16b_simulation.ipynb

{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulation (Monte Carlo Analysis)"
   ]
  },
  {
   "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.stats import norm\n",
    "from scipy.stats import multivariate_normal as mvn\n",
    "np.set_printoptions(precision=4)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simulating a single time series of returns"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inputs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Expected return\n",
    "MN = 0.06\n",
    "\n",
    "# Standard deviation\n",
    "SD = 0.10\n",
    "\n",
    "# Number of time periods\n",
    "T = 30\n",
    "\n",
    "# Number of simulations\n",
    "N_SIMS = 100"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following cell will produce different values each time you run it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.0221, -0.0195, -0.0442,  0.0154, -0.1463, -0.0199, -0.0562,\n",
       "       -0.0742,  0.001 ,  0.0651,  0.1084,  0.0459,  0.0612,  0.0895,\n",
       "        0.0664, -0.0043, -0.06  ,  0.018 ,  0.1591,  0.1223, -0.0946,\n",
       "        0.0434,  0.1611,  0.1306, -0.1213,  0.0829,  0.0534, -0.1345,\n",
       "        0.3406,  0.3467])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm.rvs(loc=MN, scale = SD, size=T)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fixing the seed (random_state) will return the same values each time\n",
    "\n",
    "- This is not what we want for simulations, but is useful for research replicability"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.1932,  0.1315, -0.0945,  0.0592,  0.1221, -0.012 ,  0.0866,\n",
       "        0.0709,  0.0604,  0.0425,  0.1033,  0.1803, -0.0365,  0.1628,\n",
       "        0.0829,  0.1045, -0.0537,  0.0735,  0.2085, -0.048 , -0.1378,\n",
       "       -0.1143,  0.0866,  0.2985,  0.1724,  0.2273,  0.0699,  0.1998,\n",
       "        0.0329,  0.1213])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm.rvs(loc=MN, scale = SD, size=T, random_state=10)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's simulate 1000 returns and perform a calculation on each time-series.\n",
    "The calculation will be a simple average of the time-series."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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",
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       "    }\n",
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       "        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>avg_ret</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.061895</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.066148</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.048310</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.065971</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.098055</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    avg_ret\n",
       "0  0.061895\n",
       "1  0.066148\n",
       "2  0.048310\n",
       "3  0.065971\n",
       "4  0.098055"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_SIMS=1000\n",
    "sims = pd.DataFrame(dtype=float,columns=['avg_ret'],index=np.arange(N_SIMS))\n",
    "for s in sims.index:\n",
    "    rets = norm.rvs(loc=MN, scale=SD, size=T)\n",
    "    sims.loc[s] = np.mean(rets)\n",
    "sims.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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    "# Plot the distribution of means\n",
    "fig = go.Figure()\n",
    "trace= go.Histogram(x=sims.avg_ret, histnorm='percent',hovertemplate=\"<br>%{y:.2}% of simulations <br><extra></extra>\")\n",
    "fig.add_trace(trace)\n",
    "# some formatting\n",
    "fig.update_traces(marker_line_width=1, marker_line_color='black')\n",
    "fig.layout.xaxis[\"title\"] = \"Estimated Average Return\"\n",
    "fig.layout.yaxis[\"title\"] = \"Percent of Simulations\"\n",
    "fig.add_vline(x=MN, line_width=4, line_dash=\"dash\", line_color=\"black\")\n",
    "fig.show()"
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   "source": [
    "We could have done the above using a function to the calculation based on the random input.  A single random realization of the calculation is obtained by calling the function."
   ]
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   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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      ]
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     "execution_count": 7,
     "metadata": {},
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    "def sim_calc(mean, sd, n_time):\n",
    "    rets = norm.rvs(loc=mean, scale = sd, size=n_time)\n",
    "    return np.mean(rets)    \n",
    "sim_calc(MN,SD,T)"
   ]
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   "source": [
    "We can collect many simulations in a dataframe by calling the function in a loop."
   ]
  },
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   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "sims = pd.DataFrame(dtype=float,columns=['avg_ret'],index=np.arange(N_SIMS))\n",
    "for s in sims.index:\n",
    "    sims.loc[s] = sim_calc(MN,SD,T)"
   ]
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   "execution_count": 9,
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          "text": "Estimated Average Return"
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   ],
   "source": [
    "# Plot the distribution of means\n",
    "fig = go.Figure()\n",
    "trace= go.Histogram(x=sims.avg_ret, histnorm='percent',hovertemplate=\"<br>%{y:.2}% of simulations <br><extra></extra>\")\n",
    "fig.add_trace(trace)\n",
    "# some formatting\n",
    "fig.update_traces(marker_line_width=1, marker_line_color='black')\n",
    "fig.layout.xaxis[\"title\"] = \"Estimated Average Return\"\n",
    "fig.layout.yaxis[\"title\"] = \"Percent of Simulations\"\n",
    "fig.add_vline(x=MN, line_width=4, line_dash=\"dash\", line_color=\"black\")\n",
    "fig.show()"
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  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The general set-up for simulation is to \n",
    "1. Identify one or more random inputs\n",
    "2. Set up a function that generates a random draw of the inputs and does some calculations to produce output(s).\n",
    "3. Run the function in step 2 many times to collect the simulated distribution of the output(s).\n",
    "4. Summarize the output distribution in some way (average value, percentiles, etc.)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's switch the calculation function to be the standard deviation of the returns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Average of simulated output:  0.0991\n"
     ]
    },
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    "N_SIMS=1000\n",
    "# Simulate a single realization and do calculation(s)\n",
    "def sim_calc(mean, sd, n_time):\n",
    "    rets = norm.rvs(loc=mean, scale=sd, size=n_time)\n",
    "    return np.std(rets,ddof=True)    \n",
    "\n",
    "# Collect N_SIMS runs of the simulation function\n",
    "sims = pd.DataFrame(dtype=float,columns=['sd_ret'],index=np.arange(N_SIMS))\n",
    "for s in sims.index:\n",
    "    sims.loc[s] = sim_calc(MN,SD,T)\n",
    "\n",
    "# Summarize simulated output\n",
    "output = sims.sd_ret\n",
    "print(f'Average of simulated output: {output.mean(): .4f}')\n",
    "\n",
    "# Plot the distribution of output\n",
    "fig = go.Figure()\n",
    "trace= go.Histogram(x=sims.sd_ret, histnorm='percent',hovertemplate=\"<br>%{y:.2}% of simulations <br><extra></extra>\")\n",
    "fig.add_trace(trace)\n",
    "fig.update_traces(marker_line_width=1, marker_line_color='black')\n",
    "fig.layout.xaxis[\"title\"] = \"Estimated SD(Return)\"\n",
    "fig.layout.yaxis[\"title\"] = \"Percent of Simulations\"\n",
    "fig.add_vline(x=SD, line_width=4, line_dash=\"dash\", line_color=\"black\")\n",
    "fig.show()\n"
   ]
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   "metadata": {},
   "source": [
    "### Simulating correlated returns\n",
    "\n",
    "To simulate multiple asset returns, we need to make sure that the simulation captures the correlation structure across returns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
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   "source": [
    "# Risk-free rate\n",
    "RF = 0.01\n",
    "\n",
    "# Expected returns\n",
    "MNS = np.array([0.05, 0.10, 0.15])\n",
    "\n",
    "# Standard deviations\n",
    "SDS = np.array([0.15, 0.20, 0.25])\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",
    "COV = np.diag(SDS) @ C @ np.diag(SDS)"
   ]
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   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A single realization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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     "data": {
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       "       [-4.8661e-03, -1.8979e-01, -1.4689e-01],\n",
       "       [ 2.3117e-01, -1.1743e-01,  3.2781e-01],\n",
       "       [ 1.7670e-01,  4.9919e-02,  1.7757e-01],\n",
       "       [-9.3892e-02,  2.0724e-01,  1.1048e-01],\n",
       "       [ 1.2839e-01,  2.6051e-03,  6.3791e-01],\n",
       "       [ 1.1767e-01,  2.2322e-01,  4.8146e-01],\n",
       "       [-3.0245e-02,  1.0886e-01, -1.8190e-01],\n",
       "       [-1.6284e-02, -4.5101e-02,  1.2753e-01],\n",
       "       [-1.6069e-01,  1.1679e-05, -2.3118e-01],\n",
       "       [ 2.0941e-01,  2.6007e-01,  5.1541e-02],\n",
       "       [-1.2026e-01, -1.2544e-01, -3.2606e-01],\n",
       "       [ 1.5048e-01,  4.5519e-01,  6.9281e-02],\n",
       "       [ 1.6539e-01,  2.2596e-01,  3.5316e-01],\n",
       "       [ 1.4823e-01,  1.2183e-01,  1.7014e-01],\n",
       "       [ 6.3549e-03, -5.6657e-02,  7.3041e-02],\n",
       "       [ 2.6080e-01,  4.1557e-01,  3.9464e-01],\n",
       "       [ 3.3199e-01,  4.5821e-01,  7.2091e-01],\n",
       "       [ 4.2153e-02,  1.5820e-01,  1.0286e-01],\n",
       "       [-4.0138e-02,  8.8329e-02,  2.4983e-02],\n",
       "       [ 7.5581e-02,  7.8853e-02,  1.9670e-01],\n",
       "       [-1.0901e-01, -1.5382e-02, -6.9122e-02],\n",
       "       [-8.4442e-02, -1.0082e-01, -1.5119e-01],\n",
       "       [ 1.3655e-01,  1.5294e-01,  3.5099e-01],\n",
       "       [-2.3929e-01, -3.9328e-01, -4.5617e-01],\n",
       "       [ 4.4972e-02,  1.1799e-01,  1.5209e-01],\n",
       "       [-6.9665e-02, -1.4365e-01,  7.4999e-02],\n",
       "       [ 2.6936e-01,  2.3862e-01,  3.6103e-01],\n",
       "       [ 8.8640e-02,  3.5736e-01,  3.6545e-01],\n",
       "       [ 7.6994e-02, -1.1746e-01, -9.1900e-02]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "mvn.rvs(MNS, COV, size=T)"
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  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Putting simulated data in a dataframe and doing calculations"
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  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "rets = pd.DataFrame(data=mvn.rvs(MNS, COV, size=T), columns=['ret1','ret2','ret3'])\n",
    "x = rets.corr()\n",
    "corr12 = x.loc['ret1', 'ret2']\n",
    "corr13 = x.loc['ret1', 'ret3']\n",
    "corr23 = x.loc['ret2', 'ret3']\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define a function to do the simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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     "execution_count": 14,
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   "source": [
    "# Simulate a single realization and do calculation(s)\n",
    "def sim_calc(means, cov, n_time):\n",
    "    n = len(means)\n",
    "    rets = pd.DataFrame(data=mvn.rvs(means, cov, size=n_time), columns=['ret' + str(i+1) for i in np.arange(n)])\n",
    "    # print(rets)\n",
    "    x = rets.corr()\n",
    "    corr12 = x.loc['ret1', 'ret2']\n",
    "    corr13 = x.loc['ret1', 'ret3']\n",
    "    return corr12, corr13\n",
    "sim_calc(MNS,COV,T)\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run the `N_SIMS` simulations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Average corr12:  0.7463\n",
      "Average corr13:  0.7436\n"
     ]
    },
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   "source": [
    "N_SIMS=1000\n",
    " \n",
    "# Collect N_SIMS runs of the simulation function\n",
    "sims = pd.DataFrame(dtype=float,columns=['corr12', 'corr13'],index=np.arange(N_SIMS))\n",
    "for s in sims.index:\n",
    "    sims.loc[s,['corr12','corr13']] = sim_calc(MNS,COV,T)\n",
    "    \n",
    "# Summarize simulated output\n",
    "output1 = sims.corr12\n",
    "output2 = sims.corr13\n",
    "print(f'Average corr12: {output1.mean(): .4f}')\n",
    "print(f'Average corr13: {output2.mean(): .4f}')\n",
    "\n",
    "# Plot the distribution of output\n",
    "fig = go.Figure()\n",
    "trace= go.Histogram(x=sims.corr12, histnorm='percent',hovertemplate=\"<br>%{y:.2}% of simulations <br><extra></extra>\")\n",
    "fig.add_trace(trace)\n",
    "fig.update_traces(marker_line_width=1, marker_line_color='black')\n",
    "fig.layout.xaxis[\"title\"] = \"Estimated Correlation(R1,R2)\"\n",
    "fig.layout.yaxis[\"title\"] = \"Percent of Simulations\"\n",
    "fig.add_vline(x=C[0, 1], line_width=4, line_dash=\"dash\", line_color=\"black\")\n",
    "fig.show()"
   ]
  }
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