421 lines
53 KiB
Plaintext
421 lines
53 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Mean Variance Frontier with Short Sales Constraints"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Load Packages and Utility Functions"
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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": 1,
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"metadata": {},
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"outputs": [
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{
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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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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"using Printf, LinearAlgebra, Optim\n",
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"\n",
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"include(\"jlFiles/printmat.jl\")"
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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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"metadata": {},
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"outputs": [],
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"source": [
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"using Plots\n",
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"\n",
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"#pyplot(size=(600,400)) #use pyplot or gr\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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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Traditional MV Calculations \n",
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"\n",
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"(no constraints)"
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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": 3,
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"metadata": {},
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"outputs": [
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{
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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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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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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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"assetNames = [\"A\",\"B\",\"C\"];\n",
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"n = length(μ)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## A Function for Calculating the (traditional) Mean-Variance Frontier\n"
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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": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"MVCalc"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"\"\"\"\n",
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" MVCalc(μstar,μ,Σ)\n",
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"\n",
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"Calculate the std and weights of a portfolio (with mean return μstar) on MVF of risky assets.\n",
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"\n",
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"# Remark\n",
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"- Only (λ,δ) and thus (w,stdRp) depend on μstar. We could therefore speed up the computations a bit\n",
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"by doing the loop over different μstar values inside the function (and thus not recalculate Σ_1,a,b,c).\n",
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"\"\"\"\n",
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"function MVCalc(μstar,μ,Σ)\n",
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" n = length(μ)\n",
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" Σ_1 = inv(Σ)\n",
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" a = μ'Σ_1*μ\n",
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" b = μ'Σ_1*ones(n)\n",
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" c = ones(n)'Σ_1*ones(n)\n",
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" λ = (c*μstar - b)/(a*c-b^2)\n",
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" δ = (a-b*μstar)/(a*c-b^2)\n",
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" w = Σ_1 *(μ*λ.+δ)\n",
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" StdRp = sqrt(w'Σ*w)\n",
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" return StdRp,w\n",
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"end"
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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": 5,
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"metadata": {},
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"outputs": [
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{
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" 1425.43,844.271 1414.49,840.667 1403.6,837.063 1392.77,833.459 1381.99,829.855 1371.26,826.251 1360.59,822.647 1349.99,819.043 1339.44,815.439 1328.95,811.835 \n",
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" 1318.53,808.231 1308.17,804.627 1297.88,801.023 1287.66,797.419 1277.52,793.815 1267.44,790.211 1257.44,786.607 1247.52,783.003 1237.68,779.399 1227.92,775.795 \n",
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" 1126.74,736.151 1118.17,732.547 1109.72,728.943 1101.4,725.339 1093.19,721.735 1085.12,718.131 1077.18,714.527 1069.38,710.923 1061.72,707.319 1054.2,703.715 \n",
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" 1046.82,700.111 1039.6,696.507 1032.54,692.903 1025.63,689.299 1018.88,685.695 1012.31,682.091 1005.9,678.487 999.669,674.883 993.618,671.279 987.75,667.675 \n",
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" 982.069,664.071 976.58,660.467 971.287,656.863 966.194,653.259 961.305,649.655 956.623,646.051 952.154,642.447 947.899,638.843 943.864,635.239 940.051,631.635 \n",
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" 936.465,628.031 933.108,624.427 929.983,620.823 927.094,617.219 924.444,613.615 922.034,610.011 919.868,606.407 917.948,602.803 916.275,599.199 914.851,595.595 \n",
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" 913.678,591.991 912.758,588.387 912.09,584.783 911.676,581.179 911.516,577.575 911.61,573.971 911.959,570.368 912.562,566.764 913.417,563.16 914.525,559.556 \n",
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" 915.885,555.952 917.493,552.348 919.35,548.744 921.454,545.14 923.801,541.536 926.389,537.932 929.217,534.328 932.281,530.724 935.579,527.12 939.107,523.516 \n",
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" 942.861,519.912 946.84,516.308 951.038,512.704 955.453,509.1 960.08,505.496 964.917,501.892 969.958,498.288 975.2,494.684 980.639,491.08 986.27,487.476 \n",
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" 992.09,483.872 998.095,480.268 1004.28,476.664 1010.64,473.06 1017.18,469.456 1023.88,465.852 1030.74,462.248 1037.77,458.644 1044.95,455.04 1052.29,451.436 \n",
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" 1141.94,411.792 1150.81,408.188 1159.8,404.584 1168.88,400.98 1178.07,397.376 1187.36,393.772 1196.74,390.168 1206.21,386.564 1215.77,382.96 1225.43,379.356 \n",
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" 1235.16,375.752 1244.98,372.148 1254.88,368.544 1264.86,364.94 1274.92,361.336 1285.05,357.732 1295.25,354.128 1305.52,350.524 1315.86,346.92 1326.26,343.316 \n",
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" 1336.73,339.712 1347.27,336.108 1357.86,332.504 1368.51,328.9 1379.22,325.296 1389.99,321.692 1400.81,318.088 1411.68,314.484 1422.61,310.88 1433.59,307.276 \n",
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" 1444.61,303.672 1455.68,300.068 1466.8,296.464 1477.97,292.86 1489.18,289.256 1500.43,285.652 1511.72,282.048 1523.06,278.444 1534.43,274.84 1545.84,271.236 \n",
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" 1557.29,267.632 1568.78,264.028 1580.31,260.424 1591.86,256.82 1603.46,253.216 1615.08,249.612 1626.74,246.008 1638.43,242.404 1650.15,238.8 1661.9,235.196 \n",
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" 1673.69,231.592 1685.5,227.988 1697.33,224.384 1709.2,220.78 1721.09,217.176 1733.01,213.572 1744.95,209.968 1756.92,206.364 1768.91,202.76 1780.93,199.156 \n",
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" 1792.97,195.552 1805.04,191.948 1817.12,188.344 1829.23,184.74 1841.36,181.136 1853.51,177.532 1865.68,173.928 1877.87,170.324 1890.08,166.72 1902.3,163.116 \n",
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" 2037.97,123.472 \n",
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]
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|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"μstar_range = range(0.04,0.15,length=201)\n",
|
|
"L = length(μstar_range)\n",
|
|
"\n",
|
|
"StdRp = fill(NaN,L)\n",
|
|
"for i = 1:L\n",
|
|
" StdRp[i] = MVCalc(μstar_range[i],μ,Σ)[1]\n",
|
|
"end\n",
|
|
"\n",
|
|
"p1 = plot( StdRp*100,μstar_range*100,\n",
|
|
" linecolor = :red,\n",
|
|
" linewidth = 2,\n",
|
|
" label = \"MVF\",\n",
|
|
" legend = :topleft,\n",
|
|
" xlim = (0,15),\n",
|
|
" ylim = (0,15),\n",
|
|
" title = \"MVF (no portfolio 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": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# The Maximization Problem\n",
|
|
"\n",
|
|
"The investor maximizes:\n",
|
|
"\n",
|
|
"$\\text{E}R_{p}-\\frac{k}{2} \\text{Var}(R_{p})$,\n",
|
|
"\n",
|
|
"where $\\text{E}R_{p}= w'\\mu$ and $\\text{Var}(R_{p}=w'\\Sigma w$\n",
|
|
"\n",
|
|
"subject to $\\Sigma_{i=1}^n w_{i} = 1$.\n",
|
|
"\n",
|
|
"We can trace out the mean-variance frontier by solving this problem for different values of $k$.\n",
|
|
"\n",
|
|
"To impose the restriction, optimise over `v` and let `w=[v;1-sum(v)]`"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 6,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"objfun1 (generic function with 1 method)"
|
|
]
|
|
},
|
|
"execution_count": 6,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"function objfun1(v,k,μ,Σ)\n",
|
|
" w = [v;1-sum(v)]\n",
|
|
" ERp = w'μ\n",
|
|
" VarRp = w'Σ*w\n",
|
|
" StdRp = sqrt(VarRp)\n",
|
|
" U = ERp - k/2*VarRp\n",
|
|
" loss = -U #minimise this\n",
|
|
" return loss, w, ERp, StdRp\n",
|
|
"end"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 7,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"A 0.313\n",
|
|
"B 0.695\n",
|
|
"C -0.008\n",
|
|
"\n",
|
|
"σₚ 0.079\n",
|
|
"μₚ 0.102\n",
|
|
"\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"k = 10\n",
|
|
"Sol = optimize(v -> objfun1(v,k,μ,Σ)[1],zeros(n-1))\n",
|
|
"\n",
|
|
"v = Optim.minimizer(Sol)\n",
|
|
"(_,w,ERp,StdRp) = objfun1(v,k,μ,Σ)\n",
|
|
"\n",
|
|
"printmat(w,rowNames=assetNames)\n",
|
|
"printmat([StdRp,ERp],rowNames=[\"σₚ\",\"μₚ\"])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# Task 1\n",
|
|
"\n",
|
|
"Redo the optimisation for the following range of $k$ values:\n",
|
|
"`k = [range(0.1,1.9,length=30);range(1,30,length=25)]`\n",
|
|
"\n",
|
|
"Then plot the resulting \"efficient frontier\" (similar to the MV plot above). Is is the same as the plot above?"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# Task 2\n",
|
|
"\n",
|
|
"Change the optimisation problem to `U = ERp + cost - k/2*VarRp` where `cost` is (sum of the *negative* portfolio weights)*0.05. This is a (crude/dramatic) attaempt to illustrate the impact of costs of short selling ($w_i<0$) assets. Plot the new \"effecient frontier\" together with the old one."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# Task 3\n",
|
|
"\n",
|
|
"Compare the portfolio weights (with/without the extra `cost`)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
}
|
|
],
|
|
"metadata": {
|
|
"@webio": {
|
|
"lastCommId": null,
|
|
"lastKernelId": null
|
|
},
|
|
"anaconda-cloud": {},
|
|
"kernelspec": {
|
|
"display_name": "Julia 1.6.3",
|
|
"language": "julia",
|
|
"name": "julia-1.6"
|
|
},
|
|
"language_info": {
|
|
"file_extension": ".jl",
|
|
"mimetype": "application/julia",
|
|
"name": "julia",
|
|
"version": "1.6.3"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
}
|