1194 lines
236 KiB
Plaintext
1194 lines
236 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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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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||
"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)"
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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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||
"# The Maximization Problem\n",
|
||
"\n",
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||
"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)]`"
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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": 6,
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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": [
|
||
"objfun1 (generic function with 1 method)"
|
||
]
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},
|
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"execution_count": 6,
|
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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": [
|
||
"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": {},
|
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"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(0.1,0.9,length=30);range(1,30,length=25)]` EDIT\n",
|
||
"\n",
|
||
"Then plot the resulting \"efficient frontier\" (similar to the MV plot above). Is is the same as the plot above?"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 8,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"k = [range(0.1,0.9,length=30);range(1,30,length=25)]\n",
|
||
"L = length(k)\n",
|
||
"\n",
|
||
"σμM = fill(NaN,L,2) #to fill with calculation results\n",
|
||
"wM = fill(NaN,L,n)\n",
|
||
"for i = 1:L\n",
|
||
" Sol = optimize(v -> objfun1(v,k[i],μ,Σ)[1],zeros(n-1))\n",
|
||
" v = Optim.minimizer(Sol)\n",
|
||
" (_,w,ERp,StdRp) = objfun1(v,k[i],μ,Σ)\n",
|
||
" σμM[i,:] = [StdRp,ERp]\n",
|
||
" wM[i,:] = w\n",
|
||
"end\n",
|
||
"\n",
|
||
"#printmat(k,σμM)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 9,
|
||
"metadata": {},
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"outputs": [
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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||
"p2 = plot( σμM[:,1]*100,σμM[:,2]*100,\n",
|
||
" linecolor = :blue,\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(p2)"
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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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"# Task 2\n",
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"\n",
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"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) attempt to illustrate the impact of costs of short selling ($w_i<0$) assets. Plot the new \"effecient frontier\" together with the old one."
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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": 10,
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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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"objfun2 (generic function with 1 method)"
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]
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},
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"execution_count": 10,
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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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||
"function objfun2(v,k,μ,Σ)\n",
|
||
" w = [v;1-sum(v)]\n",
|
||
" ERp = w'μ\n",
|
||
" VarRp = w'Σ*w\n",
|
||
" StdRp = sqrt(VarRp)\n",
|
||
" cost = sum(((w.<0).*w))*0.05 #<0,short positions reduce the return\n",
|
||
" U = ERp + cost - k/2*VarRp\n",
|
||
" loss = -U #minimise this\n",
|
||
" return loss, w, ERp, StdRp\n",
|
||
"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": 11,
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||
"metadata": {},
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"outputs": [],
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||
"source": [
|
||
"σμMb = fill(NaN,L,2) #to fill with calculation results\n",
|
||
"wMb = fill(NaN,L,n)\n",
|
||
"for i = 1:L\n",
|
||
" Sol = optimize(v -> objfun2(v,k[i],μ,Σ)[1],zeros(n-1))\n",
|
||
" v = Optim.minimizer(Sol)\n",
|
||
" (_,w,ERp,StdRp) = objfun2(v,k[i],μ,Σ)\n",
|
||
" σμMb[i,:] = [StdRp,ERp]\n",
|
||
" wMb[i,:] = w\n",
|
||
"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": 12,
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"metadata": {},
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"outputs": [
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]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"p3 = plot( [σμM[:,1] σμMb[:,1]]*100,[σμM[:,2] σμMb[:,2]]*100,\n",
|
||
" linecolor = [:blue :black],\n",
|
||
" linewidth = 2,\n",
|
||
" label = [\"MVF (trad)\" \"MVF (with cost)\"],\n",
|
||
" legend = :topleft,\n",
|
||
" xlim = (0,15),\n",
|
||
" ylim = (0,15),\n",
|
||
" title = \"MVF\",\n",
|
||
" xlabel = \"Std(Rp), %\",\n",
|
||
" ylabel = \"ERp, %\" )\n",
|
||
"scatter!(sqrt.(diag(Σ))*100,μ*100,color=:red,label=\"assets\")\n",
|
||
"display(p3)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Task 3\n",
|
||
"\n",
|
||
"Compare the portfolio weights (with/without the extra `cost`)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 13,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
" A B C A B C\n",
|
||
"0.1 33.248 7.705 -39.954 3.617 -0.000 -2.617\n",
|
||
"0.13 26.055 6.174 -31.229 2.892 0.000 -1.892\n",
|
||
"0.16 21.419 5.188 -25.607 2.429 0.000 -1.429\n",
|
||
"0.18 18.183 4.499 -21.682 2.102 0.000 -1.102\n",
|
||
"0.21 15.798 3.989 -18.788 1.866 -0.000 -0.866\n",
|
||
"0.24 13.962 3.600 -16.562 1.681 0.000 -0.681\n",
|
||
"0.27 12.509 3.292 -14.800 1.534 0.000 -0.534\n",
|
||
"0.29 11.330 3.042 -13.371 1.417 0.000 -0.417\n",
|
||
"0.32 10.354 2.831 -12.185 1.320 0.000 -0.320\n",
|
||
"0.35 9.531 2.659 -11.191 1.240 0.000 -0.240\n",
|
||
"0.38 8.831 2.508 -10.339 1.166 0.000 -0.166\n",
|
||
"0.4 8.226 2.381 -9.607 1.105 -0.000 -0.105\n",
|
||
"0.43 7.698 2.267 -8.966 1.055 0.000 -0.055\n",
|
||
"0.46 7.234 2.168 -8.402 1.005 0.000 -0.005\n",
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||
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]
|
||
},
|
||
"execution_count": 14,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"plot_array = [] #an alternative way to create subplots\n",
|
||
"for i = 1:n\n",
|
||
" push!(plot_array,\n",
|
||
" plot(k,[wM[:,i] wMb[:,i]],linecolor = [:blue :black],linestyle = [:solid :dash],title = assetNames[i],\n",
|
||
" xlabel=\"k\",label=[\"traditional\" \"with cost\"],size = (700,450),ylims=(-3,3)) )\n",
|
||
"end\n",
|
||
"plot(plot_array...) # note the \"...\" "
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": null,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": []
|
||
}
|
||
],
|
||
"metadata": {
|
||
"@webio": {
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"lastCommId": null,
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"lastKernelId": null
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},
|
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"anaconda-cloud": {},
|
||
"kernelspec": {
|
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"display_name": "Julia 1.6.4",
|
||
"language": "julia",
|
||
"name": "julia-1.6"
|
||
},
|
||
"language_info": {
|
||
"file_extension": ".jl",
|
||
"mimetype": "application/julia",
|
||
"name": "julia",
|
||
"version": "1.6.4"
|
||
}
|
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
|