Risk Minimization using Markowitz Models.ipynb

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      "metadata": {
        "id": "747ba93c"
      },
      "outputs": [],
      "source": [
        "import pandas as pd\n",
        "# ! pip install xlrd\n",
        "import numpy as np"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "3d52bae9",
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              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>Date</th>\n",
              "      <th>500096</th>\n",
              "      <th>500112</th>\n",
              "      <th>500124</th>\n",
              "      <th>500163</th>\n",
              "      <th>500164</th>\n",
              "      <th>500182</th>\n",
              "      <th>500188</th>\n",
              "      <th>500330</th>\n",
              "      <th>500387</th>\n",
              "      <th>...</th>\n",
              "      <th>540025</th>\n",
              "      <th>532784</th>\n",
              "      <th>533269</th>\n",
              "      <th>500260</th>\n",
              "      <th>531500</th>\n",
              "      <th>533273</th>\n",
              "      <th>530007</th>\n",
              "      <th>532652</th>\n",
              "      <th>500165</th>\n",
              "      <th>500870</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>2023-08-01</td>\n",
              "      <td>575.55</td>\n",
              "      <td>619.15</td>\n",
              "      <td>5557.60</td>\n",
              "      <td>2183.00</td>\n",
              "      <td>477.00</td>\n",
              "      <td>3210.05</td>\n",
              "      <td>323.00</td>\n",
              "      <td>1928.30</td>\n",
              "      <td>24151.05</td>\n",
              "      <td>...</td>\n",
              "      <td>329.85</td>\n",
              "      <td>622.90</td>\n",
              "      <td>529.20</td>\n",
              "      <td>888.90</td>\n",
              "      <td>525.00</td>\n",
              "      <td>1114.95</td>\n",
              "      <td>268.25</td>\n",
              "      <td>208.65</td>\n",
              "      <td>337.00</td>\n",
              "      <td>145.15</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>2023-07-31</td>\n",
              "      <td>577.85</td>\n",
              "      <td>615.40</td>\n",
              "      <td>5580.00</td>\n",
              "      <td>2091.05</td>\n",
              "      <td>479.95</td>\n",
              "      <td>3185.05</td>\n",
              "      <td>319.15</td>\n",
              "      <td>1915.95</td>\n",
              "      <td>23900.00</td>\n",
              "      <td>...</td>\n",
              "      <td>329.00</td>\n",
              "      <td>628.00</td>\n",
              "      <td>530.00</td>\n",
              "      <td>883.35</td>\n",
              "      <td>513.20</td>\n",
              "      <td>1119.25</td>\n",
              "      <td>263.30</td>\n",
              "      <td>204.05</td>\n",
              "      <td>330.00</td>\n",
              "      <td>147.55</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>2023-07-28</td>\n",
              "      <td>571.65</td>\n",
              "      <td>616.10</td>\n",
              "      <td>5518.75</td>\n",
              "      <td>1912.05</td>\n",
              "      <td>477.00</td>\n",
              "      <td>3156.95</td>\n",
              "      <td>321.20</td>\n",
              "      <td>1911.20</td>\n",
              "      <td>23995.00</td>\n",
              "      <td>...</td>\n",
              "      <td>308.90</td>\n",
              "      <td>619.05</td>\n",
              "      <td>518.60</td>\n",
              "      <td>881.20</td>\n",
              "      <td>506.65</td>\n",
              "      <td>1121.60</td>\n",
              "      <td>251.85</td>\n",
              "      <td>205.05</td>\n",
              "      <td>326.35</td>\n",
              "      <td>145.05</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>2023-07-27</td>\n",
              "      <td>576.50</td>\n",
              "      <td>614.30</td>\n",
              "      <td>5535.00</td>\n",
              "      <td>1990.00</td>\n",
              "      <td>478.60</td>\n",
              "      <td>3124.95</td>\n",
              "      <td>320.00</td>\n",
              "      <td>1809.95</td>\n",
              "      <td>24101.05</td>\n",
              "      <td>...</td>\n",
              "      <td>310.65</td>\n",
              "      <td>638.05</td>\n",
              "      <td>520.25</td>\n",
              "      <td>894.00</td>\n",
              "      <td>512.35</td>\n",
              "      <td>1098.60</td>\n",
              "      <td>252.05</td>\n",
              "      <td>205.60</td>\n",
              "      <td>324.00</td>\n",
              "      <td>146.95</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>2023-07-26</td>\n",
              "      <td>569.05</td>\n",
              "      <td>611.50</td>\n",
              "      <td>5426.35</td>\n",
              "      <td>1710.95</td>\n",
              "      <td>476.85</td>\n",
              "      <td>3140.00</td>\n",
              "      <td>320.00</td>\n",
              "      <td>1760.55</td>\n",
              "      <td>23731.90</td>\n",
              "      <td>...</td>\n",
              "      <td>309.50</td>\n",
              "      <td>583.00</td>\n",
              "      <td>522.05</td>\n",
              "      <td>897.40</td>\n",
              "      <td>516.85</td>\n",
              "      <td>1063.50</td>\n",
              "      <td>243.05</td>\n",
              "      <td>203.45</td>\n",
              "      <td>320.75</td>\n",
              "      <td>136.80</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "<p>5 rows × 51 columns</p>\n",
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            ],
            "text/plain": [
              "        Date  500096  500112   500124   500163  500164   500182  500188  \\\n",
              "0 2023-08-01  575.55  619.15  5557.60  2183.00  477.00  3210.05  323.00   \n",
              "1 2023-07-31  577.85  615.40  5580.00  2091.05  479.95  3185.05  319.15   \n",
              "2 2023-07-28  571.65  616.10  5518.75  1912.05  477.00  3156.95  321.20   \n",
              "3 2023-07-27  576.50  614.30  5535.00  1990.00  478.60  3124.95  320.00   \n",
              "4 2023-07-26  569.05  611.50  5426.35  1710.95  476.85  3140.00  320.00   \n",
              "\n",
              "    500330    500387  ...  540025  532784  533269  500260  531500   533273  \\\n",
              "0  1928.30  24151.05  ...  329.85  622.90  529.20  888.90  525.00  1114.95   \n",
              "1  1915.95  23900.00  ...  329.00  628.00  530.00  883.35  513.20  1119.25   \n",
              "2  1911.20  23995.00  ...  308.90  619.05  518.60  881.20  506.65  1121.60   \n",
              "3  1809.95  24101.05  ...  310.65  638.05  520.25  894.00  512.35  1098.60   \n",
              "4  1760.55  23731.90  ...  309.50  583.00  522.05  897.40  516.85  1063.50   \n",
              "\n",
              "   530007  532652  500165  500870  \n",
              "0  268.25  208.65  337.00  145.15  \n",
              "1  263.30  204.05  330.00  147.55  \n",
              "2  251.85  205.05  326.35  145.05  \n",
              "3  252.05  205.60  324.00  146.95  \n",
              "4  243.05  203.45  320.75  136.80  \n",
              "\n",
              "[5 rows x 51 columns]"
            ]
          },
          "execution_count": 2,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "df = pd.read_excel('Dataset.xlsx')\n",
        "df.head()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "c81a0af5",
      "metadata": {
        "id": "c81a0af5",
        "outputId": "01674cff-6be2-4867-8c36-ebd453eaa52b"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "number of assets =  50\n"
          ]
        }
      ],
      "source": [
        "# converting pandas dataframe to 2d numpy array\n",
        "data = df.to_numpy()\n",
        "asset_id=df.columns.to_numpy()[1:].reshape(50,1)\n",
        "# removing the date column\n",
        "data = data[:,1:]\n",
        "data=data.astype(float)\n",
        "n = data.shape[1]\n",
        "\n",
        "print(\"number of assets = \", n)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "eb80d8cd",
      "metadata": {
        "id": "eb80d8cd"
      },
      "outputs": [],
      "source": [
        "# variance-covariance matrix of 50 assets\n",
        "omega = np.cov(data,rowvar=False)\n",
        "\n",
        "# weights\n",
        "w = np.random.uniform(-1, 1, size=(n,1))\n",
        "# column vector of ones\n",
        "e = np.ones(n).reshape((50,1))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "ebf86e24",
      "metadata": {
        "id": "ebf86e24"
      },
      "outputs": [],
      "source": [
        "# expected returns of asset j\n",
        "mu_j = np.mean(np.diff(data, axis=0)/data[:-1],axis=0).reshape(-1,1)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "0f7c0880",
      "metadata": {
        "id": "0f7c0880"
      },
      "source": [
        "### A.  Markowitz Model I: minimize wTΩ w   subject to  eTw=1, short selling is allowed."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "c8617951",
      "metadata": {
        "id": "c8617951"
      },
      "outputs": [],
      "source": []
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "a7279c3f",
      "metadata": {
        "scrolled": false,
        "id": "a7279c3f"
      },
      "outputs": [],
      "source": [
        "from numpy.linalg import inv\n",
        "Lambda = -2/(e.T@inv(omega)@e)\n",
        "w = inv(omega)@e/(e.T@inv(omega)@e)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "e5f6a417",
      "metadata": {
        "id": "e5f6a417",
        "outputId": "a305a53e-d17b-4e3d-8914-26bd2c806729"
      },
      "outputs": [
        {
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              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>asset_id</th>\n",
              "      <th>weights</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>500096</td>\n",
              "      <td>0.006712</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>500112</td>\n",
              "      <td>-0.002999</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>500124</td>\n",
              "      <td>-0.000311</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>500163</td>\n",
              "      <td>0.000051</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>500164</td>\n",
              "      <td>-0.001900</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>500182</td>\n",
              "      <td>-0.000074</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>500188</td>\n",
              "      <td>0.004743</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>500330</td>\n",
              "      <td>0.000288</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>500387</td>\n",
              "      <td>-0.000105</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>500520</td>\n",
              "      <td>0.001694</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>10</th>\n",
              "      <td>500680</td>\n",
              "      <td>0.000240</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>11</th>\n",
              "      <td>500790</td>\n",
              "      <td>-0.000084</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>12</th>\n",
              "      <td>500800</td>\n",
              "      <td>-0.003293</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>13</th>\n",
              "      <td>504212</td>\n",
              "      <td>0.005464</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>14</th>\n",
              "      <td>505010</td>\n",
              "      <td>-0.000998</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>15</th>\n",
              "      <td>505200</td>\n",
              "      <td>-0.000058</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>16</th>\n",
              "      <td>505537</td>\n",
              "      <td>-0.003052</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>17</th>\n",
              "      <td>507685</td>\n",
              "      <td>0.001280</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>18</th>\n",
              "      <td>507878</td>\n",
              "      <td>0.867350</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>19</th>\n",
              "      <td>517168</td>\n",
              "      <td>0.000399</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>20</th>\n",
              "      <td>519600</td>\n",
              "      <td>-0.002367</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>21</th>\n",
              "      <td>520066</td>\n",
              "      <td>-0.003141</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>22</th>\n",
              "      <td>523367</td>\n",
              "      <td>0.000583</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>23</th>\n",
              "      <td>523395</td>\n",
              "      <td>0.000060</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>24</th>\n",
              "      <td>524715</td>\n",
              "      <td>0.001656</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25</th>\n",
              "      <td>524816</td>\n",
              "      <td>-0.001713</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>26</th>\n",
              "      <td>530555</td>\n",
              "      <td>0.014315</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>27</th>\n",
              "      <td>532218</td>\n",
              "      <td>0.172303</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>28</th>\n",
              "      <td>532286</td>\n",
              "      <td>-0.000515</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>29</th>\n",
              "      <td>532301</td>\n",
              "      <td>0.000713</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>30</th>\n",
              "      <td>532374</td>\n",
              "      <td>-0.001552</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>31</th>\n",
              "      <td>532500</td>\n",
              "      <td>-0.000078</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>32</th>\n",
              "      <td>532776</td>\n",
              "      <td>-0.013244</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>33</th>\n",
              "      <td>532977</td>\n",
              "      <td>0.000473</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>34</th>\n",
              "      <td>533106</td>\n",
              "      <td>-0.003131</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>35</th>\n",
              "      <td>533278</td>\n",
              "      <td>0.006080</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>36</th>\n",
              "      <td>534091</td>\n",
              "      <td>-0.000175</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>37</th>\n",
              "      <td>538268</td>\n",
              "      <td>0.001042</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>38</th>\n",
              "      <td>539056</td>\n",
              "      <td>-0.024664</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>39</th>\n",
              "      <td>539254</td>\n",
              "      <td>-0.000180</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>40</th>\n",
              "      <td>540025</td>\n",
              "      <td>-0.000987</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>41</th>\n",
              "      <td>532784</td>\n",
              "      <td>0.001437</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>42</th>\n",
              "      <td>533269</td>\n",
              "      <td>-0.004775</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>43</th>\n",
              "      <td>500260</td>\n",
              "      <td>0.002108</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>44</th>\n",
              "      <td>531500</td>\n",
              "      <td>-0.002708</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>45</th>\n",
              "      <td>533273</td>\n",
              "      <td>0.002298</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>46</th>\n",
              "      <td>530007</td>\n",
              "      <td>-0.005610</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>47</th>\n",
              "      <td>532652</td>\n",
              "      <td>-0.014414</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>48</th>\n",
              "      <td>500165</td>\n",
              "      <td>-0.000684</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>49</th>\n",
              "      <td>500870</td>\n",
              "      <td>0.001525</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   asset_id   weights\n",
              "0    500096  0.006712\n",
              "1    500112 -0.002999\n",
              "2    500124 -0.000311\n",
              "3    500163  0.000051\n",
              "4    500164 -0.001900\n",
              "5    500182 -0.000074\n",
              "6    500188  0.004743\n",
              "7    500330  0.000288\n",
              "8    500387 -0.000105\n",
              "9    500520  0.001694\n",
              "10   500680  0.000240\n",
              "11   500790 -0.000084\n",
              "12   500800 -0.003293\n",
              "13   504212  0.005464\n",
              "14   505010 -0.000998\n",
              "15   505200 -0.000058\n",
              "16   505537 -0.003052\n",
              "17   507685  0.001280\n",
              "18   507878  0.867350\n",
              "19   517168  0.000399\n",
              "20   519600 -0.002367\n",
              "21   520066 -0.003141\n",
              "22   523367  0.000583\n",
              "23   523395  0.000060\n",
              "24   524715  0.001656\n",
              "25   524816 -0.001713\n",
              "26   530555  0.014315\n",
              "27   532218  0.172303\n",
              "28   532286 -0.000515\n",
              "29   532301  0.000713\n",
              "30   532374 -0.001552\n",
              "31   532500 -0.000078\n",
              "32   532776 -0.013244\n",
              "33   532977  0.000473\n",
              "34   533106 -0.003131\n",
              "35   533278  0.006080\n",
              "36   534091 -0.000175\n",
              "37   538268  0.001042\n",
              "38   539056 -0.024664\n",
              "39   539254 -0.000180\n",
              "40   540025 -0.000987\n",
              "41   532784  0.001437\n",
              "42   533269 -0.004775\n",
              "43   500260  0.002108\n",
              "44   531500 -0.002708\n",
              "45   533273  0.002298\n",
              "46   530007 -0.005610\n",
              "47   532652 -0.014414\n",
              "48   500165 -0.000684\n",
              "49   500870  0.001525"
            ]
          },
          "execution_count": 7,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# (a) Optimal portfolio\n",
        "Optimal_portfolio = {'asset_id': asset_id.reshape(50),\n",
        "        'weights': w.reshape(50)}\n",
        "view=pd.DataFrame(Optimal_portfolio)\n",
        "view"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "c1b5651f",
      "metadata": {
        "id": "c1b5651f",
        "outputId": "d9d3fc9d-7a4c-45bd-dbef-97ad3203343c"
      },
      "outputs": [
        {
          "data": {
            "text/html": [
              "<div>\n",
              "<style scoped>\n",
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              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>asset_id</th>\n",
              "      <th>Investment</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>500096</td>\n",
              "      <td>671.167833</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>500112</td>\n",
              "      <td>-299.917292</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>500124</td>\n",
              "      <td>-31.054822</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>500163</td>\n",
              "      <td>5.061251</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>500164</td>\n",
              "      <td>-189.970683</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>500182</td>\n",
              "      <td>-7.399052</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>500188</td>\n",
              "      <td>474.324240</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>500330</td>\n",
              "      <td>28.777729</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>500387</td>\n",
              "      <td>-10.506000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>500520</td>\n",
              "      <td>169.414217</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>10</th>\n",
              "      <td>500680</td>\n",
              "      <td>24.042003</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>11</th>\n",
              "      <td>500790</td>\n",
              "      <td>-8.377280</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>12</th>\n",
              "      <td>500800</td>\n",
              "      <td>-329.256077</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>13</th>\n",
              "      <td>504212</td>\n",
              "      <td>546.364935</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>14</th>\n",
              "      <td>505010</td>\n",
              "      <td>-99.759968</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>15</th>\n",
              "      <td>505200</td>\n",
              "      <td>-5.834367</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>16</th>\n",
              "      <td>505537</td>\n",
              "      <td>-305.151244</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>17</th>\n",
              "      <td>507685</td>\n",
              "      <td>128.048163</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>18</th>\n",
              "      <td>507878</td>\n",
              "      <td>86734.951864</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>19</th>\n",
              "      <td>517168</td>\n",
              "      <td>39.857125</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>20</th>\n",
              "      <td>519600</td>\n",
              "      <td>-236.739203</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>21</th>\n",
              "      <td>520066</td>\n",
              "      <td>-314.122139</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>22</th>\n",
              "      <td>523367</td>\n",
              "      <td>58.318708</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>23</th>\n",
              "      <td>523395</td>\n",
              "      <td>6.046192</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>24</th>\n",
              "      <td>524715</td>\n",
              "      <td>165.587432</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25</th>\n",
              "      <td>524816</td>\n",
              "      <td>-171.298216</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>26</th>\n",
              "      <td>530555</td>\n",
              "      <td>1431.504765</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>27</th>\n",
              "      <td>532218</td>\n",
              "      <td>17230.281668</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>28</th>\n",
              "      <td>532286</td>\n",
              "      <td>-51.515110</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>29</th>\n",
              "      <td>532301</td>\n",
              "      <td>71.255820</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>30</th>\n",
              "      <td>532374</td>\n",
              "      <td>-155.190703</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>31</th>\n",
              "      <td>532500</td>\n",
              "      <td>-7.836486</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>32</th>\n",
              "      <td>532776</td>\n",
              "      <td>-1324.351428</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>33</th>\n",
              "      <td>532977</td>\n",
              "      <td>47.346727</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>34</th>\n",
              "      <td>533106</td>\n",
              "      <td>-313.144994</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>35</th>\n",
              "      <td>533278</td>\n",
              "      <td>608.020973</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>36</th>\n",
              "      <td>534091</td>\n",
              "      <td>-17.509772</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>37</th>\n",
              "      <td>538268</td>\n",
              "      <td>104.217604</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>38</th>\n",
              "      <td>539056</td>\n",
              "      <td>-2466.394242</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>39</th>\n",
              "      <td>539254</td>\n",
              "      <td>-18.028016</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>40</th>\n",
              "      <td>540025</td>\n",
              "      <td>-98.746336</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>41</th>\n",
              "      <td>532784</td>\n",
              "      <td>143.654040</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>42</th>\n",
              "      <td>533269</td>\n",
              "      <td>-477.516558</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>43</th>\n",
              "      <td>500260</td>\n",
              "      <td>210.820582</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>44</th>\n",
              "      <td>531500</td>\n",
              "      <td>-270.832994</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>45</th>\n",
              "      <td>533273</td>\n",
              "      <td>229.757113</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>46</th>\n",
              "      <td>530007</td>\n",
              "      <td>-561.017860</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>47</th>\n",
              "      <td>532652</td>\n",
              "      <td>-1441.419768</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>48</th>\n",
              "      <td>500165</td>\n",
              "      <td>-68.408688</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>49</th>\n",
              "      <td>500870</td>\n",
              "      <td>152.478315</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   asset_id    Investment\n",
              "0    500096    671.167833\n",
              "1    500112   -299.917292\n",
              "2    500124    -31.054822\n",
              "3    500163      5.061251\n",
              "4    500164   -189.970683\n",
              "5    500182     -7.399052\n",
              "6    500188    474.324240\n",
              "7    500330     28.777729\n",
              "8    500387    -10.506000\n",
              "9    500520    169.414217\n",
              "10   500680     24.042003\n",
              "11   500790     -8.377280\n",
              "12   500800   -329.256077\n",
              "13   504212    546.364935\n",
              "14   505010    -99.759968\n",
              "15   505200     -5.834367\n",
              "16   505537   -305.151244\n",
              "17   507685    128.048163\n",
              "18   507878  86734.951864\n",
              "19   517168     39.857125\n",
              "20   519600   -236.739203\n",
              "21   520066   -314.122139\n",
              "22   523367     58.318708\n",
              "23   523395      6.046192\n",
              "24   524715    165.587432\n",
              "25   524816   -171.298216\n",
              "26   530555   1431.504765\n",
              "27   532218  17230.281668\n",
              "28   532286    -51.515110\n",
              "29   532301     71.255820\n",
              "30   532374   -155.190703\n",
              "31   532500     -7.836486\n",
              "32   532776  -1324.351428\n",
              "33   532977     47.346727\n",
              "34   533106   -313.144994\n",
              "35   533278    608.020973\n",
              "36   534091    -17.509772\n",
              "37   538268    104.217604\n",
              "38   539056  -2466.394242\n",
              "39   539254    -18.028016\n",
              "40   540025    -98.746336\n",
              "41   532784    143.654040\n",
              "42   533269   -477.516558\n",
              "43   500260    210.820582\n",
              "44   531500   -270.832994\n",
              "45   533273    229.757113\n",
              "46   530007   -561.017860\n",
              "47   532652  -1441.419768\n",
              "48   500165    -68.408688\n",
              "49   500870    152.478315"
            ]
          },
          "execution_count": 8,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# (b)Investment strategy(How much to invest in which asset to get minimum risk)\n",
        "Investment = 100000.00\n",
        "Investment_strategy = {'asset_id': asset_id.reshape(50),\n",
        "        'Investment': (w*Investment).reshape(50)}\n",
        "view=pd.DataFrame(Investment_strategy)\n",
        "view"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "b98b7d7b",
      "metadata": {
        "id": "b98b7d7b",
        "outputId": "0f31eeb4-d303-442c-c483-3d5c70bf7e1f"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Total_return:  [[0.00206812]]\n",
            "variance_risk:  [[0.13148691]]\n"
          ]
        }
      ],
      "source": [
        "# Total return and variance risk at the optimal point\n",
        "Total_return = (mu_j.T@w)\n",
        "variance_risk = (w.T@omega@w)\n",
        "print(\"Total_return: \",Total_return)\n",
        "print(\"variance_risk: \",variance_risk)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "03c43291",
      "metadata": {
        "id": "03c43291"
      },
      "source": [
        "### B. Markowitz Model II: minimize wTΩ w   subject to  eTw=1, short selling is not allowed."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "d5c529a9",
      "metadata": {
        "id": "d5c529a9"
      },
      "outputs": [],
      "source": [
        "# pip install cvxpy\n",
        "import cvxpy as cp\n",
        "import numpy as np"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "39f1af78",
      "metadata": {
        "id": "39f1af78",
        "outputId": "3ba2f494-1741-40fc-e1f4-2ab8c07c3766"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "2.050540230788057"
            ]
          },
          "execution_count": 11,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "P= 2*omega\n",
        "A = e.T\n",
        "b = np.ones(1)\n",
        "G = -np.eye(n)\n",
        "h = np.zeros(n)\n",
        "x = cp.Variable(n)\n",
        "prob = cp.Problem(cp.Minimize((1/2)*cp.quad_form(x, cp.psd_wrap(P))),\n",
        "                 [G @ x <= h,\n",
        "                  A @ x == b])\n",
        "prob.solve()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "9dc29c49",
      "metadata": {
        "id": "9dc29c49",
        "outputId": "ddae92e9-6d7c-462e-96af-70ee74d61755"
      },
      "outputs": [
        {
          "data": {
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              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>asset_id</th>\n",
              "      <th>weights</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>500096</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>500112</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>500124</td>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>500163</td>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>500164</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>500182</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>500188</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>500330</td>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>500387</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>500520</td>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>10</th>\n",
              "      <td>500680</td>\n",
              "      <td>0.000932</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>11</th>\n",
              "      <td>500790</td>\n",
              "      <td>0.000084</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>12</th>\n",
              "      <td>500800</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>13</th>\n",
              "      <td>504212</td>\n",
              "      <td>0.005582</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>14</th>\n",
              "      <td>505010</td>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>15</th>\n",
              "      <td>505200</td>\n",
              "      <td>0.000018</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>16</th>\n",
              "      <td>505537</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>17</th>\n",
              "      <td>507685</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>18</th>\n",
              "      <td>507878</td>\n",
              "      <td>0.993344</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>19</th>\n",
              "      <td>517168</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>20</th>\n",
              "      <td>519600</td>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>21</th>\n",
              "      <td>520066</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>22</th>\n",
              "      <td>523367</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>23</th>\n",
              "      <td>523395</td>\n",
              "      <td>0.000041</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>24</th>\n",
              "      <td>524715</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25</th>\n",
              "      <td>524816</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>26</th>\n",
              "      <td>530555</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>27</th>\n",
              "      <td>532218</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>28</th>\n",
              "      <td>532286</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>29</th>\n",
              "      <td>532301</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>30</th>\n",
              "      <td>532374</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>31</th>\n",
              "      <td>532500</td>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>32</th>\n",
              "      <td>532776</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>33</th>\n",
              "      <td>532977</td>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>34</th>\n",
              "      <td>533106</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>35</th>\n",
              "      <td>533278</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>36</th>\n",
              "      <td>534091</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>37</th>\n",
              "      <td>538268</td>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>38</th>\n",
              "      <td>539056</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>39</th>\n",
              "      <td>539254</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>40</th>\n",
              "      <td>540025</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>41</th>\n",
              "      <td>532784</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>42</th>\n",
              "      <td>533269</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>43</th>\n",
              "      <td>500260</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>44</th>\n",
              "      <td>531500</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>45</th>\n",
              "      <td>533273</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>46</th>\n",
              "      <td>530007</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>47</th>\n",
              "      <td>532652</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>48</th>\n",
              "      <td>500165</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>49</th>\n",
              "      <td>500870</td>\n",
              "      <td>-0.000000</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   asset_id   weights\n",
              "0    500096 -0.000000\n",
              "1    500112 -0.000000\n",
              "2    500124  0.000000\n",
              "3    500163  0.000000\n",
              "4    500164 -0.000000\n",
              "5    500182 -0.000000\n",
              "6    500188 -0.000000\n",
              "7    500330  0.000000\n",
              "8    500387 -0.000000\n",
              "9    500520  0.000000\n",
              "10   500680  0.000932\n",
              "11   500790  0.000084\n",
              "12   500800 -0.000000\n",
              "13   504212  0.005582\n",
              "14   505010  0.000000\n",
              "15   505200  0.000018\n",
              "16   505537 -0.000000\n",
              "17   507685 -0.000000\n",
              "18   507878  0.993344\n",
              "19   517168 -0.000000\n",
              "20   519600  0.000000\n",
              "21   520066 -0.000000\n",
              "22   523367 -0.000000\n",
              "23   523395  0.000041\n",
              "24   524715 -0.000000\n",
              "25   524816 -0.000000\n",
              "26   530555 -0.000000\n",
              "27   532218 -0.000000\n",
              "28   532286 -0.000000\n",
              "29   532301 -0.000000\n",
              "30   532374 -0.000000\n",
              "31   532500  0.000000\n",
              "32   532776 -0.000000\n",
              "33   532977  0.000000\n",
              "34   533106 -0.000000\n",
              "35   533278 -0.000000\n",
              "36   534091 -0.000000\n",
              "37   538268  0.000000\n",
              "38   539056 -0.000000\n",
              "39   539254 -0.000000\n",
              "40   540025 -0.000000\n",
              "41   532784 -0.000000\n",
              "42   533269 -0.000000\n",
              "43   500260 -0.000000\n",
              "44   531500 -0.000000\n",
              "45   533273 -0.000000\n",
              "46   530007 -0.000000\n",
              "47   532652 -0.000000\n",
              "48   500165 -0.000000\n",
              "49   500870 -0.000000"
            ]
          },
          "execution_count": 12,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# (a) Optimal portfolio\n",
        "w = ['{:.8f}'.format(val) for val in x.value]\n",
        "w = np.array(w,dtype='float64')\n",
        "Optimal_portfolio = {'asset_id': asset_id.reshape(50),\n",
        "        'weights': w.reshape(50)}\n",
        "view=pd.DataFrame(Optimal_portfolio)\n",
        "view"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "a22bda2c",
      "metadata": {
        "id": "a22bda2c",
        "outputId": "d427a358-9d5d-4cbd-b676-c0e073352b91"
      },
      "outputs": [
        {
          "data": {
            "text/html": [
              "<div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>asset_id</th>\n",
              "      <th>Investment</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>500096</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>500112</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>500124</td>\n",
              "      <td>0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>500163</td>\n",
              "      <td>0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>500164</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>500182</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>500188</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>500330</td>\n",
              "      <td>0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>500387</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>500520</td>\n",
              "      <td>0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>10</th>\n",
              "      <td>500680</td>\n",
              "      <td>93.178</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>11</th>\n",
              "      <td>500790</td>\n",
              "      <td>8.395</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>12</th>\n",
              "      <td>500800</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>13</th>\n",
              "      <td>504212</td>\n",
              "      <td>558.172</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>14</th>\n",
              "      <td>505010</td>\n",
              "      <td>0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>15</th>\n",
              "      <td>505200</td>\n",
              "      <td>1.770</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>16</th>\n",
              "      <td>505537</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>17</th>\n",
              "      <td>507685</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>18</th>\n",
              "      <td>507878</td>\n",
              "      <td>99334.352</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>19</th>\n",
              "      <td>517168</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>20</th>\n",
              "      <td>519600</td>\n",
              "      <td>0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>21</th>\n",
              "      <td>520066</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>22</th>\n",
              "      <td>523367</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>23</th>\n",
              "      <td>523395</td>\n",
              "      <td>4.133</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>24</th>\n",
              "      <td>524715</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25</th>\n",
              "      <td>524816</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>26</th>\n",
              "      <td>530555</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>27</th>\n",
              "      <td>532218</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>28</th>\n",
              "      <td>532286</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>29</th>\n",
              "      <td>532301</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>30</th>\n",
              "      <td>532374</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>31</th>\n",
              "      <td>532500</td>\n",
              "      <td>0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>32</th>\n",
              "      <td>532776</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>33</th>\n",
              "      <td>532977</td>\n",
              "      <td>0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>34</th>\n",
              "      <td>533106</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>35</th>\n",
              "      <td>533278</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>36</th>\n",
              "      <td>534091</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>37</th>\n",
              "      <td>538268</td>\n",
              "      <td>0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>38</th>\n",
              "      <td>539056</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>39</th>\n",
              "      <td>539254</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>40</th>\n",
              "      <td>540025</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>41</th>\n",
              "      <td>532784</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>42</th>\n",
              "      <td>533269</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>43</th>\n",
              "      <td>500260</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>44</th>\n",
              "      <td>531500</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>45</th>\n",
              "      <td>533273</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>46</th>\n",
              "      <td>530007</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>47</th>\n",
              "      <td>532652</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>48</th>\n",
              "      <td>500165</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>49</th>\n",
              "      <td>500870</td>\n",
              "      <td>-0.000</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   asset_id  Investment\n",
              "0    500096      -0.000\n",
              "1    500112      -0.000\n",
              "2    500124       0.000\n",
              "3    500163       0.000\n",
              "4    500164      -0.000\n",
              "5    500182      -0.000\n",
              "6    500188      -0.000\n",
              "7    500330       0.000\n",
              "8    500387      -0.000\n",
              "9    500520       0.000\n",
              "10   500680      93.178\n",
              "11   500790       8.395\n",
              "12   500800      -0.000\n",
              "13   504212     558.172\n",
              "14   505010       0.000\n",
              "15   505200       1.770\n",
              "16   505537      -0.000\n",
              "17   507685      -0.000\n",
              "18   507878   99334.352\n",
              "19   517168      -0.000\n",
              "20   519600       0.000\n",
              "21   520066      -0.000\n",
              "22   523367      -0.000\n",
              "23   523395       4.133\n",
              "24   524715      -0.000\n",
              "25   524816      -0.000\n",
              "26   530555      -0.000\n",
              "27   532218      -0.000\n",
              "28   532286      -0.000\n",
              "29   532301      -0.000\n",
              "30   532374      -0.000\n",
              "31   532500       0.000\n",
              "32   532776      -0.000\n",
              "33   532977       0.000\n",
              "34   533106      -0.000\n",
              "35   533278      -0.000\n",
              "36   534091      -0.000\n",
              "37   538268       0.000\n",
              "38   539056      -0.000\n",
              "39   539254      -0.000\n",
              "40   540025      -0.000\n",
              "41   532784      -0.000\n",
              "42   533269      -0.000\n",
              "43   500260      -0.000\n",
              "44   531500      -0.000\n",
              "45   533273      -0.000\n",
              "46   530007      -0.000\n",
              "47   532652      -0.000\n",
              "48   500165      -0.000\n",
              "49   500870      -0.000"
            ]
          },
          "execution_count": 13,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "Investment = 100000.00\n",
        "Investment_strategy = {'asset_id': asset_id.reshape(50),\n",
        "        'Investment': (w*Investment).reshape(50)}\n",
        "view=pd.DataFrame(Investment_strategy)\n",
        "view"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "6e8201cb",
      "metadata": {
        "id": "6e8201cb",
        "outputId": "925d0c9d-f6e6-4943-bccd-3901575a6c82"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Total_return:  [0.00230565]\n",
            "variance_risk:  2.0505402309718437\n"
          ]
        }
      ],
      "source": [
        "# Total return and variance risk at the optimal point\n",
        "Total_return = (mu_j.T@w)\n",
        "variance_risk = (w.T@omega@w)\n",
        "print(\"Total_return: \",Total_return)\n",
        "print(\"variance_risk: \",variance_risk)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "2fc86819",
      "metadata": {
        "id": "2fc86819"
      },
      "source": [
        " ### C : Markowitz Model III: minimize wTΩ w   subject to  eTw=1, and total return is exactly 5%,  short selling is allowed."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "d48697cb",
      "metadata": {
        "id": "d48697cb",
        "outputId": "10f72245-bf88-4d95-88db-afd8b79bbdfa"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "lambda 1:  -1.3939431935240465\n",
            "lambda 2:  0.0031737591004078413\n"
          ]
        }
      ],
      "source": [
        "Return = .05\n",
        "A = np.vstack((e.T,mu_j.T))\n",
        "b = np.array([[1],[Return]],dtype=float)\n",
        "Lambda = -2/inv(A@inv(omega)@A.T)@b\n",
        "w = inv(omega)@A.T@inv(A@inv(omega)@A.T)@b\n",
        "print(\"lambda 1: \",Lambda[0][0])\n",
        "print(\"lambda 2: \",Lambda[1][0])"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "3e6f479b",
      "metadata": {
        "id": "3e6f479b",
        "outputId": "ed8a34ea-12b8-424d-e84e-f6001e5eaa1e"
      },
      "outputs": [
        {
          "data": {
            "text/html": [
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              "\n",
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              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>asset_id</th>\n",
              "      <th>weights</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>500096</td>\n",
              "      <td>-0.299414</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>500112</td>\n",
              "      <td>-0.242819</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>500124</td>\n",
              "      <td>-0.027311</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>500163</td>\n",
              "      <td>0.054956</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>500164</td>\n",
              "      <td>0.073917</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>500182</td>\n",
              "      <td>-0.003143</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>500188</td>\n",
              "      <td>0.337754</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>500330</td>\n",
              "      <td>0.060679</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>500387</td>\n",
              "      <td>-0.005332</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>500520</td>\n",
              "      <td>0.084718</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>10</th>\n",
              "      <td>500680</td>\n",
              "      <td>0.030291</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>11</th>\n",
              "      <td>500790</td>\n",
              "      <td>0.005661</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>12</th>\n",
              "      <td>500800</td>\n",
              "      <td>0.126470</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>13</th>\n",
              "      <td>504212</td>\n",
              "      <td>0.071826</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>14</th>\n",
              "      <td>505010</td>\n",
              "      <td>0.025864</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>15</th>\n",
              "      <td>505200</td>\n",
              "      <td>0.002165</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>16</th>\n",
              "      <td>505537</td>\n",
              "      <td>0.163759</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>17</th>\n",
              "      <td>507685</td>\n",
              "      <td>-0.383053</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>18</th>\n",
              "      <td>507878</td>\n",
              "      <td>30.002658</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>19</th>\n",
              "      <td>517168</td>\n",
              "      <td>-0.531077</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>20</th>\n",
              "      <td>519600</td>\n",
              "      <td>0.280012</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>21</th>\n",
              "      <td>520066</td>\n",
              "      <td>-0.266153</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>22</th>\n",
              "      <td>523367</td>\n",
              "      <td>-0.015483</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>23</th>\n",
              "      <td>523395</td>\n",
              "      <td>-0.006792</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>24</th>\n",
              "      <td>524715</td>\n",
              "      <td>-0.105969</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25</th>\n",
              "      <td>524816</td>\n",
              "      <td>0.133715</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>26</th>\n",
              "      <td>530555</td>\n",
              "      <td>-5.179502</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>27</th>\n",
              "      <td>532218</td>\n",
              "      <td>-27.327178</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>28</th>\n",
              "      <td>532286</td>\n",
              "      <td>0.176996</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>29</th>\n",
              "      <td>532301</td>\n",
              "      <td>-0.549871</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>30</th>\n",
              "      <td>532374</td>\n",
              "      <td>0.201403</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>31</th>\n",
              "      <td>532500</td>\n",
              "      <td>-0.010562</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>32</th>\n",
              "      <td>532776</td>\n",
              "      <td>-0.391582</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>33</th>\n",
              "      <td>532977</td>\n",
              "      <td>0.036326</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>34</th>\n",
              "      <td>533106</td>\n",
              "      <td>0.031217</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>35</th>\n",
              "      <td>533278</td>\n",
              "      <td>0.193092</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>36</th>\n",
              "      <td>534091</td>\n",
              "      <td>-0.010238</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>37</th>\n",
              "      <td>538268</td>\n",
              "      <td>0.051080</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>38</th>\n",
              "      <td>539056</td>\n",
              "      <td>1.423301</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>39</th>\n",
              "      <td>539254</td>\n",
              "      <td>-0.020235</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>40</th>\n",
              "      <td>540025</td>\n",
              "      <td>0.372440</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>41</th>\n",
              "      <td>532784</td>\n",
              "      <td>0.156335</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>42</th>\n",
              "      <td>533269</td>\n",
              "      <td>-0.019973</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>43</th>\n",
              "      <td>500260</td>\n",
              "      <td>0.213938</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>44</th>\n",
              "      <td>531500</td>\n",
              "      <td>0.049113</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>45</th>\n",
              "      <td>533273</td>\n",
              "      <td>-0.069538</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>46</th>\n",
              "      <td>530007</td>\n",
              "      <td>-0.051949</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>47</th>\n",
              "      <td>532652</td>\n",
              "      <td>2.318150</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>48</th>\n",
              "      <td>500165</td>\n",
              "      <td>0.094467</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>49</th>\n",
              "      <td>500870</td>\n",
              "      <td>-0.255130</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   asset_id    weights\n",
              "0    500096  -0.299414\n",
              "1    500112  -0.242819\n",
              "2    500124  -0.027311\n",
              "3    500163   0.054956\n",
              "4    500164   0.073917\n",
              "5    500182  -0.003143\n",
              "6    500188   0.337754\n",
              "7    500330   0.060679\n",
              "8    500387  -0.005332\n",
              "9    500520   0.084718\n",
              "10   500680   0.030291\n",
              "11   500790   0.005661\n",
              "12   500800   0.126470\n",
              "13   504212   0.071826\n",
              "14   505010   0.025864\n",
              "15   505200   0.002165\n",
              "16   505537   0.163759\n",
              "17   507685  -0.383053\n",
              "18   507878  30.002658\n",
              "19   517168  -0.531077\n",
              "20   519600   0.280012\n",
              "21   520066  -0.266153\n",
              "22   523367  -0.015483\n",
              "23   523395  -0.006792\n",
              "24   524715  -0.105969\n",
              "25   524816   0.133715\n",
              "26   530555  -5.179502\n",
              "27   532218 -27.327178\n",
              "28   532286   0.176996\n",
              "29   532301  -0.549871\n",
              "30   532374   0.201403\n",
              "31   532500  -0.010562\n",
              "32   532776  -0.391582\n",
              "33   532977   0.036326\n",
              "34   533106   0.031217\n",
              "35   533278   0.193092\n",
              "36   534091  -0.010238\n",
              "37   538268   0.051080\n",
              "38   539056   1.423301\n",
              "39   539254  -0.020235\n",
              "40   540025   0.372440\n",
              "41   532784   0.156335\n",
              "42   533269  -0.019973\n",
              "43   500260   0.213938\n",
              "44   531500   0.049113\n",
              "45   533273  -0.069538\n",
              "46   530007  -0.051949\n",
              "47   532652   2.318150\n",
              "48   500165   0.094467\n",
              "49   500870  -0.255130"
            ]
          },
          "execution_count": 16,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# (a) Optimal portfolio\n",
        "Optimal_portfolio = {'asset_id': asset_id.reshape(50),\n",
        "        'weights': w.reshape(50)}\n",
        "view=pd.DataFrame(Optimal_portfolio)\n",
        "view"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "3c704c2f",
      "metadata": {
        "id": "3c704c2f",
        "outputId": "d005d3d0-d22e-4b24-fa4a-a91be8377fdc"
      },
      "outputs": [
        {
          "data": {
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              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>asset_id</th>\n",
              "      <th>Investment</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>500096</td>\n",
              "      <td>-2.994142e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>500112</td>\n",
              "      <td>-2.428192e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>500124</td>\n",
              "      <td>-2.731116e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>500163</td>\n",
              "      <td>5.495597e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>500164</td>\n",
              "      <td>7.391662e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>500182</td>\n",
              "      <td>-3.143379e+02</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>500188</td>\n",
              "      <td>3.377538e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>500330</td>\n",
              "      <td>6.067944e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>500387</td>\n",
              "      <td>-5.332063e+02</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>500520</td>\n",
              "      <td>8.471770e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>10</th>\n",
              "      <td>500680</td>\n",
              "      <td>3.029080e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>11</th>\n",
              "      <td>500790</td>\n",
              "      <td>5.661252e+02</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>12</th>\n",
              "      <td>500800</td>\n",
              "      <td>1.264703e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>13</th>\n",
              "      <td>504212</td>\n",
              "      <td>7.182641e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>14</th>\n",
              "      <td>505010</td>\n",
              "      <td>2.586427e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>15</th>\n",
              "      <td>505200</td>\n",
              "      <td>2.164864e+02</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>16</th>\n",
              "      <td>505537</td>\n",
              "      <td>1.637590e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>17</th>\n",
              "      <td>507685</td>\n",
              "      <td>-3.830528e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>18</th>\n",
              "      <td>507878</td>\n",
              "      <td>3.000266e+06</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>19</th>\n",
              "      <td>517168</td>\n",
              "      <td>-5.310766e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>20</th>\n",
              "      <td>519600</td>\n",
              "      <td>2.800121e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>21</th>\n",
              "      <td>520066</td>\n",
              "      <td>-2.661532e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>22</th>\n",
              "      <td>523367</td>\n",
              "      <td>-1.548254e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>23</th>\n",
              "      <td>523395</td>\n",
              "      <td>-6.792209e+02</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>24</th>\n",
              "      <td>524715</td>\n",
              "      <td>-1.059685e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25</th>\n",
              "      <td>524816</td>\n",
              "      <td>1.337155e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>26</th>\n",
              "      <td>530555</td>\n",
              "      <td>-5.179502e+05</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>27</th>\n",
              "      <td>532218</td>\n",
              "      <td>-2.732718e+06</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>28</th>\n",
              "      <td>532286</td>\n",
              "      <td>1.769965e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>29</th>\n",
              "      <td>532301</td>\n",
              "      <td>-5.498707e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>30</th>\n",
              "      <td>532374</td>\n",
              "      <td>2.014027e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>31</th>\n",
              "      <td>532500</td>\n",
              "      <td>-1.056152e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>32</th>\n",
              "      <td>532776</td>\n",
              "      <td>-3.915819e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>33</th>\n",
              "      <td>532977</td>\n",
              "      <td>3.632568e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>34</th>\n",
              "      <td>533106</td>\n",
              "      <td>3.121677e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>35</th>\n",
              "      <td>533278</td>\n",
              "      <td>1.930919e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>36</th>\n",
              "      <td>534091</td>\n",
              "      <td>-1.023798e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>37</th>\n",
              "      <td>538268</td>\n",
              "      <td>5.107959e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>38</th>\n",
              "      <td>539056</td>\n",
              "      <td>1.423301e+05</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>39</th>\n",
              "      <td>539254</td>\n",
              "      <td>-2.023485e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>40</th>\n",
              "      <td>540025</td>\n",
              "      <td>3.724399e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>41</th>\n",
              "      <td>532784</td>\n",
              "      <td>1.563349e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>42</th>\n",
              "      <td>533269</td>\n",
              "      <td>-1.997299e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>43</th>\n",
              "      <td>500260</td>\n",
              "      <td>2.139385e+04</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>44</th>\n",
              "      <td>531500</td>\n",
              "      <td>4.911280e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>45</th>\n",
              "      <td>533273</td>\n",
              "      <td>-6.953775e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>46</th>\n",
              "      <td>530007</td>\n",
              "      <td>-5.194928e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>47</th>\n",
              "      <td>532652</td>\n",
              "      <td>2.318150e+05</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>48</th>\n",
              "      <td>500165</td>\n",
              "      <td>9.446683e+03</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>49</th>\n",
              "      <td>500870</td>\n",
              "      <td>-2.551302e+04</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   asset_id    Investment\n",
              "0    500096 -2.994142e+04\n",
              "1    500112 -2.428192e+04\n",
              "2    500124 -2.731116e+03\n",
              "3    500163  5.495597e+03\n",
              "4    500164  7.391662e+03\n",
              "5    500182 -3.143379e+02\n",
              "6    500188  3.377538e+04\n",
              "7    500330  6.067944e+03\n",
              "8    500387 -5.332063e+02\n",
              "9    500520  8.471770e+03\n",
              "10   500680  3.029080e+03\n",
              "11   500790  5.661252e+02\n",
              "12   500800  1.264703e+04\n",
              "13   504212  7.182641e+03\n",
              "14   505010  2.586427e+03\n",
              "15   505200  2.164864e+02\n",
              "16   505537  1.637590e+04\n",
              "17   507685 -3.830528e+04\n",
              "18   507878  3.000266e+06\n",
              "19   517168 -5.310766e+04\n",
              "20   519600  2.800121e+04\n",
              "21   520066 -2.661532e+04\n",
              "22   523367 -1.548254e+03\n",
              "23   523395 -6.792209e+02\n",
              "24   524715 -1.059685e+04\n",
              "25   524816  1.337155e+04\n",
              "26   530555 -5.179502e+05\n",
              "27   532218 -2.732718e+06\n",
              "28   532286  1.769965e+04\n",
              "29   532301 -5.498707e+04\n",
              "30   532374  2.014027e+04\n",
              "31   532500 -1.056152e+03\n",
              "32   532776 -3.915819e+04\n",
              "33   532977  3.632568e+03\n",
              "34   533106  3.121677e+03\n",
              "35   533278  1.930919e+04\n",
              "36   534091 -1.023798e+03\n",
              "37   538268  5.107959e+03\n",
              "38   539056  1.423301e+05\n",
              "39   539254 -2.023485e+03\n",
              "40   540025  3.724399e+04\n",
              "41   532784  1.563349e+04\n",
              "42   533269 -1.997299e+03\n",
              "43   500260  2.139385e+04\n",
              "44   531500  4.911280e+03\n",
              "45   533273 -6.953775e+03\n",
              "46   530007 -5.194928e+03\n",
              "47   532652  2.318150e+05\n",
              "48   500165  9.446683e+03\n",
              "49   500870 -2.551302e+04"
            ]
          },
          "execution_count": 17,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "Investment = 100000.00\n",
        "Investment_strategy = {'asset_id': asset_id.reshape(50),\n",
        "        'Investment': (w*Investment).reshape(50)}\n",
        "view=pd.DataFrame(Investment_strategy)\n",
        "view"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "c433c2c4",
      "metadata": {
        "id": "c433c2c4",
        "outputId": "c29417af-b6c4-4f62-9ada-67fa352e146f"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Total_return:  [[0.05]]\n",
            "variance_risk:  [[700.108729]]\n"
          ]
        }
      ],
      "source": [
        "# Total return and variance risk at the optimal point\n",
        "Total_return = (mu_j.T@w)\n",
        "variance_risk = (w.T@omega@w)\n",
        "print(\"Total_return: \",Total_return)\n",
        "print(\"variance_risk: \",variance_risk)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "0b02456f",
      "metadata": {
        "id": "0b02456f"
      },
      "source": [
        "### D:  Markowitz Model IV: minimize  wTΩ w subject to eTw= 1, and total return is a minimum of 5%, short selling is allowed"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "41249294",
      "metadata": {
        "id": "41249294",
        "outputId": "75e7441b-1282-4ad5-b0d5-2ed7f03de460"
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "700.10872864776"
            ]
          },
          "execution_count": 19,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "Return = .05\n",
        "P= 2*omega\n",
        "A = e.T\n",
        "b = np.ones(1)\n",
        "G = -mu_j.T\n",
        "h = np.array([-Return])\n",
        "x = cp.Variable(n)\n",
        "prob = cp.Problem(cp.Minimize((1/2)*cp.quad_form(x, cp.psd_wrap(P))),\n",
        "                 [G @ x <= h,\n",
        "                  A @ x == b])\n",
        "prob.solve()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "e5747a84",
      "metadata": {
        "id": "e5747a84",
        "outputId": "56db6d1d-9393-4f54-8c1b-c4ef518254e0"
      },
      "outputs": [
        {
          "data": {
            "text/html": [
              "<div>\n",
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              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>asset_id</th>\n",
              "      <th>weights</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>500096</td>\n",
              "      <td>-0.299414</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>500112</td>\n",
              "      <td>-0.242819</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>500124</td>\n",
              "      <td>-0.027311</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>500163</td>\n",
              "      <td>0.054956</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>500164</td>\n",
              "      <td>0.073917</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>500182</td>\n",
              "      <td>-0.003143</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>500188</td>\n",
              "      <td>0.337754</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>500330</td>\n",
              "      <td>0.060679</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>500387</td>\n",
              "      <td>-0.005332</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>500520</td>\n",
              "      <td>0.084718</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>10</th>\n",
              "      <td>500680</td>\n",
              "      <td>0.030291</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>11</th>\n",
              "      <td>500790</td>\n",
              "      <td>0.005661</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>12</th>\n",
              "      <td>500800</td>\n",
              "      <td>0.126470</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>13</th>\n",
              "      <td>504212</td>\n",
              "      <td>0.071826</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>14</th>\n",
              "      <td>505010</td>\n",
              "      <td>0.025864</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>15</th>\n",
              "      <td>505200</td>\n",
              "      <td>0.002165</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>16</th>\n",
              "      <td>505537</td>\n",
              "      <td>0.163759</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>17</th>\n",
              "      <td>507685</td>\n",
              "      <td>-0.383053</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>18</th>\n",
              "      <td>507878</td>\n",
              "      <td>30.002658</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>19</th>\n",
              "      <td>517168</td>\n",
              "      <td>-0.531077</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>20</th>\n",
              "      <td>519600</td>\n",
              "      <td>0.280012</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>21</th>\n",
              "      <td>520066</td>\n",
              "      <td>-0.266153</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>22</th>\n",
              "      <td>523367</td>\n",
              "      <td>-0.015483</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>23</th>\n",
              "      <td>523395</td>\n",
              "      <td>-0.006792</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>24</th>\n",
              "      <td>524715</td>\n",
              "      <td>-0.105969</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25</th>\n",
              "      <td>524816</td>\n",
              "      <td>0.133715</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>26</th>\n",
              "      <td>530555</td>\n",
              "      <td>-5.179502</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>27</th>\n",
              "      <td>532218</td>\n",
              "      <td>-27.327178</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>28</th>\n",
              "      <td>532286</td>\n",
              "      <td>0.176996</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>29</th>\n",
              "      <td>532301</td>\n",
              "      <td>-0.549871</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>30</th>\n",
              "      <td>532374</td>\n",
              "      <td>0.201403</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>31</th>\n",
              "      <td>532500</td>\n",
              "      <td>-0.010562</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>32</th>\n",
              "      <td>532776</td>\n",
              "      <td>-0.391582</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>33</th>\n",
              "      <td>532977</td>\n",
              "      <td>0.036326</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>34</th>\n",
              "      <td>533106</td>\n",
              "      <td>0.031217</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>35</th>\n",
              "      <td>533278</td>\n",
              "      <td>0.193092</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>36</th>\n",
              "      <td>534091</td>\n",
              "      <td>-0.010238</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>37</th>\n",
              "      <td>538268</td>\n",
              "      <td>0.051080</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>38</th>\n",
              "      <td>539056</td>\n",
              "      <td>1.423301</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>39</th>\n",
              "      <td>539254</td>\n",
              "      <td>-0.020235</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>40</th>\n",
              "      <td>540025</td>\n",
              "      <td>0.372440</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>41</th>\n",
              "      <td>532784</td>\n",
              "      <td>0.156335</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>42</th>\n",
              "      <td>533269</td>\n",
              "      <td>-0.019973</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>43</th>\n",
              "      <td>500260</td>\n",
              "      <td>0.213938</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>44</th>\n",
              "      <td>531500</td>\n",
              "      <td>0.049113</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>45</th>\n",
              "      <td>533273</td>\n",
              "      <td>-0.069538</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>46</th>\n",
              "      <td>530007</td>\n",
              "      <td>-0.051949</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>47</th>\n",
              "      <td>532652</td>\n",
              "      <td>2.318150</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>48</th>\n",
              "      <td>500165</td>\n",
              "      <td>0.094467</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>49</th>\n",
              "      <td>500870</td>\n",
              "      <td>-0.255130</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   asset_id    weights\n",
              "0    500096  -0.299414\n",
              "1    500112  -0.242819\n",
              "2    500124  -0.027311\n",
              "3    500163   0.054956\n",
              "4    500164   0.073917\n",
              "5    500182  -0.003143\n",
              "6    500188   0.337754\n",
              "7    500330   0.060679\n",
              "8    500387  -0.005332\n",
              "9    500520   0.084718\n",
              "10   500680   0.030291\n",
              "11   500790   0.005661\n",
              "12   500800   0.126470\n",
              "13   504212   0.071826\n",
              "14   505010   0.025864\n",
              "15   505200   0.002165\n",
              "16   505537   0.163759\n",
              "17   507685  -0.383053\n",
              "18   507878  30.002658\n",
              "19   517168  -0.531077\n",
              "20   519600   0.280012\n",
              "21   520066  -0.266153\n",
              "22   523367  -0.015483\n",
              "23   523395  -0.006792\n",
              "24   524715  -0.105969\n",
              "25   524816   0.133715\n",
              "26   530555  -5.179502\n",
              "27   532218 -27.327178\n",
              "28   532286   0.176996\n",
              "29   532301  -0.549871\n",
              "30   532374   0.201403\n",
              "31   532500  -0.010562\n",
              "32   532776  -0.391582\n",
              "33   532977   0.036326\n",
              "34   533106   0.031217\n",
              "35   533278   0.193092\n",
              "36   534091  -0.010238\n",
              "37   538268   0.051080\n",
              "38   539056   1.423301\n",
              "39   539254  -0.020235\n",
              "40   540025   0.372440\n",
              "41   532784   0.156335\n",
              "42   533269  -0.019973\n",
              "43   500260   0.213938\n",
              "44   531500   0.049113\n",
              "45   533273  -0.069538\n",
              "46   530007  -0.051949\n",
              "47   532652   2.318150\n",
              "48   500165   0.094467\n",
              "49   500870  -0.255130"
            ]
          },
          "execution_count": 20,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# (a) Optimal portfolio\n",
        "w = ['{:.8f}'.format(val) for val in x.value]\n",
        "w = np.array(w,dtype='float64')\n",
        "Optimal_portfolio = {'asset_id': asset_id.reshape(50),\n",
        "        'weights': w.reshape(50)}\n",
        "view=pd.DataFrame(Optimal_portfolio)\n",
        "view"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "5e9a3276",
      "metadata": {
        "id": "5e9a3276",
        "outputId": "defa466b-5016-4015-c067-668bd1f6fc25"
      },
      "outputs": [
        {
          "data": {
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              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>asset_id</th>\n",
              "      <th>Investment</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>500096</td>\n",
              "      <td>-29941.424</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>500112</td>\n",
              "      <td>-24281.919</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>500124</td>\n",
              "      <td>-2731.116</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>500163</td>\n",
              "      <td>5495.597</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>500164</td>\n",
              "      <td>7391.662</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>500182</td>\n",
              "      <td>-314.338</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>500188</td>\n",
              "      <td>33775.381</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>500330</td>\n",
              "      <td>6067.944</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>500387</td>\n",
              "      <td>-533.206</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>500520</td>\n",
              "      <td>8471.770</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>10</th>\n",
              "      <td>500680</td>\n",
              "      <td>3029.080</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>11</th>\n",
              "      <td>500790</td>\n",
              "      <td>566.125</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>12</th>\n",
              "      <td>500800</td>\n",
              "      <td>12647.032</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>13</th>\n",
              "      <td>504212</td>\n",
              "      <td>7182.641</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>14</th>\n",
              "      <td>505010</td>\n",
              "      <td>2586.427</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>15</th>\n",
              "      <td>505200</td>\n",
              "      <td>216.486</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>16</th>\n",
              "      <td>505537</td>\n",
              "      <td>16375.905</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>17</th>\n",
              "      <td>507685</td>\n",
              "      <td>-38305.278</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>18</th>\n",
              "      <td>507878</td>\n",
              "      <td>3000265.848</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>19</th>\n",
              "      <td>517168</td>\n",
              "      <td>-53107.664</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>20</th>\n",
              "      <td>519600</td>\n",
              "      <td>28001.207</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>21</th>\n",
              "      <td>520066</td>\n",
              "      <td>-26615.319</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>22</th>\n",
              "      <td>523367</td>\n",
              "      <td>-1548.254</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>23</th>\n",
              "      <td>523395</td>\n",
              "      <td>-679.221</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>24</th>\n",
              "      <td>524715</td>\n",
              "      <td>-10596.852</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25</th>\n",
              "      <td>524816</td>\n",
              "      <td>13371.548</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>26</th>\n",
              "      <td>530555</td>\n",
              "      <td>-517950.191</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>27</th>\n",
              "      <td>532218</td>\n",
              "      <td>-2732717.824</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>28</th>\n",
              "      <td>532286</td>\n",
              "      <td>17699.648</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>29</th>\n",
              "      <td>532301</td>\n",
              "      <td>-54987.073</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>30</th>\n",
              "      <td>532374</td>\n",
              "      <td>20140.269</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>31</th>\n",
              "      <td>532500</td>\n",
              "      <td>-1056.152</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>32</th>\n",
              "      <td>532776</td>\n",
              "      <td>-39158.194</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>33</th>\n",
              "      <td>532977</td>\n",
              "      <td>3632.568</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>34</th>\n",
              "      <td>533106</td>\n",
              "      <td>3121.677</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>35</th>\n",
              "      <td>533278</td>\n",
              "      <td>19309.188</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>36</th>\n",
              "      <td>534091</td>\n",
              "      <td>-1023.798</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>37</th>\n",
              "      <td>538268</td>\n",
              "      <td>5107.959</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>38</th>\n",
              "      <td>539056</td>\n",
              "      <td>142330.061</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>39</th>\n",
              "      <td>539254</td>\n",
              "      <td>-2023.485</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>40</th>\n",
              "      <td>540025</td>\n",
              "      <td>37243.992</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>41</th>\n",
              "      <td>532784</td>\n",
              "      <td>15633.490</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>42</th>\n",
              "      <td>533269</td>\n",
              "      <td>-1997.299</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>43</th>\n",
              "      <td>500260</td>\n",
              "      <td>21393.849</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>44</th>\n",
              "      <td>531500</td>\n",
              "      <td>4911.280</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>45</th>\n",
              "      <td>533273</td>\n",
              "      <td>-6953.775</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>46</th>\n",
              "      <td>530007</td>\n",
              "      <td>-5194.928</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>47</th>\n",
              "      <td>532652</td>\n",
              "      <td>231815.009</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>48</th>\n",
              "      <td>500165</td>\n",
              "      <td>9446.683</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>49</th>\n",
              "      <td>500870</td>\n",
              "      <td>-25513.016</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "   asset_id   Investment\n",
              "0    500096   -29941.424\n",
              "1    500112   -24281.919\n",
              "2    500124    -2731.116\n",
              "3    500163     5495.597\n",
              "4    500164     7391.662\n",
              "5    500182     -314.338\n",
              "6    500188    33775.381\n",
              "7    500330     6067.944\n",
              "8    500387     -533.206\n",
              "9    500520     8471.770\n",
              "10   500680     3029.080\n",
              "11   500790      566.125\n",
              "12   500800    12647.032\n",
              "13   504212     7182.641\n",
              "14   505010     2586.427\n",
              "15   505200      216.486\n",
              "16   505537    16375.905\n",
              "17   507685   -38305.278\n",
              "18   507878  3000265.848\n",
              "19   517168   -53107.664\n",
              "20   519600    28001.207\n",
              "21   520066   -26615.319\n",
              "22   523367    -1548.254\n",
              "23   523395     -679.221\n",
              "24   524715   -10596.852\n",
              "25   524816    13371.548\n",
              "26   530555  -517950.191\n",
              "27   532218 -2732717.824\n",
              "28   532286    17699.648\n",
              "29   532301   -54987.073\n",
              "30   532374    20140.269\n",
              "31   532500    -1056.152\n",
              "32   532776   -39158.194\n",
              "33   532977     3632.568\n",
              "34   533106     3121.677\n",
              "35   533278    19309.188\n",
              "36   534091    -1023.798\n",
              "37   538268     5107.959\n",
              "38   539056   142330.061\n",
              "39   539254    -2023.485\n",
              "40   540025    37243.992\n",
              "41   532784    15633.490\n",
              "42   533269    -1997.299\n",
              "43   500260    21393.849\n",
              "44   531500     4911.280\n",
              "45   533273    -6953.775\n",
              "46   530007    -5194.928\n",
              "47   532652   231815.009\n",
              "48   500165     9446.683\n",
              "49   500870   -25513.016"
            ]
          },
          "execution_count": 21,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "Investment = 100000.00\n",
        "Investment_strategy = {'asset_id': asset_id.reshape(50),\n",
        "        'Investment': (w*Investment).reshape(50)}\n",
        "view=pd.DataFrame(Investment_strategy)\n",
        "view"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "77135b6c",
      "metadata": {
        "id": "77135b6c",
        "outputId": "95fd6c82-3ed2-4cc5-a33b-7e5cd691c751"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Total_return:  [0.05]\n",
            "variance_risk:  700.1087279935554\n"
          ]
        }
      ],
      "source": [
        "# Total return and variance risk at the optimal point\n",
        "Total_return = (mu_j.T@w)\n",
        "variance_risk = (w.T@omega@w)\n",
        "print(\"Total_return: \",Total_return)\n",
        "print(\"variance_risk: \",variance_risk)"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.8.8"
    },
    "colab": {
      "provenance": []
    }
  },
  "nbformat": 4,
  "nbformat_minor": 5
}