{"id":193186,"date":"2023-07-11T09:18:49","date_gmt":"2023-07-11T13:18:49","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=193186"},"modified":"2025-02-12T10:55:32","modified_gmt":"2025-02-12T15:55:32","slug":"black-litterman-model","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/","title":{"rendered":"Black-Litterman Model"},"content":{"rendered":"\n<p>This post gives a derivation of the Black-Litterman (BL) model. BL model is&nbsp;<strong>a bespoke tactical asset allocation<\/strong>&nbsp;model&nbsp;<strong>incorporating investor&#8217;s view<\/strong>&nbsp;for tilting market equilibrium weights.&nbsp;The BL posterior expected returns can be obtained by using the&nbsp;<strong>Theil mixed estimator<\/strong>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-black-litterman-model\">Black-Litterman Model<\/h2>\n\n\n\n<p>Goldman Sachs says that&nbsp;&#8220;Fischer Black and Robert Litterman revolutionize portfolio management in 1990 with the creation of the Black-Litterman Global Asset Allocation Model \u2013 quickly adopted for optimal portfolio allocation across international equity, fixed income and currency markets&#8221;.<\/p>\n\n\n\n<p>Black-Litterman (BL) model uses the Theil mixed estimator which is a kind of GLS estimator. You can find more information regarding these estimators in the following previous posts.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/shleeai.blogspot.com\/2022\/09\/generalized-least-squares-gls-estimator.html\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Generalized Least Squares (GLS) estimator<\/strong><\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/shleeai.blogspot.com\/2022\/09\/theil-mixed-estimator.html\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Theil mixed estimator<\/strong><\/a><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-prior-expected-returns\">Prior Expected Returns<\/h3>\n\n\n\n<p>Prior expected returns relies on market equilibrium returns or market clearing returns. To find these returns, a reverse optimization is used.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"573\" height=\"48\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-1.png\" alt=\"\" class=\"wp-image-193188 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-1.png 573w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-1-300x25.png 300w\" data-sizes=\"(max-width: 573px) 100vw, 573px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 573px; aspect-ratio: 573\/48;\" \/><\/figure>\n\n\n\n<p>where<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03a0&nbsp;: the implied excess equilibrium return vector (<em>N \u00d7 1<\/em>)<\/li>\n\n\n\n<li>\u03bb&nbsp;: the risk aversion coefficient (scalar)<\/li>\n\n\n\n<li>\u03a3&nbsp;: the covariance matrix of excess returns (<em>N \u00d7 N<\/em>)<\/li>\n\n\n\n<li>w<sub>m<\/sub>&nbsp;: the market capitalization weight of assets (<em>N \u00d7 1<\/em>)<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"582\" height=\"143\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-2.png\" alt=\"\" class=\"wp-image-193191 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-2.png 582w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-2-300x74.png 300w\" data-sizes=\"(max-width: 582px) 100vw, 582px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 582px; aspect-ratio: 582\/143;\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Uncertainty of Prior Expected Returns<\/h3>\n\n\n\n<p>&#8220;Expected&#8221; is always followed by &#8220;uncertainty&#8221; or error. While the market clearing returns (\u03a0) are &#8220;estimates&#8221; and are known, the true expected returns (\u03bc) are unknown. This leads to the following equation.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"583\" height=\"48\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-3.png\" alt=\"\" class=\"wp-image-193194 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-3.png 583w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-3-300x25.png 300w\" data-sizes=\"(max-width: 583px) 100vw, 583px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 583px; aspect-ratio: 583\/48;\" \/><\/figure>\n\n\n\n<p>for some small scale factor parameter&nbsp;\u03c4\u226a1. A small&nbsp;\u03c4&nbsp;means a high confidence in equilibrium estimates. This reflects that the variance in the expected returns is smaller than the variance of the actual returns.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Investor&#8217;s Views<\/h3>\n\n\n\n<p>Views of BL model can be relative or absolute. K views are expressed as K linear combinations of the true excess return vector (\u03bc). It is a common practice to set the off diagonal elements of&nbsp;\u03a9&nbsp;to zero due to the independence assumption for each views.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"561\" height=\"47\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-4.png\" alt=\"\" class=\"wp-image-193197 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-4.png 561w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-4-300x25.png 300w\" data-sizes=\"(max-width: 561px) 100vw, 561px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 561px; aspect-ratio: 561\/47;\" \/><\/figure>\n\n\n\n<p>where&nbsp;<em>P<\/em>&nbsp;is a&nbsp;<em>K \u00d7 K<\/em>&nbsp;pick matrix and&nbsp;<em>\u03a9<\/em>&nbsp;is a&nbsp;<em>K \u00d7 K<\/em>&nbsp;matrix of confidence in the views.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Matrix Representation<\/h3>\n\n\n\n<p>Let&#8217;s collect two equations and represent it as a matrix form.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"577\" height=\"200\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-5.png\" alt=\"\" class=\"wp-image-193198 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-5.png 577w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-5-300x104.png 300w\" data-sizes=\"(max-width: 577px) 100vw, 577px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 577px; aspect-ratio: 577\/200;\" \/><\/figure>\n\n\n\n<p>where&nbsp;<em>I<\/em>&nbsp;is the&nbsp;<em>N \u00d7 N<\/em>&nbsp;identity matrix. We are familiar with this matrix representation.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">The Posterior Expected Returns<\/h3>\n\n\n\n<p>Using Theil mixed estimator (GLS estimator) which is explained in the previous posts, we can get the following posterior returns (<em>\u03bcBL<\/em>).<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"567\" height=\"75\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-6.png\" alt=\"\" class=\"wp-image-193199 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-6.png 567w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/07\/Black-Litterman-Model-SHLee-AI-Financial-Model-6-300x40.png 300w\" data-sizes=\"(max-width: 567px) 100vw, 567px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 567px; aspect-ratio: 567\/75;\" \/><\/figure>\n\n\n\n<p>This is a combined expected returns of the market equilibrium and investor&#8217;s views.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Concluding Remarks<\/h3>\n\n\n\n<p>This post delivers the derivation of the famous the Black-Litterman model using the Theil mixed estimator. For a clear understanding of BL model, I also explained the GLS estimator and Theil mixed estimator in the earlier posts. Real applications will be treated in the next post<\/p>\n\n\n\n<p><em>Originally posted on <a href=\"https:\/\/shleeai.blogspot.com\/2022\/09\/black-litterman-model.html\">SHLee AI Financial Mode<\/a>l blog.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post gives a derivation of the Black-Litterman (BL) model.<\/p>\n","protected":false},"author":662,"featured_media":63588,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[339,338,341],"tags":[15549,806,4922],"contributors-categories":[13728],"class_list":{"0":"post-193186","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-ibkr-quant-news","9":"category-quant-development","10":"tag-black-litterman-model","11":"tag-data-science","12":"tag-econometrics","13":"contributors-categories-sh-fintech-modeling"},"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v27.8) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Black-Litterman Model | IBKR Quant<\/title>\n<meta name=\"description\" content=\"This post gives a derivation of the Black-Litterman (BL) model.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.interactivebrokers.com\/campus\/wp-json\/wp\/v2\/posts\/193186\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Black-Litterman Model | IBKR Campus US\" \/>\n<meta property=\"og:description\" content=\"This post gives a derivation of the Black-Litterman (BL) model.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/\" \/>\n<meta property=\"og:site_name\" content=\"IBKR Campus US\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-11T13:18:49+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-02-12T15:55:32+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/10\/digital-cube-numbers.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"900\" \/>\n\t<meta property=\"og:image:height\" content=\"550\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Sang-Heon Lee\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Sang-Heon Lee\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\n\t    \"@context\": \"https:\\\/\\\/schema.org\",\n\t    \"@graph\": [\n\t        {\n\t            \"@type\": \"NewsArticle\",\n\t            \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/#article\",\n\t            \"isPartOf\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/\"\n\t            },\n\t            \"author\": {\n\t                \"name\": \"Sang-Heon Lee\",\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/person\\\/0a959ff9de7f0465a07baa1fe1ae0200\"\n\t            },\n\t            \"headline\": \"Black-Litterman Model\",\n\t            \"datePublished\": \"2023-07-11T13:18:49+00:00\",\n\t            \"dateModified\": \"2025-02-12T15:55:32+00:00\",\n\t            \"mainEntityOfPage\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/\"\n\t            },\n\t            \"wordCount\": 459,\n\t            \"commentCount\": 0,\n\t            \"publisher\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#organization\"\n\t            },\n\t            \"image\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/#primaryimage\"\n\t            },\n\t            \"thumbnailUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2020\\\/10\\\/digital-cube-numbers.jpg\",\n\t            \"keywords\": [\n\t                \"Black-Litterman Model\",\n\t                \"Data Science\",\n\t                \"Econometrics\"\n\t            ],\n\t            \"articleSection\": [\n\t                \"Data Science\",\n\t                \"Quant\",\n\t                \"Quant Development\"\n\t            ],\n\t            \"inLanguage\": \"en-US\",\n\t            \"potentialAction\": [\n\t                {\n\t                    \"@type\": \"CommentAction\",\n\t                    \"name\": \"Comment\",\n\t                    \"target\": [\n\t                        \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/#respond\"\n\t                    ]\n\t                }\n\t            ]\n\t        },\n\t        {\n\t            \"@type\": \"WebPage\",\n\t            \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/\",\n\t            \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/\",\n\t            \"name\": \"Black-Litterman Model | IBKR Campus US\",\n\t            \"isPartOf\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#website\"\n\t            },\n\t            \"primaryImageOfPage\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/#primaryimage\"\n\t            },\n\t            \"image\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/#primaryimage\"\n\t            },\n\t            \"thumbnailUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2020\\\/10\\\/digital-cube-numbers.jpg\",\n\t            \"datePublished\": \"2023-07-11T13:18:49+00:00\",\n\t            \"dateModified\": \"2025-02-12T15:55:32+00:00\",\n\t            \"description\": \"This post gives a derivation of the Black-Litterman (BL) model.\",\n\t            \"inLanguage\": \"en-US\",\n\t            \"potentialAction\": [\n\t                {\n\t                    \"@type\": \"ReadAction\",\n\t                    \"target\": [\n\t                        \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/\"\n\t                    ]\n\t                }\n\t            ]\n\t        },\n\t        {\n\t            \"@type\": \"ImageObject\",\n\t            \"inLanguage\": \"en-US\",\n\t            \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/black-litterman-model\\\/#primaryimage\",\n\t            \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2020\\\/10\\\/digital-cube-numbers.jpg\",\n\t            \"contentUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2020\\\/10\\\/digital-cube-numbers.jpg\",\n\t            \"width\": 900,\n\t            \"height\": 550,\n\t            \"caption\": \"world\"\n\t        },\n\t        {\n\t            \"@type\": \"WebSite\",\n\t            \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#website\",\n\t            \"url\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/\",\n\t            \"name\": \"IBKR Campus US\",\n\t            \"description\": \"Financial Education from Interactive Brokers\",\n\t            \"publisher\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#organization\"\n\t            },\n\t            \"potentialAction\": [\n\t                {\n\t                    \"@type\": \"SearchAction\",\n\t                    \"target\": {\n\t                        \"@type\": \"EntryPoint\",\n\t                        \"urlTemplate\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/?s={search_term_string}\"\n\t                    },\n\t                    \"query-input\": {\n\t                        \"@type\": \"PropertyValueSpecification\",\n\t                        \"valueRequired\": true,\n\t                        \"valueName\": \"search_term_string\"\n\t                    }\n\t                }\n\t            ],\n\t            \"inLanguage\": \"en-US\"\n\t        },\n\t        {\n\t            \"@type\": \"Organization\",\n\t            \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#organization\",\n\t            \"name\": \"Interactive Brokers\",\n\t            \"alternateName\": \"IBKR\",\n\t            \"url\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/\",\n\t            \"logo\": {\n\t                \"@type\": \"ImageObject\",\n\t                \"inLanguage\": \"en-US\",\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/logo\\\/image\\\/\",\n\t                \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2024\\\/05\\\/ibkr-campus-logo.jpg\",\n\t                \"contentUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2024\\\/05\\\/ibkr-campus-logo.jpg\",\n\t                \"width\": 669,\n\t                \"height\": 669,\n\t                \"caption\": \"Interactive Brokers\"\n\t            },\n\t            \"image\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/logo\\\/image\\\/\"\n\t            },\n\t            \"publishingPrinciples\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/about-ibkr-campus\\\/\",\n\t            \"ethicsPolicy\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/cyber-security-notice\\\/\"\n\t        },\n\t        {\n\t            \"@type\": \"Person\",\n\t            \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/person\\\/0a959ff9de7f0465a07baa1fe1ae0200\",\n\t            \"name\": \"Sang-Heon Lee\",\n\t            \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/author\\\/sang-heonlee\\\/\"\n\t        }\n\t    ]\n\t}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Black-Litterman Model | IBKR Quant","description":"This post gives a derivation of the Black-Litterman (BL) model.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.interactivebrokers.com\/campus\/wp-json\/wp\/v2\/posts\/193186\/","og_locale":"en_US","og_type":"article","og_title":"Black-Litterman Model | IBKR Campus US","og_description":"This post gives a derivation of the Black-Litterman (BL) model.","og_url":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/","og_site_name":"IBKR Campus US","article_published_time":"2023-07-11T13:18:49+00:00","article_modified_time":"2025-02-12T15:55:32+00:00","og_image":[{"width":900,"height":550,"url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/10\/digital-cube-numbers.jpg","type":"image\/jpeg"}],"author":"Sang-Heon Lee","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Sang-Heon Lee","Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"NewsArticle","@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/#article","isPartOf":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/"},"author":{"name":"Sang-Heon Lee","@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/person\/0a959ff9de7f0465a07baa1fe1ae0200"},"headline":"Black-Litterman Model","datePublished":"2023-07-11T13:18:49+00:00","dateModified":"2025-02-12T15:55:32+00:00","mainEntityOfPage":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/"},"wordCount":459,"commentCount":0,"publisher":{"@id":"https:\/\/ibkrcampus.com\/campus\/#organization"},"image":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/#primaryimage"},"thumbnailUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/10\/digital-cube-numbers.jpg","keywords":["Black-Litterman Model","Data Science","Econometrics"],"articleSection":["Data Science","Quant","Quant Development"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/","url":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/","name":"Black-Litterman Model | IBKR Campus US","isPartOf":{"@id":"https:\/\/ibkrcampus.com\/campus\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/#primaryimage"},"image":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/#primaryimage"},"thumbnailUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/10\/digital-cube-numbers.jpg","datePublished":"2023-07-11T13:18:49+00:00","dateModified":"2025-02-12T15:55:32+00:00","description":"This post gives a derivation of the Black-Litterman (BL) model.","inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/black-litterman-model\/#primaryimage","url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/10\/digital-cube-numbers.jpg","contentUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/10\/digital-cube-numbers.jpg","width":900,"height":550,"caption":"world"},{"@type":"WebSite","@id":"https:\/\/ibkrcampus.com\/campus\/#website","url":"https:\/\/ibkrcampus.com\/campus\/","name":"IBKR Campus US","description":"Financial Education from Interactive Brokers","publisher":{"@id":"https:\/\/ibkrcampus.com\/campus\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/ibkrcampus.com\/campus\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/ibkrcampus.com\/campus\/#organization","name":"Interactive Brokers","alternateName":"IBKR","url":"https:\/\/ibkrcampus.com\/campus\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/logo\/image\/","url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/05\/ibkr-campus-logo.jpg","contentUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/05\/ibkr-campus-logo.jpg","width":669,"height":669,"caption":"Interactive Brokers"},"image":{"@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/logo\/image\/"},"publishingPrinciples":"https:\/\/www.interactivebrokers.com\/campus\/about-ibkr-campus\/","ethicsPolicy":"https:\/\/www.interactivebrokers.com\/campus\/cyber-security-notice\/"},{"@type":"Person","@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/person\/0a959ff9de7f0465a07baa1fe1ae0200","name":"Sang-Heon Lee","url":"https:\/\/www.interactivebrokers.com\/campus\/author\/sang-heonlee\/"}]}},"jetpack_featured_media_url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/10\/digital-cube-numbers.jpg","_links":{"self":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/posts\/193186","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/users\/662"}],"replies":[{"embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/comments?post=193186"}],"version-history":[{"count":0,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/posts\/193186\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/media\/63588"}],"wp:attachment":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/media?parent=193186"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/categories?post=193186"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/tags?post=193186"},{"taxonomy":"contributors-categories","embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/contributors-categories?post=193186"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}