",
    "\n",
    "   \\begin{aligned}\n",
    "     \\mathcal{P} \\quad -\\text{maximize} \\quad\n",
    "       -x_0 - s_0 x_1 - \\sum_{j = 2}^n p_j x_j &\\\\\n",
    "     \\text{subject to} \\quad\n",
    "       -x_0 - s_1(i) x_1 - \\sum_{j = 2}^n h_j(s_1(i)) x_j &\\leq -g(s_1(i))\n",
    "           \\quad i = 1, \\ldots, m.\n",
    "   \\end{aligned}\n",
    "\n",
    "Observe that the primal only has inequality constraints with free variables\n",
    ":math:`\\left\\{ x_j \\right\\}_{j = 0}^n`.  This implies that the dual consists of\n",
    "restricted variables and equality constraints.\n",
    "\n",
    ".. math::\n",
    "\n",
    "   \\begin{aligned}\n",
    "     \\mathcal{D} \\quad \\text{minimize} \\quad\n",
    "       \\sum_i -g(s_1(i)) y_i &\\\\\n",
    "     \\text{subject to} \\quad\n",
    "       \\sum_i (-1) y_i &= -1\\\\\n",
    "       \\sum_i -s_1(i) y_i &= -s_0\\\\\n",
    "       \\sum_i -h_j(s_1(i)) y_i &= -p_j\\\\\n",
    "       y_i &\\geq 0 \\quad i = 1, \\ldots, m\n",
    "   \\end{aligned}\n",
    "   \\quad \\equiv \\quad\n",
    "   \\begin{aligned}\n",
    "     \\mathcal{D} \\quad -\\text{maximize} \\quad\n",
    "       \\sum_i g(s_1(i)) y_i &\\\\\n",
    "     \\text{subject to} \\quad\n",
    "       \\sum_i y_i &= 1\\\\\n",
    "       \\sum_i s_1(i) y_i &= s_0\\\\\n",
    "       \\sum_i h_j(s_1(i)) y_i &= p_j\\\\\n",
    "       y_i &\\geq 0 \\quad i = 1, \\ldots, m.\n",
    "   \\end{aligned}"
   ]
  }
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