Tutorial_05_Finance.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Some Typical Financial Calculations\n",
    "\n",
    "This notebook (a) estimates CAPM equations and autocorrelations; (b) implements a simple trading strategy; (c) calculates Value at Risk using a simple model for time-varying volatility; (d) calculates the Black-Scholes option price and implied volatility; (e) calculates and draws the mean-variance frontier (w/w.o short selling restrictions)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load Packages and Extra Functions\n",
    "\n",
    "The [Roots](https://github.com/JuliaMath/Roots.jl) package solves non-linear equations and the [StatsBase](https://github.com/JuliaStats/StatsBase.jl) package has methods for estimating autocorrelations etc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "printyellow (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Printf, Dates, DelimitedFiles, LinearAlgebra, Roots, Distributions, StatsBase\n",
    "\n",
    "include(\"jlFiles/printmat.jl\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "\n",
    "#pyplot(size=(600,400))        #use pyplot or gr\n",
    "gr(size=(480,320))\n",
    "default(fmt = :svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Load Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1979-01-01    10.190     4.180\n",
      "1979-02-01    -2.820    -3.410\n",
      "1979-03-01    10.900     5.750\n",
      "1979-04-01     2.470     0.050\n",
      "\n"
     ]
    }
   ],
   "source": [
    "x   = readdlm(\"Data/MyData.csv\",',',skipstart=1)        #monthly return data\n",
    "ym  = round.(Int,x[:,1])     #yearmonth, like 200712\n",
    "Rme = x[:,2]                 #market excess return\n",
    "Rf  = x[:,3]                 #interest rate\n",
    "R   = x[:,4]                 #return small growth stocks\n",
    "Re  = R - Rf                 #excess returns\n",
    "T   = size(Rme,1)\n",
    "\n",
    "dN = Date.(string.(ym),\"yyyymm\")  #convert to string and then Julia Date\n",
    "printmat([dN[1:4] Re[1:4] Rme[1:4]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# CAPM\n",
    "\n",
    "The CAPM regression is\n",
    "\n",
    "$R_{it}^{e}  =\\alpha_{i}+\\beta_{i}R_{mt}^{e}+\\varepsilon_{it}$,\n",
    "\n",
    "where $R_{it}^{e}$ is the excess return of asset $i$ and $R_{mt}^{e}$ is the market excess return. Theory says that $\\alpha=0$, which is easily tested."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      coeff       std    t-stat\n",
      "α    -0.504     0.304    -1.656\n",
      "β     1.341     0.066    20.427\n",
      "\n",
      "      R2:      0.519\n",
      "no. of observations:    388    \n"
     ]
    }
   ],
   "source": [
    "x    = [ones(T) Rme]             #regressors\n",
    "y    = copy(Re)                  #to get standard OLS notation\n",
    "b    = x\\y                       #OLS\n",
    "u    = y - x*b                   #residuals\n",
    "covb = inv(x'x)*var(u)           #cov(b), see any textbook\n",
    "stdb = sqrt.(diag(covb))         #std(b)\n",
    "R2   = 1 - var(u)/var(y)\n",
    "\n",
    "printmat([b stdb b./stdb],colNames=[\"coeff\",\"std\",\"t-stat\"],rowNames=[\"α\",\"β\"])\n",
    "printlnPs(\"R2: \",R2)\n",
    "printlnPs(\"no. of observations: \",T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Return Autocorrelation\n",
    "\n",
    "That is, the correlation of $R_{t}^{e}$ and $R_{t-s}^{e}$. \n",
    "\n",
    "It can be shown that the t-stat of an autocorrelation is $\\sqrt{T}$ times the autocorrelation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Autocorrelations (different lags) of the excess returns in Re\n",
      "lag  autocorr    t-stat\n",
      "1       0.216     4.253\n",
      "2       0.002     0.046\n",
      "3      -0.018    -0.359\n",
      "4      -0.065    -1.289\n",
      "5      -0.027    -0.536\n",
      "\n"
     ]
    }
   ],
   "source": [
    "plags = 1:5\n",
    "xCorr = autocor(Re,plags)         #using the StatsBase package\n",
    "\n",
    "println(\"Autocorrelations (different lags) of the excess returns in Re\")\n",
    "printmat([xCorr sqrt(T)*xCorr],colNames=[\"autocorr\",\"t-stat\"],rowNames=string.(plags),cell00=\"lag\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A Trading Strategy\n",
    "\n",
    "The next cell implements a very simple momentum trading strategy. \n",
    "\n",
    "1. If $R_{t-1}^{e}\\ge0$, then we hold the market index and shorten the riskfree from $t-1$ to $t$. This means that we will earn $R_{t}^{e}$.\n",
    "\n",
    "2. Instead,  if $R_{t-1}^{e}<0$, then we do the opposite. This means that we will earn $-R_{t}^{e}$. \n",
    "\n",
    "This simple strategy could be coded without using a loop, but \"vectorization\" does not speed up much. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The annualized mean excess return of the strategy and a passive portfolio are:     26.733     3.329\n",
      "The annualized Sharpe ratios are:      0.932     0.112\n"
     ]
    }
   ],
   "source": [
    "(w,Rp)  = (fill(NaN,T),fill(NaN,T))\n",
    "for t = 2:T\n",
    "  w[t]  = (Re[t-1] < 0)*(-1) + (Re[t-1] >= 0)*1       #w is -1 or 1\n",
    "  Rp[t] = w[t]*Re[t] \n",
    "end\n",
    "\n",
    "μ = [mean(Rp[2:end]) mean(Re[2:end])]\n",
    "σ = [std(Rp[2:end])  std(Re[2:end])]\n",
    "\n",
    "printlnPs(\"The annualized mean excess return of the strategy and a passive portfolio are: \",μ*12)\n",
    "printlnPs(\"The annualized Sharpe ratios are: \",sqrt(12)*μ./σ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Value at Risk\n",
    "\n",
    "The next cell constructs an simple estimate of $\\sigma_t^2$ as a backward looking moving average (the RiskMetrics approach):\n",
    "\n",
    "$\\sigma_t^2 = \\lambda \\sigma_{t-1}^2 + (1-\\lambda) (R_{t-1} -\\mu)^2$,\n",
    "where $\\mu$ is the average return (for all data).\n",
    "\n",
    "Then, we calculate the 95% VaR by assuming a $N(\\mu,\\sigma_t^2)$ distribution:\n",
    "\n",
    "$\\textrm{VaR}_{t} = - (\\mu-1.64\\sigma_t)$.\n",
    "\n",
    "If the model is correct, then $-R_t > \\text{VaR}_{t}$ should only happen 5% of the times."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "μ = mean(Rme)\n",
    "\n",
    "λ = 0.95                         #weight on old volatility\n",
    "σ² = fill(var(Rme),T)            #RiskMetrics approach to estimate variance\n",
    "for t = 2:T\n",
    "  σ²[t] = λ*σ²[t-1] + (1-λ)*(Rme[t-1]-μ)^2\n",
    "end\n",
    "\n",
    "VaR95 = -(μ .- 1.64*sqrt.(σ²));      #VaR at 95% level"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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       "  356.645,451.699 360.592,469.275 364.671,488.285 368.619,442.39 372.698,439.801 376.776,434.81 380.461,454.855 384.54,468.056 388.487,480.797 392.566,499.56 \n",
       "  396.514,510.439 400.592,524.558 404.671,493.919 408.619,457.077 412.698,462.349 416.645,474.837 420.724,478.821 424.803,484.392 428.487,467.128 432.566,480.869 \n",
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       "  516.778,555.951 520.857,566.382 524.673,557.841 528.752,574.782 532.699,590.003 536.778,564.68 540.725,580.529 544.804,585.297 548.883,513.592 552.831,529.887 \n",
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       "  596.91,628.213 600.857,616.025 604.936,621.198 608.884,603.871 612.963,610.35 617.042,625.928 620.726,603.488 624.805,601.962 628.752,614.147 632.831,613.678 \n",
       "  636.779,629.115 640.858,593.923 644.937,580.211 648.884,523.496 652.963,528.159 656.911,545.612 660.99,550.226 665.069,448.969 668.753,457.42 672.832,475.539 \n",
       "  676.779,488.346 680.858,506.858 684.806,515.782 688.885,523.961 692.964,535.609 696.911,544.316 700.99,164.687 704.937,152.975 709.016,159.193 713.095,178.689 \n",
       "  716.911,195.232 720.99,216.948 724.938,242.519 729.017,266.716 732.964,280.386 737.043,302.118 741.122,314.845 745.069,333.679 749.148,356.086 753.096,372.488 \n",
       "  757.175,393.559 761.254,392.65 764.938,407.712 769.017,427.867 772.964,438.329 777.043,453.214 780.991,470.19 785.07,456.231 789.149,475.134 793.096,492.54 \n",
       "  797.175,496.215 801.122,514.513 805.201,532.216 809.28,492.129 812.965,510.648 817.044,527.595 820.991,530.306 825.07,497.933 829.017,514.059 833.096,527.8 \n",
       "  837.175,454.12 841.123,439.482 845.202,454.356 849.149,450.737 853.228,467.937 857.307,475.448 860.991,460.212 865.07,476.85 869.018,495.306 873.097,506.204 \n",
       "  877.044,499.444 881.123,506.905 885.202,522.869 889.15,536.566 893.229,553.526 897.176,550.229 901.255,485.685 905.334,503.505 909.15,521.64 913.229,529.86 \n",
       "  917.176,547.353 921.255,564.505 925.203,573.66 929.282,581.213 933.361,589.252 937.308,605.258 941.387,620.926 945.334,625.894 949.413,640.274 953.492,654.933 \n",
       "  957.177,669.328 961.255,680.378 965.203,681.313 969.282,689.924 973.229,703.408 977.308,715.65 981.387,717.014 985.335,729.138 989.414,740.519 993.361,744.161 \n",
       "  997.44,754.771 1001.52,759.797 1005.2,757.878 1009.28,731.785 1013.23,744.313 1017.31,756.532 1021.26,750.608 1025.34,756.493 1029.41,754.279 1033.36,755.922 \n",
       "  1037.44,767.557 1041.39,750.219 1045.47,762.227 1049.55,772.602 1053.23,772.244 1057.31,780.088 1061.26,788.442 1065.34,792.348 1069.28,797.369 1073.36,795.129 \n",
       "  1077.44,806.032 1081.39,806.724 1085.47,810.225 1089.41,805.27 1093.49,815.675 1097.57,821.318 1101.39,830.961 1105.47,840.971 1109.41,847.179 1113.49,852.303 \n",
       "  1117.44,856.239 1121.52,801.643 1125.6,805.124 1129.55,789.8 1133.63,800.695 1137.57,768.629 1141.65,773.679 1145.73,760.411 1149.42,770.602 1153.49,742.247 \n",
       "  1157.44,741.448 1161.52,708.563 1165.47,707.652 1169.55,670.647 1173.63,661.076 1177.57,650.471 1181.65,643.912 1185.6,654.201 1189.68,668.171 1193.76,681.94 \n",
       "  1197.44,651.575 1201.52,647.867 1205.47,662.525 1209.55,662.884 1213.49,671.941 1217.57,673.577 1221.65,430.239 1225.6,428.565 1229.68,416.25 1233.63,415.535 \n",
       "  1237.71,414.539 1241.78,428.682 1245.47,431.364 1249.55,445.563 1253.5,453.06 1257.57,465.374 1261.52,471.151 1265.6,476.853 1269.68,492.551 1273.63,502.105 \n",
       "  1277.71,497.391 1281.65,509.57 1285.73,482.362 1289.81,480.784 1293.63,495.826 1297.71,498.883 1301.65,476.356 1305.73,474.846 1309.68,479.68 1313.76,492.119 \n",
       "  1317.84,475.825 1321.79,463.356 1325.86,472.045 1329.81,391.228 1333.89,411.73 1337.97,426.357 1341.65,358.608 1345.73,334.997 1349.68,320.194 1353.76,343.139 \n",
       "  1357.71,360.639 1361.79,377.214 1365.86,365.324 1369.81,317.166 1373.89,337.574 1377.84,325.337 1381.92,347.419 1386,365.805 1389.68,381.528 1393.76,392.682 \n",
       "  1397.71,389.756 1401.79,408.563 1405.73,384.595 1409.81,349.669 1413.89,371.878 1417.84,313.187 1421.92,305.749 1425.86,309.473 1429.94,308.209 1434.02,325.207 \n",
       "  1437.71,344.598 1441.79,366.862 1445.73,347.957 1449.81,347.777 1453.76,369.426 1457.84,389.187 1461.92,407.912 1465.87,426.721 1469.94,424.905 1473.89,444.453 \n",
       "  1477.97,452.424 1482.05,469.901 1485.87,488.418 1489.94,504.499 1493.89,514.527 1497.97,532.074 1501.92,547.708 1506,546.781 1510.08,563.822 1514.02,578.929 \n",
       "  1518.1,594.238 1522.05,594.144 1526.13,602.708 1530.21,606.917 1533.89,620.331 1537.97,629.284 1541.92,632.937 1546,638.855 1549.95,653.637 1554.02,655.061 \n",
       "  1558.1,667.117 1562.05,681.259 1566.13,685.201 1570.08,687.763 1574.16,701.036 1578.23,703.441 1581.92,715.25 1586,727.133 1589.95,739.644 1594.02,730.599 \n",
       "  1597.97,741.802 1602.05,752.261 1606.13,761.377 1610.08,772.01 1614.16,773.652 1618.1,782.633 1622.18,793.873 1626.26,803.677 1629.95,806.111 1634.02,816.659 \n",
       "  1637.97,814.037 1642.05,812.159 1646,813.569 1650.08,798.564 1654.16,809.377 1658.1,805.215 1662.18,811.843 1666.13,773.468 1670.21,782.623 1674.29,729.07 \n",
       "  1678.1,731.018 1682.18,739.46 1686.13,728.291 1690.21,737.69 1694.16,662.989 1698.24,672.537 1702.31,686.404 1706.26,583.873 1710.34,310.937 1714.29,278.666 \n",
       "  1718.37,301.278 1722.45,278.855 1726.13,230.061 1730.21,214.593 1734.16,175.205 1738.24,179.789 1742.18,205.839 1746.26,196.745 1750.34,218.811 1754.29,234.879 \n",
       "  1758.37,252.574 1762.32,260.484 1766.39,281.52 1770.47,293.266 1774.16,311.193 1778.24,311.447 1782.18,333.257 1786.26,305.665 1790.21,306.395 1794.29,301.52 \n",
       "  1798.37,308.235 1802.32,282.17 1806.39,299.02 1810.34,322.513 1814.42,319.541 1818.5,341.153 1822.18,356.179 1826.26,378.152 \n",
       "  \"/>\n",
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xTicksLoc = [Date(1980);Date(1990);Date(2000);Date(2010)]\n",
    "xTicksLab = Dates.format.(xTicksLoc,\"Y\")\n",
    "\n",
    "p1 = plot( dN,VaR95,\n",
    "           color = :blue,\n",
    "           legend = false,\n",
    "           xticks = (xTicksLoc,xTicksLab),\n",
    "           ylim = (0,11),\n",
    "           title = \"1-month Value at Risk (95%)\",\n",
    "           ylabel = \"%\",\n",
    "           annotation = (Date(1982),1,text(\"(for US equity market)\",8,:left)) )\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Options"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Black-Scholes Option Price\n",
    "\n",
    "Let $S$ be the the current spot price of an asset and $y$ be the interest rate.\n",
    "\n",
    "The Black-Scholes formula for a European call option with strike price $K$ and time to expiration $m$ is\n",
    "\n",
    "$C  =S\\Phi(d_{1})  -e^{-ym}K\\Phi(d_{2})$, where\n",
    "\n",
    "$d_{1} =\\frac{\\ln(S/K)+(y+\\sigma^{2}/2)m}{\\sigma\\sqrt{m}} \\ \\text{ and } \\ d_{2}=d_{1}-\\sigma\\sqrt{m}$ \n",
    "\n",
    "and where $\\Phi(d)$ denotes the probability of $x\\leq d$ when $x$ has an $N(0,1)$ distribution. All variables except the volatility ($\\sigma$) are directly observable. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OptionBlackSPs"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Φ(x) = cdf(Normal(0,1),x)     #a one-line function, Pr(z<=x) for N(0,1)\n",
    "\n",
    "\"\"\"\n",
    "Calculate Black-Scholes european call option price\n",
    "\"\"\"\n",
    "function OptionBlackSPs(S,K,m,y,σ)\n",
    "\n",
    "  d1 = ( log(S/K) + (y+1/2*σ^2)*m ) / (σ*sqrt(m))\n",
    "  d2 = d1 - σ*sqrt(m)\n",
    "  c  = S*Φ(d1) - exp(-y*m)*K*Φ(d2)\n",
    "  return c\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "         \n",
      "call price according to Black-Scholes:      1.358\n"
     ]
    }
   ],
   "source": [
    "σ = 0.4\n",
    "c1 = OptionBlackSPs(10,10,0.5,0.1,σ)\n",
    "printlnPs(\"\\n\",\"call price according to Black-Scholes: \",c1)\n",
    "\n",
    "K = range(7,stop=13,length=51)\n",
    "c = OptionBlackSPs.(10,K,0.5,0.1,σ);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = plot( K,c,\n",
    "           color = :red,\n",
    "           legend = false,\n",
    "           title = \"Black-Scholes call option price\",\n",
    "           xlabel = \"strike price\",\n",
    "           ylabel = \"option price\" )\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Implied Volatility\n",
    "\n",
    "is the $\\sigma$ value that makes the Black-Scholes equation give the same option price as observed on the market. It is often interpreted as the \"market uncertainty.\"\n",
    "\n",
    "The next cell uses the call option price calculated above as the market price. The implied volatility should then equal the volatility used above (this is a way to check your coding).\n",
    "\n",
    "The next few cells instead use some data on options on German government bonds. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Implied volatility:      0.400, compare with: 0.4\n"
     ]
    }
   ],
   "source": [
    "                                #solve for implied vol\n",
    "iv = find_zero(σ->OptionBlackSPs(10,10,0.5,0.1,σ)-c1,(0.00001,5))\n",
    "\n",
    "printlnPs(\"Implied volatility: \",iv,\", compare with: $σ\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "21"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#  LIFFE Bunds option data, trade date April 6, 1994\n",
    "K = [                        #strike prices; Mx1 vector\n",
    "      92.00;  94.00;  94.50;  95.00;  95.50;  96.00;  96.50;  97.00;\n",
    "      97.50;  98.00;  98.50;  99.00;  99.50;  100.0;  100.5;  101.0;\n",
    "     101.5;  102.0;  102.5;  103.0;  103.5 ];\n",
    "C = [                        #call prices; Mx1 vector\n",
    "      5.13;    3.25;    2.83;    2.40;    2.00;    1.64;    1.31;    1.02;\n",
    "      0.770;   0.570;   0.400;   0.280;   0.190;   0.130;  0.0800;  0.0500;\n",
    "      0.0400;  0.0300;  0.0200;  0.0100;  0.0100 ];\n",
    "S = 97.05                #spot price\n",
    "m = 48/365               #time to expiry in years\n",
    "y = 0.0                  #Interest rate: LIFFE=>no discounting\n",
    "N = length(K)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Strike and iv for data: \n",
      "    92.000     0.094\n",
      "    94.000     0.081\n",
      "    94.500     0.081\n",
      "    95.000     0.078\n",
      "    95.500     0.075\n",
      "    96.000     0.074\n",
      "    96.500     0.072\n",
      "    97.000     0.071\n",
      "    97.500     0.070\n",
      "    98.000     0.069\n",
      "    98.500     0.068\n",
      "    99.000     0.067\n",
      "    99.500     0.067\n",
      "   100.000     0.068\n",
      "   100.500     0.067\n",
      "   101.000     0.067\n",
      "   101.500     0.070\n",
      "   102.000     0.073\n",
      "   102.500     0.074\n",
      "   103.000     0.072\n",
      "   103.500     0.077\n",
      "\n"
     ]
    }
   ],
   "source": [
    "iv = fill(NaN,N)       #looping over strikes\n",
    "for i = 1:N\n",
    "  iv[i] = find_zero(sigma->OptionBlackSPs(S,K[i],m,y,sigma)-C[i],(0.00001,5))\n",
    "end\n",
    "\n",
    "println(\"Strike and iv for data: \")\n",
    "printmat([K iv])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = plot( K,iv,\n",
    "           color = :red,\n",
    "           legend =false,\n",
    "           title = \"Implied volatility\",\n",
    "           xlabel = \"strike price\",\n",
    "           annotation = (98,0.09,text(\"Bunds options April 6, 1994\",8,:left)) )\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mean-Variance Frontier\n",
    "\n",
    "Given a vector of average returns ($\\mu$) and a variance-covariance matrix ($\\Sigma$), the mean-variance frontier shows the lowest possible portfolio uncertainty for a given expected portfolio return (denoted $\\mu\\text{star}$ below).\n",
    "\n",
    "It is thus the solution to a quadratic minimization problem. The cells below will use the explicit (matrix) formulas for this solution, but we often have to resort to numerical methods when there are portfolio restrictions.\n",
    "\n",
    "It is typically plotted with the portfolio standard deviation on the horizontal axis and the portfolio expected return on the vertical axis.\n",
    "\n",
    "We calculate and plot two different mean-variance frontiers: (1) when we only consider risky assets; (2) when we also consider a risk-free asset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "μ: \n",
      "     0.115\n",
      "     0.095\n",
      "     0.060\n",
      "\n",
      "Σ: \n",
      "     0.017     0.003     0.006\n",
      "     0.003     0.006     0.000\n",
      "     0.006     0.000     0.010\n",
      "\n",
      "Rf: \n",
      "     0.030\n",
      "\n"
     ]
    }
   ],
   "source": [
    "μ = [11.5; 9.5; 6]/100          #expected returns\n",
    "Σ  = [166  34  58;              #covariance matrix\n",
    "       34  64   4;\n",
    "       58   4 100]/100^2\n",
    "Rf = 0.03                       #riskfree return (an interest rate)\n",
    "\n",
    "println(\"μ: \")\n",
    "printmat(μ)\n",
    "println(\"Σ: \")\n",
    "printmat(Σ)\n",
    "println(\"Rf: \")\n",
    "printmat(Rf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MVCalcRf (generic function with 1 method)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function MVCalc(μstar,μ,Σ)           #calculate the std of a portfolio on MVF of risky assets\n",
    "    n    = length(μ)                  #μstar is a scalar, μ a vector and Σ a matrix\n",
    "    oneV = ones(n) \n",
    "    Σ_1  = inv(Σ)\n",
    "    A    = μ'Σ_1*μ\n",
    "    B    = μ'Σ_1*oneV\n",
    "    C    = oneV'Σ_1*oneV\n",
    "    λ    = (C*μstar - B)/(A*C-B^2)\n",
    "    δ    = (A-B*μstar)/(A*C-B^2)\n",
    "    w    = Σ_1 *(μ*λ + δ*oneV)\n",
    "    StdRp = sqrt(w'Σ*w)\n",
    "    return StdRp,w\n",
    "end\n",
    "\n",
    "function MVCalcRf(μstar,μ,Σ,Rf)           #calculates the std of a portfolio on MVF of (Risky,Riskfree)\n",
    "    n     = length(μ)\n",
    "    μe    = μ .- Rf                        #expected excess returns\n",
    "    Σ_1   = inv(Σ)\n",
    "    w     = (μstar - Rf)/(μe'Σ_1*μe) * Σ_1 *μe\n",
    "    StdRp = sqrt(w'Σ*w)\n",
    "    return StdRp,w\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "μstar = range(Rf,stop=0.15,length=201)\n",
    "L     = length(μstar)\n",
    "\n",
    "StdRp  = fill(NaN,L)                      #risky assets only \n",
    "for i = 1:L                               #loop over different expected portfolio returns\n",
    "    StdRp[i] = MVCalc(μstar[i],μ,Σ)[1]    #[1] to get the first output\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
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       "  1078.83,438.003 1087.59,434.071 1096.5,430.139 1105.57,426.208 1114.78,422.276 1124.14,418.344 1133.63,414.413 1143.25,410.481 1153.01,406.55 1162.89,402.618 \n",
       "  1172.9,398.686 1183.02,394.755 1193.25,390.823 1203.6,386.891 1214.06,382.96 1224.62,379.028 1235.28,375.097 1246.04,371.165 1256.9,367.233 1267.85,363.302 \n",
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       "  1514.91,280.737 1527.32,276.806 1539.78,272.874 1552.29,268.942 1564.84,265.011 1577.44,261.079 1590.08,257.148 1602.75,253.216 1615.47,249.284 1628.23,245.353 \n",
       "  1641.02,241.421 1653.85,237.489 1666.72,233.558 1679.62,229.626 1692.56,225.695 1705.52,221.763 1718.52,217.831 1731.56,213.9 1744.62,209.968 1757.71,206.036 \n",
       "  1770.83,202.105 1783.98,198.173 1797.16,194.241 1810.36,190.31 1823.59,186.378 1836.85,182.447 1850.13,178.515 1863.44,174.583 1876.77,170.652 1890.12,166.72 \n",
       "  1903.5,162.788 1916.89,158.857 1930.31,154.925 1943.76,150.994 1957.22,147.062 1970.7,143.13 1984.2,139.199 1997.72,135.267 2011.26,131.335 2024.82,127.404 \n",
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     "metadata": {},
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   ],
   "source": [
    "p1 = plot( StdRp*100,μstar*100,\n",
    "           linecolor = :red,\n",
    "           xlim = (0,15),\n",
    "           ylim = (0,15),\n",
    "           label = \"MVF\",\n",
    "           legend = :topleft,\n",
    "           title = \"MVF, only risky assets\",\n",
    "           xlabel = \"Std(Rp), %\",\n",
    "           ylabel = \"ERp, %\" )\n",
    "scatter!(sqrt.(diag(Σ))*100,μ*100,color=:red,label=\"assets\")\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "StdRpRf  = fill(NaN,L)       #with riskfree too\n",
    "for i = 1:L\n",
    "    StdRpRf[i] = MVCalcRf(μstar[i],μ,Σ,Rf)[1]\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
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       "  1505.45,870.482 1493.12,866.55 1480.84,862.619 1468.62,858.687 1456.45,854.755 1444.33,850.824 1432.27,846.892 1420.27,842.961 1408.32,839.029 1396.44,835.097 \n",
       "  1384.63,831.166 1372.88,827.234 1361.2,823.302 1349.58,819.371 1338.04,815.439 1326.58,811.508 1315.19,807.576 1303.88,803.644 1292.65,799.713 1281.5,795.781 \n",
       "  1270.44,791.849 1259.47,787.918 1248.6,783.986 1237.81,780.054 1227.13,776.123 1216.54,772.191 1206.06,768.26 1195.69,764.328 1185.42,760.396 1175.27,756.465 \n",
       "  1165.24,752.533 1155.33,748.601 1145.54,744.67 1135.89,740.738 1126.36,736.807 1116.97,732.875 1107.73,728.943 1098.63,725.012 1089.68,721.08 1080.88,717.148 \n",
       "  1072.25,713.217 1063.77,709.285 1055.47,705.353 1047.34,701.422 1039.39,697.49 1031.62,693.559 1024.04,689.627 1016.65,685.695 1009.47,681.764 1002.49,677.832 \n",
       "  995.717,673.9 989.161,669.969 982.826,666.037 976.717,662.106 970.84,658.174 965.199,654.242 959.801,650.311 954.649,646.379 949.749,642.447 945.107,638.516 \n",
       "  940.726,634.584 936.612,630.653 932.769,626.721 929.201,622.789 925.912,618.858 922.906,614.926 920.186,610.994 917.756,607.063 915.619,603.131 913.777,599.199 \n",
       "  912.232,595.268 910.987,591.336 910.043,587.405 909.401,583.473 909.062,579.541 909.026,575.61 909.293,571.678 909.864,567.746 910.737,563.815 911.912,559.883 \n",
       "  913.386,555.952 915.158,552.02 917.226,548.088 919.587,544.157 922.238,540.225 925.177,536.293 928.4,532.362 931.902,528.43 935.681,524.498 939.732,520.567 \n",
       "  944.05,516.635 948.631,512.704 953.471,508.772 958.563,504.84 963.904,500.909 969.488,496.977 975.31,493.045 981.365,489.114 987.647,485.182 994.151,481.251 \n",
       "  1000.87,477.319 1007.8,473.387 1014.94,469.456 1022.28,465.524 1029.81,461.592 1037.54,457.661 1045.45,453.729 1053.54,449.797 1061.8,445.866 1070.23,441.934 \n",
       "  1078.83,438.003 1087.59,434.071 1096.5,430.139 1105.57,426.208 1114.78,422.276 1124.14,418.344 1133.63,414.413 1143.25,410.481 1153.01,406.55 1162.89,402.618 \n",
       "  1172.9,398.686 1183.02,394.755 1193.25,390.823 1203.6,386.891 1214.06,382.96 1224.62,379.028 1235.28,375.097 1246.04,371.165 1256.9,367.233 1267.85,363.302 \n",
       "  1278.89,359.37 1290.01,355.438 1301.22,351.507 1312.51,347.575 1323.89,343.643 1335.33,339.712 1346.86,335.78 1358.45,331.849 1370.12,327.917 1381.85,323.985 \n",
       "  1393.65,320.054 1405.52,316.122 1417.44,312.19 1429.43,308.259 1441.48,304.327 1453.58,300.396 1465.74,296.464 1477.96,292.532 1490.22,288.601 1502.54,284.669 \n",
       "  1514.91,280.737 1527.32,276.806 1539.78,272.874 1552.29,268.942 1564.84,265.011 1577.44,261.079 1590.08,257.148 1602.75,253.216 1615.47,249.284 1628.23,245.353 \n",
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = plot( [StdRp StdRpRf]*100,[μstar μstar]*100,\n",
    "           legend = nothing,\n",
    "           linestyle = [:solid :dash],\n",
    "           linecolor = [:red :blue],\n",
    "           xlim = (0,15),\n",
    "           ylim = (0,15), \n",
    "           title = \"MVF, risky and riskfree assets\",\n",
    "           xlabel = \"Std(Rp), %\",\n",
    "           ylabel = \"ERp, %\" )\n",
    "scatter!(sqrt.(diag(Σ))*100,μ*100,color=:red)\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mean-Variance Frontier without Short Selling (extra)\n",
    "\n",
    "The code below solves (numerically) the following minimization problem \n",
    "\n",
    "$\\min \\text{Var}(R_p) \\: \\text{ s.t. } \\: \\text{E}R_p = \\mu^*$,\n",
    " \n",
    "and where we also require $w_i\\ge 0$ and $\\sum_{i=1}^{n}w_{i}=1$.\n",
    "\n",
    "To solve this, we use the [JuMP](https://github.com/JuliaOpt/JuMP.jl) package (version >= 0.19) together with [Ipopt](https://github.com/JuliaOpt/Ipopt.jl)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "using JuMP, Ipopt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MeanVarNoSSPs (generic function with 1 method)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function MeanVarNoSSPs(μ,Σ,μstar)   #MV with no short-sales, numerical minimization\n",
    "    n = length(μ)\n",
    "    if minimum(μ) <= μstar <= maximum(μ)      #try only if feasible\n",
    "        model = Model(optimizer_with_attributes(Ipopt.Optimizer,\"print_level\" => 1))\n",
    "        @variable(model,w[i=1:n] >= 0.0)       #no short sales\n",
    "        @objective(model,Min,w'*Σ*w)           #minimize portfolio variance \n",
    "        @constraint(model,sum(w) == 1.0)       #w sums to 1\n",
    "        @constraint(model,w'*μ == μstar)       #mean equals μstar\n",
    "        optimize!(model)\n",
    "        if has_values(model)\n",
    "            w_p = value.(w)\n",
    "            StdRp = sqrt(w_p'Σ*w_p)\n",
    "        end\n",
    "    else\n",
    "        (w_p,StdRp) = (NaN,NaN)\n",
    "    end\n",
    "    return StdRp,w_p\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "******************************************************************************\n",
      "This program contains Ipopt, a library for large-scale nonlinear optimization.\n",
      " Ipopt is released as open source code under the Eclipse Public License (EPL).\n",
      "         For more information visit https://github.com/coin-or/Ipopt\n",
      "******************************************************************************\n",
      "\n"
     ]
    }
   ],
   "source": [
    "Std_no_ss = fill(NaN,length(μstar))\n",
    "for i = 1:length(μstar)                   #risky assets only, no short sales\n",
    "    Std_no_ss[i] = MeanVarNoSSPs(μ,Σ,μstar[i])[1]\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = plot( [StdRp Std_no_ss]*100,[μstar μstar]*100,\n",
    "           linecolor = [:red :green],\n",
    "           linestyle = [:solid :dash],\n",
    "           linewidth = 2, \n",
    "           label = [\"no constraints\" \"no short sales\"],\n",
    "           xlim = (0,15),\n",
    "           ylim = (0,15),\n",
    "           legend = :topleft,\n",
    "           title = \"MVF (with/without constraints)\",\n",
    "           xlabel = \"Std(Rp), %\",\n",
    "           ylabel = \"ERp, %\" )\n",
    "scatter!(sqrt.(diag(Σ))*100,μ*100,color=:red,label=\"assets\")\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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