SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Portfolio optimization</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Portfolio optimization</span></span>

</h1>

</header>

</p>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1126788409">

/* start https://en.wikipedia.org/ */

.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1246091330">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r886047488">

/* start https://en.wikipedia.org/ */

.mw-parser-output .nobold{font-weight:normal}

/* end https://en.wikipedia.org/ */

</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="background:#cce;margin-bottom:0.4em;"><a href="Personal_finance" title="Personal finance">Personal finance</a></th></tr><tr><td class="sidebar-image" style="background:white;margin-bottom:0.4em;"><span typeof="mw:File"></span></td></tr><tr><th class="sidebar-heading" style="background:#e0e0ff;">

<a href="Credit_(finance)" class="mw-redirect" title="Credit (finance)">Credit</a> <b>·</b> <a href="Debt" title="Debt">Debt</a></th></tr><tr><td class="sidebar-content hlist" style="padding:0.2em 0 0.8em;">

<ul><li><a href="Mortgage_loan" class="mw-redirect" title="Mortgage loan">Mortgage</a></li>

<li><a href="Charge_card" title="Charge card">Charge card</a></li>

<li><a href="Credit_card" title="Credit card">Credit card</a></li>

<li><a href="Unsecured_debt" title="Unsecured debt">Unsecured personal loan</a></li>

<li><a href="Student_loan" title="Student loan">Student loan</a></li>

<li><a href="Title_loan" title="Title loan">Title loan</a></li>

<li><a href="Payday_loan" title="Payday loan">Payday loan</a></li>

<li><a href="Refund_anticipation_loan" title="Refund anticipation loan">Refund anticipation loan</a></li>

<li><a href="Refinancing" title="Refinancing">Refinancing</a></li>

<li><a href="Debt_consolidation" title="Debt consolidation">Debt consolidation</a></li>

<li><a href="Debt_rescheduling" title="Debt rescheduling">Debt rescheduling</a></li>

<li><a href="Bankruptcy" title="Bankruptcy">Bankruptcy</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="background:#e0e0ff;">

<a href="Employment_contract" title="Employment contract">Employment contract</a></th></tr><tr><td class="sidebar-content hlist" style="padding:0.2em 0 0.8em;">

<ul><li><a href="Salary" title="Salary">Salary</a></li>

<li><a href="Salary_packaging" title="Salary packaging">Salary packaging</a></li>

<li><a href="Employee_stock_ownership" title="Employee stock ownership">Employee stock ownership</a></li>

<li><a href="Employee_benefits" title="Employee benefits">Employee benefits</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="background:#e0e0ff;">

<a href="Retirement" title="Retirement">Retirement</a></th></tr><tr><td class="sidebar-content hlist" style="padding:0.2em 0 0.8em;">

<ul><li><a href="Filial_responsibility_laws" title="Filial responsibility laws">Filial responsibility laws</a></li>

<li><a href="Pension" title="Pension">Pension</a>

<ul><li><a href="Pension_systems_by_country" title="Pension systems by country">By country</a></li>

<li><a href="Defined_benefit_pension_plan" title="Defined benefit pension plan">Defined benefit</a></li>

<li><a href="Defined_contribution_plan" title="Defined contribution plan">Defined contribution</a></li>

<li><a href="Social_pension" title="Social pension">Social pension</a></li></ul></li></ul></td>

</tr><tr><th class="sidebar-heading" style="background:#e0e0ff;">

<a href="Personal_budget" title="Personal budget">Personal budget</a> and <a href="Investment" title="Investment">investment</a></th></tr><tr><td class="sidebar-content hlist" style="padding:0.2em 0 0.8em;">

<ul><li><a href="Active_management" title="Active management">Active management</a></li>

<li><a href="Alternative_investment" title="Alternative investment">Alternative investment</a></li>

<li><a href="Asset" title="Asset">Asset</a>

<ul><li><a href="Asset_allocation" title="Asset allocation">allocation</a></li>

<li><a href="Asset_(economics)" title="Asset (economics)">economics</a></li>

<li><a href="Capital_appreciation" title="Capital appreciation">growth</a></li></ul></li>

<li><a href="Bond_(finance)" title="Bond (finance)">Bond (finance)</a></li>

<li><a href="Diversification_(finance)" title="Diversification (finance)">Diversification (finance)</a></li>

<li><a href="Equity_(finance)" title="Equity (finance)">Equity (finance)</a></li>

<li><a href="Estate_planning" title="Estate planning">Estate planning</a></li>

<li><a href="Finance" title="Finance">Financial</a>

<ul><li><a href="Financial_adviser" title="Financial adviser">adviser</a></li>

<li><a href="Financial_asset" title="Financial asset">asset</a></li>

<li><a href="Financial_independence" title="Financial independence">independence</a></li>

<li><a href="Financial_literacy" title="Financial literacy">literacy</a></li>

<li><a href="Financial_planner" title="Financial planner">planner</a></li></ul></li>

<li><a href="Fundamental_analysis" title="Fundamental analysis">Fundamental analysis</a></li>

<li><a href="Government_bond" title="Government bond">Government bond</a></li>

<li><a href="Growth_investing" title="Growth investing">Growth investing</a></li>

<li><a href="Growth_stock" title="Growth stock">Growth stock</a></li>

<li><a href="Hedge_(finance)" title="Hedge (finance)">Hedge (finance)</a></li>

<li><a href="Impact_investing" title="Impact investing">Impact investing</a></li>

<li><a href="Investment_advisory" title="Investment advisory">Investment advisory</a></li>

<li><a href="Investment_performance" title="Investment performance">Investment performance</a></li>

<li><a href="Investment_style" title="Investment style">Investment style</a></li>

<li><a href="Investor_profile" title="Investor profile">Investor profile</a></li>

<li><a href="Market_risk" title="Market risk">Market risk</a></li>

<li><a href="Net_worth" title="Net worth">Net worth</a></li>

<li><a href="Passive_management" title="Passive management">Passive management</a></li>

<li><a href="Robo-advisor" title="Robo-advisor">Robo-advisor</a></li>

<li><a href="Saving" title="Saving">Saving</a>

<ul><li><a href="Savings_account" title="Savings account">Savings account</a></li></ul></li>

<li><a href="Speculation" title="Speculation">Speculation</a></li>

<li><a href="Stock" title="Stock">Stock</a></li>

<li><a href="Stockbroker" title="Stockbroker">Stockbroker</a></li>

<li><a href="Target_date_fund" title="Target date fund">Target date fund</a></li>

<li><a href="Wealth" title="Wealth">Wealth</a>

<ul><li><a href="List_of_countries_by_wealth_per_adult" title="List of countries by wealth per adult">List of countries by wealth per adult</a></li></ul></li></ul></td>

</tr><tr><th class="sidebar-heading" style="background:#e0e0ff;">

<ul><li><a href="Asset_pricing" title="Asset pricing">Asset pricing</a></li>

<li><a href="Citizen's_dividend" title="Citizen's dividend">Citizen's dividend</a></li>

<li><a href="Cooperative" title="Cooperative">Cooperative</a></li>

<li><a href="Credit_union" title="Credit union">Credit union</a></li>

<li><a href="Economic_bubble" title="Economic bubble">Economic bubble</a></li>

<li><a href="Financial_deepening" title="Financial deepening">Financial deepening</a></li>

<li><a href="Financial_inclusion" title="Financial inclusion">Financial inclusion</a></li>

<li><a href="Financial_risk" title="Financial risk">Financial risk</a></li>

<li><a href="Financial_technology" title="Financial technology">Financial technology</a></li>

<li><a href="Guaranteed_minimum_income" title="Guaranteed minimum income">Guaranteed minimum income</a></li>

<li><a href="Market_impact" title="Market impact">Market impact</a></li>

<li><a href="Risk-return_spectrum" class="mw-redirect" title="Risk-return spectrum">Risk-return spectrum</a></li>

<li><a href="Social_dividend" title="Social dividend">Social dividend</a></li>

<li><a href="Structured_product" title="Structured product">Structured product</a></li>

<li><a href="Sustainability" title="Sustainability">Sustainability</a></li>

<li><a href="Systematic_risk" title="Systematic risk">Systematic risk</a></li>

<li><a href="Universal_basic_income" title="Universal basic income">Universal basic income</a></li>

<li><a href="Volatility_(finance)" title="Volatility (finance)">Volatility (finance)</a></li>

<li><a href="Watered_stock" title="Watered stock">Watered stock</a></li></ul></td>

</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></td></tr></tbody></table>

<p><b>Portfolio optimization</b> is the process of selecting an optimal <a href="Portfolio_(finance)" title="Portfolio (finance)">portfolio</a> (<a href="Asset" title="Asset">asset</a> distribution), out of a set of considered portfolios, according to some <a href="Cost_functional" class="mw-redirect" title="Cost functional">objective</a>. The objective typically maximizes factors such as <a href="Expected_return" title="Expected return">expected return</a>, and minimizes costs like <a href="Financial_risk" title="Financial risk">financial risk</a>, resulting in a <a href="Multi-objective_optimization" title="Multi-objective optimization">multi-objective optimization</a> problem. Factors being considered may range from tangible (such as <a href="Asset" title="Asset">assets</a>, <a href="Liability_(financial_accounting)" title="Liability (financial accounting)">liabilities</a>, <a href="Earnings" title="Earnings">earnings</a> or other <a href="Fundamental_analysis" title="Fundamental analysis">fundamentals</a>) to intangible (such as selective <a href="Divestment" title="Divestment">divestment</a>).

</p>

<div class="mw-heading mw-heading2"><h2 id="Modern_portfolio_theory">Modern portfolio theory</h2></div>

<p><a href="Modern_portfolio_theory" title="Modern portfolio theory">Modern portfolio theory</a> was introduced in a 1952 doctoral thesis by <a href="Harry_Markowitz" title="Harry Markowitz">Harry Markowitz</a>, where the <a href="Markowitz_model" title="Markowitz model">Markowitz model</a> was first defined.<sup id="cite_ref-markowitz1952_1-0" class="reference"><a href="#cite_note-markowitz1952-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The model assumes that an investor aims to maximize a portfolio's expected return contingent on a prescribed amount of risk. Portfolios that meet this criterion, i.e., maximize the expected return given a prescribed amount of risk, are known as efficient portfolios. By definition, any other portfolio yielding a higher amount of expected return must also have excessive risk. This results in a trade-off between the desired expected return and allowable risk. This risk-expected return relationship of efficient portfolios is graphically represented by a curve known as the <a href="Efficient_frontier" title="Efficient frontier">efficient frontier</a>. All efficient portfolios, each represented by a point on the efficient frontier, are <a href="Diversification_(finance)" title="Diversification (finance)">well-diversified</a>. While ignoring higher moments of the return can lead to significant over-investment in risky securities, especially when volatility is high,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> the optimization of portfolios when return <a href="Probability_distribution" title="Probability distribution">distributions</a> are non-<a href="Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian</a> is mathematically challenging.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="Hierarchical_Risk_Parity" title="Hierarchical Risk Parity">Hierarchical Risk Parity</a> is a sophisticated approach to portfolio optimization introduced in 2016 as an alternative to the traditional mean-variance optimization model developed by <a href="Harry_Markowitz" title="Harry Markowitz">Harry Markowitz</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Optimization_methods">Optimization methods</h2></div>

<p>The portfolio optimization problem is specified as a <a href="Constrained_optimization" title="Constrained optimization">constrained</a> utility-maximization problem. Common formulations of portfolio <a href="Utility" title="Utility">utility</a> functions define it as the expected portfolio return (net of transaction and financing costs) minus a cost of risk. The latter component, the cost of risk, is defined as the portfolio risk multiplied by a <a href="Risk_aversion" title="Risk aversion">risk aversion</a> parameter (or unit price of risk). For return distributions that are <a href="Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian</a>, this is equivalent to maximizing a certain <a href="Quantile_function" title="Quantile function">quantile</a> of the return, where the corresponding probability is dictated by the risk aversion parameter. Practitioners often add additional <a href="Constrained_optimization" title="Constrained optimization">constraints</a> to improve diversification and further limit risk. Examples of such constraints are asset, sector, and region portfolio weight limits.

</p>

<div class="mw-heading mw-heading3"><h3 id="Specific_approaches">Specific approaches</h3></div>

<p>Portfolio optimization often takes place in two stages: optimizing weights of asset classes to hold, and optimizing weights of assets within the same asset class. An example of the former would be choosing the proportions placed in equities versus bonds, while an example of the latter would be choosing the proportions of the stock sub-portfolio placed in stocks X, Y, and Z. Equities and bonds have fundamentally different financial characteristics and have different <a href="Systematic_risk" title="Systematic risk">systematic risk</a> and hence can be viewed as separate asset classes; holding some of the portfolio in each class provides some diversification, and holding various specific assets within each class affords further diversification. By using such a two-step procedure one eliminates non-systematic risks both on the individual asset and the asset class level. For the specific formulas for efficient portfolios,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> see <a href="Mutual_fund_separation_theorem#Portfolio_separation_in_mean-variance_analysis" title="Mutual fund separation theorem">Portfolio separation in mean-variance analysis</a>.

</p><p>One approach to portfolio optimization is to specify a <a href="Von_Neumann%E2%80%93Morgenstern_utility_function" class="mw-redirect" title="Von Neumann–Morgenstern utility function">von Neumann–Morgenstern utility function</a> defined over final portfolio wealth; the expected value of utility is to be maximized. To reflect a preference for higher rather than lower returns, this objective function is <a href="First_derivative" class="mw-redirect" title="First derivative">increasing</a> in wealth, and to reflect risk aversion it is <a href="Concave_function" title="Concave function">concave</a>. For realistic utility functions in the presence of many assets that can be held, this approach, while theoretically the most defensible, can be computationally intensive.

</p><p><a href="Harry_Markowitz" title="Harry Markowitz">Harry Markowitz</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> developed the "critical line method", a general procedure for <a href="Quadratic_programming" title="Quadratic programming">quadratic programming</a> that can handle additional linear constraints and upper and lower bounds on holdings. Moreover, in this context, the approach provides a method for determining the entire set of efficient portfolios. Its application here was later explicated by <a href="William_F._Sharpe" title="William F. Sharpe">William Sharpe</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Mathematical_tools">Mathematical tools</h3></div>

<p>The complexity and scale of optimizing portfolios over many assets means that the work is generally done by computer. Central to this optimization is the construction of the <a href="Covariance_matrix" title="Covariance matrix">covariance matrix</a> for the rates of return on the assets in the portfolio.

</p><p>Techniques include:

</p>

<ul><li><a href="Linear_programming" title="Linear programming">Linear programming</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Quadratic_programming" title="Quadratic programming">Quadratic programming</a></li>

<li><a href="Nonlinear_programming" title="Nonlinear programming">Nonlinear programming</a></li>

<li><a href="Mixed_integer_programming" class="mw-redirect" title="Mixed integer programming">Mixed integer programming</a></li>

<li><a href="Metaheuristic" title="Metaheuristic">Meta-heuristic methods</a><sup id="cite_ref-Performance_Investigation_and_Comparison_of_Two_Evolutionary_Algorithms_in_Portfolio_Optimization:_Genetic_and_Particle_Swarm_Optimization_11-0" class="reference"><a href="#cite_note-Performance_Investigation_and_Comparison_of_Two_Evolutionary_Algorithms_in_Portfolio_Optimization:_Genetic_and_Particle_Swarm_Optimization-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Stochastic_programming" title="Stochastic programming">Stochastic programming</a> for multistage portfolio optimization<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Copula_(probability_theory)" class="mw-redirect" title="Copula (probability theory)">Copula based methods</a><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Principal_component_analysis#Quantitative_finance" title="Principal component analysis">Principal component-based methods</a></li>

<li><a href="Deterministic_global_optimization" title="Deterministic global optimization">Deterministic global optimization</a></li>

<li><a href="Genetic_algorithm" title="Genetic algorithm">Genetic algorithm</a><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Optimization_constraints">Optimization constraints</h2></div>

<p>Portfolio optimization is usually done subject to constraints, such as regulatory constraints, or illiquidity. These constraints can lead to portfolio weights that focus on a small sub-sample of assets within the portfolio. When the portfolio optimization process is subject to other constraints such as taxes, transaction costs, and management fees, the optimization process may result in an under-diversified portfolio.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Regulation_and_taxes">Regulation and taxes</h3></div>

<p>Investors may be forbidden by law to hold some assets. In some cases, unconstrained portfolio optimization would lead to <a href="Short-selling" class="mw-redirect" title="Short-selling">short-selling</a> of some assets. However short-selling can be forbidden. Sometimes it is impractical to hold an asset because the associated tax cost is too high. In such cases appropriate constraints must be imposed on the optimization process.

</p>

<div class="mw-heading mw-heading3"><h3 id="Transaction_costs">Transaction costs</h3></div>

<p><a href="Transaction_cost" title="Transaction cost">Transaction costs</a> are the costs of trading to change the portfolio weights. Since the optimal portfolio changes with time, there is an incentive to re-optimize frequently. However, too frequent trading would incur too-frequent transactions costs; so the optimal strategy is to find the frequency of re-optimization and trading that appropriately trades off the avoidance of transaction costs with the avoidance of sticking with an out-of-date set of portfolio proportions. This is related to the topic of <a href="Tracking_error" title="Tracking error">tracking error</a>, by which stock proportions deviate over time from some benchmark in the absence of re-balancing.

</p>

<div class="mw-heading mw-heading3"><h3 id="Concentration_risk">Concentration risk</h3></div>

<p><a href="Concentration_risk" title="Concentration risk">Concentration risk</a> refers to the risk caused by holding an exposure to a single position or sector that is large enough to cause material losses to the overall portfolio when adverse events occur. If the portfolio is optimized without any constraints with regards to concentration risk, the optimal portfolio can be any risky-asset portfolio, and therefore there is nothing to prevent it from being a portfolio that invests solely in a single asset. Managing concentration risk should be part of a comprehensive risk management framework<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and to achieve a reduction in such a risk it is possible to add constraints that force upper bound limits to the weight that can be attributed to any single component of the optimal portfolio.

</p>

<div class="mw-heading mw-heading2"><h2 id="Improving_portfolio_optimization">Improving portfolio optimization</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Correlations_and_risk_evaluation">Correlations and risk evaluation</h3></div>

<p>Different approaches to portfolio optimization measure risk differently. In addition to the traditional measure, <a href="Standard_deviation" title="Standard deviation">standard deviation</a>, or its square (<a href="Variance" title="Variance">variance</a>), which are not <a href="Robust_statistic" class="mw-redirect" title="Robust statistic">robust</a> risk measures, other measures include the <a href="Sortino_ratio" title="Sortino ratio">Sortino ratio</a>, <a href="Expected_shortfall" title="Expected shortfall">CVaR (Conditional Value at Risk)</a>, and <a href="Statistical_dispersion" title="Statistical dispersion">statistical dispersion</a>.

</p><p>Investment is a forward-looking activity, and thus the <a href="Covariance" title="Covariance">covariances</a> of returns must be forecast rather than observed.

<a href="Black-Litterman" class="mw-redirect" title="Black-Litterman">Black-Litterman</a> is often used here. This model <sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> takes the market-implied (i.e. historical) returns and covariances, and through a <a href="Bayesian" class="mw-redirect" title="Bayesian">Bayesian</a> approach, updates these <a href="Prior_probability" title="Prior probability">prior results</a> with the portfolio manager's "views" on certain assets, to produce a <a href="Posterior_probability" title="Posterior probability">posterior estimate</a> of the returns and the covariance matrix. These may then be passed through an optimizer. (Alternatively, the model-implied weights are optimal in the sense of achieving the returns matching the manager's "views".)

</p><p>Portfolio optimization assumes the investor may have some <a href="Risk_aversion" title="Risk aversion">risk aversion</a> and the stock prices may exhibit significant differences between their historical or forecast values and what is experienced.

In particular, financial crises are characterized by a significant increase in correlation of stock price movements which may seriously degrade the benefits of diversification.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>

</p><p>In a mean-variance optimization framework, accurate estimation of the <a href="Variance-covariance_matrix" class="mw-redirect" title="Variance-covariance matrix">variance-covariance matrix</a> is paramount.

Quantitative techniques that use <a href="Monte_Carlo_method" title="Monte Carlo method">Monte-Carlo simulation</a> with the Gaussian copula and well-specified marginal distributions are effective.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>

Allowing the modeling process to allow for empirical characteristics in stock returns such as <a href="Autoregression" class="mw-redirect" title="Autoregression">autoregression</a>, asymmetric volatility, <a href="Skewness" title="Skewness">skewness</a>, and <a href="Kurtosis" title="Kurtosis">kurtosis</a> is important.

Not accounting for these attributes can lead to severe estimation error in the correlations, variances and covariances that have negative biases (as much as 70% of the true values).<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>

</p><p>Other optimization strategies that focus on minimizing tail-risk (e.g., <a href="Value_at_risk" title="Value at risk">value at risk</a>, <a href="Conditional_value_at_risk" class="mw-redirect" title="Conditional value at risk">conditional value at risk</a>) in investment portfolios are popular among risk averse investors.

To minimize exposure to tail risk, forecasts of asset returns using Monte-Carlo simulation with vine copulas to allow for lower (left) tail dependence (e.g., Clayton, Rotated Gumbel) across large portfolios of assets are most suitable.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>

<a href="Tail_risk_parity" title="Tail risk parity">(Tail)</a> <a href="Risk_parity" title="Risk parity">risk parity</a> focuses on allocation of risk, rather than allocation of capital.

</p><p>Hedge fund managers have been applying "full-scale optimization" whereby any investor utility function can be used to optimize a portfolio.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>

It is purported that such a methodology is more practical and suitable for modern investors whose risk preferences involve reducing <a href="Tail_risk" title="Tail risk">tail risk</a>, minimizing negative skewness and <a href="Fat_tail" class="mw-redirect" title="Fat tail">fat tails</a> in the returns distribution of the investment portfolio.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>

Where such methodologies involve the use of higher-moment utility functions, it is necessary to use a methodology that allows for forecasting of a <a href="Joint_distribution" class="mw-redirect" title="Joint distribution">joint distribution</a> that accounts for asymmetric dependence.

A suitable methodology that allows for the joint distribution to incorporate asymmetric dependence is the Clayton Canonical Vine Copula. See <a href="Copula_(probability_theory)" class="mw-redirect" title="Copula (probability theory)">Copula (probability theory) § Quantitative finance</a>.

</p><p>Some modern machine learning approaches to portfolio construction, such as <a href="Hierarchical_Risk_Parity" title="Hierarchical Risk Parity">Hierarchical Risk Parity</a> (HRP), utilize graph-based methods to improve out-of-sample performance relative to traditional mean-variance portfolios. These methods estimate a maximum spanning tree from the asset covariance matrix, aiming to capture the essential structure of asset dependencies while pruning weaker, potentially noisy links. This hierarchical clustering process can yield more robust and interpretable portfolio allocations.

</p>

<div class="mw-heading mw-heading3"><h3 id="Cooperation_in_portfolio_optimization">Cooperation in portfolio optimization</h3></div>

<p>A group of investors, instead of investing individually, may choose to invest their total capital into the joint portfolio, and then divide the (uncertain) investment profit in a way which suits best their <a href="Utility" title="Utility">utility</a>/risk preferences. It turns out that, at least in the expected utility model,<sup id="cite_ref-Xia_24-0" class="reference"><a href="#cite_note-Xia-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> and mean-deviation model,<sup id="cite_ref-games_25-0" class="reference"><a href="#cite_note-games-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> each investor can usually get a share which he/she values strictly more than his/her optimal portfolio from the individual investment.

</p>

<ul><li><a href="Outline_of_finance#Portfolio_theory" title="Outline of finance">Outline of finance § Portfolio theory</a> for related articles</li>

<li><a href="Asset_allocation" title="Asset allocation">Asset allocation</a></li>

<li><a href="Chance-constrained_portfolio_selection" title="Chance-constrained portfolio selection">Chance-constrained portfolio selection</a></li>

<li><a href="Hierarchical_Risk_Parity" title="Hierarchical Risk Parity">Hierarchical Risk Parity</a></li>

<li><a href="Intertemporal_portfolio_choice" title="Intertemporal portfolio choice">Intertemporal portfolio choice</a></li>

<li><a href="Financial_risk_management#Investment_management" title="Financial risk management">Financial risk management § Investment management</a></li>

<li><a href="List_of_genetic_algorithm_applications#Finance_and_Economics" title="List of genetic algorithm applications">List of genetic algorithm applications § Finance and Economics</a></li>

<li><a href="Machine_learning#Applications" title="Machine learning">Machine learning § Applications</a></li>

<li><a href="Marginal_conditional_stochastic_dominance" title="Marginal conditional stochastic dominance">Marginal conditional stochastic dominance</a>, a way of showing that a portfolio is not efficient</li>

<li><a href="Merton's_portfolio_problem" title="Merton's portfolio problem">Merton's portfolio problem</a></li>

<li><a href="Mutual_fund_separation_theorem" title="Mutual fund separation theorem">Mutual fund separation theorem</a>, giving a property of mean-variance efficient portfolios</li>

<li><a href="Portfolio_theory" class="mw-redirect" title="Portfolio theory">Portfolio theory</a>, for the formulas</li>

<li><a href="Risk_parity" title="Risk parity">Risk parity</a> / <a href="Tail_risk_parity" title="Tail risk parity">Tail risk parity</a></li>

<li><a href="Stochastic_portfolio_theory" title="Stochastic portfolio theory">Stochastic portfolio theory</a></li>

<li><a href="Universal_portfolio_algorithm" title="Universal portfolio algorithm">Universal portfolio algorithm</a>, giving the first online portfolio selection algorithm</li>

<li><a href="Resampled_efficient_frontier" title="Resampled efficient frontier">Resampled efficient frontier</a>, accounting for the <a href="Uncertainty" title="Uncertainty">uncertainty</a> of the risk and return estimates using <a href="Resampling_(statistics)" title="Resampling (statistics)">resampling</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFMarkowitz,_H.M.1952" class="citation journal cs1">Markowitz, H.M. (March 1952). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://onlinelibrary.wiley.com/doi/10.1111/j.1540-6261.1952.tb01525.x/full">"Portfolio Selection"</a></span>. <i>The Journal of Finance</i>. <b>7</b> (1): <span class="nowrap">77–</span>91. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2975974">10.2307/2975974</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2975974">2975974</a>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFMarkowitz,_H.M.1959" class="citation book cs1">Markowitz, H.M. (1959). <a rel="nofollow" class="external text" href="http://cowles.econ.yale.edu/P/cm/m16/index.htm"><i>Portfolio Selection: Efficient Diversification of Investments</i></a>. New York: John Wiley & Sons.</cite> (reprinted by Yale University Press, 1970, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-300-01372-6</bdi>; 2nd ed. Basil Blackwell, 1991, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-55786-108-5</bdi>)</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFCvitanićPolimenisZapatero2008" class="citation journal cs1">Cvitanić, Jakša; Polimenis, Vassilis; Zapatero, Fernando (1 January 2008). "Optimal portfolio allocation with higher moments". <i>Annals of Finance</i>. <b>4</b> (1): <span class="nowrap">1–</span>28. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10436-007-0071-5">10.1007/s10436-007-0071-5</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1614-2446">1614-2446</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16514619">16514619</a>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFKimGiacomettiRachevFabozzi2012" class="citation journal cs1">Kim, Young Shin; Giacometti, Rosella; Rachev, Svetlozar; Fabozzi, Frank J.; Mignacca, Domenico (21 November 2012). <a rel="nofollow" class="external text" href="https://publikationen.bibliothek.kit.edu/1000029307/2233536">"Measuring financial risk and portfolio optimization with a non-Gaussian multivariate model"</a>. <i>Annals of Operations Research</i>. <b>201</b> (1): <span class="nowrap">325–</span>343. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10479-012-1229-8">10.1007/s10479-012-1229-8</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:45585936">45585936</a>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFLópez_de_Prado2016" class="citation journal cs1">López de Prado, Marcos (31 May 2016). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://pm-research.com/lookup/doi/10.3905/jpm.2016.42.4.059">"Building Diversified Portfolios that Outperform Out of Sample"</a></span>. <i>The Journal of Portfolio Management</i>. <b>42</b> (4): <span class="nowrap">59–</span>69. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3905%2Fjpm.2016.42.4.059">10.3905/jpm.2016.42.4.059</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0095-4918">0095-4918</a>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Merton, Robert. September 1972. "An analytic derivation of the efficient portfolio frontier," <i><a href="Journal_of_Financial_and_Quantitative_Analysis" title="Journal of Financial and Quantitative Analysis">Journal of Financial and Quantitative Analysis</a></i> 7, 1851–1872.</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFMarkowitz1956" class="citation journal cs1"><a href="Harry_Markowitz" title="Harry Markowitz">Markowitz, Harry</a> (1956). "The optimization of a quadratic function subject to linear constraints". <i><a href="Naval_Research_Logistics_Quarterly" class="mw-redirect" title="Naval Research Logistics Quarterly">Naval Research Logistics Quarterly</a></i>. <b>3</b> (<span class="nowrap">1–</span>2): <span class="nowrap">111–</span>133. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fnav.3800030110">10.1002/nav.3800030110</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://web.stanford.edu/~wfsharpe/mia/opt/mia_opt3.htm">The Critical Line Method</a> in William Sharpe, <i>Macro-Investment Analysis</i> (online text)</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFRockafellarUryasev2000" class="citation journal cs1">Rockafellar, R. Tyrrell; Uryasev, Stanislav (2000). <a rel="nofollow" class="external text" href="https://sites.math.washington.edu/~rtr/papers/rtr179-CVaR1.pdf">"Optimization of conditional value-at-risk"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Risk</i>. <b>2</b> (3): <span class="nowrap">21–</span>42. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.21314%2FJOR.2000.038">10.21314/JOR.2000.038</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:854622">854622</a>.</cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFKapsosZymlerChristofidesRustem2014" class="citation journal cs1">Kapsos, Michalis; Zymler, Steve; <a href="Nicos_Christofides" title="Nicos Christofides">Christofides, Nicos</a>; Rustem, Berç (Summer 2014). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/~yuying/Courses/CS870_2012/Omega_paper_Short_Cm.pdf">"Optimizing the Omega Ratio using Linear Programming"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Computational Finance</i>. <b>17</b> (4): <span class="nowrap">49–</span>57. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.21314%2FJCF.2014.283">10.21314/JCF.2014.283</a>.</cite></span>

</li>

<li id="cite_note-Performance_Investigation_and_Comparison_of_Two_Evolutionary_Algorithms_in_Portfolio_Optimization:_Genetic_and_Particle_Swarm_Optimization-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Performance_Investigation_and_Comparison_of_Two_Evolutionary_Algorithms_in_Portfolio_Optimization:_Genetic_and_Particle_Swarm_Optimization_11-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFTalebiMolaei,_Sheikh2010" class="citation book cs1">Talebi, Arash; Molaei, Sheikh (17 September 2010). "Performance investigation and comparison of two evolutionary algorithms in portfolio optimization: Genetic and particle swarm optimization". <i>2010 2nd IEEE International Conference on Information and Financial Engineering</i>. pp. <span class="nowrap">430–</span>437. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ficife.2010.5609394">10.1109/icife.2010.5609394</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4244-6927-7</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17386345">17386345</a>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFShapiroDentchevaRuszczyński2009" class="citation book cs1">Shapiro, Alexander; <a href="Darinka_Dentcheva" title="Darinka Dentcheva">Dentcheva, Darinka</a>; <a href="Andrzej_Piotr_Ruszczy%C5%84ski" title="Andrzej Piotr Ruszczyński">Ruszczyński, Andrzej</a> (2009). <a rel="nofollow" class="external text" href="http://www2.isye.gatech.edu/people/faculty/Alex_Shapiro/SPbook.pdf"><i>Lectures on stochastic programming: Modeling and theory</i></a> <span class="cs1-format">(PDF)</span>. MPS/SIAM Series on Optimization. Vol. 9. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) and the Mathematical Programming Society (MPS). pp. xvi+436. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-89871-687-0</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2562798">2562798</a>.</cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFZhuWelsch2018" class="citation journal cs1">Zhu, Zhe; Welsch, Roy E. (2018). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F17-AOAS1087">"Robust dependence modeling for high-dimensional covariance matrices with financial applications"</a>. <i>Ann. Appl. Stat</i>. <b>12</b> (2): <span class="nowrap">1228–</span>1249. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F17-AOAS1087">10.1214/17-AOAS1087</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:23490041">23490041</a>.</cite></span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Sefiane, Slimane and Benbouziane, Mohamed (2012). <a rel="nofollow" class="external text" href="http://www-math.unice.fr/publis/delmoral_lezaud.ps"><i>Portfolio Selection Using Genetic Algorithm</i></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160429142443/http://www-math.unice.fr/publis/delmoral_lezaud.ps">Archived</a> 29 April 2016 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Journal of Applied Finance & Banking, Vol. 2, No. 4 (2012): pp. 143–154.</span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFHumphreyBensonLowLee2015" class="citation journal cs1">Humphrey, J.; Benson, K.; Low, R.K.Y.; Lee, W.L. (2015). <a rel="nofollow" class="external text" href="http://espace.library.uq.edu.au/view/UQ:374981/UQ374981_PostPrint.pdf">"Is diversification always optimal?"</a> <span class="cs1-format">(PDF)</span>. <i>Pacific Basin Finance Journal</i>. <b>35</b> (B): B. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.pacfin.2015.09.003">10.1016/j.pacfin.2015.09.003</a>.</cite></span>

</li>

<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.finra.org/investors/insights/concentration-risk">"Concentrate on Concentration Risk | FINRA.org"</a>. <i>www.finra.org</i>. 15 June 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">16 March</span> 2024</span>.</cite></span>

</li>

<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Robert Martin (2018). <a rel="nofollow" class="external text" href="https://pyportfolioopt.readthedocs.io/en/latest/BlackLitterman.html">Black-Litterman Allocation</a></span>

</li>

<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite id="CITEREFChuaKrizmanPage2009" class="citation journal cs1">Chua, D.; Krizman, M.; Page, S. (2009). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.cfainstitute.org/learning/products/publications/dig/Pages/dig.v40.n2.5.aspx">"The Myth of Diversification"</a></span>. <i>Journal of Portfolio Management</i>. <b>36</b> (1): <span class="nowrap">26–</span>35. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3905%2FJPM.2009.36.1.026">10.3905/JPM.2009.36.1.026</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:154921810">154921810</a>.</cite></span>

</li>

<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite id="CITEREFLowFaffAas2016" class="citation journal cs1">Low, R.K.Y.; Faff, R.; Aas, K. (2016). <a rel="nofollow" class="external text" href="http://espace.library.uq.edu.au/view/UQ:377912/UQ377912_OA.pdf">"Enhancing mean–variance portfolio selection by modeling distributional asymmetries"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Economics and Business</i>. <b>85</b>: <span class="nowrap">49–</span>72. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jeconbus.2016.01.003">10.1016/j.jeconbus.2016.01.003</a>.</cite></span>

</li>

<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite id="CITEREFFantazzinni2009" class="citation journal cs1">Fantazzinni, D. (2009). "The effects of misspecified marginals and copulas on computing the value at risk: A Monte Carlo study". <i>Computational Statistics & Data Analysis</i>. <b>53</b> (6): <span class="nowrap">2168–</span>2188. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.csda.2008.02.002">10.1016/j.csda.2008.02.002</a>.</cite></span>

</li>

<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><cite id="CITEREFLowAlcockFaffBrailsford2013" class="citation journal cs1">Low, R.K.Y.; Alcock, J.; Faff, R.; Brailsford, T. (2013). <a rel="nofollow" class="external text" href="https://espace.library.uq.edu.au/view/UQ:297895/EC15UQ297895.pdf">"Canonical vine copulas in the context of modern portfolio management: Are they worth it?"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Banking & Finance</i>. <b>37</b> (8): 3085. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jbankfin.2013.02.036">10.1016/j.jbankfin.2013.02.036</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:154138333">154138333</a>.</cite></span>

</li>

<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite id="CITEREFChuaKritzmanPage2009" class="citation journal cs1">Chua, David; Kritzman, Mark; Page, Sebastien (2009). "The Myth of Diversification". <i>Journal of Portfolio Management</i>. <b>36</b> (1): <span class="nowrap">26–</span>35. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3905%2FJPM.2009.36.1.026">10.3905/JPM.2009.36.1.026</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:154921810">154921810</a>.</cite></span>

</li>

<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><cite id="CITEREFAdlerKritzman2007" class="citation journal cs1">Adler, Tim; Kritzman, Mark (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.2469%2Fdig.v37.n3.4799">"Mean-Variance versus Full-Scale Optimization: In and Out of Sample"</a>. <i>Journal of Asset Management</i>. <b>7</b> (5): <span class="nowrap">71–</span>73. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2469%2Fdig.v37.n3.4799">10.2469/dig.v37.n3.4799</a></span>.</cite></span>

</li>

<li id="cite_note-Xia-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Xia_24-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFXia2004" class="citation journal cs1">Xia, Jianming (2004). "Multi-agent investment in incomplete markets". <i>Finance and Stochastics</i>. <b>8</b> (2): <span class="nowrap">241–</span>259. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00780-003-0115-2">10.1007/s00780-003-0115-2</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7162635">7162635</a>.</cite></span>

</li>

<li id="cite_note-games-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-games_25-0">^</a></b></span> <span class="reference-text">Grechuk, B., Molyboha, A., Zabarankin, M. (2013). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/229565567_Cooperative_Games_With_General_Deviation_Measures">"Cooperative games with general deviation measures"</a>, Mathematical Finance, 23(2), 339–365.</span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFBaker,_H._KentFilbeck,_Greg2015" class="citation book cs1">Baker, H. Kent; Filbeck, Greg (2015). <i>Investment Risk Management</i>. Oxford Academic. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0199331963</bdi>.</cite></li>

<li><cite id="CITEREFFabozziSergio_M._FocardiPetter_N._Kolm2004" class="citation book cs1">Fabozzi, Frank J.; Sergio M. Focardi; Petter N. Kolm (2004). <i>Financial Modeling of the Equity Market: From CAPM to Cointegration</i>. Hoboken, NJ: <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-69900-4</bdi>.</cite></li>

<li><a href="Frank_Fabozzi" class="mw-redirect" title="Frank Fabozzi">Fabozzi, Frank J.</a>; Petter N. Kolm; Dessislava Pachamanova; Sergio M. Focardi (2007). <a rel="nofollow" class="external text" href="https://www.wiley.com/en-us/Robust+Portfolio+Optimization+and+Management-p-9780471921226"><i>Robust Portfolio Optimization and Management</i></a>. Hoboken, New Jersey: John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-92122-6</bdi></li>

<li><cite id="CITEREFGrinold,_RichardKahn,_Ronald1999" class="citation book cs1">Grinold, Richard; Kahn, Ronald (1999). <i>Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk</i> (2nd ed.). McGraw Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0070248823</bdi>.</cite></li>

<li><cite id="CITEREFHarvey,_CampbellRattray,_SandyVan_Hemert,Otto2021" class="citation book cs1"><a href="Campbell_Harvey" title="Campbell Harvey">Harvey, Campbell</a>; Rattray, Sandy; Van Hemert,Otto (2021). <i>Strategic Risk Management: Designing Portfolios and Managing Risk</i>. Wiley Finance. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1119773917</bdi>.</cite></li>

<li><cite id="CITEREFMaginn,_John_L.Tuttle,_Donald_L.Pinto,_Jerald_E.McLeavey,Dennis_W.2007" class="citation book cs1">Maginn, John L.; Tuttle, Donald L.; Pinto, Jerald E.; McLeavey,Dennis W. (2007). <i>Managing Investment Portfolios: A Dynamic Process</i> (3rd ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0470080146</bdi>.</cite></li>

<li><cite id="CITEREFPaleologo,_Giuseppe_A.2021" class="citation book cs1">Paleologo, Giuseppe A. (2021). <i>Advanced Portfolio Management: A Quant's Guide for Fundamental Investors</i> (1st ed.). Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1119789796</bdi>.</cite></li>

<li><cite id="CITEREFRasmussen,_M.2003" class="citation book cs1">Rasmussen, M. (2003). <i>Quantitative Portfolio Optimisation, Asset Allocation and Risk Management</i>. Palgrave Macmillan. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1403904584</bdi>.</cite></li>

<li><cite id="CITEREFSchulmerich,_MarcusLeporcher,_Yves-MichelEu,_Ching-Hwa2015" class="citation book cs1">Schulmerich, Marcus; Leporcher, Yves-Michel; Eu, Ching-Hwa (2015). <i>Applied Asset and Risk Management</i>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3642554438</bdi>.</cite></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Financial_risk_and_financial_risk_management192" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Financial_risk_and_financial_risk_management192" style="font-size:114%;margin:0 4em"><a href="Financial_risk" title="Financial risk">Financial risk</a> and <a href="Financial_risk_management" title="Financial risk management">financial risk management</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Categories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Credit_risk" title="Credit risk">Credit risk</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Consumer_credit_risk" title="Consumer credit risk">Consumer credit risk</a></li>

<li><a href="Sovereign_credit_risk" title="Sovereign credit risk">Sovereign credit risk</a></li>

<li><a href="Settlement_risk" title="Settlement risk">Settlement risk</a></li>

<li><a href="Default_(finance)" title="Default (finance)">Default risk</a></li>

<li><a href="Concentration_risk" title="Concentration risk">Concentration risk</a></li>

<li><a href="Credit_derivative" title="Credit derivative">Credit derivative</a></li>

<li><a href="Securitization" title="Securitization">Securitization</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Market_risk" title="Market risk">Market risk</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Commodity_risk" title="Commodity risk">Commodity risk</a> (e.g. <a href="Volume_risk" title="Volume risk">Volume risk</a>, <a href="Basis_risk" title="Basis risk">Basis risk</a>, <a href="Shape_risk" title="Shape risk">Shape risk</a>, <a href="Holding_period_risk" title="Holding period risk">Holding period risk</a>, <a href="Price_area_risk" class="mw-redirect" title="Price area risk">Price area risk</a>)</li>

<li><a href="Equity_risk" title="Equity risk">Equity risk</a></li>

<li><a href="Valuation_risk" title="Valuation risk">Valuation risk</a></li>

<li><a href="Margining_risk" title="Margining risk">Margining risk</a></li>

<li><a href="Interest_rate_risk" title="Interest rate risk">Interest rate risk</a></li>

<li><a href="Inflation_risk" class="mw-redirect" title="Inflation risk">Inflation risk</a></li>

<li><a href="Volatility_risk" title="Volatility risk">Volatility risk</a></li>

<li><a href="Liquidity_risk" title="Liquidity risk">Liquidity risk</a> (e.g. <a href="Refinancing_risk" title="Refinancing risk">Refinancing risk</a>, <a href="Deposit_risk" title="Deposit risk">Deposit risk</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Operational_risk" title="Operational risk">Operational risk</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Operational_risk_management" title="Operational risk management">Operational risk management</a></li>

<li><a href="Business_risk" class="mw-redirect" title="Business risk">Business risk</a></li>

<li><a href="Model_risk" title="Model risk">Model risk</a></li>

<li><a href="Reputational_risk" class="mw-redirect" title="Reputational risk">Reputational risk</a></li>

<li><a href="Country_risk" title="Country risk">Country risk</a></li>

<li><a href="Political_risk" title="Political risk">Political risk</a></li>

<li><a href="Legal_risk" title="Legal risk">Legal risk</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Execution_risk" class="mw-redirect" title="Execution risk">Execution risk</a></li>

<li><a href="Profit_risk" title="Profit risk">Profit risk</a></li>

<li><a href="Systemic_risk" title="Systemic risk">Systemic risk</a></li>

<li><a href="Non-financial_risk" title="Non-financial risk">Non-financial risk</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Financial_risk_modeling" title="Financial risk modeling">Modeling</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arbitrage_pricing_theory" title="Arbitrage pricing theory">Arbitrage pricing theory</a></li>

<li><a href="Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes model</a></li>

<li><a href="Replicating_portfolio" title="Replicating portfolio">Replicating portfolio</a></li>

<li><a href="Cashflow_matching" title="Cashflow matching">Cash flow matching</a></li>

<li><a href="Expected_shortfall" title="Expected shortfall">Conditional Value-at-Risk (CVaR)</a></li>

<li><a href="Copula_(probability_theory)" class="mw-redirect" title="Copula (probability theory)">Copula</a></li>

<li><a href="Drawdown_(economics)" title="Drawdown (economics)">Drawdown</a></li>

<li><a href="First-hitting-time_model" title="First-hitting-time model">First-hitting-time model</a></li>

<li><a href="Immunization_(finance)" title="Immunization (finance)">Interest rate immunization</a></li>

<li><a href="Market_portfolio" title="Market portfolio">Market portfolio</a></li>

<li><a href="Modern_portfolio_theory" title="Modern portfolio theory">Modern portfolio theory</a></li>

<li><a href="Omega_ratio" title="Omega ratio">Omega ratio</a></li>

<li><a href="Risk-adjusted_return_on_capital" title="Risk-adjusted return on capital">RAROC</a></li>

<li><a href="Risk_parity" title="Risk parity">Risk parity</a></li>

<li><a href="Sharpe_ratio" title="Sharpe ratio">Sharpe ratio</a></li>

<li><a href="Sortino_ratio" title="Sortino ratio">Sortino ratio</a></li>

<li><a href="Survival_analysis" title="Survival analysis">Survival analysis</a> (<a href="Proportional_hazards_model" title="Proportional hazards model">Proportional hazards model</a>)</li>

<li><a href="Tracking_error" title="Tracking error">Tracking error</a></li>

<li><a href="Value_at_risk" title="Value at risk">Value-at-Risk (VaR)</a> and extensions (<a href="Profit_at_risk" title="Profit at risk">Profit at risk</a>, <a href="Margin_at_risk" title="Margin at risk">Margin at risk</a>, <a href="Liquidity_at_risk" title="Liquidity at risk">Liquidity at risk</a>, <a href="Cash_flow_at_risk" class="mw-redirect" title="Cash flow at risk">Cash flow at risk</a>, <a href="Earnings_at_risk" title="Earnings at risk">Earnings at risk</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Asset_allocation" title="Asset allocation">Asset allocation</a></li>

<li><a href="Asset_and_liability_management" title="Asset and liability management">Asset and liability management</a></li>

<li><a href="Asset_pricing" title="Asset pricing">Asset pricing</a></li>

<li><a href="Capital_asset" title="Capital asset">Capital asset</a></li>

<li><a href="Capital_structure" title="Capital structure">Capital structure</a></li>

<li><a href="Corporate_finance" title="Corporate finance">Corporate finance</a></li>

<li><a href="Cost_of_capital" title="Cost of capital">Cost of capital</a></li>

<li><a href="Diversification_(finance)" title="Diversification (finance)">Diversification</a></li>

<li><a href="Economic_bubble" title="Economic bubble">Economic bubble</a></li>

<li><a href="Enterprise_value" title="Enterprise value">Enterprise value</a></li>

<li><a href="Exchange_traded_fund" class="mw-redirect" title="Exchange traded fund">Exchange traded fund</a></li>

<li><a href="Expected_return" title="Expected return">Expected return</a></li>

<li><a href="Finance" title="Finance">Financial</a>

<ul><li><a href="Financial_adviser" title="Financial adviser">adviser</a></li>

<li><a href="Financial_analysis" title="Financial analysis">analysis</a></li>

<li><a href="Financial_analyst" title="Financial analyst">analyst</a></li>

<li><a href="Financial_asset" title="Financial asset">asset</a></li>

<li><a href="Financial_betting" title="Financial betting">betting</a></li>

<li><a href="Financial_crime" title="Financial crime">crime</a></li>

<li><a href="Financial_engineering" title="Financial engineering">engineering</a></li>

<li><a href="Financial_social_work" title="Financial social work">social work</a></li></ul></li>

<li><a href="Fundamental_analysis" title="Fundamental analysis">Fundamental analysis</a></li>

<li><a href="Growth_investing" title="Growth investing">Growth investing</a></li>

<li><a href="Hazard" title="Hazard">Hazard</a></li>

<li><a href="Hedge_(finance)" title="Hedge (finance)">Hedge</a></li>

<li><a href="Investment_management" title="Investment management">Investment management</a></li>

<li><a href="Systematic_risk" title="Systematic risk">Systematic risk</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Moral_hazard" title="Moral hazard">Moral hazard</a></li>

<li><a href="Risk-return_spectrum" class="mw-redirect" title="Risk-return spectrum">Risk-return spectrum</a></li>

<li><a href="Speculation" title="Speculation">Speculation</a></li>

<li><a href="Speculative_attack" title="Speculative attack">Speculative attack</a></li>

<li><a href="Statistical_finance" title="Statistical finance">Statistical finance</a></li>

<li><a href="Strategic_financial_management" title="Strategic financial management">Strategic financial management</a></li>

<li><a href="Stress_test_(financial)" title="Stress test (financial)">Stress test (financial)</a></li>

<li><a href="Structured_finance" title="Structured finance">Structured finance</a></li>

<li><a href="Structured_product" title="Structured product">Structured product</a></li>

<li><a href="Systemic_risk" title="Systemic risk">Systemic risk</a></li>

<li><a href="Toxic_asset" title="Toxic asset">Toxic asset</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Financial_economics" title="Financial economics">Financial economics</a></li>

<li><a href="Investment_management" title="Investment management">Investment management</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="General_areas_of_finance28" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="General_areas_of_finance28" style="font-size:114%;margin:0 4em">General areas of <a href="Finance" title="Finance">finance</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alternative_investment" title="Alternative investment">Alternative investment</a></li>

<li><a href="Angel_investor" title="Angel investor">Angel investor</a></li>

<li><a href="Super_angel" title="Super angel">Super angel</a></li>

<li><a href="Asset_(economics)" title="Asset (economics)">Asset (economics)</a></li>

<li><a href="Asset_allocation" title="Asset allocation">Asset allocation</a></li>

<li><a href="Bond_(finance)" title="Bond (finance)">Bond (finance)</a></li>

<li><a href="Bull_(stock_market_speculator)" title="Bull (stock market speculator)">Bull (stock market speculator)</a></li>

<li><a href="Capital_appreciation" title="Capital appreciation">Asset growth</a></li>

<li><a href="Capital_asset" title="Capital asset">Capital asset</a></li>

<li><a href="Capital_management" title="Capital management">Capital management</a></li>

<li><a href="Capital_structure" title="Capital structure">Capital structure</a></li>

<li><a href="Climate_finance" title="Climate finance">Climate finance</a></li>

<li><a href="Computational_finance" title="Computational finance">Computational finance</a></li>

<li><a href="Corporate_finance" title="Corporate finance">Corporate finance</a></li>

<li><a href="Cost_of_capital" title="Cost of capital">Cost of capital</a></li>

<li><a href="Disinvestment" title="Disinvestment">Disinvestment</a></li>

<li><a href="Diversification_(finance)" title="Diversification (finance)">Diversification (finance)</a></li>

<li><a href="Divestment" title="Divestment">Divestment</a></li>

<li><a href="Eco-investing" title="Eco-investing">Eco-investing</a></li>

<li><a href="Enterprise_risk_management" title="Enterprise risk management">Enterprise risk management</a></li>

<li><a href="Environmental_finance" title="Environmental finance">Environmental finance</a></li>

<li><a href="Equity_(finance)" title="Equity (finance)">Equity (finance)</a></li>

<li><a href="Exchange-traded_fund" title="Exchange-traded fund">Exchange-traded fund</a></li>

<li><a href="Experimental_finance" title="Experimental finance">Experimental finance</a></li>

<li><a href="Finance" title="Finance">Financial</a>

<ul><li><a href="Financial_analysis" title="Financial analysis">analysis</a></li>

<li><a href="Financial_analyst" title="Financial analyst">analyst</a></li>

<li><a href="Financial_asset" title="Financial asset">asset</a></li>

<li><a href="Financial_crime" title="Financial crime">crime</a></li>

<li><a href="Financial_deepening" title="Financial deepening">deepening</a></li>

<li><a href="Financial_economics" title="Financial economics">economics</a></li>

<li><a href="Financial_engineering" title="Financial engineering">engineering</a></li>

<li><a href="Financial_inclusion" title="Financial inclusion">inclusion</a></li>

<li><a href="Financial_institution" title="Financial institution">institutions</a></li>

<li><a href="Financial_management" title="Financial management">management</a></li>

<li><a href="Financial_market" title="Financial market">market</a></li>

<li><a href="Financial_planner" title="Financial planner">planner</a></li>

<li><a href="Financial_regulation" title="Financial regulation">regulation</a></li>

<li><a href="Financial_services" title="Financial services">services</a></li>

<li><a href="Financial_social_work" title="Financial social work">social work</a></li>

<li><a href="Financial_system" title="Financial system">system</a></li></ul></li>

<li><a href="Financial_technology" title="Financial technology">Financial technology</a></li>

<li><a href="Fundamental_analysis" title="Fundamental analysis">Fundamental analysis</a></li>

<li><a href="Government_bond" title="Government bond">Government bond</a></li>

<li><a href="Greater_fool_theory" title="Greater fool theory">Greater fool theory</a></li>

<li><a href="Growth_investing" title="Growth investing">Growth investing</a></li>

<li><a href="Growth_stock" title="Growth stock">Growth stock</a></li>

<li><a href="Hedge_(finance)" title="Hedge (finance)">Hedge (finance)</a></li>

<li><a href="History_of_banking" title="History of banking">History of banking</a></li>

<li><a href="History_of_money" title="History of money">History of money</a></li>

<li><a href="Impact_investing" title="Impact investing">Impact investing</a></li>

<li><a href="International_finance" title="International finance">International finance</a></li>

<li><a href="Investment_advisory" title="Investment advisory">Investment advisory</a></li>

<li><a href="Investment_banking" title="Investment banking">Investment banking</a></li>

<li><a href="Investment_management" title="Investment management">Investment management</a></li>

<li><a href="Investment_performance" title="Investment performance">Investment performance</a></li>

<li><a href="Investor_profile" title="Investor profile">Investor profile</a></li>

<li><a href="Market_risk" title="Market risk">Market risk</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mutual_fund" title="Mutual fund">Mutual fund</a></li>

<li><a href="Over-the-counter_(finance)" title="Over-the-counter (finance)">Over-the-counter</a></li>

<li><a href="Pension_fund" title="Pension fund">Pension fund</a></li>

<li><a href="Personal_finance" title="Personal finance">Personal finance</a></li>

<li><a href="Position_of_trust" title="Position of trust">Position of trust</a></li>

<li><a href="Public_finance" title="Public finance">Public finance</a></li>

<li><a href="Quantitative_behavioral_finance" title="Quantitative behavioral finance">Quantitative behavioral finance</a></li>

<li><a href="Quantum_finance" title="Quantum finance">Quantum finance</a></li>

<li><a href="Risk%E2%80%93return_spectrum" title="Risk–return spectrum">Risk–return spectrum</a></li>

<li><a href="Social_finance" title="Social finance">Social finance</a></li>

<li><a href="Speculation" title="Speculation">Speculation</a></li>

<li><a href="Statistical_finance" title="Statistical finance">Statistical finance</a></li>

<li><a href="Stock_exchange" title="Stock exchange">Stock exchange</a></li>

<li><a href="Stockbroker" title="Stockbroker">Stockbroker</a></li>

<li><a href="Strategic_financial_management" title="Strategic financial management">Strategic financial management</a></li>

<li><a href="Statistical_finance" title="Statistical finance">Statistical finance</a></li>

<li><a href="Stock" title="Stock">Stock</a></li>

<li><a href="Structured_finance" title="Structured finance">Structured finance</a></li>

<li><a href="Structured_product" title="Structured product">Structured product</a></li>

<li><a href="Sustainability" title="Sustainability">Sustainability</a></li>

<li><a href="Sustainable_Development_Goals" title="Sustainable Development Goals">Sustainable Development Goals</a></li>

<li><a href="Sustainable_finance" title="Sustainable finance">Sustainable finance</a></li>

<li><a href="Swap_(finance)" title="Swap (finance)">Swap (finance)</a></li>

<li><a href="Systematic_risk" title="Systematic risk">Systematic risk</a></li>

<li><a href="Toxic_asset" title="Toxic asset">Toxic asset</a></li>

<li><a href="Valuation_using_discounted_cash_flows" title="Valuation using discounted cash flows">Valuation using discounted cash flows</a></li>

<li><a href="Watered_stock" title="Watered stock">Watered stock</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-09" href="https://en.wikipedia.org/wiki/?title=Portfolio_optimization&oldid=1294706248">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>