206 lines
766 KiB
Plaintext
206 lines
766 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "markdown",
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"source": [
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"## Load Packages"
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],
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"metadata": {}
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"source": [
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"using Printf, Distributions\n",
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"include(\"jlFiles/printmat.jl\")"
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],
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"outputs": [
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{
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"output_type": "execute_result",
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"data": {
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"text/plain": [
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"printyellow (generic function with 1 method)"
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]
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},
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"metadata": {},
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"execution_count": 1
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}
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],
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"metadata": {
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"tags": []
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}
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"source": [
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"using Plots\n",
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"gr(size=(480,320))\n",
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"default(fmt = :svg)"
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],
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"outputs": [],
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"metadata": {
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"tags": []
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}
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},
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{
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"cell_type": "markdown",
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"source": [
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"# MV Frontier by Simulations\n",
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"\n",
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"We illustrate the mean-variance frontier (really, the $\\sigma \\times \\mu$ frontier) by calculating the portfolio mean and standard deviation for a large number of randomly drawn portfolios (in each case the portfolio weights should sum to one).\n",
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"\n",
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"As always, we have \n",
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"$\\text{E}R_p = w'\\mu$ and \n",
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"\n",
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"$\\text{Var}(R_p) = w'\\Sigma w$.\n",
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"\n",
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"The information about $\\mu$ and $\\Sigma$ is found below."
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],
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"metadata": {}
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"source": [
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"μ = [11.5; 9.5; 6]/100 #expected returns\n",
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"Σ = [166 34 58; #covariance matrix\n",
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" 34 64 4;\n",
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" 58 4 100]/100^2\n",
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"\n",
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"n = length(μ)"
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],
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"outputs": [
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{
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"output_type": "execute_result",
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"data": {
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"text/plain": [
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"3"
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]
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},
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"metadata": {},
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"execution_count": 3
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}
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],
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"metadata": {}
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"source": [
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"using Random\n",
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"Random.seed!(123)\n",
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"\n",
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"NSim = 2000\n",
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"(ERp,StdRp) = (fill(NaN,NSim),fill(NaN,NSim))\n",
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"\n",
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"for i = 1:NSim\n",
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" w = randn(n-1) #draw n-1 random portfolio weights\n",
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" w = [w;1-sum(w)] #make the n weights sum to 1\n",
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" ERp[i] = w'μ\n",
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" StdRp[i] = sqrt(w'Σ*w)\n",
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"end"
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],
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"outputs": [],
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"metadata": {}
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"source": [
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"p1 = scatter(StdRp,ERp,\n",
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" title = \"MV frontier by bootstrap\",\n",
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" xlabel = \"std\",\n",
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" ylabel = \"mean\",\n",
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" xlims = [0,0.2],\n",
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" ylims = [0.0,0.15],\n",
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" legend = false )\n",
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"display(p1)"
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],
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"outputs": [
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{
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"output_type": "display_data",
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"data": {
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},
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"metadata": {}
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}
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],
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"metadata": {}
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"source": [
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"NSim = 2000\n",
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"(ERp,StdRp) = (fill(NaN,NSim),fill(NaN,NSim))\n",
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"\n",
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"for i = 1:NSim\n",
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" w = rand(n-1) #draw n-1 random portfolio weights on [0,1]\n",
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" w = [w;1-sum(w)] #make the n weights sum to 1\n",
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" if all(w.>=0) #don't allow short sales\n",
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" ERp[i] = w'μ\n",
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" StdRp[i] = sqrt(w'Σ*w)\n",
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" end \n",
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"end"
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],
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"outputs": [],
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"metadata": {}
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"source": [
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"scatter!(p1,StdRp,ERp,\n",
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" title = \"MV frontier by bootstrap\",\n",
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" xlabel = \"std\",\n",
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" ylabel = \"mean\",\n",
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" xlims = [0,0.2],\n",
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" ylims = [0,0.15],\n",
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" legend = false,\n",
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" color= :red )"
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],
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"outputs": [
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{
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"output_type": "execute_result",
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"data": {
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