13a_portfolios_theory.ipynb

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 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Optimal portfolios"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import plotly.graph_objects as go\n",
    "from scipy.optimize import minimize\n",
    "from cvxopt import matrix\n",
    "from cvxopt.solvers import qp as Solver, options as SolverOptions"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Inputs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.0225    -0.0002625]\n",
      " [-0.0002625  0.001225 ]]\n"
     ]
    }
   ],
   "source": [
    "##### Inputs\n",
    "# Risk-free rate\n",
    "RF = 0.02\n",
    "\n",
    "# Expected returns\n",
    "MNS = np.array([0.06, 0.035])\n",
    "\n",
    "# Standard deviations\n",
    "SDS = np.array([0.15, 0.035])\n",
    "\n",
    "# Correlations\n",
    "C  = np.identity(2)\n",
    "C[0, 1] = C[1, 0] = -0.05\n",
    "\n",
    "# Covariance matrix\n",
    "COV = np.diag(SDS) @ C @ np.diag(SDS)\n",
    "print(COV)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tangency portfolio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangency portfolio weights: [0.13203413 0.86796587]\n"
     ]
    }
   ],
   "source": [
    "n = len(MNS)\n",
    "def f(w):\n",
    "    mn = w @ MNS\n",
    "    sd = np.sqrt(w @ COV @ w)\n",
    "    return -(mn - RF) / sd\n",
    "# Initial guess (equal-weighted)\n",
    "w0 = (1/n)*np.ones(n)\n",
    "# Constraint: fully-invested portfolio\n",
    "A = np.ones(n)\n",
    "b = 1\n",
    "cons = [{\"type\": \"eq\", \"fun\": lambda x: A @ x - b}]\n",
    "# No short-sale constraint\n",
    "bnds = [(None, None) for i in range(n)] \n",
    "# Optimization\n",
    "TOL = 10**(-10)\n",
    "wgts = minimize(f, w0, bounds=bnds, constraints=cons, options={'ftol':TOL}).x\n",
    "\n",
    "print(f'Tangency portfolio weights: {wgts}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tangency portfolio: theoretical solution without short-sale constraint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangency portfolio weights: [0.13203429 0.86796571]\n"
     ]
    }
   ],
   "source": [
    "w = np.linalg.solve(COV, MNS - RF)\n",
    "wgts = w / np.sum(w)\n",
    "print(f'Tangency portfolio weights: {wgts}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Portfolio Expected Return:\t  0.0383\n",
      "Portfolio Standard Deviation:\t  0.0354\n",
      "Portfolio Sharpe Ratio:\t\t  0.5166\n"
     ]
    }
   ],
   "source": [
    "#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}')\n",
    "\n",
    "# Portfolio sharpe ratio\n",
    "port_sr = (port_expret - RF)/port_sd\n",
    "print(f'Portfolio Sharpe Ratio:\\t\\t {port_sr: ,.4f}')"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Capital allocation and risk aversion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def capital_allocation(mean, sd, rf, risk_aversion):\n",
    "    return (mean - rf) / (risk_aversion * (sd**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For risk aversion of  10.0, invest  145.8% in the risky asset\n"
     ]
    }
   ],
   "source": [
    "RAVER = 10\n",
    "wgt_risky = capital_allocation(port_expret, port_sd, RF, RAVER)\n",
    "print(f'For risk aversion of {RAVER: .1f}, invest {wgt_risky: ,.1%} in the risky asset')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For risk aversion of  10.0, Portfolio Expected Return:\t\t  0.0467\n",
      "For risk aversion of  10.0, Portfolio Standard Deviation:\t  0.0517\n"
     ]
    }
   ],
   "source": [
    "# Portfolio expected return\n",
    "optport_expret = wgt_risky*port_expret + (1-wgt_risky)*RF\n",
    "print(f'For risk aversion of {RAVER: .1f}, Portfolio Expected Return:\\t\\t {optport_expret: ,.4f}')\n",
    "\n",
    "# Portfolio standard deviation\n",
    "optport_sd     = np.abs(wgt_risky)*port_sd\n",
    "print(f'For risk aversion of {RAVER: .1f}, Portfolio Standard Deviation:\\t {optport_sd: ,.4f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Capital allocation line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>wgt_tangency</th>\n",
       "      <th>wgt_rf</th>\n",
       "      <th>port_expret</th>\n",
       "      <th>port_sd</th>\n",
       "      <th>risk_aversion</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>46</th>\n",
       "      <td>2.30</td>\n",
       "      <td>-1.30</td>\n",
       "      <td>0.062092</td>\n",
       "      <td>0.081478</td>\n",
       "      <td>6.340410</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>47</th>\n",
       "      <td>2.35</td>\n",
       "      <td>-1.35</td>\n",
       "      <td>0.063007</td>\n",
       "      <td>0.083249</td>\n",
       "      <td>6.205507</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>48</th>\n",
       "      <td>2.40</td>\n",
       "      <td>-1.40</td>\n",
       "      <td>0.063922</td>\n",
       "      <td>0.085021</td>\n",
       "      <td>6.076226</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49</th>\n",
       "      <td>2.45</td>\n",
       "      <td>-1.45</td>\n",
       "      <td>0.064837</td>\n",
       "      <td>0.086792</td>\n",
       "      <td>5.952221</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>2.50</td>\n",
       "      <td>-1.50</td>\n",
       "      <td>0.065752</td>\n",
       "      <td>0.088563</td>\n",
       "      <td>5.833177</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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       "    wgt_tangency  wgt_rf  port_expret   port_sd  risk_aversion\n",
       "46          2.30   -1.30     0.062092  0.081478       6.340410\n",
       "47          2.35   -1.35     0.063007  0.083249       6.205507\n",
       "48          2.40   -1.40     0.063922  0.085021       6.076226\n",
       "49          2.45   -1.45     0.064837  0.086792       5.952221\n",
       "50          2.50   -1.50     0.065752  0.088563       5.833177"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "WGTS = np.arange(0,2.51,0.05)\n",
    "cal = pd.DataFrame(dtype='float',columns=['wgt_tangency', 'wgt_rf', 'port_expret', 'port_sd','risk_aversion'], index=np.arange(len(WGTS)))\n",
    "cal.wgt_tangency = WGTS\n",
    "cal.wgt_rf = 1-WGTS\n",
    "cal.port_expret = cal.wgt_tangency*port_expret + cal.wgt_rf*RF\n",
    "cal.port_sd = np.abs(cal.wgt_tangency)*port_sd\n",
    "cal.risk_aversion = (port_expret - RF) / (cal.wgt_tangency * port_sd**2)\n",
    "cal.tail()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The full picture\n",
    "Note we will now also calculate the sharpe ratio for each frontier portfolio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Calculate frontier portfolios (from last time)\n",
    "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",
    "NUM_TARGETS = 30\n",
    "df = pd.DataFrame(dtype='float',columns=['target_expret','w1','w2','port_expret','port_sd','sharpe'], index=np.arange(NUM_TARGETS))\n",
    "df.target_expret = np.linspace(MNS.min(), MNS.max(),NUM_TARGETS)\n",
    "for i in df.index:\n",
    "    wgts = frontier(MNS, COV, df.loc[i,'target_expret'])\n",
    "    df.loc[i,['w1','w2']] = wgts\n",
    "    df.loc[i,'port_expret'] = wgts @ MNS\n",
    "    df.loc[i,'port_sd'] = np.sqrt(wgts @ COV @ wgts)\n",
    "df.sharpe = (df.port_expret - RF)/df.port_sd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
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            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "yaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "zaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           }
          },
          "shapedefaults": {
           "line": {
            "color": "#2a3f5f"
           }
          },
          "ternary": {
           "aaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "baxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          }
         }
        },
        "xaxis": {
         "range": [
          0.02358456210235941,
          0.1875
         ],
         "tickformat": ".1%",
         "title": {
          "text": "Standard Deviation"
         }
        },
        "yaxis": {
         "range": [
          0.024500000000000004,
          0.075
         ],
         "tickformat": ".1%",
         "title": {
          "text": "Expected Return"
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = go.Figure()\n",
    "\n",
    "# Plot frontier\n",
    "string =  \"Stock: %{customdata[0]:.1%}<br>\"\n",
    "string += \"Bond: %{customdata[1]:.1%}<br>\"\n",
    "string += \"Sharpe ratio: %{customdata[2]:.4f}<br>\"\n",
    "string += \"<extra></extra>\"\n",
    "trace= go.Scatter(x=df.port_sd, y=df.port_expret,mode=\"lines\",\n",
    "    customdata=df[['w1','w2','sharpe']],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='Stock')\n",
    "fig.add_trace(trace1)\n",
    "trace2= go.Scatter(x=[SDS[1]], y=[MNS[1]],mode=\"markers\", marker=dict(size=10, color=\"red\"), name='Bond')\n",
    "fig.add_trace(trace2)\n",
    "\n",
    "# Plot tangency\n",
    "trace= go.Scatter(x=[port_sd], y=[port_expret] ,\n",
    "    mode=\"markers\", marker=dict(size=10, color=\"black\"),\n",
    "    customdata=np.append(wgts,port_sr).reshape(1,3),hovertemplate=string, name='Tangency')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Plot CAL\n",
    "string =  \"Tangency: %{customdata[0]:.1%}<br>\"\n",
    "string += \"Risk-free: %{customdata[1]:.1%}<br>\"\n",
    "string += \"Optimal if risk aversion is: %{customdata[2]:.1f}<br>\"\n",
    "string += \"<extra></extra>\"\n",
    "trace= go.Scatter(x=cal.port_sd, y=cal.port_expret,mode=\"lines\",marker=dict(color=\"black\"),\n",
    "    customdata=cal[['wgt_tangency','wgt_rf','risk_aversion']],hovertemplate=string, name='CAL')\n",
    "fig.add_trace(trace)\n",
    "\n",
    "# Plot optimal location on CAL for risk aversion\n",
    "string =  \"Tangency: %{customdata[0]:.1%}<br>\"\n",
    "string += \"Risk-free: %{customdata[1]:.1%}<br>\"\n",
    "string += \"Optimal if risk aversion is: %{customdata[2]:.1f}<br>\"\n",
    "string += \"<extra></extra>\"\n",
    "trace= go.Scatter(x=[optport_sd], y=[optport_expret],\n",
    "    mode=\"markers\", marker=dict(size=15, color=\"red\", symbol='star'),\n",
    "    customdata=[[wgt_risky, 1-wgt_risky, RAVER]],hovertemplate=string, name='Optimal Portfolio')\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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