{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "[![Binder](img/badge-binder.svg)](https://mybinder.org/v2/gh/nhirschey/teaching/gh-pages?filepath=stockslongrun.ipynb)\u0026emsp;\n",
    "[![Script](img/badge-script.svg)](/Teaching//stockslongrun.fsx)\u0026emsp;\n",
    "[![Notebook](img/badge-notebook.svg)](/Teaching//stockslongrun.ipynb)\n",
    "\n",
    "# Stocks for the long run\n",
    "\n",
    "To explore working with returns, we\u0027re going to simulate return data and look at how\n",
    "our portfolio would grow.\n",
    "\n",
    "We will do analysis using relatively simple syntax common\n",
    "to many programming languages to make learning easier.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 1, "outputs": [],
   "source": [
    "#r \"nuget: FSharp.Stats, 0.5.0\"\n",
    "#r \"nuget: Plotly.NET, 3.*\"\n",
    "#r \"nuget: Plotly.NET.Interactive, 3.*\"\n",
    "\n",
    "open FSharp.Stats\n",
    "\n",
    "open Plotly.NET\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "From 1/1871-1/2023, the US market return was annualized\n",
    "7.5% with a 14% standard deviation (Robert Schiller data).\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 2, "outputs": [
          {
           "data": {
            "text/plain": ["0.11281501405599485",
"",
"-0.08408038022942653",
"",
"-0.1347073356587039",
"",
"0.13671097786971176",
"",
"0.10692195358626277",
"",
"0.16073449907483076"]
        },
           "execution_count": 2,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "let rnorm = Distributions.Continuous.Normal.Init 0.075 0.14\n",
    "\n",
    "for i = 0 to 5 do\n",
    "    printfn $\"{rnorm.Sample()}\"\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Put sampled values in a list.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 3, "outputs": [],
   "source": [
    "let returns = [ for i = 1 to 1000 do rnorm.Sample() ]\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Plot the distribution of returns.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 4, "outputs": [],
   "source": [
    "let hist = Chart.Histogram(returns)\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": null, "outputs": [
          {
           "data": {
            "text/html": ["\u003cdiv id=\"12e70b21-e90a-4187-8247-7b18e91b9035\"\u003e\u003c!-- Plotly chart will be drawn inside this DIV --\u003e\u003c/div\u003e",
"",
"\u003cscript type=\"text/javascript\"\u003e",
"",
"",
"",
"            var renderPlotly_12e70b21e90a418782477b18e91b9035 = function() {",
"",
"            var fsharpPlotlyRequire = requirejs.config({context:\u0027fsharp-plotly\u0027,paths:{plotly:\u0027https://cdn.plot.ly/plotly-2.6.3.min\u0027}}) || require;",
"",
"            fsharpPlotlyRequire([\u0027plotly\u0027], function(Plotly) {",
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"",
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"            if ((typeof(requirejs) !==  typeof(Function)) || (typeof(requirejs.config) !== typeof(Function))) {",
"",
"                var script = document.createElement(\"script\");",
"",
"                script.setAttribute(\"src\", \"https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js\");",
"",
"                script.onload = function(){",
"",
"                    renderPlotly_12e70b21e90a418782477b18e91b9035();",
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"\u003c/script\u003e",
"",
""]
        },
           "execution_count": null,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "hist\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 6, "outputs": [
          {
           "data": {
            "text/plain": ["val percentiles: (float * float) list =",
"",
"  [(0.0, -0.523598482); (0.01, -0.236118288); (0.05, -0.1491462588);",
"",
"   (0.5, 0.07794851357); (0.95, 0.3105842283); (0.99, 0.4150182609);",
"",
"   (1.0, 0.5562842221)]"]
        },
           "execution_count": 6,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "let percentiles =\n",
    "    [ for p in [0.0; 0.01; 0.05; 0.5; 0.95; 0.99; 1.0] do \n",
    "        let pctl = Quantile.compute p returns\n",
    "        p, pctl ]\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 7, "outputs": [
          {
           "data": {
            "text/plain": ["percentile 0.00 is -0.5235984819519119",
"",
"percentile 0.01 is -0.23611828795794754",
"",
"percentile 0.05 is -0.14914625875278156",
"",
"percentile 0.50 is 0.07794851357062386",
"",
"percentile 0.95 is 0.31058422825831894",
"",
"percentile 0.99 is 0.41501826087627836",
"",
"percentile 1.00 is 0.5562842220620063"]
        },
           "execution_count": 7,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "for (p, pctl) in percentiles do\n",
    "    printfn $\"percentile %.2f{p} is {pctl}\"\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "A draw of 10-years of returns.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 8, "outputs": [
          {
           "data": {
            "text/plain": ["val draw10y: float list =",
"",
"  [0.07090658771; 0.1290453357; 0.2513317343; 0.03518658973; 0.2223930737;",
"",
"   -0.06832573973; 0.1606886181; 0.07021140711; -0.03060338974;",
"",
"   -0.006223749925]"]
        },
           "execution_count": 8,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "let draw10y = [ for i = 1 to 10 do rnorm.Sample() ]\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Compute the cumulative product of the returns.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 9, "outputs": [
          {
           "data": {
            "text/plain": ["r=0.071",
"",
"    cumprod=1.071",
"",
"r=0.129",
"",
"    cumprod=1.209",
"",
"r=0.251",
"",
"    cumprod=1.513",
"",
"r=0.035",
"",
"    cumprod=1.566",
"",
"r=0.222",
"",
"    cumprod=1.915",
"",
"r=-0.068",
"",
"    cumprod=1.784",
"",
"r=0.161",
"",
"    cumprod=2.07",
"",
"r=0.07",
"",
"    cumprod=2.216",
"",
"r=-0.031",
"",
"    cumprod=2.148",
"",
"r=-0.006",
"",
"    cumprod=2.135"]
        },
           "execution_count": 9,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "let mutable cumprod = 1.0\n",
    "for r in draw10y do\n",
    "    cumprod \u003c- cumprod * (1.0 + r)\n",
    "    printfn $\"r={round 3 r}\"\n",
    "    printfn $\"    cumprod={round 3 cumprod}\"\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Let\u0027s plot the cumulative return.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 10, "outputs": [
          {
           "data": {
            "text/plain": ["val draw10yCR: (int * float) list =",
"",
"  [(1, 1.070906588); (2, 1.209102088); (3, 1.512987812); (4, 1.566224694);",
"",
"   (5, 1.914542218); (6, 1.783729704); (7, 2.070354766); (8, 2.215717287);",
"",
"   (9, 2.147908827); (10, 2.13454078)]"]
        },
           "execution_count": 10,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "let draw10yCR = \n",
    "    let mutable year = 0\n",
    "    let mutable cumprod = 1.0\n",
    "    [ for r in draw10y do\n",
    "        cumprod \u003c- cumprod * (1.0 + r)\n",
    "        year \u003c- year + 1\n",
    "        year, cumprod ]\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Functional version of the cumulative return calculation.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 11, "outputs": [
          {
           "data": {
            "text/plain": ["val it: (int * float) list =",
"",
"  [(1, 1.070906588); (2, 1.209102088); (3, 1.512987812); (4, 1.566224694);",
"",
"   (5, 1.914542218); (6, 1.783729704); (7, 2.070354766); (8, 2.215717287);",
"",
"   (9, 2.147908827); (10, 2.13454078)]"]
        },
           "execution_count": 11,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "((0, 1.0), draw10y)\n",
    "||\u003e List.scan (fun (year, cumprod) r -\u003e \n",
    "    (year + 1, cumprod * (1.0 + r)))\n",
    "|\u003e List.tail\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Plot the cumulative return.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 12, "outputs": [],
   "source": [
    "let draw10yCrPlot = Chart.Line(draw10yCR)\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": null, "outputs": [
          {
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"            var renderPlotly_f8289d6d9cb04d198504c62f08862e9b = function() {",
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"            var fsharpPlotlyRequire = requirejs.config({context:\u0027fsharp-plotly\u0027,paths:{plotly:\u0027https://cdn.plot.ly/plotly-2.6.3.min\u0027}}) || require;",
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        },
           "execution_count": null,
           "metadata": {},
           "output_type": "execute_result"
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   "source": [
    "draw10yCrPlot\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "\u003e **Practice**: Sample 20 years of returns and plot the cumulative return.\n",
    "\u003e \n",
    "\n"
   ]
  }
,
  {
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     "language": "fsharp"
    },
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   },
   "execution_count": 29, "outputs": [],
   "source": [
    "// Answer here\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Let\u0027s simulate 1000 draws of 30 years of returns.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 15, "outputs": [],
   "source": [
    "let draw1k = \n",
    "    [ for i = 1 to 1_000 do\n",
    "        [ for y = 1 to 30 do rnorm.Sample() ] ]\n",
    "\n",
    "let draw1kCR =\n",
    "    [ for life in draw1k do\n",
    "        let mutable accRet = 1.0\n",
    "        let mutable year = 0\n",
    "        [ for r in life do\n",
    "            accRet \u003c- accRet * (1.0 + r)\n",
    "            year \u003c- year+1\n",
    "            year, accRet ] ]\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Look at a few observtions from the first draw.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 16, "outputs": [
          {
           "data": {
            "text/plain": ["val it: (int * float) list =",
"",
"  [(1, 1.176291981); (2, 1.270733795); (3, 1.499237526); (4, 1.923948337);",
"",
"   (5, 2.026791218); (6, 1.558203089)]"]
        },
           "execution_count": 16,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "draw1kCR[0][0..5]\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Plot the first draw\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": null, "outputs": [
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   "source": [
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           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "[ Chart.Line(draw1kCR[0])\n",
    "  Chart.Line(draw1kCR[1]) ]\n",
    "|\u003e Chart.combine  \n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Plot all 1000 draws together\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": null, "outputs": [
          {
           "data": {
            "text/html": ["\u003cdiv id=\"e803396c-6d2a-4246-8578-30723a42b29e\"\u003e\u003c!-- Plotly chart will be drawn inside this DIV --\u003e\u003c/div\u003e",
"",
"\u003cscript type=\"text/javascript\"\u003e",
"",
"",
"",
"            var renderPlotly_e803396c6d2a4246857830723a42b29e = function() {",
"",
"            var fsharpPlotlyRequire = requirejs.config({context:\u0027fsharp-plotly\u0027,paths:{plotly:\u0027https://cdn.plot.ly/plotly-2.6.3.min\u0027}}) || require;",
"",
"            fsharpPlotlyRequire([\u0027plotly\u0027], function(Plotly) {",
"",
"",
"",
"            var data = 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"",
"            Plotly.newPlot(\u0027e803396c-6d2a-4246-8578-30723a42b29e\u0027, data, layout, config);",
"",
"});",
"",
"            };",
"",
"            if ((typeof(requirejs) !==  typeof(Function)) || (typeof(requirejs.config) !== typeof(Function))) {",
"",
"                var script = document.createElement(\"script\");",
"",
"                script.setAttribute(\"src\", \"https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js\");",
"",
"                script.onload = function(){",
"",
"                    renderPlotly_e803396c6d2a4246857830723a42b29e();",
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"\u003c/script\u003e",
"",
""]
        },
           "execution_count": null,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "[ for x in draw1kCR do Chart.Line(x) ]\n",
    "|\u003e Chart.combine\n",
    "|\u003e Chart.withLayoutStyle(ShowLegend=false)\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "A more useful version may be to plot the values at the end of the last year.\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 21, "outputs": [],
   "source": [
    "let terminalValues = \n",
    "    [ for x in draw1kCR do \n",
    "        let (yr, ret) = x[x.Length-1]\n",
    "        ret ]\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": null, "outputs": [
          {
           "data": {
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"            fsharpPlotlyRequire([\u0027plotly\u0027], function(Plotly) {",
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"",
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"            Plotly.newPlot(\u002729ec5f33-2676-464e-bfb9-5756ce5ad1ca\u0027, data, layout, config);",
"",
"});",
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"            if ((typeof(requirejs) !==  typeof(Function)) || (typeof(requirejs.config) !== typeof(Function))) {",
"",
"                var script = document.createElement(\"script\");",
"",
"                script.setAttribute(\"src\", \"https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js\");",
"",
"                script.onload = function(){",
"",
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"\u003c/script\u003e",
"",
""]
        },
           "execution_count": null,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "terminalValues |\u003e Chart.Histogram\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "What is the chance we lose money?\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 23, "outputs": [
          {
           "data": {
            "text/plain": ["chance lose money=0.004"]
        },
           "execution_count": 23,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "let nLoseMoney =\n",
    "    terminalValues\n",
    "    |\u003e List.filter (fun x -\u003e x \u003c 1.0)\n",
    "    |\u003e List.length\n",
    "    |\u003e float\n",
    "\n",
    "let chanceLose = nLoseMoney / (float terminalValues.Length)    \n",
    "printfn $\"chance lose money=%.3f{chanceLose}\"\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "What is the chance we double our money?\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 24, "outputs": [
          {
           "data": {
            "text/plain": ["chance double money=0.945"]
        },
           "execution_count": 24,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "let nDoubleMoney =\n",
    "    terminalValues\n",
    "    |\u003e List.filter (fun x -\u003e x \u003e= 2.0)\n",
    "    |\u003e List.length\n",
    "    |\u003e float\n",
    "\n",
    "let chanceDouble = nDoubleMoney / (float terminalValues.Length)    \n",
    "printfn $\"chance double money=%.3f{chanceDouble}\"\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "\u003e **Practice**: What is the chance we earn at least a 10% compound rate of return per year?\n",
    "\u003e \n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 29, "outputs": [],
   "source": [
    "// Answer here\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "Now let\u0027s say we have 1m EUR and we want to live off of 50k EUR per year.\n",
    "What is the chance that we can do that?\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 26, "outputs": [],
   "source": [
    "let expenses = 50_000.0\n",
    "let initialWealth = 1_000_000.0\n",
    "let wealthEvolution =\n",
    "    [ for life in draw1k do\n",
    "        let mutable wealth = initialWealth\n",
    "        [ for r in life do\n",
    "            // We\u0027ll take expenses out at the start of the year.\n",
    "            if wealth \u003e expenses then\n",
    "                wealth \u003c- (wealth - expenses) * (1.0 + r)\n",
    "            else\n",
    "                wealth \u003c- 0.0\n",
    "            wealth ] ]\n",
    "\n",
    "let terminalWealth = [ for x in wealthEvolution do x[x.Length-1] ]\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "What\u0027s the chance that we don\u0027t have enough money?\n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 27, "outputs": [
          {
           "data": {
            "text/plain": ["chance broke=0.158"]
        },
           "execution_count": 27,
           "metadata": {},
           "output_type": "execute_result"
          }],
   "source": [
    "let nBroke =\n",
    "    terminalWealth\n",
    "    |\u003e List.filter (fun x -\u003e x \u003c= 0.0)\n",
    "    |\u003e List.length\n",
    "    |\u003e float\n",
    "\n",
    "let chanceBroke = nBroke / (float terminalWealth.Length)    \n",
    "\n",
    "printfn $\"chance broke=%.3f{chanceBroke}\"\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "\u003e **Practice**: What is our chance of going broke if the market\u0027s standard deviation is 20% per year?\n",
    "\u003e \n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 29, "outputs": [],
   "source": [
    "// Answer here\n"
   ]
  }
,
  {
   "cell_type": "markdown",
   "metadata": {},
   
   "source": [
    "\u003e **Practice**: What is our chance of going broke if the market\u0027s\n",
    "return is 5% per year?\n",
    "\u003e \n",
    "\n"
   ]
  }
,
  {
   "cell_type": "code",
   "metadata": {
    "dotnet_interactive": {
     "language": "fsharp"
    },
    "polyglot_notebook": {
     "kernelName": "fsharp"
    }
   },
   "execution_count": 29, "outputs": [],
   "source": [
    "// Answer here\n"
   ]
  }
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