12_diversification.ipynb

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   "source": [
    "# Diversification"
   ]
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   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import plotly.graph_objects as go\n",
    "from cvxopt import matrix\n",
    "from cvxopt.solvers import qp as Solver, options as SolverOptions"
   ]
  },
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   "attachments": {},
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   "source": [
    "### Inputs"
   ]
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   "outputs": [
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     "name": "stdout",
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     "text": [
      "[[0.0225    0.0185625 0.023625 ]\n",
      " [0.0185625 0.027225  0.0259875]\n",
      " [0.023625  0.0259875 0.0441   ]]\n"
     ]
    }
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   "source": [
    "##### Inputs\n",
    "# Expected returns\n",
    "MNS = np.array([0.06, 0.065, 0.08])\n",
    "\n",
    "# Standard deviations\n",
    "SDS = np.array([0.15, 0.165, 0.21])\n",
    "\n",
    "# Correlations\n",
    "C  = np.identity(3)\n",
    "C[0, 1] = C[1, 0] = 0.75\n",
    "C[0, 2] = C[2, 0] = 0.75\n",
    "C[1, 2] = C[2, 1] = 0.75\n",
    "\n",
    "# Covariance matrix\n",
    "COV = np.diag(SDS) @ C @ np.diag(SDS)\n",
    "\n",
    "print(COV)"
   ]
  },
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     "text": [
      "Portfolio Expected Return:\t  0.0675\n",
      "Portfolio Standard Deviation:\t  0.1583\n"
     ]
    }
   ],
   "source": [
    "# Portfolio weights\n",
    "WGTS = np.array([0.25, 0.5, 0.25])\n",
    "\n",
    "#Portfolio expected return\n",
    "port_expret = WGTS @ MNS\n",
    "print(f'Portfolio Expected Return:\\t {port_expret: ,.4f}')\n",
    "\n",
    "# Portfolio standard deviation\n",
    "port_sd = np.sqrt(WGTS @ COV @ WGTS)\n",
    "print(f'Portfolio Standard Deviation:\\t {port_sd: ,.4f}')"
   ]
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   "source": [
    "### Investment opportunity set\n",
    "First, let's consider the set of just investing in asset 1 and asset 2"
   ]
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   "source": [
    "WGT1 = np.arange(0,1.05,0.05)\n",
    "df = pd.DataFrame(dtype='float',columns = ['w1','w2','w3','port_expret','port_sd'], index=np.arange(len(WGT1)))\n",
    "df.w1 = WGT1\n",
    "df.w2 = 1-WGT1\n",
    "df.w3 = 0\n",
    "df.port_expret = [[w,1-w,0] @ MNS for w in WGT1]\n",
    "df.port_sd = [np.sqrt([w,1-w,0] @ COV @ [w,1-w,0]) for w in WGT1]"
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    "fig = go.Figure()\n",
    "\n",
    "# Plot portfolios\n",
    "trace= go.Scatter(x=df.port_sd, y=df.port_expret, name='Portfolios')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Plot underlying assets\n",
    "trace1= go.Scatter(x=[SDS[0]], y=[MNS[0]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 1')\n",
    "fig.add_trace(trace1)\n",
    "trace2= go.Scatter(x=[SDS[1]], y=[MNS[1]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 2')\n",
    "fig.add_trace(trace2)\n",
    "\n",
    "# Formatting\n",
    "fig.layout.yaxis[\"title\"] = \"Expected Return\"\n",
    "fig.layout.xaxis[\"title\"] = \"Standard Deviation\"\n",
    "fig.update_yaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(range=[0.85 * df[\"port_sd\"].min(), 1.25 * df[\"port_sd\"].max()])\n",
    "fig.update_yaxes(range=[0.85 * df[\"port_expret\"].min(), 1.25 * df[\"port_expret\"].max()])\n",
    "fig.update_layout(legend=dict(yanchor=\"top\", y=0.99, xanchor=\"left\", x=0.01))\n",
    "fig.show()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's add some of asset 3 now.  We will simulate weights that are non-negative and sum to 1.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.30804601, 0.47178612, 0.22016787])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def simulate_weights(n):\n",
    "    ''' Simulate non-negative weights that sum to 1.'''\n",
    "    x = np.random.exponential(scale=1.0, size=n)\n",
    "    return x / sum(x)\n",
    "simulate_weights(3)\n",
    "# simulate_weights(3).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "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>w1</th>\n",
       "      <th>w2</th>\n",
       "      <th>w3</th>\n",
       "      <th>port_expret</th>\n",
       "      <th>port_sd</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.549491</td>\n",
       "      <td>0.330444</td>\n",
       "      <td>0.120065</td>\n",
       "      <td>0.064054</td>\n",
       "      <td>0.149407</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.317636</td>\n",
       "      <td>0.145703</td>\n",
       "      <td>0.536661</td>\n",
       "      <td>0.071462</td>\n",
       "      <td>0.171423</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.645725</td>\n",
       "      <td>0.113278</td>\n",
       "      <td>0.240997</td>\n",
       "      <td>0.065386</td>\n",
       "      <td>0.154207</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.016917</td>\n",
       "      <td>0.766479</td>\n",
       "      <td>0.216604</td>\n",
       "      <td>0.068164</td>\n",
       "      <td>0.165389</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.344663</td>\n",
       "      <td>0.536064</td>\n",
       "      <td>0.119272</td>\n",
       "      <td>0.065066</td>\n",
       "      <td>0.152475</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         w1        w2        w3  port_expret   port_sd\n",
       "0  0.549491  0.330444  0.120065     0.064054  0.149407\n",
       "1  0.317636  0.145703  0.536661     0.071462  0.171423\n",
       "2  0.645725  0.113278  0.240997     0.065386  0.154207\n",
       "3  0.016917  0.766479  0.216604     0.068164  0.165389\n",
       "4  0.344663  0.536064  0.119272     0.065066  0.152475"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num_sims = 500\n",
    "df = pd.DataFrame(dtype='float',columns = ['w1','w2','w3','port_expret','port_sd'], index=np.arange(num_sims))\n",
    "for i in df.index:\n",
    "    wgts = simulate_weights(3)\n",
    "    df.loc[i,['w1','w2','w3']] = wgts\n",
    "    df.loc[i,'port_expret'] = wgts @ MNS\n",
    "    df.loc[i,'port_sd'] = np.sqrt(wgts @ COV @ wgts)\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.plotly.v1+json": {
       "config": {
        "plotlyServerURL": "https://plot.ly"
       },
       "data": [
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   "source": [
    "fig = go.Figure()\n",
    "\n",
    "# Plot portfolios\n",
    "trace= go.Scatter(x=df.port_sd, y=df.port_expret,mode=\"markers\", name='Portfolios')\n",
    "fig.add_trace(trace)\n",
    "fig.layout.yaxis[\"title\"] = \"Expected Return\"\n",
    "fig.layout.xaxis[\"title\"] = \"Standard Deviation\"\n",
    "\n",
    "# Plot underlying assets\n",
    "trace1= go.Scatter(x=[SDS[0]], y=[MNS[0]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 1')\n",
    "fig.add_trace(trace1)\n",
    "trace2= go.Scatter(x=[SDS[1]], y=[MNS[1]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 2')\n",
    "fig.add_trace(trace2)\n",
    "trace3= go.Scatter(x=[SDS[2]], y=[MNS[2]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 3')\n",
    "fig.add_trace(trace3)\n",
    "\n",
    "# Formatting\n",
    "fig.update_yaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(range=[0.7 * df[\"port_sd\"].min(), 1.25 * df[\"port_sd\"].max()])\n",
    "fig.update_yaxes(range=[0.7 * df[\"port_expret\"].min(), 1.25 * df[\"port_expret\"].max()])\n",
    "fig.update_layout(legend=dict(yanchor=\"top\", y=0.99, xanchor=\"left\", x=0.01))\n",
    "fig.show()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Allowing short sales"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "def simulate_weights(n):\n",
    "    ''' Simulate weights that sum to 1.'''\n",
    "    x = np.random.uniform(low=-0.5, high=2, size=n-1)\n",
    "    x = np.append(x, 1-x.sum())\n",
    "    return x \n",
    "simulate_weights(3)\n",
    "# simulate_weights(3).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>w1</th>\n",
       "      <th>w2</th>\n",
       "      <th>w3</th>\n",
       "      <th>port_expret</th>\n",
       "      <th>port_sd</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-0.036979</td>\n",
       "      <td>1.665248</td>\n",
       "      <td>-0.628268</td>\n",
       "      <td>0.055761</td>\n",
       "      <td>0.193309</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.192450</td>\n",
       "      <td>0.661827</td>\n",
       "      <td>-0.854277</td>\n",
       "      <td>0.046224</td>\n",
       "      <td>0.166981</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.239937</td>\n",
       "      <td>1.736158</td>\n",
       "      <td>-0.976095</td>\n",
       "      <td>0.049159</td>\n",
       "      <td>0.204192</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.245192</td>\n",
       "      <td>0.290382</td>\n",
       "      <td>0.464426</td>\n",
       "      <td>0.070740</td>\n",
       "      <td>0.167909</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1.173444</td>\n",
       "      <td>0.972220</td>\n",
       "      <td>-1.145664</td>\n",
       "      <td>0.041948</td>\n",
       "      <td>0.188518</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
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       "         w1        w2        w3  port_expret   port_sd\n",
       "0 -0.036979  1.665248 -0.628268     0.055761  0.193309\n",
       "1  1.192450  0.661827 -0.854277     0.046224  0.166981\n",
       "2  0.239937  1.736158 -0.976095     0.049159  0.204192\n",
       "3  0.245192  0.290382  0.464426     0.070740  0.167909\n",
       "4  1.173444  0.972220 -1.145664     0.041948  0.188518"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "num_sims = 500\n",
    "df = pd.DataFrame(dtype='float',columns = ['w1','w2','w3','port_expret','port_sd'], index=np.arange(num_sims))\n",
    "for i in df.index:\n",
    "    wgts = simulate_weights(3)\n",
    "    df.loc[i,['w1','w2','w3']] = wgts\n",
    "    df.loc[i,'port_expret'] = wgts @ MNS\n",
    "    df.loc[i,'port_sd'] = np.sqrt(wgts @ COV @ wgts)\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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          "text": "Expected Return"
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   "source": [
    "fig = go.Figure()\n",
    "# Plot random portfolios\n",
    "string =  \"Asset 1: %{customdata[0]:.1%}<br>\"\n",
    "string += \"Asset 2: %{customdata[1]:.1%}<br>\"\n",
    "string += \"Asset 3: %{customdata[2]:.1%}<br>\"\n",
    "string += \"<extra></extra>\"\n",
    "trace= go.Scatter(x=df.port_sd, y=df.port_expret,mode=\"markers\",\n",
    "    customdata=df[['w1','w2','w3']],hovertemplate=string, name='Random Portfolios')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Plot underlying assets\n",
    "trace1= go.Scatter(x=[SDS[0]], y=[MNS[0]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 1')\n",
    "fig.add_trace(trace1)\n",
    "trace2= go.Scatter(x=[SDS[1]], y=[MNS[1]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 2')\n",
    "fig.add_trace(trace2)\n",
    "trace3= go.Scatter(x=[SDS[2]], y=[MNS[2]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 3')\n",
    "fig.add_trace(trace3)\n",
    "\n",
    "# Formatting\n",
    "fig.layout.yaxis[\"title\"] = \"Expected Return\"\n",
    "fig.layout.xaxis[\"title\"] = \"Standard Deviation\"\n",
    "fig.update_yaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(range=[0.7 * df[\"port_sd\"].min(), 1.25 * df[\"port_sd\"].max()])\n",
    "fig.update_yaxes(range=[0.7 * df[\"port_expret\"].min(), 1.25 * df[\"port_expret\"].max()])\n",
    "fig.update_layout(legend=dict(yanchor=\"top\", y=0.99, xanchor=\"left\", x=0.01))\n",
    "fig.show()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Efficient portfolios"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `cvxopt.solvers.qp` solves problems of the general form:\n",
    "\\begin{align*}\n",
    "    \\underset{w}{\\text{min  }}& \\frac{1}{2} w' Q w + p'w \\\\\n",
    "     \\text{subject to  } & Gw \\le h \\\\\n",
    "                        & Aw = b \\\\\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
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     "execution_count": 12,
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     "output_type": "execute_result"
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   "source": [
    "def frontier(means, cov, target):\n",
    "    n = len(means)\n",
    "    Q = matrix(cov, tc=\"d\")\n",
    "    p = matrix(np.zeros(n), (n, 1), tc=\"d\")\n",
    "    # Constraint: short-sales allowed\n",
    "    G = matrix(np.zeros((n,n)), tc=\"d\")\n",
    "    h = matrix(np.zeros(n), (n, 1), tc=\"d\")\n",
    "    # Fully-invested constraint + E[r] = target\n",
    "    A = matrix(np.vstack((np.ones(n), means)), (2, n), tc=\"d\")\n",
    "    b = matrix([1, target], (2, 1), tc=\"d\")\n",
    "    sol = Solver(Q, p, G, h, A, b)\n",
    "    wgts = np.array(sol[\"x\"]).flatten() if sol[\"status\"] == \"optimal\" else np.array(n * [np.nan])\n",
    "    return wgts\n",
    "SolverOptions['show_progress'] = False\n",
    "\n",
    "TARGET_EXP_RET = 0.07\n",
    "frontier(MNS, COV, TARGET_EXP_RET)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can trace out the frontier by solving for the minimum variance portfolio for a number of expected return targets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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       "      <th>target_expret</th>\n",
       "      <th>w1</th>\n",
       "      <th>w2</th>\n",
       "      <th>w3</th>\n",
       "      <th>port_expret</th>\n",
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       "   target_expret        w1        w2        w3  port_expret   port_sd\n",
       "0       0.050000  1.151715  0.464380 -0.616095     0.050000  0.154641\n",
       "1       0.052632  1.033676  0.446327 -0.480003     0.052632  0.148910\n",
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       "3       0.057895  0.797598  0.410221 -0.207818     0.057895  0.143819\n",
       "4       0.060526  0.679558  0.392168 -0.071726     0.060526  0.144683"
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     "execution_count": 13,
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   "source": [
    "NUM_TARGETS = 20\n",
    "df = pd.DataFrame(dtype='float',columns=['target_expret','w1','w2','w3','port_expret','port_sd'], index=np.arange(NUM_TARGETS))\n",
    "df.target_expret = np.linspace(0.05, 0.10,NUM_TARGETS)\n",
    "for i in df.index:\n",
    "    wgts = frontier(MNS, COV, df.loc[i,'target_expret'])\n",
    "    df.loc[i,['w1','w2','w3']] = wgts\n",
    "    df.loc[i,'port_expret'] = wgts @ MNS\n",
    "    df.loc[i,'port_sd'] = np.sqrt(wgts @ COV @ wgts)\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
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    "fig = go.Figure()\n",
    "# Plot frontier\n",
    "string =  \"Asset 1: %{customdata[0]:.1%}<br>\"\n",
    "string += \"Asset 2: %{customdata[1]:.1%}<br>\"\n",
    "string += \"Asset 3: %{customdata[2]:.1%}<br>\"\n",
    "string += \"<extra></extra>\"\n",
    "trace= go.Scatter(x=df.port_sd, y=df.port_expret,mode=\"lines\",\n",
    "    customdata=df[['w1','w2','w3']],hovertemplate=string, name='Frontier')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Plot underlying assets\n",
    "trace1= go.Scatter(x=[SDS[0]], y=[MNS[0]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 1')\n",
    "fig.add_trace(trace1)\n",
    "trace2= go.Scatter(x=[SDS[1]], y=[MNS[1]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 2')\n",
    "fig.add_trace(trace2)\n",
    "trace3= go.Scatter(x=[SDS[2]], y=[MNS[2]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 3')\n",
    "fig.add_trace(trace3)\n",
    "\n",
    "# Formatting\n",
    "fig.layout.yaxis[\"title\"] = \"Expected Return\"\n",
    "fig.layout.xaxis[\"title\"] = \"Standard Deviation\"\n",
    "fig.update_yaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(range=[0.7 * df[\"port_sd\"].min(), 1.25 * df[\"port_sd\"].max()])\n",
    "fig.update_yaxes(range=[0.7 * df[\"port_expret\"].min(), 1.25 * df[\"port_expret\"].max()])\n",
    "fig.update_layout(legend=dict(yanchor=\"top\", y=0.99, xanchor=\"left\", x=0.01))\n",
    "fig.show()"
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   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Global minimum variance\n",
    "\n",
    "Note there is no longer a target expected return constraint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "import numpy as np\n",
    "from cvxopt import matrix\n",
    "from cvxopt.solvers import qp as Solver, options as SolverOptions\n",
    "\n",
    "def gmv(means, cov):\n",
    "    n = len(means)\n",
    "    Q = matrix(cov, tc=\"d\")\n",
    "    p = matrix(np.zeros(n), (n, 1), tc=\"d\")\n",
    "    # Constraint: short-sales allowed\n",
    "    G = matrix(np.zeros((n,n)), tc=\"d\")\n",
    "    h = matrix(np.zeros(n), (n, 1), tc=\"d\")\n",
    "    # Constraint: fully-invested portfolio\n",
    "    A = matrix(np.ones(n), (1, n), tc=\"d\")\n",
    "    b = matrix([1], (1, 1), tc=\"d\")\n",
    "    sol = Solver(Q, p, G, h, A, b)\n",
    "    wgts = np.array(sol[\"x\"]).flatten() if sol[\"status\"] == \"optimal\" else np.array(n * [np.nan])\n",
    "    return wgts\n",
    "SolverOptions['show_progress'] = False\n",
    "\n",
    "wgts_gmv = gmv(MNS, COV)\n",
    "wgts_gmv"
   ]
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  {
   "cell_type": "code",
   "execution_count": 16,
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    "fig = go.Figure()\n",
    "\n",
    "# Plot frontier\n",
    "string =  \"Asset 1: %{customdata[0]:.1%}<br>\"\n",
    "string += \"Asset 2: %{customdata[1]:.1%}<br>\"\n",
    "string += \"Asset 3: %{customdata[2]:.1%}<br>\"\n",
    "string += \"<extra></extra>\"\n",
    "trace= go.Scatter(x=df.port_sd, y=df.port_expret,mode=\"lines\",\n",
    "    customdata=df[['w1','w2','w3']],hovertemplate=string, name='Frontier')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Plot underlying assets\n",
    "trace1= go.Scatter(x=[SDS[0]], y=[MNS[0]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 1')\n",
    "fig.add_trace(trace1)\n",
    "trace2= go.Scatter(x=[SDS[1]], y=[MNS[1]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 2')\n",
    "fig.add_trace(trace2)\n",
    "trace3= go.Scatter(x=[SDS[2]], y=[MNS[2]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Asset 3')\n",
    "fig.add_trace(trace3)\n",
    "\n",
    "# Plot GMV\n",
    "trace= go.Scatter(x=[np.sqrt(wgts_gmv @ COV @ wgts_gmv)], y=[wgts_gmv @ MNS] ,\n",
    "    mode=\"markers\", marker=dict(size=10, color=\"black\"),\n",
    "    customdata=wgts_gmv.reshape(1,3),hovertemplate=string, name='GMV')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Formatting\n",
    "fig.layout.yaxis[\"title\"] = \"Expected Return\"\n",
    "fig.layout.xaxis[\"title\"] = \"Standard Deviation\"\n",
    "fig.update_yaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(tickformat=\".1%\")\n",
    "fig.update_xaxes(range=[0.7 * df[\"port_sd\"].min(), 1.25 * df[\"port_sd\"].max()])\n",
    "fig.update_yaxes(range=[0.7 * df[\"port_expret\"].min(), 1.25 * df[\"port_expret\"].max()])\n",
    "fig.update_layout(legend=dict(yanchor=\"top\", y=0.99, xanchor=\"left\", x=0.01))\n",
    "fig.show()"
   ]
  }
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