03_saving_real_nominal.ipynb

{
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    "# Saving for retirement with inflation\n",
    "- Let's assume that we want to consume a constant **real** amount in retirement\n",
    "  - our standard of living will be unchanged each year\n",
    "- We can either finance retirement in nominal terms or in real terms\n",
    "\n",
    "We need a few relationships to work on this.  We'll assume a constant inflation rate and that the base year is time 0.\n",
    "\n",
    "- First, what is the relationship between real and nominal cash flows at a point in time $t$?\n",
    "$$ \\text{Nominal CF}_t = \\text{Real CF}_t \\cdot (1+\\text{inflation})^t. $$\n",
    "\n",
    "- Second, what is the relationship between real and nominal rates of return?\n",
    "$$ 1+ \\text{nominal rate} = (1+\\text{real rate})(1+\\text{inflation}).$$"
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Requirement already satisfied: numpy_financial in d:\\bbcx-investments-quarto\\quarto\\venv\\lib\\site-packages (1.0.0)\n",
      "Requirement already satisfied: numpy>=1.15 in d:\\bbcx-investments-quarto\\quarto\\venv\\lib\\site-packages (from numpy_financial) (1.23.2)\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "[notice] A new release of pip available: 22.2.1 -> 23.3.2\n",
      "[notice] To update, run: python.exe -m pip install --upgrade pip\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "!pip install numpy_financial\n",
    "import numpy_financial as npf\n",
    "import plotly.graph_objects as go\n",
    "pd.options.display.float_format = '${:,.2f}'.format"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Nominal                           Real                        \n",
       "    begbal capgain dep/with endbal begbal capgain dep/with endbal\n",
       "1      NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "2      NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "3      NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "4      NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "5      NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "6      NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "7      NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "8      NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "9      NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "10     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "11     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "12     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "13     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "14     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "15     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "16     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "17     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "18     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "19     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "20     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "21     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "22     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "23     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "24     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "25     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "26     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "27     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "28     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "29     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "30     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "31     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "32     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "33     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "34     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "35     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "36     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "37     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "38     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "39     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "40     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "41     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "42     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "43     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "44     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "45     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "46     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "47     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "48     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "49     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN\n",
       "50     NaN     NaN      NaN    NaN    NaN     NaN      NaN    NaN"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We'll set up 2 bank accounts (1 in nominal terms and 1 in real terms)\n",
    "NOMINAL    = 0.05\n",
    "INFLATION  = 0.02\n",
    "N_SAVING   = 30\n",
    "N_WITHDRAW = 20\n",
    "PV  = 0.0\n",
    "WITHDRAWAL = 100000\n",
    "\n",
    "column_index = pd.MultiIndex.from_product([['Nominal','Real'],['begbal','capgain','dep/with','endbal']])\n",
    "idx = np.arange(1,N_SAVING + N_WITHDRAW + 1)\n",
    "\n",
    "acct = pd.DataFrame(dtype=float,columns=column_index,index=idx)\n",
    "acct"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.94%\n"
     ]
    }
   ],
   "source": [
    "# Let's start by saving a constant real amount each year.\n",
    "# This means that we can solve the problem entirely in real terms.\n",
    "real = (1+NOMINAL)/(1+INFLATION)-1\n",
    "print(f'{real:,.2%}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1,495,870.98\n"
     ]
    }
   ],
   "source": [
    "# First, we find how much we'd need to have saved as of t=N_SAVING:\n",
    "end_savings_bal = npf.pv(real,N_WITHDRAW,-WITHDRAWAL,0)\n",
    "print(f'{end_savings_bal:,.2f}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "31,742.87\n"
     ]
    }
   ],
   "source": [
    "# Second, we solve a PMT problem to determine the amount we need to save:\n",
    "\n",
    "pmt = npf.pmt(real,N_SAVING,PV,-end_savings_bal)\n",
    "print(f'{pmt:,.2f}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "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 tr th {\n",
       "        text-align: left;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr>\n",
       "      <th></th>\n",
       "      <th colspan=\"4\" halign=\"left\">Nominal</th>\n",
       "      <th colspan=\"4\" halign=\"left\">Real</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th></th>\n",
       "      <th>begbal</th>\n",
       "      <th>capgain</th>\n",
       "      <th>dep/with</th>\n",
       "      <th>endbal</th>\n",
       "      <th>begbal</th>\n",
       "      <th>capgain</th>\n",
       "      <th>dep/with</th>\n",
       "      <th>endbal</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$0.00</td>\n",
       "      <td>$0.00</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$31,742.87</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$933.61</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$64,419.35</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$64,419.35</td>\n",
       "      <td>$1,894.69</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$98,056.91</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$98,056.91</td>\n",
       "      <td>$2,884.03</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$132,683.80</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$132,683.80</td>\n",
       "      <td>$3,902.46</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$168,329.14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$168,329.14</td>\n",
       "      <td>$4,950.86</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$205,022.86</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$205,022.86</td>\n",
       "      <td>$6,030.08</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$242,795.81</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$242,795.81</td>\n",
       "      <td>$7,141.05</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$281,679.74</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$281,679.74</td>\n",
       "      <td>$8,284.70</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$321,707.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$321,707.30</td>\n",
       "      <td>$9,461.98</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$362,912.15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$362,912.15</td>\n",
       "      <td>$10,673.89</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$405,328.91</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$405,328.91</td>\n",
       "      <td>$11,921.44</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$448,993.21</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$448,993.21</td>\n",
       "      <td>$13,205.68</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$493,941.76</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$493,941.76</td>\n",
       "      <td>$14,527.70</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$540,212.33</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$540,212.33</td>\n",
       "      <td>$15,888.60</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$587,843.80</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$587,843.80</td>\n",
       "      <td>$17,289.52</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$636,876.19</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$636,876.19</td>\n",
       "      <td>$18,731.65</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$687,350.71</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$687,350.71</td>\n",
       "      <td>$20,216.20</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$739,309.78</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$739,309.78</td>\n",
       "      <td>$21,744.41</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$792,797.05</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$792,797.05</td>\n",
       "      <td>$23,317.56</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$847,857.48</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$847,857.48</td>\n",
       "      <td>$24,936.98</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$904,537.34</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$904,537.34</td>\n",
       "      <td>$26,604.04</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$962,884.24</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$962,884.24</td>\n",
       "      <td>$28,320.12</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,022,947.24</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,022,947.24</td>\n",
       "      <td>$30,086.68</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,084,776.79</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,084,776.79</td>\n",
       "      <td>$31,905.20</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,148,424.86</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,148,424.86</td>\n",
       "      <td>$33,777.20</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,213,944.93</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,213,944.93</td>\n",
       "      <td>$35,704.26</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,281,392.06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,281,392.06</td>\n",
       "      <td>$37,688.00</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,350,822.93</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,350,822.93</td>\n",
       "      <td>$39,730.09</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,422,295.88</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,422,295.88</td>\n",
       "      <td>$41,832.23</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,495,870.98</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,495,870.98</td>\n",
       "      <td>$43,996.21</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,439,867.19</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>32</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,439,867.19</td>\n",
       "      <td>$42,349.04</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,382,216.23</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>33</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,382,216.23</td>\n",
       "      <td>$40,653.42</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,322,869.64</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>34</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,322,869.64</td>\n",
       "      <td>$38,907.93</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,261,777.57</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>35</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,261,777.57</td>\n",
       "      <td>$37,111.11</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,198,888.68</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>36</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,198,888.68</td>\n",
       "      <td>$35,261.43</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,134,150.11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>37</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,134,150.11</td>\n",
       "      <td>$33,357.36</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,067,507.47</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>38</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$1,067,507.47</td>\n",
       "      <td>$31,397.28</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$998,904.75</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>39</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$998,904.75</td>\n",
       "      <td>$29,379.55</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$928,284.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>40</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$928,284.30</td>\n",
       "      <td>$27,302.48</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$855,586.78</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>41</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$855,586.78</td>\n",
       "      <td>$25,164.32</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$780,751.09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>42</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$780,751.09</td>\n",
       "      <td>$22,963.27</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$703,714.36</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>43</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$703,714.36</td>\n",
       "      <td>$20,697.48</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$624,411.84</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>44</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$624,411.84</td>\n",
       "      <td>$18,365.05</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$542,776.90</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>45</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$542,776.90</td>\n",
       "      <td>$15,964.03</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$458,740.92</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>46</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$458,740.92</td>\n",
       "      <td>$13,492.38</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$372,233.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>47</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$372,233.30</td>\n",
       "      <td>$10,948.04</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$283,181.34</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>48</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$283,181.34</td>\n",
       "      <td>$8,328.86</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$191,510.20</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$191,510.20</td>\n",
       "      <td>$5,632.65</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$97,142.86</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>$97,142.86</td>\n",
       "      <td>$2,857.14</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$0.00</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Nominal                                  Real                          \\\n",
       "    begbal capgain dep/with endbal        begbal    capgain     dep/with   \n",
       "1      NaN     NaN      NaN    NaN         $0.00      $0.00   $31,742.87   \n",
       "2      NaN     NaN      NaN    NaN    $31,742.87    $933.61   $31,742.87   \n",
       "3      NaN     NaN      NaN    NaN    $64,419.35  $1,894.69   $31,742.87   \n",
       "4      NaN     NaN      NaN    NaN    $98,056.91  $2,884.03   $31,742.87   \n",
       "5      NaN     NaN      NaN    NaN   $132,683.80  $3,902.46   $31,742.87   \n",
       "6      NaN     NaN      NaN    NaN   $168,329.14  $4,950.86   $31,742.87   \n",
       "7      NaN     NaN      NaN    NaN   $205,022.86  $6,030.08   $31,742.87   \n",
       "8      NaN     NaN      NaN    NaN   $242,795.81  $7,141.05   $31,742.87   \n",
       "9      NaN     NaN      NaN    NaN   $281,679.74  $8,284.70   $31,742.87   \n",
       "10     NaN     NaN      NaN    NaN   $321,707.30  $9,461.98   $31,742.87   \n",
       "11     NaN     NaN      NaN    NaN   $362,912.15 $10,673.89   $31,742.87   \n",
       "12     NaN     NaN      NaN    NaN   $405,328.91 $11,921.44   $31,742.87   \n",
       "13     NaN     NaN      NaN    NaN   $448,993.21 $13,205.68   $31,742.87   \n",
       "14     NaN     NaN      NaN    NaN   $493,941.76 $14,527.70   $31,742.87   \n",
       "15     NaN     NaN      NaN    NaN   $540,212.33 $15,888.60   $31,742.87   \n",
       "16     NaN     NaN      NaN    NaN   $587,843.80 $17,289.52   $31,742.87   \n",
       "17     NaN     NaN      NaN    NaN   $636,876.19 $18,731.65   $31,742.87   \n",
       "18     NaN     NaN      NaN    NaN   $687,350.71 $20,216.20   $31,742.87   \n",
       "19     NaN     NaN      NaN    NaN   $739,309.78 $21,744.41   $31,742.87   \n",
       "20     NaN     NaN      NaN    NaN   $792,797.05 $23,317.56   $31,742.87   \n",
       "21     NaN     NaN      NaN    NaN   $847,857.48 $24,936.98   $31,742.87   \n",
       "22     NaN     NaN      NaN    NaN   $904,537.34 $26,604.04   $31,742.87   \n",
       "23     NaN     NaN      NaN    NaN   $962,884.24 $28,320.12   $31,742.87   \n",
       "24     NaN     NaN      NaN    NaN $1,022,947.24 $30,086.68   $31,742.87   \n",
       "25     NaN     NaN      NaN    NaN $1,084,776.79 $31,905.20   $31,742.87   \n",
       "26     NaN     NaN      NaN    NaN $1,148,424.86 $33,777.20   $31,742.87   \n",
       "27     NaN     NaN      NaN    NaN $1,213,944.93 $35,704.26   $31,742.87   \n",
       "28     NaN     NaN      NaN    NaN $1,281,392.06 $37,688.00   $31,742.87   \n",
       "29     NaN     NaN      NaN    NaN $1,350,822.93 $39,730.09   $31,742.87   \n",
       "30     NaN     NaN      NaN    NaN $1,422,295.88 $41,832.23   $31,742.87   \n",
       "31     NaN     NaN      NaN    NaN $1,495,870.98 $43,996.21 $-100,000.00   \n",
       "32     NaN     NaN      NaN    NaN $1,439,867.19 $42,349.04 $-100,000.00   \n",
       "33     NaN     NaN      NaN    NaN $1,382,216.23 $40,653.42 $-100,000.00   \n",
       "34     NaN     NaN      NaN    NaN $1,322,869.64 $38,907.93 $-100,000.00   \n",
       "35     NaN     NaN      NaN    NaN $1,261,777.57 $37,111.11 $-100,000.00   \n",
       "36     NaN     NaN      NaN    NaN $1,198,888.68 $35,261.43 $-100,000.00   \n",
       "37     NaN     NaN      NaN    NaN $1,134,150.11 $33,357.36 $-100,000.00   \n",
       "38     NaN     NaN      NaN    NaN $1,067,507.47 $31,397.28 $-100,000.00   \n",
       "39     NaN     NaN      NaN    NaN   $998,904.75 $29,379.55 $-100,000.00   \n",
       "40     NaN     NaN      NaN    NaN   $928,284.30 $27,302.48 $-100,000.00   \n",
       "41     NaN     NaN      NaN    NaN   $855,586.78 $25,164.32 $-100,000.00   \n",
       "42     NaN     NaN      NaN    NaN   $780,751.09 $22,963.27 $-100,000.00   \n",
       "43     NaN     NaN      NaN    NaN   $703,714.36 $20,697.48 $-100,000.00   \n",
       "44     NaN     NaN      NaN    NaN   $624,411.84 $18,365.05 $-100,000.00   \n",
       "45     NaN     NaN      NaN    NaN   $542,776.90 $15,964.03 $-100,000.00   \n",
       "46     NaN     NaN      NaN    NaN   $458,740.92 $13,492.38 $-100,000.00   \n",
       "47     NaN     NaN      NaN    NaN   $372,233.30 $10,948.04 $-100,000.00   \n",
       "48     NaN     NaN      NaN    NaN   $283,181.34  $8,328.86 $-100,000.00   \n",
       "49     NaN     NaN      NaN    NaN   $191,510.20  $5,632.65 $-100,000.00   \n",
       "50     NaN     NaN      NaN    NaN    $97,142.86  $2,857.14 $-100,000.00   \n",
       "\n",
       "                  \n",
       "          endbal  \n",
       "1     $31,742.87  \n",
       "2     $64,419.35  \n",
       "3     $98,056.91  \n",
       "4    $132,683.80  \n",
       "5    $168,329.14  \n",
       "6    $205,022.86  \n",
       "7    $242,795.81  \n",
       "8    $281,679.74  \n",
       "9    $321,707.30  \n",
       "10   $362,912.15  \n",
       "11   $405,328.91  \n",
       "12   $448,993.21  \n",
       "13   $493,941.76  \n",
       "14   $540,212.33  \n",
       "15   $587,843.80  \n",
       "16   $636,876.19  \n",
       "17   $687,350.71  \n",
       "18   $739,309.78  \n",
       "19   $792,797.05  \n",
       "20   $847,857.48  \n",
       "21   $904,537.34  \n",
       "22   $962,884.24  \n",
       "23 $1,022,947.24  \n",
       "24 $1,084,776.79  \n",
       "25 $1,148,424.86  \n",
       "26 $1,213,944.93  \n",
       "27 $1,281,392.06  \n",
       "28 $1,350,822.93  \n",
       "29 $1,422,295.88  \n",
       "30 $1,495,870.98  \n",
       "31 $1,439,867.19  \n",
       "32 $1,382,216.23  \n",
       "33 $1,322,869.64  \n",
       "34 $1,261,777.57  \n",
       "35 $1,198,888.68  \n",
       "36 $1,134,150.11  \n",
       "37 $1,067,507.47  \n",
       "38   $998,904.75  \n",
       "39   $928,284.30  \n",
       "40   $855,586.78  \n",
       "41   $780,751.09  \n",
       "42   $703,714.36  \n",
       "43   $624,411.84  \n",
       "44   $542,776.90  \n",
       "45   $458,740.92  \n",
       "46   $372,233.30  \n",
       "47   $283,181.34  \n",
       "48   $191,510.20  \n",
       "49    $97,142.86  \n",
       "50         $0.00  "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Fill out the real side of the balance sheet\n",
    "acct[('Real','dep/with')] = np.where(acct.index <= N_SAVING, pmt,-WITHDRAWAL)\n",
    "for t in acct.index:\n",
    "    if t==1:\n",
    "        acct.loc[t,('Real','begbal')] = PV\n",
    "    else:\n",
    "        acct.loc[t,('Real','begbal')] = acct.loc[t-1,('Real','endbal')]\n",
    "    acct.loc[t,('Real','capgain')] = acct.loc[t,('Real','begbal')]*real\n",
    "    acct.loc[t,('Real','endbal')]  = acct.loc[t,('Real','begbal')] + acct.loc[t,('Real','capgain')] + acct.loc[t,('Real','dep/with')] \n",
    "acct"
   ]
  },
  {
   "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 tr th {\n",
       "        text-align: left;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr>\n",
       "      <th></th>\n",
       "      <th colspan=\"4\" halign=\"left\">Nominal</th>\n",
       "      <th colspan=\"4\" halign=\"left\">Real</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th></th>\n",
       "      <th>begbal</th>\n",
       "      <th>capgain</th>\n",
       "      <th>dep/with</th>\n",
       "      <th>endbal</th>\n",
       "      <th>begbal</th>\n",
       "      <th>capgain</th>\n",
       "      <th>dep/with</th>\n",
       "      <th>endbal</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>$0.00</td>\n",
       "      <td>$0.00</td>\n",
       "      <td>$32,377.73</td>\n",
       "      <td>$32,377.73</td>\n",
       "      <td>$0.00</td>\n",
       "      <td>$0.00</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$31,742.87</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>$32,377.73</td>\n",
       "      <td>$1,618.89</td>\n",
       "      <td>$33,025.28</td>\n",
       "      <td>$67,021.89</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$933.61</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$64,419.35</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>$67,021.89</td>\n",
       "      <td>$3,351.09</td>\n",
       "      <td>$33,685.79</td>\n",
       "      <td>$104,058.77</td>\n",
       "      <td>$64,419.35</td>\n",
       "      <td>$1,894.69</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$98,056.91</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>$104,058.77</td>\n",
       "      <td>$5,202.94</td>\n",
       "      <td>$34,359.50</td>\n",
       "      <td>$143,621.21</td>\n",
       "      <td>$98,056.91</td>\n",
       "      <td>$2,884.03</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$132,683.80</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>$143,621.21</td>\n",
       "      <td>$7,181.06</td>\n",
       "      <td>$35,046.69</td>\n",
       "      <td>$185,848.97</td>\n",
       "      <td>$132,683.80</td>\n",
       "      <td>$3,902.46</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$168,329.14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>$185,848.97</td>\n",
       "      <td>$9,292.45</td>\n",
       "      <td>$35,747.63</td>\n",
       "      <td>$230,889.04</td>\n",
       "      <td>$168,329.14</td>\n",
       "      <td>$4,950.86</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$205,022.86</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>$230,889.04</td>\n",
       "      <td>$11,544.45</td>\n",
       "      <td>$36,462.58</td>\n",
       "      <td>$278,896.07</td>\n",
       "      <td>$205,022.86</td>\n",
       "      <td>$6,030.08</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$242,795.81</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>$278,896.07</td>\n",
       "      <td>$13,944.80</td>\n",
       "      <td>$37,191.83</td>\n",
       "      <td>$330,032.71</td>\n",
       "      <td>$242,795.81</td>\n",
       "      <td>$7,141.05</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$281,679.74</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>$330,032.71</td>\n",
       "      <td>$16,501.64</td>\n",
       "      <td>$37,935.67</td>\n",
       "      <td>$384,470.01</td>\n",
       "      <td>$281,679.74</td>\n",
       "      <td>$8,284.70</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$321,707.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>$384,470.01</td>\n",
       "      <td>$19,223.50</td>\n",
       "      <td>$38,694.38</td>\n",
       "      <td>$442,387.89</td>\n",
       "      <td>$321,707.30</td>\n",
       "      <td>$9,461.98</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$362,912.15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>$442,387.89</td>\n",
       "      <td>$22,119.39</td>\n",
       "      <td>$39,468.27</td>\n",
       "      <td>$503,975.55</td>\n",
       "      <td>$362,912.15</td>\n",
       "      <td>$10,673.89</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$405,328.91</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>$503,975.55</td>\n",
       "      <td>$25,198.78</td>\n",
       "      <td>$40,257.63</td>\n",
       "      <td>$569,431.96</td>\n",
       "      <td>$405,328.91</td>\n",
       "      <td>$11,921.44</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$448,993.21</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>$569,431.96</td>\n",
       "      <td>$28,471.60</td>\n",
       "      <td>$41,062.79</td>\n",
       "      <td>$638,966.34</td>\n",
       "      <td>$448,993.21</td>\n",
       "      <td>$13,205.68</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$493,941.76</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>$638,966.34</td>\n",
       "      <td>$31,948.32</td>\n",
       "      <td>$41,884.04</td>\n",
       "      <td>$712,798.70</td>\n",
       "      <td>$493,941.76</td>\n",
       "      <td>$14,527.70</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$540,212.33</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>$712,798.70</td>\n",
       "      <td>$35,639.94</td>\n",
       "      <td>$42,721.72</td>\n",
       "      <td>$791,160.36</td>\n",
       "      <td>$540,212.33</td>\n",
       "      <td>$15,888.60</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$587,843.80</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>$791,160.36</td>\n",
       "      <td>$39,558.02</td>\n",
       "      <td>$43,576.16</td>\n",
       "      <td>$874,294.53</td>\n",
       "      <td>$587,843.80</td>\n",
       "      <td>$17,289.52</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$636,876.19</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>$874,294.53</td>\n",
       "      <td>$43,714.73</td>\n",
       "      <td>$44,447.68</td>\n",
       "      <td>$962,456.94</td>\n",
       "      <td>$636,876.19</td>\n",
       "      <td>$18,731.65</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$687,350.71</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>$962,456.94</td>\n",
       "      <td>$48,122.85</td>\n",
       "      <td>$45,336.63</td>\n",
       "      <td>$1,055,916.42</td>\n",
       "      <td>$687,350.71</td>\n",
       "      <td>$20,216.20</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$739,309.78</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>$1,055,916.42</td>\n",
       "      <td>$52,795.82</td>\n",
       "      <td>$46,243.37</td>\n",
       "      <td>$1,154,955.60</td>\n",
       "      <td>$739,309.78</td>\n",
       "      <td>$21,744.41</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$792,797.05</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>$1,154,955.60</td>\n",
       "      <td>$57,747.78</td>\n",
       "      <td>$47,168.23</td>\n",
       "      <td>$1,259,871.62</td>\n",
       "      <td>$792,797.05</td>\n",
       "      <td>$23,317.56</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$847,857.48</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>$1,259,871.62</td>\n",
       "      <td>$62,993.58</td>\n",
       "      <td>$48,111.60</td>\n",
       "      <td>$1,370,976.80</td>\n",
       "      <td>$847,857.48</td>\n",
       "      <td>$24,936.98</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$904,537.34</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>$1,370,976.80</td>\n",
       "      <td>$68,548.84</td>\n",
       "      <td>$49,073.83</td>\n",
       "      <td>$1,488,599.47</td>\n",
       "      <td>$904,537.34</td>\n",
       "      <td>$26,604.04</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$962,884.24</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>$1,488,599.47</td>\n",
       "      <td>$74,429.97</td>\n",
       "      <td>$50,055.31</td>\n",
       "      <td>$1,613,084.75</td>\n",
       "      <td>$962,884.24</td>\n",
       "      <td>$28,320.12</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,022,947.24</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>$1,613,084.75</td>\n",
       "      <td>$80,654.24</td>\n",
       "      <td>$51,056.41</td>\n",
       "      <td>$1,744,795.39</td>\n",
       "      <td>$1,022,947.24</td>\n",
       "      <td>$30,086.68</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,084,776.79</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>$1,744,795.39</td>\n",
       "      <td>$87,239.77</td>\n",
       "      <td>$52,077.54</td>\n",
       "      <td>$1,884,112.71</td>\n",
       "      <td>$1,084,776.79</td>\n",
       "      <td>$31,905.20</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,148,424.86</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>$1,884,112.71</td>\n",
       "      <td>$94,205.64</td>\n",
       "      <td>$53,119.09</td>\n",
       "      <td>$2,031,437.43</td>\n",
       "      <td>$1,148,424.86</td>\n",
       "      <td>$33,777.20</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,213,944.93</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>$2,031,437.43</td>\n",
       "      <td>$101,571.87</td>\n",
       "      <td>$54,181.47</td>\n",
       "      <td>$2,187,190.78</td>\n",
       "      <td>$1,213,944.93</td>\n",
       "      <td>$35,704.26</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,281,392.06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>$2,187,190.78</td>\n",
       "      <td>$109,359.54</td>\n",
       "      <td>$55,265.10</td>\n",
       "      <td>$2,351,815.42</td>\n",
       "      <td>$1,281,392.06</td>\n",
       "      <td>$37,688.00</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,350,822.93</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>$2,351,815.42</td>\n",
       "      <td>$117,590.77</td>\n",
       "      <td>$56,370.40</td>\n",
       "      <td>$2,525,776.59</td>\n",
       "      <td>$1,350,822.93</td>\n",
       "      <td>$39,730.09</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,422,295.88</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>$2,525,776.59</td>\n",
       "      <td>$126,288.83</td>\n",
       "      <td>$57,497.81</td>\n",
       "      <td>$2,709,563.24</td>\n",
       "      <td>$1,422,295.88</td>\n",
       "      <td>$41,832.23</td>\n",
       "      <td>$31,742.87</td>\n",
       "      <td>$1,495,870.98</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>$2,709,563.24</td>\n",
       "      <td>$135,478.16</td>\n",
       "      <td>$-184,758.88</td>\n",
       "      <td>$2,660,282.52</td>\n",
       "      <td>$1,495,870.98</td>\n",
       "      <td>$43,996.21</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,439,867.19</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>32</th>\n",
       "      <td>$2,660,282.52</td>\n",
       "      <td>$133,014.13</td>\n",
       "      <td>$-188,454.06</td>\n",
       "      <td>$2,604,842.58</td>\n",
       "      <td>$1,439,867.19</td>\n",
       "      <td>$42,349.04</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,382,216.23</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>33</th>\n",
       "      <td>$2,604,842.58</td>\n",
       "      <td>$130,242.13</td>\n",
       "      <td>$-192,223.14</td>\n",
       "      <td>$2,542,861.57</td>\n",
       "      <td>$1,382,216.23</td>\n",
       "      <td>$40,653.42</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,322,869.64</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>34</th>\n",
       "      <td>$2,542,861.57</td>\n",
       "      <td>$127,143.08</td>\n",
       "      <td>$-196,067.60</td>\n",
       "      <td>$2,473,937.05</td>\n",
       "      <td>$1,322,869.64</td>\n",
       "      <td>$38,907.93</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,261,777.57</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>35</th>\n",
       "      <td>$2,473,937.05</td>\n",
       "      <td>$123,696.85</td>\n",
       "      <td>$-199,988.96</td>\n",
       "      <td>$2,397,644.94</td>\n",
       "      <td>$1,261,777.57</td>\n",
       "      <td>$37,111.11</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,198,888.68</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>36</th>\n",
       "      <td>$2,397,644.94</td>\n",
       "      <td>$119,882.25</td>\n",
       "      <td>$-203,988.73</td>\n",
       "      <td>$2,313,538.46</td>\n",
       "      <td>$1,198,888.68</td>\n",
       "      <td>$35,261.43</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,134,150.11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>37</th>\n",
       "      <td>$2,313,538.46</td>\n",
       "      <td>$115,676.92</td>\n",
       "      <td>$-208,068.51</td>\n",
       "      <td>$2,221,146.87</td>\n",
       "      <td>$1,134,150.11</td>\n",
       "      <td>$33,357.36</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$1,067,507.47</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>38</th>\n",
       "      <td>$2,221,146.87</td>\n",
       "      <td>$111,057.34</td>\n",
       "      <td>$-212,229.88</td>\n",
       "      <td>$2,119,974.34</td>\n",
       "      <td>$1,067,507.47</td>\n",
       "      <td>$31,397.28</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$998,904.75</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>39</th>\n",
       "      <td>$2,119,974.34</td>\n",
       "      <td>$105,998.72</td>\n",
       "      <td>$-216,474.48</td>\n",
       "      <td>$2,009,498.58</td>\n",
       "      <td>$998,904.75</td>\n",
       "      <td>$29,379.55</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$928,284.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>40</th>\n",
       "      <td>$2,009,498.58</td>\n",
       "      <td>$100,474.93</td>\n",
       "      <td>$-220,803.97</td>\n",
       "      <td>$1,889,169.54</td>\n",
       "      <td>$928,284.30</td>\n",
       "      <td>$27,302.48</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$855,586.78</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>41</th>\n",
       "      <td>$1,889,169.54</td>\n",
       "      <td>$94,458.48</td>\n",
       "      <td>$-225,220.05</td>\n",
       "      <td>$1,758,407.97</td>\n",
       "      <td>$855,586.78</td>\n",
       "      <td>$25,164.32</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$780,751.09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>42</th>\n",
       "      <td>$1,758,407.97</td>\n",
       "      <td>$87,920.40</td>\n",
       "      <td>$-229,724.45</td>\n",
       "      <td>$1,616,603.92</td>\n",
       "      <td>$780,751.09</td>\n",
       "      <td>$22,963.27</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$703,714.36</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>43</th>\n",
       "      <td>$1,616,603.92</td>\n",
       "      <td>$80,830.20</td>\n",
       "      <td>$-234,318.94</td>\n",
       "      <td>$1,463,115.18</td>\n",
       "      <td>$703,714.36</td>\n",
       "      <td>$20,697.48</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$624,411.84</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>44</th>\n",
       "      <td>$1,463,115.18</td>\n",
       "      <td>$73,155.76</td>\n",
       "      <td>$-239,005.31</td>\n",
       "      <td>$1,297,265.63</td>\n",
       "      <td>$624,411.84</td>\n",
       "      <td>$18,365.05</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$542,776.90</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>45</th>\n",
       "      <td>$1,297,265.63</td>\n",
       "      <td>$64,863.28</td>\n",
       "      <td>$-243,785.42</td>\n",
       "      <td>$1,118,343.49</td>\n",
       "      <td>$542,776.90</td>\n",
       "      <td>$15,964.03</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$458,740.92</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>46</th>\n",
       "      <td>$1,118,343.49</td>\n",
       "      <td>$55,917.17</td>\n",
       "      <td>$-248,661.13</td>\n",
       "      <td>$925,599.53</td>\n",
       "      <td>$458,740.92</td>\n",
       "      <td>$13,492.38</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$372,233.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>47</th>\n",
       "      <td>$925,599.53</td>\n",
       "      <td>$46,279.98</td>\n",
       "      <td>$-253,634.35</td>\n",
       "      <td>$718,245.16</td>\n",
       "      <td>$372,233.30</td>\n",
       "      <td>$10,948.04</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$283,181.34</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>48</th>\n",
       "      <td>$718,245.16</td>\n",
       "      <td>$35,912.26</td>\n",
       "      <td>$-258,707.04</td>\n",
       "      <td>$495,450.38</td>\n",
       "      <td>$283,181.34</td>\n",
       "      <td>$8,328.86</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$191,510.20</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49</th>\n",
       "      <td>$495,450.38</td>\n",
       "      <td>$24,772.52</td>\n",
       "      <td>$-263,881.18</td>\n",
       "      <td>$256,341.72</td>\n",
       "      <td>$191,510.20</td>\n",
       "      <td>$5,632.65</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$97,142.86</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>$256,341.72</td>\n",
       "      <td>$12,817.09</td>\n",
       "      <td>$-269,158.80</td>\n",
       "      <td>$-0.00</td>\n",
       "      <td>$97,142.86</td>\n",
       "      <td>$2,857.14</td>\n",
       "      <td>$-100,000.00</td>\n",
       "      <td>$0.00</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         Nominal                                                 Real  \\\n",
       "          begbal     capgain     dep/with        endbal        begbal   \n",
       "1          $0.00       $0.00   $32,377.73    $32,377.73         $0.00   \n",
       "2     $32,377.73   $1,618.89   $33,025.28    $67,021.89    $31,742.87   \n",
       "3     $67,021.89   $3,351.09   $33,685.79   $104,058.77    $64,419.35   \n",
       "4    $104,058.77   $5,202.94   $34,359.50   $143,621.21    $98,056.91   \n",
       "5    $143,621.21   $7,181.06   $35,046.69   $185,848.97   $132,683.80   \n",
       "6    $185,848.97   $9,292.45   $35,747.63   $230,889.04   $168,329.14   \n",
       "7    $230,889.04  $11,544.45   $36,462.58   $278,896.07   $205,022.86   \n",
       "8    $278,896.07  $13,944.80   $37,191.83   $330,032.71   $242,795.81   \n",
       "9    $330,032.71  $16,501.64   $37,935.67   $384,470.01   $281,679.74   \n",
       "10   $384,470.01  $19,223.50   $38,694.38   $442,387.89   $321,707.30   \n",
       "11   $442,387.89  $22,119.39   $39,468.27   $503,975.55   $362,912.15   \n",
       "12   $503,975.55  $25,198.78   $40,257.63   $569,431.96   $405,328.91   \n",
       "13   $569,431.96  $28,471.60   $41,062.79   $638,966.34   $448,993.21   \n",
       "14   $638,966.34  $31,948.32   $41,884.04   $712,798.70   $493,941.76   \n",
       "15   $712,798.70  $35,639.94   $42,721.72   $791,160.36   $540,212.33   \n",
       "16   $791,160.36  $39,558.02   $43,576.16   $874,294.53   $587,843.80   \n",
       "17   $874,294.53  $43,714.73   $44,447.68   $962,456.94   $636,876.19   \n",
       "18   $962,456.94  $48,122.85   $45,336.63 $1,055,916.42   $687,350.71   \n",
       "19 $1,055,916.42  $52,795.82   $46,243.37 $1,154,955.60   $739,309.78   \n",
       "20 $1,154,955.60  $57,747.78   $47,168.23 $1,259,871.62   $792,797.05   \n",
       "21 $1,259,871.62  $62,993.58   $48,111.60 $1,370,976.80   $847,857.48   \n",
       "22 $1,370,976.80  $68,548.84   $49,073.83 $1,488,599.47   $904,537.34   \n",
       "23 $1,488,599.47  $74,429.97   $50,055.31 $1,613,084.75   $962,884.24   \n",
       "24 $1,613,084.75  $80,654.24   $51,056.41 $1,744,795.39 $1,022,947.24   \n",
       "25 $1,744,795.39  $87,239.77   $52,077.54 $1,884,112.71 $1,084,776.79   \n",
       "26 $1,884,112.71  $94,205.64   $53,119.09 $2,031,437.43 $1,148,424.86   \n",
       "27 $2,031,437.43 $101,571.87   $54,181.47 $2,187,190.78 $1,213,944.93   \n",
       "28 $2,187,190.78 $109,359.54   $55,265.10 $2,351,815.42 $1,281,392.06   \n",
       "29 $2,351,815.42 $117,590.77   $56,370.40 $2,525,776.59 $1,350,822.93   \n",
       "30 $2,525,776.59 $126,288.83   $57,497.81 $2,709,563.24 $1,422,295.88   \n",
       "31 $2,709,563.24 $135,478.16 $-184,758.88 $2,660,282.52 $1,495,870.98   \n",
       "32 $2,660,282.52 $133,014.13 $-188,454.06 $2,604,842.58 $1,439,867.19   \n",
       "33 $2,604,842.58 $130,242.13 $-192,223.14 $2,542,861.57 $1,382,216.23   \n",
       "34 $2,542,861.57 $127,143.08 $-196,067.60 $2,473,937.05 $1,322,869.64   \n",
       "35 $2,473,937.05 $123,696.85 $-199,988.96 $2,397,644.94 $1,261,777.57   \n",
       "36 $2,397,644.94 $119,882.25 $-203,988.73 $2,313,538.46 $1,198,888.68   \n",
       "37 $2,313,538.46 $115,676.92 $-208,068.51 $2,221,146.87 $1,134,150.11   \n",
       "38 $2,221,146.87 $111,057.34 $-212,229.88 $2,119,974.34 $1,067,507.47   \n",
       "39 $2,119,974.34 $105,998.72 $-216,474.48 $2,009,498.58   $998,904.75   \n",
       "40 $2,009,498.58 $100,474.93 $-220,803.97 $1,889,169.54   $928,284.30   \n",
       "41 $1,889,169.54  $94,458.48 $-225,220.05 $1,758,407.97   $855,586.78   \n",
       "42 $1,758,407.97  $87,920.40 $-229,724.45 $1,616,603.92   $780,751.09   \n",
       "43 $1,616,603.92  $80,830.20 $-234,318.94 $1,463,115.18   $703,714.36   \n",
       "44 $1,463,115.18  $73,155.76 $-239,005.31 $1,297,265.63   $624,411.84   \n",
       "45 $1,297,265.63  $64,863.28 $-243,785.42 $1,118,343.49   $542,776.90   \n",
       "46 $1,118,343.49  $55,917.17 $-248,661.13   $925,599.53   $458,740.92   \n",
       "47   $925,599.53  $46,279.98 $-253,634.35   $718,245.16   $372,233.30   \n",
       "48   $718,245.16  $35,912.26 $-258,707.04   $495,450.38   $283,181.34   \n",
       "49   $495,450.38  $24,772.52 $-263,881.18   $256,341.72   $191,510.20   \n",
       "50   $256,341.72  $12,817.09 $-269,158.80        $-0.00    $97,142.86   \n",
       "\n",
       "                                          \n",
       "      capgain     dep/with        endbal  \n",
       "1       $0.00   $31,742.87    $31,742.87  \n",
       "2     $933.61   $31,742.87    $64,419.35  \n",
       "3   $1,894.69   $31,742.87    $98,056.91  \n",
       "4   $2,884.03   $31,742.87   $132,683.80  \n",
       "5   $3,902.46   $31,742.87   $168,329.14  \n",
       "6   $4,950.86   $31,742.87   $205,022.86  \n",
       "7   $6,030.08   $31,742.87   $242,795.81  \n",
       "8   $7,141.05   $31,742.87   $281,679.74  \n",
       "9   $8,284.70   $31,742.87   $321,707.30  \n",
       "10  $9,461.98   $31,742.87   $362,912.15  \n",
       "11 $10,673.89   $31,742.87   $405,328.91  \n",
       "12 $11,921.44   $31,742.87   $448,993.21  \n",
       "13 $13,205.68   $31,742.87   $493,941.76  \n",
       "14 $14,527.70   $31,742.87   $540,212.33  \n",
       "15 $15,888.60   $31,742.87   $587,843.80  \n",
       "16 $17,289.52   $31,742.87   $636,876.19  \n",
       "17 $18,731.65   $31,742.87   $687,350.71  \n",
       "18 $20,216.20   $31,742.87   $739,309.78  \n",
       "19 $21,744.41   $31,742.87   $792,797.05  \n",
       "20 $23,317.56   $31,742.87   $847,857.48  \n",
       "21 $24,936.98   $31,742.87   $904,537.34  \n",
       "22 $26,604.04   $31,742.87   $962,884.24  \n",
       "23 $28,320.12   $31,742.87 $1,022,947.24  \n",
       "24 $30,086.68   $31,742.87 $1,084,776.79  \n",
       "25 $31,905.20   $31,742.87 $1,148,424.86  \n",
       "26 $33,777.20   $31,742.87 $1,213,944.93  \n",
       "27 $35,704.26   $31,742.87 $1,281,392.06  \n",
       "28 $37,688.00   $31,742.87 $1,350,822.93  \n",
       "29 $39,730.09   $31,742.87 $1,422,295.88  \n",
       "30 $41,832.23   $31,742.87 $1,495,870.98  \n",
       "31 $43,996.21 $-100,000.00 $1,439,867.19  \n",
       "32 $42,349.04 $-100,000.00 $1,382,216.23  \n",
       "33 $40,653.42 $-100,000.00 $1,322,869.64  \n",
       "34 $38,907.93 $-100,000.00 $1,261,777.57  \n",
       "35 $37,111.11 $-100,000.00 $1,198,888.68  \n",
       "36 $35,261.43 $-100,000.00 $1,134,150.11  \n",
       "37 $33,357.36 $-100,000.00 $1,067,507.47  \n",
       "38 $31,397.28 $-100,000.00   $998,904.75  \n",
       "39 $29,379.55 $-100,000.00   $928,284.30  \n",
       "40 $27,302.48 $-100,000.00   $855,586.78  \n",
       "41 $25,164.32 $-100,000.00   $780,751.09  \n",
       "42 $22,963.27 $-100,000.00   $703,714.36  \n",
       "43 $20,697.48 $-100,000.00   $624,411.84  \n",
       "44 $18,365.05 $-100,000.00   $542,776.90  \n",
       "45 $15,964.03 $-100,000.00   $458,740.92  \n",
       "46 $13,492.38 $-100,000.00   $372,233.30  \n",
       "47 $10,948.04 $-100,000.00   $283,181.34  \n",
       "48  $8,328.86 $-100,000.00   $191,510.20  \n",
       "49  $5,632.65 $-100,000.00    $97,142.86  \n",
       "50  $2,857.14 $-100,000.00         $0.00  "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Fill our the nominal side of the balance sheet\n",
    "acct[('Nominal','dep/with')] = acct[('Real','dep/with')] *(1+INFLATION)**acct.index\n",
    "for t in acct.index:\n",
    "    if t==1:\n",
    "        acct.loc[t,('Nominal','begbal')] = PV\n",
    "    else:\n",
    "        acct.loc[t,('Nominal','begbal')] = acct.loc[t-1,('Nominal','endbal')]\n",
    "    acct.loc[t,('Nominal','capgain')] = acct.loc[t,('Nominal','begbal')]*NOMINAL\n",
    "    acct.loc[t,('Nominal','endbal')]  = acct.loc[t,('Nominal','begbal')] + acct.loc[t,('Nominal','capgain')] + acct.loc[t,('Nominal','dep/with')] \n",
    "acct"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Excercise: what if we had wanted to save in constant nominal terms?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Uncertain returns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.17315865,  0.1115279 , -0.11454003,  0.03916162,  0.1021336 ,\n",
       "       -0.03200856,  0.06655116,  0.05085485,  0.04042914,  0.02253998,\n",
       "        0.08330262,  0.16030374, -0.05650657,  0.14282741,  0.06286301,\n",
       "        0.08451376, -0.07366022,  0.05351369,  0.1884537 , -0.06798049,\n",
       "       -0.15777283, -0.13433723,  0.06660702,  0.27849673,  0.15236913,\n",
       "        0.20726222,  0.04991492,  0.17979964,  0.0128752 ,  0.10132042])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "norm.rvs(loc=0.04, scale = 0.1, size=30, random_state=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "911801.0357697688"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "MEAN       = 0.03\n",
    "SD         = 0.10\n",
    "N_SAVING   = 30\n",
    "N_WITHDRAW = 20\n",
    "PMT        = 32000.0\n",
    "WITHDRAWAL = 100000.0\n",
    "PV         = 0.0\n",
    "\n",
    "acct = pd.DataFrame(dtype=float,columns=['begbal','return','capgain','dep/with','endbal'],index=np.arange(1,N_SAVING + N_WITHDRAW + 1))\n",
    "acct['return'] = norm.rvs(loc=MEAN, scale = SD, size=N_SAVING+N_WITHDRAW, random_state=10)\n",
    "\n",
    "acct['dep/with'] = np.where(acct.index <= N_SAVING, PMT,-WITHDRAWAL)\n",
    "for t in acct.index:\n",
    "    if t==1:\n",
    "        acct.loc[t,'begbal'] = PV\n",
    "    else:\n",
    "        acct.loc[t,'begbal'] = acct.loc[t-1,'endbal']\n",
    "    acct.loc[t,'capgain'] = acct.loc[t,'begbal']*acct.loc[t,'return']\n",
    "    acct.loc[t,'endbal']  = acct.loc[t,'begbal'] + acct.loc[t,'capgain'] + acct.loc[t,'dep/with'] \n",
    "acct.loc[N_SAVING + N_WITHDRAW,'endbal']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But this was just a single possible realization.  We want to know what the **distribution** of possible balances look like.\n",
    "\n",
    "Let's run the simulation above 1000 times and collect the ending balance for each month."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "N_SIMS=1000\n",
    "sims = pd.DataFrame(dtype=float,columns=['endbal'],index=np.arange(N_SIMS))\n",
    "def simulate(MEAN,SD):\n",
    "    acct = pd.DataFrame(dtype=float,columns=['begbal','return','capgain','dep/with','endbal'],index=np.arange(1,N_SAVING + N_WITHDRAW + 1))\n",
    "    acct['return'] = norm.rvs(loc=MEAN, scale = SD, size=N_SAVING+N_WITHDRAW)\n",
    "\n",
    "    acct['dep/with'] = np.where(acct.index <= N_SAVING, PMT,-WITHDRAWAL)\n",
    "    for t in acct.index:\n",
    "        if t==1:\n",
    "            acct.loc[t,'begbal'] = PV\n",
    "        else:\n",
    "            acct.loc[t,'begbal'] = acct.loc[t-1,'endbal']\n",
    "        acct.loc[t,'capgain'] = acct.loc[t,'begbal']*acct.loc[t,'return']\n",
    "        acct.loc[t,'endbal']  = acct.loc[t,'begbal'] + acct.loc[t,'capgain'] + acct.loc[t,'dep/with'] \n",
    "    return acct.loc[N_SAVING + N_WITHDRAW,'endbal']\n",
    "for s in sims.index:\n",
    "    sims.loc[s] = simulate(MEAN,SD)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.387\n"
     ]
    }
   ],
   "source": [
    "# How many times did we run out of money?\n",
    "frac_pos = (sims.endbal >0).mean()\n",
    "print(frac_pos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
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           "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": {
         "title": {
          "text": "Ending Balance"
         }
        },
        "yaxis": {
         "title": {
          "text": "Percent of Simulations"
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the distribution of ending balances\n",
    "fig = go.Figure()\n",
    "trace= go.Histogram(x=sims.endbal, 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\"] = \"Ending Balance\"\n",
    "fig.layout.yaxis[\"title\"] = \"Percent of Simulations\"\n",
    "fig.add_vline(x=0, line_width=4, line_dash=\"dash\", line_color=\"black\")\n",
    "fig.add_annotation(x=sims.endbal.max()*0.7, y=8,\n",
    "            text=\"Ending balance positive in <br>\"+f'{frac_pos:.1%} of simulations', showarrow=False)\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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