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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Markowitz Efficient Frontier · ParametricOptInterface.jl</title><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script><link href="../../assets/favicon.ico" rel="icon" type="image/x-icon"/></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../../">ParametricOptInterface.jl</a></span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><a class="tocitem" href="../../manual/">Manual</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../example/">Basic Examples</a></li><li><a class="tocitem" href="../benders/">Benders Quantile Regression</a></li><li class="is-active"><a class="tocitem" href>Markowitz Efficient Frontier</a></li></ul></li><li><a class="tocitem" href="../../reference/">Reference</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>Markowitz Efficient Frontier</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Markowitz Efficient Frontier</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/jump-dev/ParametricOptInterface.jl/blob/master/docs/src/Examples/markowitz.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Markowitz-Efficient-Frontier"><a class="docs-heading-anchor" href="#Markowitz-Efficient-Frontier">Markowitz Efficient Frontier</a><a id="Markowitz-Efficient-Frontier-1"></a><a class="docs-heading-anchor-permalink" href="#Markowitz-Efficient-Frontier" title="Permalink"></a></h1><p>In this example, we solve the classical portfolio problem where we introduce the weight parameter <span>$\gamma$</span> and maximize <span>$\gamma \text{ risk} - \text{expected return}$</span>. By updating the values of <span>$\gamma$</span> we trace the efficient frontier.</p><p>Given the prices changes with mean <span>$\mu$</span> and covariance <span>$\Sigma$</span>, we can construct the classical portfolio problem:</p><p class="math-container">\[\begin{array}{ll}
\text{maximize} &amp; \gamma* x^T \mu - x^T \Sigma x \\
\text{subject to} &amp; \| x \|_1 = 1 \\
&amp; x \succeq 0
\end{array}\]</p><p>The problem data was gotten from the example <a href="https://jump.dev/Convex.jl/dev/examples/portfolio_optimization/portfolio_optimization2/">portfolio optimization</a></p><pre><code class="language-julia hljs">using ParametricOptInterface, MathOptInterface, JuMP, Ipopt
using LinearAlgebra, Plots

const POI = ParametricOptInterface
const MOI = MathOptInterface

# generate problem data
μ = [11.5; 9.5; 6] / 100 #expected returns
Σ = [
166 34 58 #covariance matrix
34 64 4
58 4 100
] / 100^2
</code></pre><p>We first build the model with <span>$\gamma$</span> as parameter in POI</p><pre><code class="language-julia hljs">function first_model(μ,Σ)
cached = MOI.Bridges.full_bridge_optimizer(
MOIU.CachingOptimizer(
MOIU.UniversalFallback(MOIU.Model{Float64}()),
Ipopt.Optimizer(),
),
Float64,
)
optimizer = POI.Optimizer(cached)
portfolio = direct_model(optimizer)
set_silent(portfolio)

N = length(μ)
@variable(portfolio, x[1:N] &gt;= 0)
@variable(portfolio, γ in POI.Parameter(0.0))

@objective(portfolio, Max, γ*dot(μ,x) - x&#39; * Σ * x)
@constraint(portfolio, sum(x) == 1)
optimize!(portfolio)

return portfolio
end</code></pre><p>Then, we update the <span>$\gamma$</span> value in the model</p><pre><code class="language-julia hljs">function update_model!(portfolio,γ_value)
γ = portfolio[:γ]
MOI.set(portfolio, POI.ParameterValue(), γ, γ_value)
optimize!(portfolio)
return portfolio
end</code></pre><p>Collecting all the return and risk resuls for each <span>$\gamma$</span></p><pre><code class="language-julia hljs">function add_to_dict(portfolios_values,portfolio,μ,Σ)
γ = portfolio[:γ]
γ_value = value(γ)
x = portfolio[:x]
x_value = value.(x)
portfolio_return = dot(μ,x_value)
portfolio_deviation = x_value&#39; * Σ * x_value
portfolios_values[γ_value] = (portfolio_return,portfolio_deviation)
end</code></pre><p>Run the portfolio optimization for different values of <span>$\gamma$</span></p><pre><code class="language-julia hljs">portfolio = first_model(μ,Σ)
portfolios_values = Dict()
# Create a reference to the model to change it later
portfolio_ref = [portfolio]
add_to_dict(portfolios_values,portfolio,μ,Σ)

for γ_value in 0.02:0.02:1.0
portfolio_ref[] = update_model!(portfolio_ref[],γ_value)
add_to_dict(portfolios_values,portfolio_ref[],μ,Σ)
end</code></pre><p>Plot the efficient frontier</p><pre><code class="language-julia hljs">portfolios_values = sort(portfolios_values,by=x-&gt;x[1])
portfolios_values_matrix = hcat([[v[1],v[2]] for v in values(portfolios_values)]...)&#39;
plot(portfolios_values_matrix[:,2],portfolios_values_matrix[:,1],legend=false,
xlabel=&quot;Standard Deviation&quot;, ylabel = &quot;Return&quot;, title = &quot;Efficient Frontier&quot;)</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../benders/">« Benders Quantile Regression</a><a class="docs-footer-nextpage" href="../../reference/">Reference »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 23 October 2023 00:53">Monday 23 October 2023</span>. Using Julia version 1.6.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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