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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Return expectations while diversifying</span> </div> </a> <ul id="toc-Return_expectations_while_diversifying-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Amount_of_diversification" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Amount_of_diversification"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Amount of diversification</span> </div> </a> <ul id="toc-Amount_of_diversification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effect_of_diversification_on_variance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Effect_of_diversification_on_variance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Effect of diversification on variance</span> </div> </a> <ul id="toc-Effect_of_diversification_on_variance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diversification_with_correlated_returns_via_an_equally_weighted_portfolio" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diversification_with_correlated_returns_via_an_equally_weighted_portfolio"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Diversification with correlated returns via an equally weighted portfolio</span> </div> </a> <ul id="toc-Diversification_with_correlated_returns_via_an_equally_weighted_portfolio-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diversifiable_and_non-diversifiable_risk" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diversifiable_and_non-diversifiable_risk"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Diversifiable and non-diversifiable risk</span> </div> </a> <ul id="toc-Diversifiable_and_non-diversifiable_risk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-An_empirical_example_relating_diversification_to_risk_reduction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#An_empirical_example_relating_diversification_to_risk_reduction"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>An empirical example relating diversification to risk reduction</span> </div> </a> <ul id="toc-An_empirical_example_relating_diversification_to_risk_reduction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Corporate_diversification_strategies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Corporate_diversification_strategies"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Corporate diversification strategies</span> </div> </a> <ul id="toc-Corporate_diversification_strategies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fallacy_of_time_diversification" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fallacy_of_time_diversification"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Fallacy of time diversification</span> </div> </a> <ul id="toc-Fallacy_of_time_diversification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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mw-first-heading"><span class="mw-page-title-main">Diversification (finance)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://az.wikipedia.org/wiki/Diversifikasiya" title="Diversifikasiya – Azerbaijani" lang="az" hreflang="az" data-title="Diversifikasiya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%86%D0%B8%D1%8F_(%D1%84%D0%B8%D0%BD%D0%B0%D0%BD%D1%81%D0%B8)" title="Диверсификация (финанси) – Bulgarian" lang="bg" hreflang="bg" data-title="Диверсификация (финанси)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diverzifikace" title="Diverzifikace – Czech" lang="cs" hreflang="cs" data-title="Diverzifikace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Diversifikation_(Wirtschaft)" title="Diversifikation (Wirtschaft) – German" lang="de" hreflang="de" data-title="Diversifikation (Wirtschaft)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Diversificaci%C3%B3n" title="Diversificación – Spanish" lang="es" hreflang="es" data-title="Diversificación" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%86%D9%88%D8%B9%E2%80%8C%D8%A8%D8%AE%D8%B4%DB%8C" title="تنوع‌بخشی – Persian" lang="fa" hreflang="fa" data-title="تنوع‌بخشی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Diversification_(finance)" title="Diversification (finance) – French" lang="fr" hreflang="fr" data-title="Diversification (finance)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/%C3%89ags%C3%BAl%C3%BA_(Airgeadas)" title="Éagsúlú (Airgeadas) – Irish" lang="ga" hreflang="ga" data-title="Éagsúlú (Airgeadas)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%EA%B0%81%ED%99%94_(%EA%B8%88%EC%9C%B5)" title="다각화 (금융) – Korean" lang="ko" hreflang="ko" data-title="다각화 (금융)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%AB%D5%BE%D5%A5%D6%80%D5%BD%D5%AB%D6%86%D5%AB%D5%AF%D5%A1%D6%81%D5%AB%D5%A1" title="Դիվերսիֆիկացիա – Armenian" lang="hy" hreflang="hy" data-title="Դիվերսիֆիկացիա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Diversifikasi_(ekonomi)" title="Diversifikasi (ekonomi) – Indonesian" lang="id" hreflang="id" data-title="Diversifikasi (ekonomi)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Diversificazione_(finanza)" title="Diversificazione (finanza) – Italian" lang="it" hreflang="it" data-title="Diversificazione (finanza)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%86%D0%B8%D1%8F" title="Диверсификация – Kazakh" lang="kk" hreflang="kk" data-title="Диверсификация" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%86%D0%B8%D1%8F" title="Диверсификация – Kyrgyz" lang="ky" hreflang="ky" data-title="Диверсификация" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Diversifik%C4%81cija" title="Diversifikācija – Latvian" lang="lv" hreflang="lv" data-title="Diversifikācija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Diversifikavimas" title="Diversifikavimas – Lithuanian" lang="lt" hreflang="lt" data-title="Diversifikavimas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Diverzifik%C3%A1ci%C3%B3" title="Diverzifikáció – Hungarian" lang="hu" hreflang="hu" data-title="Diverzifikáció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Diversificatie" title="Diversificatie – Dutch" lang="nl" hreflang="nl" data-title="Diversificatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%86%E6%95%A3%E6%8A%95%E8%B3%87" title="分散投資 – Japanese" lang="ja" hreflang="ja" data-title="分散投資" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Diversifikatsiya" title="Diversifikatsiya – Uzbek" lang="uz" hreflang="uz" data-title="Diversifikatsiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dywersyfikacja" title="Dywersyfikacja – Polish" lang="pl" hreflang="pl" data-title="Dywersyfikacja" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B8%D1%84%D0%B8%D0%BA%D0%B0%D1%86%D0%B8%D1%8F" title="Диверсификация – Russian" lang="ru" hreflang="ru" data-title="Диверсификация" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Diverzifik%C3%A1cia" title="Diverzifikácia – Slovak" lang="sk" hreflang="sk" data-title="Diverzifikácia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Diversifiering" title="Diversifiering – Swedish" lang="sv" hreflang="sv" data-title="Diversifiering" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B8%D1%84%D1%96%D0%BA%D0%B0%D1%86%D1%96%D1%8F" title="Диверсифікація – Ukrainian" lang="uk" hreflang="uk" data-title="Диверсифікація" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90a_d%E1%BA%A1ng_h%C3%B3a_(%C4%91%E1%BA%A7u_t%C6%B0)" title="Đa dạng hóa (đầu tư) – Vietnamese" lang="vi" hreflang="vi" data-title="Đa dạng hóa (đầu tư)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A4%9A%E8%A7%92%E6%8A%95%E8%B3%87" title="多角投資 – Cantonese" lang="yue" hreflang="yue" data-title="多角投資" data-language-autonym="粵語" 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style="background:#e9e9ff;font-weight:normal;font-style:italic;"> General</th></tr><tr><td class="sidebar-content hlist" style="padding-left:0.5em;padding-right:0.5em;"> <ul><li><a href="/wiki/Accounting" title="Accounting">Accounting</a></li> <li><a href="/wiki/Audit" title="Audit">Audit</a></li> <li><a href="/wiki/Capital_budgeting" title="Capital budgeting">Capital budgeting</a></li> <li><a href="/wiki/Capital_structure" title="Capital structure">Capital structure</a></li> <li><a href="/wiki/Corporate_finance" title="Corporate finance">Corporate finance</a></li> <li><a href="/wiki/Credit_rating_agency" title="Credit rating agency">Credit rating agency</a></li> <li><a href="/wiki/Enterprise_risk_management" title="Enterprise risk management">Enterprise risk management</a></li> <li><a href="/wiki/Enterprise_value" title="Enterprise value">Enterprise value</a></li> <li><a href="/wiki/Financial_risk_management" title="Financial risk management">Risk management</a></li> <li><a href="/wiki/Financial_statement" title="Financial statement">Financial statements</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#e9e9ff;font-weight:normal;font-style:italic;"> Transactions</th></tr><tr><td class="sidebar-content hlist" style="padding-left:0.5em;padding-right:0.5em;"> <ul><li><a href="/wiki/Leveraged_buyout" title="Leveraged buyout">Leveraged buyout</a></li> <li><a href="/wiki/Mergers_and_acquisitions" title="Mergers and acquisitions">Mergers and acquisitions</a></li> <li><a href="/wiki/Structured_finance" title="Structured finance">Structured finance</a></li> <li><a href="/wiki/Venture_capital" title="Venture capital">Venture capital</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#e9e9ff;font-weight:normal;font-style:italic;"> <a href="/wiki/Corporate_tax" title="Corporate tax">Taxation</a></th></tr><tr><td class="sidebar-content hlist" style="padding-left:0.5em;padding-right:0.5em;"> <ul><li><a href="/wiki/Base_erosion_and_profit_shifting" title="Base erosion and profit shifting">Base erosion and profit shifting</a> (BEPS)</li> <li><a href="/wiki/Corporate_tax_haven" class="mw-redirect" title="Corporate tax haven">Corporate tax haven</a></li> <li><a href="/wiki/Tax_inversion" title="Tax inversion">Tax inversion</a></li> <li><a href="/wiki/Tax_haven" title="Tax haven">Tax haven</a></li> <li><a href="/wiki/Transfer_pricing" title="Transfer pricing">Transfer pricing</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Personal_finance" title="Personal finance">Personal</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Credit_(finance)" class="mw-redirect" title="Credit (finance)">Credit</a> / <a href="/wiki/Consumer_debt" title="Consumer 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style="background:#e9e9ff;font-weight:normal;font-style:italic;;display:block;margin-top:0.15em;"> <a href="/wiki/Government_spending" title="Government spending">Government spending</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Government_final_consumption_expenditure" title="Government final consumption expenditure">Final consumption expenditure</a></li> <li><div class="hlist"><ul><li><a href="/wiki/Government_operations" class="mw-redirect" title="Government operations">Operations</a></li><li><a href="/wiki/Redistribution_of_income_and_wealth" title="Redistribution of income and wealth">Redistribution</a></li></ul></div></li> <li><a href="/wiki/Transfer_payment" title="Transfer payment">Transfer payment</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#e9e9ff;font-weight:normal;font-style:italic;"> <a href="/wiki/Government_revenue" title="Government revenue">Government revenue</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;;padding-left:0.5em;padding-right:0.5em;"> <ul><li><div class="hlist"><ul><li><a href="/wiki/Tax" title="Tax">Taxation</a></li><li><a href="/wiki/Deficit_spending" title="Deficit spending">Deficit spending</a></li></ul></div></li> <li><div class="hlist"><ul><li><a href="/wiki/Government_budget" title="Government budget">Budget</a> (<a href="/wiki/Government_budget_balance" title="Government budget balance">balance</a>)</li><li><a href="/wiki/Government_debt" title="Government debt">Debt</a></li></ul></div></li> <li><a href="/wiki/Non-tax_revenue" title="Non-tax revenue">Non-tax revenue</a></li> <li><a href="/wiki/Warrant_of_payment" title="Warrant of payment">Warrant of payment</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Bank" title="Bank">Banking</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Central_bank" title="Central bank">Central bank</a></li> <li><a href="/wiki/Deposit_account" title="Deposit account">Deposit account</a></li> <li><a href="/wiki/Fractional-reserve_banking" title="Fractional-reserve banking">Fractional-reserve</a></li> <li><a href="/wiki/Full-reserve_banking" title="Full-reserve banking">Full-reserve</a></li> <li><a href="/wiki/Investment_banking" title="Investment banking">Investment banking</a></li> <li><a href="/wiki/Loan" title="Loan">Loan</a></li> <li><a href="/wiki/Money_supply" title="Money supply">Money supply</a></li></ul> </div> <ul><li><a href="/wiki/Lists_of_banks" title="Lists of banks">Lists of banks</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#e9e9ff;font-weight:normal;font-style:italic;;display:block;margin-top:0.15em;"> <a href="/wiki/Bank_regulation" class="mw-redirect" title="Bank regulation">Bank regulation</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Banking_license" title="Banking license">Banking license</a></li> <li><a href="/wiki/Basel_Accords" title="Basel Accords">Basel Accords</a></li> <li><a href="/wiki/Bank_for_International_Settlements" title="Bank for International Settlements">Bank for International Settlements</a></li> <li><a href="/wiki/Financial_Stability_Board" title="Financial Stability Board">Financial Stability Board</a></li> <li><a href="/wiki/Deposit_insurance" title="Deposit insurance">Deposit insurance</a></li> <li><a href="/wiki/Separation_of_investment_and_retail_banking" title="Separation of investment and retail banking">Separation of investment and retail banking</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Financial_regulation" title="Financial regulation">Regulation</a> <b>·</b> <a href="/wiki/Financial_law" title="Financial law">Financial law</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/International_Financial_Reporting_Standards" title="International Financial Reporting Standards">International Financial Reporting Standards</a></li> <li><a href="/wiki/ISO_31000" title="ISO 31000">ISO 31000</a></li> <li><a href="/wiki/Professional_certification_in_financial_services" title="Professional certification in financial services">Professional certification</a></li> <li><a href="/wiki/Fund_governance" title="Fund governance">Fund governance</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#efefef;;color: var(--color-base)"><a href="/wiki/Economic_history" title="Economic history">Economic history</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/History_of_private_equity_and_venture_capital" title="History of private equity and venture capital">Private equity and venture capital</a></li> <li><a href="/wiki/Recession" title="Recession">Recession</a></li> <li><a href="/wiki/Stock_market_bubble" title="Stock market bubble">Stock market bubble</a></li> <li><a href="/wiki/Stock_market_crash" title="Stock market crash">Stock market crash</a></li> <li><a href="/wiki/Accounting_scandals" title="Accounting scandals">Accounting scandals</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><a href="/wiki/Outline_of_finance" title="Outline of finance">Outline</a></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Emblem-money.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/16px-Emblem-money.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/24px-Emblem-money.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Emblem-money.svg/32px-Emblem-money.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span> </span><a href="/wiki/Portal:Business_and_Economics" class="mw-redirect" title="Portal:Business and Economics">Business and Economics portal</a></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Bills_and_coins.svg/16px-Bills_and_coins.svg.png" decoding="async" width="16" height="8" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Bills_and_coins.svg/24px-Bills_and_coins.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Bills_and_coins.svg/32px-Bills_and_coins.svg.png 2x" data-file-width="960" data-file-height="465" /></span></span> </span><a href="/wiki/Portal:Money" title="Portal:Money">Money portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.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}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Finance_sidebar" title="Template:Finance sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Finance_sidebar" title="Template talk:Finance sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Finance_sidebar" title="Special:EditPage/Template:Finance sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Finance" title="Finance">finance</a>, <b>diversification</b> is the process of allocating capital in a way that reduces the exposure to any one particular asset or risk. A common path towards diversification is to reduce <a href="/wiki/Financial_risk" title="Financial risk">risk</a> or <a href="/wiki/Volatility_(finance)" title="Volatility (finance)">volatility</a> by <a href="/wiki/Investment" title="Investment">investing</a> in a variety of <a href="/wiki/Asset" title="Asset">assets</a>. If asset prices do not change in perfect synchrony, a diversified <a href="/wiki/Portfolio_(finance)" title="Portfolio (finance)">portfolio</a> will have less variance than the <a href="/wiki/Weighted_mean" class="mw-redirect" title="Weighted mean">weighted average</a> variance of its constituent assets, and often less volatility than the least volatile of its constituents.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Diversification is one of two general techniques for reducing investment risk. The other is <a href="/wiki/Hedge_(finance)" title="Hedge (finance)">hedging</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest example of diversification is provided by the proverb "<b>Don't put all your eggs in one basket</b>". Dropping the basket will break all the eggs. Placing each egg in a different basket is more diversified. There is more risk of losing one egg, but less risk of losing all of them. On the other hand, having a lot of baskets may increase costs. </p><p>In finance, an example of an undiversified portfolio is to hold only one stock. This is risky; it is not unusual for a single stock to go down 50% in one year. It is less common for a portfolio of 20 stocks to go down that much, especially if they are selected at random. If the stocks are selected from a variety of industries, company sizes and asset types it is even less likely to experience a 50% drop since it will mitigate any trends in that industry, company class, or asset type. </p><p>Since the mid-1970s, it has also been argued that geographic diversification would generate superior risk-adjusted returns for large <a href="/wiki/Institutional_investor" title="Institutional investor">institutional investors</a> by reducing overall portfolio risk while capturing some of the higher rates of return offered by the <a href="/wiki/Emerging_markets" class="mw-redirect" title="Emerging markets">emerging markets</a> of Asia and Latin America.<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><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> </p> <div class="mw-heading mw-heading2"><h2 id="Return_expectations_while_diversifying">Return expectations while diversifying</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=2" title="Edit section: Return expectations while diversifying"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the prior <a href="/wiki/Expected_value" title="Expected value">expectations</a> of the returns on all assets in the portfolio are identical, the <a href="/wiki/Expected_return" title="Expected return">expected return</a> on a diversified portfolio will be identical to that on an undiversified portfolio. Some assets will do better than others; but since one does not know in advance which assets will perform better, this fact cannot be exploited in advance. The return on a diversified portfolio can never exceed that of the top-performing investment, and indeed will always be lower than the highest return (unless all returns are identical). Conversely, the diversified portfolio's return will always be higher than that of the worst-performing investment. So by diversifying, one loses the chance of having invested solely in the single asset that comes out best, but one also avoids having invested solely in the asset that comes out worst. That is the role of diversification: it narrows the range of possible outcomes. Diversification need not either help or hurt expected returns, unless the alternative non-diversified portfolio has a higher expected return.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Amount_of_diversification">Amount of diversification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=3" title="Edit section: Amount of diversification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is no magic number of stocks that is diversified versus not. Sometimes quoted is 30, although it can be as low as 10, provided they are carefully chosen. This is based on a result from John Evans and Stephen Archer.<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> Similarly, a 1985 book reported that most value from diversification comes from the first 15 or 20 different stocks in a portfolio.<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> More stocks give lower price volatility. </p><p>Given the advantages of diversification, many experts <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag possibly uses too-vague attribution or weasel words. (March 2014)">who?</span></a></i>]</sup> recommend maximum diversification, also known as "buying the <a href="/wiki/Market_portfolio" title="Market portfolio">market portfolio</a>". Identifying that portfolio is not straightforward. The earliest definition comes from the <a href="/wiki/Capital_asset_pricing_model" title="Capital asset pricing model">capital asset pricing model</a> which argues the maximum diversification comes from buying a <i>pro rata</i> share of all available <a href="/wiki/Asset" title="Asset">assets</a>. This is the idea underlying <a href="/wiki/Index_fund" title="Index fund">index funds</a>. </p><p>Diversification has no maximum so long as more assets are available.<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> Every equally weighted, uncorrelated asset added to a portfolio can add to that portfolio's measured diversification. When assets are not uniformly uncorrelated, a weighting approach that puts assets in proportion to their relative correlation can maximize the available diversification. </p><p>"Risk parity" is an alternative idea. This weights assets in inverse proportion to risk, so the portfolio has equal risk in all asset classes. This is justified both on theoretical grounds, and with the pragmatic argument that future risk is much easier to forecast than either future market price or future economic footprint.<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> "Correlation parity" is an extension of risk parity, and is the solution whereby each asset in a portfolio has an equal correlation with the portfolio, and is therefore the "most diversified portfolio". Risk parity is the special case of correlation parity when all pair-wise correlations are equal.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Effect_of_diversification_on_variance">Effect of diversification on variance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=4" title="Edit section: Effect of diversification on variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One simple measure of <a href="/wiki/Financial_risk" title="Financial risk">financial risk</a> is <a href="/wiki/Variance" title="Variance">variance</a> of the <a href="/wiki/Rate_of_return" title="Rate of return">return</a> on the portfolio. Diversification can lower the variance of a portfolio's return below what it would be if the entire portfolio were invested in the asset with the lowest variance of return, even if the assets' returns are uncorrelated. For example, let asset X have stochastic return <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and asset Y have stochastic return <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>, with respective return variances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0340dabe0c0edc91ba97a6169d9defbe3fa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.843ex;" alt="{\displaystyle \sigma _{x}^{2}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690d699292b74ee1b030d5720a17595baaea5227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\displaystyle \sigma _{y}^{2}}"></span>. If the fraction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> of a one-unit (e.g. one-million-dollar) portfolio is placed in asset X and the fraction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ccb1d1caf362976d80524abd2c5b3d9e58b1344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.072ex; height:2.509ex;" alt="{\displaystyle 1-q}"></span> is placed in Y, the stochastic portfolio return is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qx+(1-q)y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qx+(1-q)y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c0dbb7ee7946876bbf4d91707fdb91f0d6045e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.277ex; height:2.843ex;" alt="{\displaystyle qx+(1-q)y}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> are uncorrelated, the variance of portfolio return is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{var}}(qx+(1-q)y)=q^{2}\sigma _{x}^{2}+(1-q)^{2}\sigma _{y}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>var</mtext> </mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{var}}(qx+(1-q)y)=q^{2}\sigma _{x}^{2}+(1-q)^{2}\sigma _{y}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bff15f024c708d3c7ac4f9a15370986922b0a454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.281ex; height:3.343ex;" alt="{\displaystyle {\text{var}}(qx+(1-q)y)=q^{2}\sigma _{x}^{2}+(1-q)^{2}\sigma _{y}^{2}}"></span>. The variance-minimizing value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167455d2a2b370630091fb1522c633b3ce158d29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.734ex; height:3.176ex;" alt="{\displaystyle q=\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]}"></span>, which is strictly between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>. Using this value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> in the expression for the variance of portfolio return gives the latter as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}^{2}\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}^{2}\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21332ce10f785024bf50efeb295fa11d4a4863c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.066ex; height:3.176ex;" alt="{\displaystyle \sigma _{x}^{2}\sigma _{y}^{2}/[\sigma _{x}^{2}+\sigma _{y}^{2}]}"></span>, which is less than what it would be at either of the undiversified values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/785938d022f0b0b0bf4b3afa5e1cedceab7a3874" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654c2d5dc1a26e0af36dc0deb5fd252c6178977a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q=0}"></span> (which respectively give portfolio return variance of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0340dabe0c0edc91ba97a6169d9defbe3fa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.843ex;" alt="{\displaystyle \sigma _{x}^{2}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690d699292b74ee1b030d5720a17595baaea5227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\displaystyle \sigma _{y}^{2}}"></span>). Note that the favorable effect of diversification on portfolio variance would be enhanced if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> were negatively correlated but diminished (though not eliminated) if they were positively correlated. </p><p>In general, the presence of more assets in a portfolio leads to greater diversification benefits, as can be seen by considering portfolio variance as a function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, the number of assets. For example, if all assets' returns are mutually uncorrelated and have identical variances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0340dabe0c0edc91ba97a6169d9defbe3fa1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.843ex;" alt="{\displaystyle \sigma _{x}^{2}}"></span>, portfolio variance is minimized by holding all assets in the equal proportions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e10667bad240500f5044257143510127e03d69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.72ex; height:2.843ex;" alt="{\displaystyle 1/n}"></span>.<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> Then the portfolio return's variance equals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{var}}[(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>var</mtext> </mrow> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{var}}[(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b9478b1bbc704b139c3a71be97bee1c0418c37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.056ex; height:2.843ex;" alt="{\displaystyle {\text{var}}[(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n}]}"></span> = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(1/n^{2})\sigma _{x}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(1/n^{2})\sigma _{x}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1a8b547a856e65b708fa0e86c172de93aca952" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.478ex; height:3.176ex;" alt="{\displaystyle n(1/n^{2})\sigma _{x}^{2}}"></span> = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}^{2}/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}^{2}/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43aaf952cd185d51ede49afed2ced12ecc85cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.057ex; height:3.009ex;" alt="{\displaystyle \sigma _{x}^{2}/n}"></span>, which is monotonically decreasing in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. </p><p>The latter analysis can be adapted to show why <i>adding</i> uncorrelated volatile assets to a portfolio,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><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> thereby increasing the portfolio's size, is not diversification, which involves subdividing the portfolio among many smaller investments. In the case of adding investments, the portfolio's return is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}+x_{2}+\dots +x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}+x_{2}+\dots +x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb1d7cb59a14495aeced61520c5619f03df1a57c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.56ex; height:2.343ex;" alt="{\displaystyle x_{1}+x_{2}+\dots +x_{n}}"></span> instead of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bffe4ecdf33561377995413add7619e59dac8214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.108ex; height:2.843ex;" alt="{\displaystyle (1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n},}"></span> and the variance of the portfolio return if the assets are uncorrelated is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{var}}[x_{1}+x_{2}+\dots +x_{n}]=\sigma _{x}^{2}+\sigma _{x}^{2}+\dots +\sigma _{x}^{2}=n\sigma _{x}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>var</mtext> </mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>n</mi> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{var}}[x_{1}+x_{2}+\dots +x_{n}]=\sigma _{x}^{2}+\sigma _{x}^{2}+\dots +\sigma _{x}^{2}=n\sigma _{x}^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/633222056c8ff242c4da0fc1c0b65212a1906500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.638ex; height:3.009ex;" alt="{\displaystyle {\text{var}}[x_{1}+x_{2}+\dots +x_{n}]=\sigma _{x}^{2}+\sigma _{x}^{2}+\dots +\sigma _{x}^{2}=n\sigma _{x}^{2},}"></span> which is <i>increasing</i> in <i>n</i> rather than decreasing. Thus, for example, when an insurance company adds more and more uncorrelated policies to its portfolio, this expansion does not itself represent diversification—the diversification occurs in the spreading of the insurance company's risks over a large number of part-owners of the company. </p> <div class="mw-heading mw-heading2"><h2 id="Diversification_with_correlated_returns_via_an_equally_weighted_portfolio">Diversification with correlated returns via an equally weighted portfolio</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=5" title="Edit section: Diversification with correlated returns via an equally weighted portfolio"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The expected return on a portfolio is a weighted average of the expected returns on each individual asset: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [R_{P}]=\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [R_{P}]=\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f892f670bf130a4a19c1a0cdb28f929af85e3ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.452ex; height:6.843ex;" alt="{\displaystyle \mathbb {E} [R_{P}]=\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> is the proportion of the investor's total invested wealth in asset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>. </p><p>The variance of the portfolio return is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \underbrace {{\text{Var}}(R_{P})} _{\equiv \sigma _{P}^{2}}=\mathbb {E} [R_{P}-\mathbb {E} [R_{P}]]^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Var</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≡</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </munder> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">]</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {{\text{Var}}(R_{P})} _{\equiv \sigma _{P}^{2}}=\mathbb {E} [R_{P}-\mathbb {E} [R_{P}]]^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76570ac1713f93bcd4ff96aa3cea7a6a7103841b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:28.646ex; height:7.509ex;" alt="{\displaystyle \underbrace {{\text{Var}}(R_{P})} _{\equiv \sigma _{P}^{2}}=\mathbb {E} [R_{P}-\mathbb {E} [R_{P}]]^{2}.}"></span></dd></dl> <p>Inserting in the expression for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [R_{P}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [R_{P}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d914f1024fb3a51f8f3e145e075dd198894df76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.074ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [R_{P}]}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}R_{i}-\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]\right]^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}R_{i}-\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]\right]^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a583effdf9099092fc5a905410ce07a0211836a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.409ex; height:8.009ex;" alt="{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}R_{i}-\sum _{i=1}^{n}x_{i}\mathbb {E} [R_{i}]\right]^{2}.}"></span></dd></dl> <p>Rearranging: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}(R_{i}-\mathbb {E} [R_{i}])\right]^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}(R_{i}-\mathbb {E} [R_{i}])\right]^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5821da5b5e1c3cad0565e10bc6ab5ab158e04b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.7ex; height:8.009ex;" alt="{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}(R_{i}-\mathbb {E} [R_{i}])\right]^{2}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}\sum _{j=1}^{n}x_{i}x_{j}(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}\sum _{j=1}^{n}x_{i}x_{j}(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a600910fc58ea5d8f8f92dd7e442f46329e50e40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.855ex; height:7.676ex;" alt="{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}\sum _{j=1}^{n}x_{i}x_{j}(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}^{2}(R_{i}-\mathbb {E} [R_{i}])^{2}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}^{2}(R_{i}-\mathbb {E} [R_{i}])^{2}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f02954db2f51380ea1fd6626bcaeb7370a584d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:72.901ex; height:7.843ex;" alt="{\displaystyle \sigma _{P}^{2}=\mathbb {E} \left[\sum _{i=1}^{n}x_{i}^{2}(R_{i}-\mathbb {E} [R_{i}])^{2}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P}^{2}=\sum _{i=1}^{n}x_{i}^{2}\underbrace {\mathbb {E} \left[R_{i}-\mathbb {E} [R_{i}]\right]^{2}} _{\equiv \sigma _{i}^{2}}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}\underbrace {\mathbb {E} \left[(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]} _{\equiv \sigma _{ij}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <msup> <mrow> <mo>[</mo> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≡</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </munder> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≡</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P}^{2}=\sum _{i=1}^{n}x_{i}^{2}\underbrace {\mathbb {E} \left[R_{i}-\mathbb {E} [R_{i}]\right]^{2}} _{\equiv \sigma _{i}^{2}}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}\underbrace {\mathbb {E} \left[(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]} _{\equiv \sigma _{ij}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed383b814403b1b18d1e8efbfeaae43d153803f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; margin-right: -0.028ex; width:72.547ex; height:9.009ex;" alt="{\displaystyle \sigma _{P}^{2}=\sum _{i=1}^{n}x_{i}^{2}\underbrace {\mathbb {E} \left[R_{i}-\mathbb {E} [R_{i}]\right]^{2}} _{\equiv \sigma _{i}^{2}}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}\underbrace {\mathbb {E} \left[(R_{i}-\mathbb {E} [R_{i}])(R_{j}-\mathbb {E} [R_{j}])\right]} _{\equiv \sigma _{ij}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P}^{2}=\sum _{i=1}^{n}x_{i}^{2}\sigma _{i}^{2}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}\sigma _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P}^{2}=\sum _{i=1}^{n}x_{i}^{2}\sigma _{i}^{2}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}\sigma _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/226327008585ef7a0f0c9e42591453fe0e1e851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.305ex; height:7.343ex;" alt="{\displaystyle \sigma _{P}^{2}=\sum _{i=1}^{n}x_{i}^{2}\sigma _{i}^{2}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}x_{i}x_{j}\sigma _{ij}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598d48f2fb8e418938d86428befb815f50096bdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\displaystyle \sigma _{i}^{2}}"></span> is the variance on asset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43acbf52cc4d4f83f187ceaa49f045114b71772e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.804ex; height:2.343ex;" alt="{\displaystyle \sigma _{ij}}"></span> is the covariance between assets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>. </p><p>In an equally weighted portfolio, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=x_{j}={\frac {1}{n}},\forall i,j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=x_{j}={\frac {1}{n}},\forall i,j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fba3e26635971ae9b472ee7ec41359365a69d763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.917ex; height:5.176ex;" alt="{\displaystyle x_{i}=x_{j}={\frac {1}{n}},\forall i,j}"></span>. The portfolio variance then becomes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P}^{2}={\frac {1}{n^{2}}}\ {n}{\bar {\sigma }}_{i}^{2}+n(n-1){\frac {1}{n}}{\frac {1}{n}}{\bar {\sigma }}_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P}^{2}={\frac {1}{n^{2}}}\ {n}{\bar {\sigma }}_{i}^{2}+n(n-1){\frac {1}{n}}{\frac {1}{n}}{\bar {\sigma }}_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d846fb330f3410a4b68a83e4e91ccc6d7c4e4a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.247ex; height:5.509ex;" alt="{\displaystyle \sigma _{P}^{2}={\frac {1}{n^{2}}}\ {n}{\bar {\sigma }}_{i}^{2}+n(n-1){\frac {1}{n}}{\frac {1}{n}}{\bar {\sigma }}_{ij}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\sigma }}_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\sigma }}_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce2240079c5f51e3f95f84e155a5338d535e131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.807ex; height:2.676ex;" alt="{\displaystyle {\bar {\sigma }}_{ij}}"></span> is the average of the covariances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43acbf52cc4d4f83f187ceaa49f045114b71772e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.804ex; height:2.343ex;" alt="{\displaystyle \sigma _{ij}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\neq j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\neq j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95aeb406bb427ac96806bc00c30c91d31b858be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.676ex;" alt="{\displaystyle i\neq j}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\sigma }}_{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\sigma }}_{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422cc721281f8b8189dae4e1c7ebba01560f4829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:3.176ex;" alt="{\displaystyle {\bar {\sigma }}_{i}^{2}}"></span> is the average of the variances. Simplifying, we obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P}^{2}={\frac {1}{n}}{\bar {\sigma }}_{i}^{2}+{\frac {n-1}{n}}{\bar {\sigma }}_{ij}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P}^{2}={\frac {1}{n}}{\bar {\sigma }}_{i}^{2}+{\frac {n-1}{n}}{\bar {\sigma }}_{ij}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c829498663477579b723022ce761da99da30dd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.035ex; height:5.176ex;" alt="{\displaystyle \sigma _{P}^{2}={\frac {1}{n}}{\bar {\sigma }}_{i}^{2}+{\frac {n-1}{n}}{\bar {\sigma }}_{ij}.}"></span></dd></dl> <p>As the number of assets grows we get the asymptotic formula: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\rightarrow \infty }\sigma _{P}^{2}={\bar {\sigma }}_{ij}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }\sigma _{P}^{2}={\bar {\sigma }}_{ij}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd544980b0b1b7b139ba61df6a4433a5d3b72ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.006ex; height:4.009ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }\sigma _{P}^{2}={\bar {\sigma }}_{ij}.}"></span></dd></dl> <p>Thus, in an equally weighted portfolio, the portfolio variance tends to the average of covariances between securities as the number of securities becomes arbitrarily large. </p> <div class="mw-heading mw-heading2"><h2 id="Diversifiable_and_non-diversifiable_risk">Diversifiable and non-diversifiable risk</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=6" title="Edit section: Diversifiable and non-diversifiable risk"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Capital_asset_pricing_model" title="Capital asset pricing model">capital asset pricing model</a> introduced the concepts of diversifiable and non-diversifiable risk. Synonyms for diversifiable risk are idiosyncratic risk, unsystematic risk, and security-specific risk. Synonyms for non-diversifiable risk are <a href="/wiki/Systematic_risk" title="Systematic risk">systematic risk</a>, <a href="/wiki/Beta_(finance)" title="Beta (finance)">beta</a> risk and <a href="/wiki/Market_risk" title="Market risk">market risk</a>. </p><p>If one buys all the stocks in the <a href="/wiki/S%26P_500" title="S&P 500">S&P 500</a> one is obviously exposed only to movements in that <a href="/wiki/Index_(economics)" title="Index (economics)">index</a>. If one buys a single stock in the S&P 500, one is exposed both to index movements and movements in the stock based on its underlying company. The first risk is called "non-diversifiable", because it exists however many S&P 500 stocks are bought. The second risk is called "diversifiable", because it can be reduced by diversifying among stocks. </p><p>In the presence of per-asset investment fees, there is also the possibility of <a href="/w/index.php?title=Overdiversifying&action=edit&redlink=1" class="new" title="Overdiversifying (page does not exist)">overdiversifying</a> to the point that the portfolio's performance will suffer because the fees outweigh the gains from diversification. </p><p>The capital asset pricing model argues that investors should only be compensated for non-diversifiable risk. Other financial models allow for multiple sources of non-diversifiable risk, but also insist that diversifiable risk should not carry any extra expected return. Still other models do not accept this contention.<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> </p> <div class="mw-heading mw-heading2"><h2 id="An_empirical_example_relating_diversification_to_risk_reduction">An empirical example relating diversification to risk reduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=7" title="Edit section: An empirical example relating diversification to risk reduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1977 <a href="/wiki/Ed_Elton" class="mw-redirect" title="Ed Elton">Edwin Elton</a> and <a href="/wiki/Martin_J._Gruber" title="Martin J. Gruber">Martin Gruber</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> worked out an empirical example of the gains from diversification. Their approach was to consider a population of 3,290 securities available for possible inclusion in a portfolio, and to consider the average risk over all possible randomly chosen <i>n</i>-asset portfolios with equal amounts held in each included asset, for various values of <i>n</i>. Their results are summarized in the following table. </p><p>The result for <i>n</i>=30 is close to <i>n</i>=1,000, and even four stocks provide most of the reduction in risk compared with one stock. </p> <table class="wikitable"> <tbody><tr> <th>Number of Stocks in Portfolio</th> <th>Average Standard Deviation of Annual Portfolio Returns</th> <th>Ratio of Portfolio Standard Deviation to Standard Deviation of a Single Stock </th></tr> <tr> <td width="90">1</td> <td width="150">49.24%</td> <td width="200">1.00 </td></tr> <tr> <td>2</td> <td>37.36</td> <td>0.76 </td></tr> <tr> <td>4</td> <td>29.69</td> <td>0.60 </td></tr> <tr> <td>6</td> <td>26.64</td> <td>0.54 </td></tr> <tr> <td>8</td> <td>24.98</td> <td>0.51 </td></tr> <tr> <td>10</td> <td>23.93</td> <td>0.49 </td></tr> <tr> <td>20</td> <td>21.68</td> <td>0.44 </td></tr> <tr> <td>30</td> <td>20.87</td> <td>0.42 </td></tr> <tr> <td>40</td> <td>20.46</td> <td>0.42 </td></tr> <tr> <td>50</td> <td>20.20</td> <td>0.41 </td></tr> <tr> <td>400</td> <td>19.29</td> <td>0.39 </td></tr> <tr> <td>500</td> <td>19.27</td> <td>0.39 </td></tr> <tr> <td>1,000</td> <td>19.21</td> <td>0.39 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Corporate_diversification_strategies">Corporate diversification strategies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=8" title="Edit section: Corporate diversification strategies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In corporate portfolio models, diversification is thought of as being vertical or horizontal. Horizontal diversification is thought of as expanding a product line or acquiring related companies. Vertical diversification is synonymous with integrating the supply chain or amalgamating distributions channels. </p><p>Non-incremental diversification is a strategy followed by conglomerates, where the individual business lines have little to do with one another, yet the company is attaining diversification from exogenous risk factors to stabilize and provide opportunity for active management of diverse resources. </p> <div class="mw-heading mw-heading2"><h2 id="Fallacy_of_time_diversification">Fallacy of time diversification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=9" title="Edit section: Fallacy of time diversification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The argument is often made that time reduces variance in a portfolio: a "time diversification". A common belief is younger investors should avoid bonds and emphasize stocks, due to the belief investors will have time to recover from any downturns. Yet this belief has flaws, as John Norstad explains: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>This kind of statement makes the implicit assumption that given enough time good returns will cancel out any possible bad returns. While the basic argument that the standard deviations of the annualized returns decrease as the time horizon increases is true, it is also misleading, and it fatally misses the point, because for an investor concerned with the value of his portfolio at the end of a period of time, it is the total return that matters, not the annualized return. Because of the effects of compounding, the standard deviation of the total return actually increases with time horizon. Thus, if we use the traditional measure of uncertainty as the standard deviation of return over the time period in question, uncertainty increases with time.<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></blockquote> <p>Three notable contributions to the literature on the fallacy of time diversification have been from <a href="/wiki/Paul_Samuelson" title="Paul Samuelson">Paul Samuelson</a>,<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> <a href="/wiki/Zvi_Bodie" title="Zvi Bodie">Zvi Bodie</a>,<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> and Mark Kritzman.<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> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=10" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Diversification is mentioned in the <a href="/wiki/Bible" title="Bible">Bible</a>, in the book of <a href="/wiki/Ecclesiastes" title="Ecclesiastes">Ecclesiastes</a> which was written in approximately 935 B.C.:<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> </p> <dl><dd>But divide your investments among many places,</dd> <dd>for you do not know what risks might lie ahead.<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></dd></dl> <p>Diversification is also mentioned in the <a href="/wiki/Talmud" title="Talmud">Talmud</a>. The formula given there is to split one's assets into thirds: one third in business (buying and selling things), one third kept liquid (e.g. gold coins), and one third in land (<a href="/wiki/Real_estate" title="Real estate">real estate</a>). This strategy of splitting wealth equally among available options is now known as "naive diversification", "Talmudic diversification" or "1/n diversification", a concept which has earned renewed attention since the year 2000 due to research showing it may offer advantages in some scenarios.<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><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> </p><p>Diversification is mentioned in Shakespeare's <i><a href="/wiki/Merchant_of_Venice" class="mw-redirect" title="Merchant of Venice">Merchant of Venice</a></i> (ca. 1599):<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> </p> <dl><dd>My ventures are not in one bottom trusted,</dd> <dd>Nor to one place; nor is my whole estate</dd> <dd>Upon the fortune of this present year:</dd> <dd>Therefore, my merchandise makes me not sad.</dd></dl> <p>Modern understanding of diversification dates back to the influential work of economist <a href="/wiki/Harry_Markowitz" title="Harry Markowitz">Harry Markowitz</a> in the 1950s,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> whose work pioneered <a href="/wiki/Modern_portfolio_theory" title="Modern portfolio theory">modern portfolio theory</a> (see <a href="/wiki/Markowitz_model" title="Markowitz model">Markowitz model</a>). </p><p>An earlier precedent for diversification was economist <a href="/wiki/John_Maynard_Keynes" title="John Maynard Keynes">John Maynard Keynes</a>, who managed the endowment of <a href="/wiki/King%27s_College,_Cambridge" title="King's College, Cambridge">King's College, Cambridge</a> from the 1920s to his 1946 death with a stock-selection strategy similar to what was later called <a href="/wiki/Value_investing" title="Value investing">value investing</a>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> While diversification in the modern sense was "not easily available in Keynes's day"<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> and Keynes typically held a small number of assets compared to later investment theories, he nonetheless is recognized as a pioneer of financial diversification. Keynes came to recognize the importance, "if possible", he wrote, of holding assets with "opposed risks [...] since they are likely to move in opposite directions when there are general fluctuations"<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Keynes was a pioneer of "international diversification" due to substantial holdings in non-U.K. stocks, up to 75%, and avoiding <a href="/wiki/Equity_home_bias_puzzle" title="Equity home bias puzzle">home bias</a> at a time when university endowments in the U.S. and U.K. were invested almost entirely in domestic assets.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Asset_allocation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Asset_allocation.png/230px-Asset_allocation.png" decoding="async" width="230" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Asset_allocation.png/345px-Asset_allocation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Asset_allocation.png/460px-Asset_allocation.png 2x" data-file-width="543" data-file-height="287" /></a><figcaption><b><a href="https://en.wikibooks.org/wiki/Asset_Allocation" class="extiw" title="b:Asset Allocation">Asset Allocation on Wikibook</a></b></figcaption></figure> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Coherent_risk_measure" title="Coherent risk measure">Coherent risk measure</a></li> <li><a href="/wiki/Dollar_cost_averaging" title="Dollar cost averaging">Dollar cost averaging</a></li> <li><a href="/wiki/Equity_repositioning" title="Equity repositioning">Equity repositioning</a></li> <li><a href="/wiki/Financial_correlation" title="Financial correlation">Financial correlation</a></li> <li><a href="/wiki/Outline_of_finance" title="Outline of finance">Outline of finance</a></li> <li><a href="/wiki/Modern_portfolio_theory" title="Modern portfolio theory">Modern portfolio theory</a></li> <li><a href="/wiki/Systematic_risk" title="Systematic risk">Systematic risk</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.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("//upload.wikimedia.org/wikipedia/commons/6/65/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("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/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}}</style><cite id="CITEREFO'SullivanSheffrin2003" class="citation book cs1"><a href="/wiki/Arthur_O%27Sullivan_(economist)" title="Arthur O'Sullivan (economist)">O'Sullivan, Arthur</a>; <a href="/wiki/Steven_M._Sheffrin" title="Steven M. Sheffrin">Sheffrin, Steven M.</a> (2003). <i>Economics: Principles in Action</i>. Upper Saddle River, New Jersey: Pearson Prentice Hall. p. 273. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-063085-3" title="Special:BookSources/0-13-063085-3"><bdi>0-13-063085-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Economics%3A+Principles+in+Action&rft.place=Upper+Saddle+River%2C+New+Jersey&rft.pages=273&rft.pub=Pearson+Prentice+Hall&rft.date=2003&rft.isbn=0-13-063085-3&rft.aulast=O%27Sullivan&rft.aufirst=Arthur&rft.au=Sheffrin%2C+Steven+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversification+%28finance%29" class="Z3988"></span></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"><span class="languageicon">(in French)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100508172054/http://canadianeuropean.com/yahoo_site_admin/assets/docs/FONDS_DASIE-PACIFIQUE_REVUE_AF_APR_09.95131642.pdf">"see M. Nicolas J. Firzli, "Asia-Pacific Funds as Diversification Tools for Institutional Investors", Revue Analyse Financière/The French Society of Financial Analysts (SFAF)"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://www.canadianeuropean.com/yahoo_site_admin/assets/docs/FONDS_DASIE-PACIFIQUE_REVUE_AF_APR_09.95131642.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2010-05-08<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-04-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=see+M.+Nicolas+J.+Firzli%2C+%22Asia-Pacific+Funds+as+Diversification+Tools+for+Institutional+Investors%22%2C+Revue+Analyse+Financi%C3%A8re%2FThe+French+Society+of+Financial+Analysts+%28SFAF%29&rft_id=http%3A%2F%2Fwww.canadianeuropean.com%2Fyahoo_site_admin%2Fassets%2Fdocs%2FFONDS_DASIE-PACIFIQUE_REVUE_AF_APR_09.95131642.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversification+%28finance%29" class="Z3988"></span></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"><span class="languageicon">(in English)</span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.insead.edu/facultyresearch/centres/global_private_equity_initiative/publications/documents/INSEAD_AsiaPEReport_long.pdf">"see Michael Prahl, "Asian Private Equity – Will it Deliver on its Promise?", INSEAD Global Private Equity Initiative (GPEI)"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2011-06-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=see+Michael+Prahl%2C+%22Asian+Private+Equity+%E2%80%93+Will+it+Deliver+on+its+Promise%3F%22%2C+INSEAD+Global+Private+Equity+Initiative+%28GPEI%29&rft_id=http%3A%2F%2Fwww.insead.edu%2Ffacultyresearch%2Fcentres%2Fglobal_private_equity_initiative%2Fpublications%2Fdocuments%2FINSEAD_AsiaPEReport_long.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversification+%28finance%29" class="Z3988"></span></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">Goetzmann, William N. <a rel="nofollow" class="external text" href="http://viking.som.yale.edu/will/finman540/classnotes/class2.html">An Introduction to Investment Theory</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170331163028/http://viking.som.yale.edu/will/finman540/classnotes/class2.html">Archived</a> 2017-03-31 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. II. Portfolios of Assets. Retrieved on November 20, 2008.</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"><a rel="nofollow" class="external text" href="http://financialfuse.com/investment-guide-beginners-introduction/">Investment Guide Beginners Introduction</a></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_LoriePeter_DoddMary_Kimpton1985" class="citation book cs1">James Lorie; Peter Dodd; Mary Kimpton (1985). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/stockmarkettheor0000lori"><i>The Stock Market: Theories and Evidence</i></a></span> (2nd ed.). Dow Jones-Irwin. p. <a rel="nofollow" class="external text" href="https://archive.org/details/stockmarkettheor0000lori/page/85">85</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780870946189" title="Special:BookSources/9780870946189"><bdi>9780870946189</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Stock+Market%3A+Theories+and+Evidence&rft.pages=85&rft.edition=2nd&rft.pub=Dow+Jones-Irwin&rft.date=1985&rft.isbn=9780870946189&rft.au=James+Lorie&rft.au=Peter+Dodd&rft.au=Mary+Kimpton&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstockmarkettheor0000lori&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversification+%28finance%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStatman1987" class="citation journal cs1">Statman, Meir (1987). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2330969">"How Many Stocks Make a Diversified Portfolio?"</a>. <i>The Journal of Financial and Quantitative Analysis</i>. <b>22</b> (3): 353–363. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2330969">10.2307/2330969</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2330969">2330969</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Financial+and+Quantitative+Analysis&rft.atitle=How+Many+Stocks+Make+a+Diversified+Portfolio%3F&rft.volume=22&rft.issue=3&rft.pages=353-363&rft.date=1987&rft_id=info%3Adoi%2F10.2307%2F2330969&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2330969%23id-name%3DJSTOR&rft.aulast=Statman&rft.aufirst=Meir&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2330969&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversification+%28finance%29" class="Z3988"></span></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">Asness, Cliff; <a href="/wiki/David_Kabiller" title="David Kabiller">David Kabiller</a> and Michael Mendelson <a rel="nofollow" class="external text" href="http://www.iimagazine.com/Popups/PrintArticle.aspx?ArticleID=2486929">Using Derivatives and Leverage To Improve Portfolio Performance, Institutional Investor, May 13, 2010</a>. Retrieved on June 21, 2010.</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">Schoen, Robert <a rel="nofollow" class="external text" href="https://www.putnam.com/literature/pdf/SU830.pdf">Parity Strategies and Maximum Diversification, Putnam Investments, June, 2013</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150402181913/https://www.putnam.com/literature/pdf/SU830.pdf">Archived</a> 2015-04-02 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</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">Samuelson, Paul, <a rel="nofollow" class="external text" href="https://www.gwern.net/docs/statistics/decision/1967-samuelson.pdf">"General Proof that Diversification Pays"</a>, <i>Journal of Financial and Quantitative Analysis</i> 2, March 1967, 1-13.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Samuelson, Paul, <a rel="nofollow" class="external text" href="https://www.casact.org/pubs/forum/94sforum/94sf049.pdf">"Risk and uncertainty: A fallacy of large numbers"</a>, <i>Scientia</i> 98, 1963, 108-113.</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">Ross, Stephen, <a rel="nofollow" class="external text" href="https://www.gwern.net/docs/statistics/decision/1999-ross.pdf">"Adding risks: Samuelson's fallacy of large numbers revisited"</a> <i><a href="/wiki/Journal_of_Financial_and_Quantitative_Analysis" title="Journal of Financial and Quantitative Analysis">Journal of Financial and Quantitative Analysis</a></i> 34, September 1999, 323-339.</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">.<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFamaMerton_H._Miller1972" class="citation book cs1"><a href="/wiki/Eugene_Fama" title="Eugene Fama">Fama, Eugene F.</a>; Merton H. 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Holt Rinehart & Winston. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-15-504266-7" title="Special:BookSources/978-0-15-504266-7"><bdi>978-0-15-504266-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Finance&rft.pub=Holt+Rinehart+%26+Winston&rft.date=1972-06&rft.isbn=978-0-15-504266-7&rft.aulast=Fama&rft.aufirst=Eugene+F.&rft.au=Merton+H.+Miller&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryoffinance00fama&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiversification+%28finance%29" class="Z3988"></span></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">E. J. Elton and M. J. 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Talmud and Markowitz Diversification Strategies: Evidence from the Nigerian Stock Market. Accounting and Finance Research Vol. 3, No. 2; 2014</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"><i>The Only Guide to a Winning Investment Strategy You'll Ever Need</i></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarkowitz1952" class="citation journal cs1">Markowitz, Harry M. (1952). 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The Economics of Keynes in Historical Context: An Intellectual History of the General Theory, Palgrave Macmillan UK, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780230288775" title="Special:BookSources/9780230288775">9780230288775</a>, p. 316</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><a href="/wiki/Kenneth_L._Fisher" class="mw-redirect" title="Kenneth L. Fisher">Kenneth L. Fisher</a> (2007). 100 Minds That Made the Market. 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Conference held September 27–28, 2012.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diversification_(finance)&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.stanford.edu/~wfsharpe/mia/mia.htm">Macro-Investment Analysis</a>, Prof. <a href="/wiki/William_F._Sharpe" title="William F. Sharpe">William F. Sharpe</a>, <a href="/wiki/Stanford_University" title="Stanford University">Stanford University</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080805124142/http://viking.som.yale.edu/will/finman540/classnotes/notes.html">An Introduction to Investment Theory</a>, Prof. William N. 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risk">Business risk</a></li> <li><a href="/wiki/Model_risk" title="Model risk">Model risk</a></li> <li><a href="/wiki/Reputational_risk" class="mw-redirect" title="Reputational risk">Reputational risk</a></li> <li><a href="/wiki/Country_risk" title="Country risk">Country risk</a></li> <li><a href="/wiki/Political_risk" title="Political risk">Political risk</a></li> <li><a href="/wiki/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="/wiki/Execution_risk" class="mw-redirect" title="Execution risk">Execution risk</a></li> <li><a href="/wiki/Profit_risk" title="Profit risk">Profit risk</a></li> <li><a href="/wiki/Systemic_risk" title="Systemic risk">Systemic risk</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Arbitrage_pricing_theory" title="Arbitrage pricing theory">Arbitrage pricing theory</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes model</a></li> <li><a href="/wiki/Replicating_portfolio" title="Replicating portfolio">Replicating portfolio</a></li> <li><a href="/wiki/Cashflow_matching" title="Cashflow matching">Cash flow matching</a></li> <li><a href="/wiki/Expected_shortfall" title="Expected shortfall">Conditional Value-at-Risk (CVaR)</a></li> <li><a href="/wiki/Copula_(probability_theory)" class="mw-redirect" title="Copula (probability theory)">Copula</a></li> <li><a href="/wiki/Drawdown_(economics)" title="Drawdown (economics)">Drawdown</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First-hitting-time model</a></li> <li><a href="/wiki/Immunization_(finance)" title="Immunization (finance)">Interest rate immunization</a></li> <li><a href="/wiki/Market_portfolio" title="Market portfolio">Market portfolio</a></li> <li><a href="/wiki/Modern_portfolio_theory" title="Modern portfolio theory">Modern portfolio theory</a></li> <li><a href="/wiki/Omega_ratio" title="Omega ratio">Omega ratio</a></li> <li><a href="/wiki/Risk-adjusted_return_on_capital" title="Risk-adjusted return on capital">RAROC</a></li> <li><a href="/wiki/Risk-free_interest_rate" class="mw-redirect" title="Risk-free interest rate">Risk-free rate</a></li> <li><a href="/wiki/Risk_parity" title="Risk parity">Risk parity</a></li> <li><a href="/wiki/Sharpe_ratio" title="Sharpe ratio">Sharpe ratio</a></li> <li><a href="/wiki/Sortino_ratio" title="Sortino ratio">Sortino ratio</a></li> <li><a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a> (<a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards model</a>)</li> <li><a href="/wiki/Tracking_error" title="Tracking error">Tracking error</a></li> <li><a href="/wiki/Value_at_risk" title="Value at risk">Value-at-Risk (VaR)</a> and extensions (<a href="/wiki/Profit_at_risk" title="Profit at risk">Profit at risk</a>, <a href="/wiki/Margin_at_risk" title="Margin at risk">Margin at risk</a>, <a href="/wiki/Liquidity_at_risk" title="Liquidity at risk">Liquidity at risk</a>, <a href="/wiki/Cash_flow_at_risk" class="mw-redirect" title="Cash flow at risk">Cash flow at risk</a>, <a href="/wiki/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="/wiki/Asset_allocation" title="Asset allocation">Asset allocation</a></li> <li><a href="/wiki/Asset_and_liability_management" title="Asset and liability management">Asset and liability management</a></li> <li><a href="/wiki/Asset_pricing" title="Asset pricing">Asset pricing</a></li> <li><a href="/wiki/Bad_debt" title="Bad debt">Bad debt</a></li> <li><a href="/wiki/Capital_asset" title="Capital asset">Capital asset</a></li> <li><a href="/wiki/Capital_structure" title="Capital structure">Capital structure</a></li> <li><a href="/wiki/Corporate_finance" title="Corporate finance">Corporate finance</a></li> <li><a href="/wiki/Cost_of_capital" title="Cost of capital">Cost of capital</a></li> <li><a class="mw-selflink selflink">Diversification</a></li> <li><a href="/wiki/Economic_bubble" title="Economic bubble">Economic bubble</a></li> <li><a href="/wiki/Enterprise_value" title="Enterprise value">Enterprise value</a></li> <li><a href="/wiki/Environmental,_social,_and_governance" title="Environmental, social, and governance">ESG</a></li> <li><a href="/wiki/Exchange_traded_fund" class="mw-redirect" title="Exchange traded fund">Exchange traded fund</a></li> <li><a href="/wiki/Expected_return" title="Expected return">Expected return</a></li> <li><a href="/wiki/Finance" title="Finance">Financial</a> <ul><li><a href="/wiki/Financial_adviser" title="Financial adviser">adviser</a></li> <li><a href="/wiki/Financial_analysis" title="Financial analysis">analysis</a></li> <li><a href="/wiki/Financial_analyst" title="Financial analyst">analyst</a></li> <li><a href="/wiki/Financial_asset" title="Financial asset">asset</a></li> <li><a href="/wiki/Financial_betting" title="Financial betting">betting</a></li> <li><a href="/wiki/Financial_crime" title="Financial crime">crime</a></li> <li><a href="/wiki/Financial_engineering" title="Financial engineering">engineering</a></li> <li><a href="/wiki/Financial_law" title="Financial law">law</a></li> <li><a href="/wiki/Financial_risk" title="Financial risk">risk</a></li> <li><a href="/wiki/Financial_social_work" title="Financial social work">social work</a></li></ul></li> <li><a href="/wiki/Fundamental_analysis" title="Fundamental analysis">Fundamental analysis</a></li> <li><a href="/wiki/Growth_investing" title="Growth investing">Growth investing</a></li> <li><a href="/wiki/Hazard" title="Hazard">Hazard</a></li> <li><a href="/wiki/Hedge_(finance)" title="Hedge (finance)">Hedge</a></li> <li><a href="/wiki/Investment_management" title="Investment management">Investment management</a></li> <li><a href="/wiki/Risk" title="Risk">Risk</a></li> <li><a href="/wiki/Risk_pool" title="Risk pool">Risk pool</a></li> <li><a href="/wiki/Risk_of_ruin" title="Risk of ruin">Risk of ruin</a></li> <li><a href="/wiki/Systematic_risk" title="Systematic risk">Systematic risk</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Moral_hazard" title="Moral hazard">Moral hazard</a></li> <li><a href="/wiki/Risk-return_spectrum" class="mw-redirect" title="Risk-return spectrum">Risk-return spectrum</a></li> <li><a href="/wiki/Speculation" title="Speculation">Speculation</a></li> <li><a href="/wiki/Speculative_attack" title="Speculative attack">Speculative attack</a></li> <li><a href="/wiki/Statistical_finance" title="Statistical finance">Statistical finance</a></li> <li><a href="/wiki/Strategic_financial_management" title="Strategic financial management">Strategic financial management</a></li> <li><a href="/wiki/Stress_test_(financial)" title="Stress test (financial)">Stress test (financial)</a></li> <li><a href="/wiki/Structured_finance" title="Structured finance">Structured finance</a></li> <li><a href="/wiki/Structured_product" title="Structured product">Structured product</a></li> <li><a href="/wiki/Systemic_risk" title="Systemic risk">Systemic risk</a></li> <li><a href="/wiki/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="/wiki/Financial_economics" title="Financial economics">Financial economics</a></li> <li><a href="/wiki/Investment_management" title="Investment management">Investment management</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q731453#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" 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="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q731453#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="font-size:114%;margin:0 4em"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q731453#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4012539-7">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Diversification in industry"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85038578">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Diversification (économie politique)"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119719241">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Diversification (économie politique)"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119719241">BnF data</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007557950805171">Israel</a></span></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><span class="uid"><a rel="nofollow" class="external text" href="http://esu.com.ua/search_articles.php?id=24218">Encyclopedia of Modern Ukraine</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Diversification_(finance)&oldid=1233784482">https://en.wikipedia.org/w/index.php?title=Diversification_(finance)&oldid=1233784482</a>"</div></div> 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