{"id":62315,"date":"2026-07-28T22:39:53","date_gmt":"2026-07-28T22:39:53","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=62315"},"modified":"2026-07-28T22:39:54","modified_gmt":"2026-07-28T22:39:54","slug":"mean-variance-and-covariance","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/","title":{"rendered":"Mean, Variance and Covariance"},"content":{"rendered":"

Investors seek to manage portfolio risk while maintaining returns. This involves understanding portfolio risk components. Diversification, particularly with assets having low correlations, can mitigate risk without necessarily lowering returns. Portfolio return is the weighted average of individual asset returns. Portfolio risk, however, isn’t simply a weighted average due to correlations among assets. Covariance and correlation measures help quantify portfolio risk, simplifying the computation of variance for a two-asset portfolio.<\/p>\n

The computation of mean, variance, and covariance statistics allows portfolio managers to compare the underlying securities’ return-risk characteristics and potential portfolio impact. These metrics are quantitatively determined and rely on historical price or return data. While we can compute the historical profile, this does not necessarily mean the relationship between assets or their return-risk profile will remain the same in the future.<\/p>\n

Mean<\/strong><\/h2>\n

The mean of a set of values or measurements is the sum of all the measurements divided by the sum of all the measurements in the set:<\/p>\n

$$ \\text{Mean} = \\frac{\\sum_{i=1}^{n} x_{i}}{n} $$<\/p>\n

If we compute the population’s mean, we call it the parametric or population mean, denoted by ? (read \u201cmu\u201d). If we get the mean of the sample, we call it the sample mean, denoted by the x bar.<\/p>\n

Population vs. Sample<\/strong><\/h4>\n

A population refers to the summation of all the elements of interest to the researcher.<\/p>\n

    \n
  • Examples: The number of people in a country, the number of hedge funds in the U.S., or even the total number of CFA candidates in a given year.<\/li>\n<\/ul>\n

    A sample is just a set of elements that represent the population as a whole. By analyzing sample data, we are able to make conclusions about the entire population.<\/p>\n

      \n
    • For example, if we sample the returns of 30 hedge funds spread across the U.S., we can use the results to make reasonable conclusions about the market as a whole (well over 10,000 hedge funds).<\/li>\n<\/ul>\n

      Variance<\/strong><\/h2>\n

      Variance is a measure of dispersion around the mean and is statistically defined as the average squared deviation from the mean. It is noted using the symbol ?\u00b2.<\/p>\n

      $$ \\sigma^2 = \\frac{\\sum_{i=1}^{N} (X_{i} – \\mu)^2}{N} $$<\/p>\n

      Where ? is the population mean, and N is the population size.<\/p>\n

      The standard deviation,\u00a0?,<\/b> is the square root of the variance and is commonly referred to as the volatility of the asset. Essentially, it is a measure of how far, on average, the observations are from the mean. A population\u2019s variance is given by:<\/p>\n

      The population standard deviation equals the square root of the population variance. The sample variance is given by:<\/p>\n

      $$ S^2 = \\frac{\\sum_{i=1}^{N} (X_{i} – \\bar{X})^2}{n-1} $$<\/p>\n

      Where X-bar is the sample mean, and n is the sample size.<\/p>\n

      Note that the sample standard deviation equals the square root of the sample variance.<\/p>\n

      Covariance<\/strong><\/h2>\n

      Covariance is a measure of how closely two assets move together. In covariance, we focus on the relationship between the deviations of some two variables rather than the deviation from the mean of one variable.<\/p>\n

      If the means of random variables \\(X\\) and \\(Y\\) are known, then the covariance between the two random variables can be determined as follows:<\/p>\n

      $$ { \\hat { \\sigma } }_{ xy }=\\frac { 1 }{ n } \\sum _{ i=1 }^{ n }{ \\left( { x }_{ i }-{ \\mu }_{ x } \\right) } \\left( { y }_{ i }-{ \\mu }_{ y } \\right) $$<\/p>\n

      If we do not know the means, then the equation changes to:<\/p>\n

      $$ { \\hat { \\sigma } }_{ xy }=\\frac { 1 }{ n-1 } \\sum _{ i=1 }^{ n }{ \\left( { x }_{ i }-{ \\hat { \\mu } }_{ x } \\right) } \\left( { y }_{ i }-{ \\hat { \\mu } }_{ y } \\right) $$<\/p>\n

      Correlation<\/h3>\n

      Correlation is a concept that is closely related to covariance in the following way:<\/p>\n

      $$ { \\rho }_{ xy }=\\frac { { \\sigma }_{ xy } }{ { \\sigma }_{ x }{ \\sigma }_{ y } } $$<\/p>\n

      Correlation ranges between +1 and -1 and is much easier to interpret than covariance. Two variables are perfectly correlated if their correlation is equal to +1. Note that they are uncorrelated if their correlation equals 0 and move in perfectly opposite directions if their correlation equals -1.<\/p>\n

      \n

      Question<\/strong><\/h2>\n

      In a two-asset portfolio, which combination of assets would result in the most diversified portfolio?<\/p>\n

      A. Correlation coefficient = 0.75.<\/p>\n

      B. Correlation coefficient = -0.2.<\/p>\n

      C. Correlation coefficient = 0.<\/p>\n

      Solution<\/strong><\/p>\n

      The correct answer is B<\/strong>.<\/p>\n

      A diversified portfolio is produced, and portfolio risk is lowered within a two-asset portfolio by combining negatively correlated assets.<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"

      Investors seek to manage portfolio risk while maintaining returns. This involves understanding portfolio risk components. Diversification, particularly with assets having low correlations, can mitigate risk without necessarily lowering returns. Portfolio return is the weighted average of individual asset returns. Portfolio…<\/p>\n","protected":false},"author":15,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-62315","post","type-post","status-publish","format-standard","hentry","category-uncategorized","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"\nMean, Variance and Covariance - AnalystPrep | CFA\u00ae Exam Study Notes<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Mean, Variance and Covariance - AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"og:description\" content=\"Investors seek to manage portfolio risk while maintaining returns. This involves understanding portfolio risk components. Diversification, particularly with assets having low correlations, can mitigate risk without necessarily lowering returns. Portfolio return is the weighted average of individual asset returns. Portfolio...\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/\" \/>\n<meta property=\"og:site_name\" content=\"AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2026-07-28T22:39:53+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-07-28T22:39:54+00:00\" \/>\n<meta name=\"author\" content=\"Kosikos Tuitoek\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Kosikos Tuitoek\" \/>\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\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/uncategorized\\\/mean-variance-and-covariance\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/uncategorized\\\/mean-variance-and-covariance\\\/\"},\"author\":{\"name\":\"Kosikos Tuitoek\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/#\\\/schema\\\/person\\\/73df713e3b6e82ee139e1eff20cebe20\"},\"headline\":\"Mean, Variance and Covariance\",\"datePublished\":\"2026-07-28T22:39:53+00:00\",\"dateModified\":\"2026-07-28T22:39:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/uncategorized\\\/mean-variance-and-covariance\\\/\"},\"wordCount\":706,\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/uncategorized\\\/mean-variance-and-covariance\\\/\",\"url\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/uncategorized\\\/mean-variance-and-covariance\\\/\",\"name\":\"Mean, Variance and Covariance - AnalystPrep | CFA\u00ae Exam Study Notes\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/#website\"},\"datePublished\":\"2026-07-28T22:39:53+00:00\",\"dateModified\":\"2026-07-28T22:39:54+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/#\\\/schema\\\/person\\\/73df713e3b6e82ee139e1eff20cebe20\"},\"breadcrumb\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/uncategorized\\\/mean-variance-and-covariance\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/uncategorized\\\/mean-variance-and-covariance\\\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/uncategorized\\\/mean-variance-and-covariance\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Mean, Variance and Covariance\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/#website\",\"url\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/\",\"name\":\"AnalystPrep | CFA\u00ae Exam Study Notes\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/#\\\/schema\\\/person\\\/73df713e3b6e82ee139e1eff20cebe20\",\"name\":\"Kosikos Tuitoek\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g\",\"caption\":\"Kosikos Tuitoek\"},\"url\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/author\\\/kosikos-tuitoek-enockanalystprep-com\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Mean, Variance and Covariance - AnalystPrep | CFA\u00ae Exam Study Notes","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:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/","og_locale":"en_US","og_type":"article","og_title":"Mean, Variance and Covariance - AnalystPrep | CFA\u00ae Exam Study Notes","og_description":"Investors seek to manage portfolio risk while maintaining returns. This involves understanding portfolio risk components. Diversification, particularly with assets having low correlations, can mitigate risk without necessarily lowering returns. Portfolio return is the weighted average of individual asset returns. Portfolio...","og_url":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/","og_site_name":"AnalystPrep | CFA\u00ae Exam Study Notes","article_published_time":"2026-07-28T22:39:53+00:00","article_modified_time":"2026-07-28T22:39:54+00:00","author":"Kosikos Tuitoek","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Kosikos Tuitoek","Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/"},"author":{"name":"Kosikos Tuitoek","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20"},"headline":"Mean, Variance and Covariance","datePublished":"2026-07-28T22:39:53+00:00","dateModified":"2026-07-28T22:39:54+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/"},"wordCount":706,"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/","name":"Mean, Variance and Covariance - AnalystPrep | CFA\u00ae Exam Study Notes","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website"},"datePublished":"2026-07-28T22:39:53+00:00","dateModified":"2026-07-28T22:39:54+00:00","author":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20"},"breadcrumb":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/mean-variance-and-covariance\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/cfa-level-1-exam\/"},{"@type":"ListItem","position":2,"name":"Mean, Variance and Covariance"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/","name":"AnalystPrep | CFA\u00ae Exam Study Notes","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20","name":"Kosikos Tuitoek","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g","caption":"Kosikos Tuitoek"},"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/kosikos-tuitoek-enockanalystprep-com\/"}]}},"_links":{"self":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/62315","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/users\/15"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/comments?post=62315"}],"version-history":[{"count":1,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/62315\/revisions"}],"predecessor-version":[{"id":62316,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/62315\/revisions\/62316"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/media?parent=62315"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/categories?post=62315"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/tags?post=62315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}