index.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Figuring Interest with Computers\n",
    "\n",
    "* In this notebook, we will learn to use R to compute the value over time of a balance of \\$1,000,000 that receives 6% interest, compounded monthly.\n",
    "* The little boxes you see all over this page are called \"cells\". They contain things that the computer can execute. To execute them, hold <kbd>Shift</kbd> while pressing <kbd>Enter</kbd>.\n",
    "* We will also experiment with how the value of the balance changes if we compound at different intervals.\n",
    "* This first cell adjusts some settings. Just hit <kbd>Shift</kbd>+<kbd>Enter</kbd> on it to have it take effect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [],
   "source": [
    "options(digits=9, repr.plot.width=5, repr.plot.height=4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Like a Pocket Calculator\n",
    "\n",
    "* You can use the computer like a pocket calculator, just by entering formulas inside of a `print(...)` statement.\n",
    "* You can also see \"comments\" in action here. A comment is a piece of code that the computer just ignores. We use it to keep track of the number of years."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1005000\n"
     ]
    }
   ],
   "source": [
    "# after 1 month\n",
    "print(1000000 + 1000000 * 0.06/12)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(You can disregard the `[1]` that is printed before the answer.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1010025\n"
     ]
    }
   ],
   "source": [
    "# after 2 months\n",
    "print(1005000 + 1005000 * 0.06/12)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Carry this out for six more years, so that you know the value after eight years:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1015075.12\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "# after 3 months\n",
    "print(1010025 + 1010025 * 0.06/12)\n",
    "\n",
    "# (We're using \"print\" to get consistent display/rounding between here and the version below.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1020150.5\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "# after 4 months\n",
    "print(1015075.12 + 1015075.12 * 0.06/12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1025251.25\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "# after 5 months\n",
    "print(1020150.5 + 1020150.5 * 0.06/12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1030377.51\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "# after 6 months\n",
    "print(1025251.25 + 1025251.25 * 0.06/12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1035529.4\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "# after 7 months\n",
    "print(1030377.51 + 1030377.51 * 0.06/12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1040707.05\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "# after 8 months\n",
    "print(1035529.4 + 1035529.4 * 0.06/12)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But copy-pasting the last value from the output to the next input is boring and error-prone, worse because we have to do it twice. We can probably do better."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using Variables\n",
    "\n",
    "* To save results of previous calculations, we can use *variables*.\n",
    "* We can make up any name we want for them, as long as it consists of only letters (`a`-`z`, `A`-`Z`), numbers (`0`-`9`), and underscores (`_`). You should use no other symbols in your variable names.\n",
    "* The values of variables \"survive\" from one cell to the next."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "month0 <- 1000000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "month1 <- month0 + month0 * 0.06/12"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We no longer get to see the value (because we saved it in a variable). We have to explicitly request to see it. We can do that like so:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1005000\n"
     ]
    }
   ],
   "source": [
    "month1 <- month0 + month0 * 0.06/12\n",
    "print(month1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1010025\n"
     ]
    }
   ],
   "source": [
    "month2 <- month1 + month1 * 0.06/12\n",
    "print(month2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Carry this out for six more years, so that you can compare with your value after eight years from above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1015075.12\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "month3 <- month2 + month2 * 0.06/12\n",
    "print(month3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1020150.5\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "month4 <- month3 + month3 * 0.06/12\n",
    "print(month4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1025251.25\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "month5 <- month4 + month4 * 0.06/12\n",
    "print(month5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1030377.51\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "month6 <- month5 + month5 * 0.06/12\n",
    "print(month6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1035529.4\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "month7 <- month6 + month6 * 0.06/12\n",
    "print(month7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1040707.04\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "month8 <- month7 + month7 * 0.06/12\n",
    "print(month8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Is this value the same as above? Exactly? Why?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now have *all* the values of the balance in computer memory at the same time. We can ask to see them all at once:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1e+06\n",
      "[1] 1005000\n",
      "[1] 1010025\n",
      "[1] 1015075.12\n",
      "[1] 1020150.5\n",
      "[1] 1025251.25\n",
      "[1] 1030377.51\n",
      "[1] 1035529.4\n",
      "[1] 1040707.04\n"
     ]
    }
   ],
   "source": [
    "print(month0)\n",
    "print(month1)\n",
    "print(month2)\n",
    "print(month3)\n",
    "print(month4)\n",
    "print(month5)\n",
    "print(month6)\n",
    "print(month7)\n",
    "print(month8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is still boring and error prone. Can we do better still?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using Variables that Change\n",
    "\n",
    "It turns out that we can *change* the value of a variable, even after we've already set it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1e+06\n",
      "[1] 1005000\n"
     ]
    }
   ],
   "source": [
    "balance <- 1000000\n",
    "print(balance)\n",
    "balance <- 1000000 + 1000000 * 0.06/12\n",
    "print(balance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's the value that was set last that counts. It turns out that that applies even *while* we're setting a new value of the same variable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1e+06\n",
      "[1] 1005000\n"
     ]
    }
   ],
   "source": [
    "balance <- 1000000\n",
    "print(balance)\n",
    "balance <- balance + balance * 0.06/12\n",
    "print(balance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That means we can make this much less annoying to carry out, using copy-and-paste. Use the cell above as a pattern to calculate the balance after eight months:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1040707.04\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "balance <- 1000000\n",
    "\n",
    "balance <- balance + balance * 0.06/12\n",
    "balance <- balance + balance * 0.06/12\n",
    "balance <- balance + balance * 0.06/12\n",
    "balance <- balance + balance * 0.06/12\n",
    "\n",
    "balance <- balance + balance * 0.06/12\n",
    "balance <- balance + balance * 0.06/12\n",
    "balance <- balance + balance * 0.06/12\n",
    "balance <- balance + balance * 0.06/12\n",
    "\n",
    "print(balance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* The value of `balance` changes throughout the calculation. Add `print` statements to see its value after each update.\n",
    "\n",
    "---\n",
    "We're still having to type the interest on every line. That's inconvenient if we would like to change it, to see what would happen if the interest rate or the compounding interval changes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using Variables for More Things\n",
    "\n",
    "We can simply use variables for those things as well. Let's use these values.\n",
    "\n",
    "SET-UP:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [],
   "source": [
    "starting_balance <- 1000000\n",
    "interest_rate <- 0.06\n",
    "compounding_interval <- (1/12)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Modify the snippet of code from above to make use of these variables, and check that it still computes the same answer:\n",
    "\n",
    "COMPUTATION:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1040707.04\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "\n",
    "balance <- starting_balance\n",
    "\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "\n",
    "print(balance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now use modified versions of the two cells above (\"SET-UP\" and \"COMPUTATION\") to figure what the balance is after eight months if interest is compounded six times a year (i.e. every two months):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [],
   "source": [
    "#clear\n",
    "starting_balance <- 100000\n",
    "interest_rate <- 0.06\n",
    "compounding_interval <- (1/6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 104060.401\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "balance <- starting_balance\n",
    "\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "balance <- balance + balance * interest_rate * compounding_interval\n",
    "\n",
    "print(balance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* How different is the result from monthly compounding?\n",
    "* Does that match what you expect?\n",
    "\n",
    "----\n",
    "\n",
    "It is sort of annoying that we can now easily change everything about the computation *except* how often the compounding is carried out. But it turns out that there is a widget for that, too."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Loops\n",
    "\n",
    "*   Type in\n",
    "    ```\n",
    "    for (i in 1:17) {\n",
    "        print(\"Hello\")\n",
    "    }\n",
    "    ```\n",
    "    and execute it. What happens?\n",
    "*   Change the `17` to a different value, and print two things per round of execution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"Hello\"\n",
      "[1] \"there!\"\n",
      "[1] \"Hello\"\n",
      "[1] \"there!\"\n",
      "[1] \"Hello\"\n",
      "[1] \"there!\"\n",
      "[1] \"Hello\"\n",
      "[1] \"there!\"\n",
      "[1] \"Hello\"\n",
      "[1] \"there!\"\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "for (i in 1:5) {\n",
    "    print(\"Hello\")\n",
    "    print(\"there!\")\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is called a *loop*.\n",
    "\n",
    "* Use the loop pattern and plug in the compounding calculation for the *loop body*.\n",
    "* Combine three lines from the cell marked \"SET-UP\" together with the cell marked \"COMPUTATION\" and replace the repeated lines with a loop.\n",
    "* Make sure to print the `balance` after each update.\n",
    "* Check that you still compute the same result as above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1005000\n",
      "[1] 1010025\n",
      "[1] 1015075.12\n",
      "[1] 1020150.5\n",
      "[1] 1025251.25\n",
      "[1] 1030377.51\n",
      "[1] 1035529.4\n",
      "[1] 1040707.04\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "starting_balance <- 1000000\n",
    "interest_rate <- 0.06\n",
    "compounding_interval <- (1/12)\n",
    "\n",
    "balance <- starting_balance\n",
    "\n",
    "for (i in 1:8) {\n",
    "    balance <- balance + balance * interest_rate * compounding_interval\n",
    "    print(balance)\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The \"how-often\" bit of a loop can also be a variable. We'll define a variable `how_often_per_year` to say how often interest is compounded each year.\n",
    "\n",
    "* Change the computation above to use it, making sure to keep the overall time to eight months.\n",
    "* Create a new variable `compoundings` to represent the number of times compounding happens.\n",
    "* Check that for \"once a month\" you still get the same answer as above.\n",
    "* Experiment with higher values of `how_often_per_year`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {},
   "outputs": [],
   "source": [
    "how_often_per_year <- 12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 1005000\n",
      "[1] 1010025\n",
      "[1] 1015075.12\n",
      "[1] 1020150.5\n",
      "[1] 1025251.25\n",
      "[1] 1030377.51\n",
      "[1] 1035529.4\n",
      "[1] 1040707.04\n"
     ]
    }
   ],
   "source": [
    "#clear\n",
    "starting_balance <- 1000000\n",
    "interest_rate <- 0.06\n",
    "compounding_interval <- 1/how_often_per_year\n",
    "\n",
    "balance <- starting_balance\n",
    "\n",
    "compoundings <- (8 / 12) * how_often_per_year  # run for eight months\n",
    "for (i in 1:compoundings) {\n",
    "    balance <- balance + balance * interest_rate * compounding_interval\n",
    "    print(balance)\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*   Does your modification do what you expect? How can you be sure? Can you be sure?\n",
    "*   Do you observe something you don't expect? Do you have a sense of what might be happening?\n",
    "*   Once you're confident that the code does what you expect, change the value of `how_often_per_year` to a very large number.\n",
    "    * How long does the computation take?\n",
    "    * What happens to the final balance? Does it keep growing with more frequent compounding? Can it become arbitrarily big?\n",
    "\n",
    "To better understand and compare what happens to these values over time in different scenarios, it would be great if we could plot them in a graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Containers and Plotting\n",
    "\n",
    "It would be nice if the computer could also draw a graph of the balance over time. To do that, we will need all the values of `balance` present at one time, not at different times. To do that, we can use a container. Those are created like so:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>100</li>\n",
       "\t<li>1300</li>\n",
       "\t<li>1500</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 100\n",
       "\\item 1300\n",
       "\\item 1500\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 100\n",
       "2. 1300\n",
       "3. 1500\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]  100 1300 1500"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "c(100, 1300, 1500)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also make names for them, just like numbers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1000</li>\n",
       "\t<li>1300</li>\n",
       "\t<li>1500</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1000\n",
       "\\item 1300\n",
       "\\item 1500\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1000\n",
       "2. 1300\n",
       "3. 1500\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 1000 1300 1500"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "data <- c(1000, 1300, 1500)\n",
    "data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also add onto them very simply:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1000</li>\n",
       "\t<li>1300</li>\n",
       "\t<li>1500</li>\n",
       "\t<li>2000</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1000\n",
       "\\item 1300\n",
       "\\item 1500\n",
       "\\item 2000\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1000\n",
       "2. 1300\n",
       "3. 1500\n",
       "4. 2000\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 1000 1300 1500 2000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "data <- c(1000, 1300, 1500)\n",
    "c(data, 2000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result of that operation can then be put back into a variable, including the same one:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1000</li>\n",
       "\t<li>1300</li>\n",
       "\t<li>1500</li>\n",
       "\t<li>2000</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1000\n",
       "\\item 1300\n",
       "\\item 1500\n",
       "\\item 2000\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1000\n",
       "2. 1300\n",
       "3. 1500\n",
       "4. 2000\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 1000 1300 1500 2000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "data <- c(1000, 1300, 1500)\n",
    "data <- c(data, 2000)\n",
    "\n",
    "data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can then ask the computer to plot the data in the container:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlgAAAHgCAMAAABOyeNrAAAC3FBMVEUAAAABAQECAgIDAwME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\n",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Change the latest version of your computation (the one with varying `how_often_per_year`) to include plotting by introducing a `balances` variable that is a container that gets filled up with all past balances. Make sure to include the first one!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlgAAAHgCAMAAABOyeNrAAAC7lBMVEUAAAABAQECAgIDAwME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\n/wOrx75Fs/K5KgAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#clear\n",
    "how_often_per_year <- 12\n",
    "starting_balance <- 1000000\n",
    "interest_rate <- 0.06\n",
    "compounding_interval <- 1/how_often_per_year\n",
    "\n",
    "balance <- starting_balance\n",
    "balances <- c(starting_balance)\n",
    "\n",
    "compoundings <- (8 / 12) * how_often_per_year  # run for eight months\n",
    "for (i in 1:compoundings) {\n",
    "    balance <- balance + balance * interest_rate * compounding_interval\n",
    "    balances <- c(balances, balance)\n",
    "}\n",
    "\n",
    "plot(balances)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is a little annoying that the labeling of the x-axis is just the \"Index\", i.e. the number of the entry in the container. Fortunately, we can fix that by passing *another* container (that has to have the same length as the first one) that includes x-axis positions:\n",
    "\n",
    "**Note** that the x-axis positions often do but do not *have to* have equal spacing!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlgAAAHgCAMAAABOyeNrAAAC6FBMVEUAAAABAQECAgIDAwME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\nICxIEBYk/g8E3NqGY672bgAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xpos <- c(1, 3, 10, 20)\n",
    "\n",
    "data <- c(1000, 1300, 1500)\n",
    "data <- c(data, 2000)\n",
    "\n",
    "plot(xpos, data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Modify your computation from above to also keep track of time, and\n",
    "* Add code to build a container `months` that, at the end, has the same number of entries as `balances` that keeps track of the time in months."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlgAAAHgCAMAAABOyeNrAAAC9FBMVEUAAAABAQECAgIDAwME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\nASX8P70LH75ur0yqAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#clear\n",
    "how_often_per_year <- 200\n",
    "starting_balance <- 1000000\n",
    "interest_rate <- 0.06\n",
    "compounding_interval <- 1/how_often_per_year\n",
    "\n",
    "balance <- starting_balance\n",
    "balances <- c(starting_balance)\n",
    "month <- 0\n",
    "months <- c(0)\n",
    "\n",
    "compoundings <- 10 * how_often_per_year\n",
    "for (i in 1:compoundings) {\n",
    "    balance <- balance + balance * interest_rate * compounding_interval\n",
    "    balances <- c(balances, balance)\n",
    "    month <- month + 12/how_often_per_year\n",
    "    months <- c(months, month)\n",
    "}\n",
    "\n",
    "plot(months, balances)\n",
    "\n",
    "# FIXME: Is this too big a jump?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Does the graph you obtain look like an exponential function? Should it?\n",
    "* What happens if you let the computation run for more months? 10 years? 20 years?\n",
    "* How does the final balance change if you modify the number of compoundings?\n",
    "* How many individual compounding steps has the computer taken for a 20-year, 5-times-a-month compunding calculation?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1200"
      ],
      "text/latex": [
       "1200"
      ],
      "text/markdown": [
       "1200"
      ],
      "text/plain": [
       "[1] 1200"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#clear\n",
    "20*12*5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# To-do\n",
    "\n",
    "* computing slopes"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}
© 2019 GitHub, Inc.