{"id":27515,"date":"2021-08-25T07:33:10","date_gmt":"2021-08-25T07:33:10","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=27515"},"modified":"2026-02-10T06:06:27","modified_gmt":"2026-02-10T06:06:27","slug":"covariance-of-portfolio-returns-given-a-joint-probability-distribution","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/covariance-of-portfolio-returns-given-a-joint-probability-distribution\/","title":{"rendered":"Covariance of Portfolio Returns Given a Joint Probability Distribution"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/covariance-of-portfolio-returns-given-a-joint-probability-distribution\/#qapage-question-1\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/covariance-of-portfolio-returns-given-a-joint-probability-distribution\/#question-1\",\n    \"name\": \"Given the joint probability distribution, what is the covariance between TY and Ford returns?\",\n    \"text\": \"Given the joint probability distribution of returns for TY and Ford across strong, average, and weak economic environments, the covariance between TY and Ford returns is closest to which of the following?\\nA. 0.054.\\nB. 0.1542.\\nC. 0.1442.\",\n    \"answerCount\": 1,\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/covariance-of-portfolio-returns-given-a-joint-probability-distribution\/#answer-1\",\n      \"text\": \"C. 0.1442. The covariance is calculated by first determining the expected returns for TY and Ford, then summing the probability-weighted products of each return\u2019s deviation from its expected value across all economic states.\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/PsZrsg3ZUDE?si=3Np-zjmkQwNlCMdE\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\n<p>Covariance between variables can be calculated in two ways. One method is the historical sample covariance between two random variables \\(X_i\\)&nbsp;and \\(Y_i\\). It is based on a sample of past data of size \\(n\\) and is given by:<\/p>\n\n\n\n<p>$$\\text{Cov}_{X_i,Y_i}=\\frac{\\sum_{i=1}^{n}{(X_i -\\bar{X})(Y_i -\\bar{Y})}}{n-1}$$<\/p>\n\n\n\n<p>Alternatively, covariance can be defined as the probability-weighted average of the cross-products of each random variable\u2019s deviation from its own expected value. That is:<\/p>\n\n\n\n<p>$$\\text{Cov}_{X_i,Y_i}=E\\left[(X_i -\\bar{X})(Y_i -\\bar{Y})\\right]$$<\/p>\n\n\n\n<!-- TOP CTA \u2013 Full Width Outline Button -->\n<div style=\"margin:24px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:block;\n       width:100%;\n       padding:14px 0;\n       border:2px solid #3b6fd8;\n       border-radius:50px;\n       font-size:18px;\n       font-weight:500;\n       text-align:center;\n       text-decoration:none;\n       color:#3b6fd8;\n       background-color:#f4f6f9;\n       box-sizing:border-box;\n     \">\n     Practice covariance questions with free trial access.\n  <\/a>\n<\/div>\n\n\n\n<p>Consider the following example:<\/p>\n<p><!--more--><\/p>\n<h4><strong>Example: Calculating the Covariance #1<\/strong><\/h4>\n<p>Suppose we wish to find the variance of each asset and the covariance between the returns of ABC and XYZ, given that the amount invested in each company is $1,000.<\/p>\n<p>This table is used to calculate the expected returns:<\/p>\n<p>$$ \\begin{array}{c|c|c|c} &amp; \\textbf{Strong Economy} &amp; \\textbf{Normal Economy} &amp; \\textbf{Week Economy} \\\\ \\hline \\text{Probability} &amp; {15\\%} &amp; {60\\%} &amp; {25\\%} \\\\ \\hline \\text{ABC Returns} &amp; {40\\%} &amp; {20\\%} &amp; {0\\%} \\\\ \\hline \\text{XYZ Returns} &amp; {20\\%} &amp; {15\\%} &amp; {4\\%} \\\\ \\end{array} $$<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>For us to find the covariance, we must calculate the expected return of each asset as well as their variances. The assets&#8217; weights are:<\/p>\n<p>$$ \\text W_{\\text{ABC}}=\\cfrac {1000}{2000} = 0.5 $$<\/p>\n<p>$$ \\text W_{\\text{XYZ}}=\\cfrac {1000}{2000} = 0.5 $$<\/p>\n<p>Next, we should calculate the individual expected returns:<\/p>\n<p>$$ \\text E(\\text R_{\\text{ABC}}) = 0.15 \u00d7 0.40 + 0.60 \u00d7 0.2 + 0.25 \u00d7 0.00 = 0.18 $$<\/p>\n<p>$$ \\text E(\\text R_{\\text{XYZ}}) = 0.15 \u00d7 0.2 + 0.60 \u00d7 0.15 + 0.25 \u00d7 0.04 = 0.13 $$<\/p>\n<p>Finally, we can compute the covariance between the returns of the two assets:<\/p>\n<p>$$ \\begin{align*}<br \/>\\text{Cov}(\\text R_{\\text{ABC},\\text{XYZ}}) &amp;= 0.15(0.40 \u2013 0.18)(0.20 \u2013 0.13) \\\\<br \/>&amp; + 0.6(0.20 \u2013 0.18)(0.15 \u2013 0.13) \\\\<br \/>&amp; + 0.25(0.00 \u2013 0.18)(0.04 \u2013 0.13) \\\\<br \/>&amp; = 0.0066<br \/>\\end{align*} $$<\/p>\n<p><strong>Example: Calculating the Covariance #2<\/strong><\/p>\n<p>A portfolio manager is considering the following two possible economic growth of a country and the joint variability of returns on two stocks in a portfolio:<\/p>\n<p>$$\\begin{array}{l|c|c}<br \/>\\textbf {Economic Growth } &amp; \\bf {&lt;4 \\%} &amp; \\bf {&gt;4 \\%} \\\\<br \/>\\hline \\text { Probability } &amp; 40 \\% &amp; 60 \\% \\\\<br \/>\\hline \\text { Return of Stock A } &amp; 2.3 \\% &amp; 8 \\% \\\\<br \/>\\hline \\text { Return of Stock B } &amp; 6.5 \\% &amp; 3 \\% \\\\<br \/>\\end{array}<br \/>$$<\/p>\n<p>What is the covariance between the return of Stock A and Stock B?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Expected return of Stock A \\(= (40\\% \u00d7 2.3\\%) + (60\\% \u00d7 8\\%) = 5.72\\%\\)<\/p>\n<p>Expected return of Stock B \\(= (40\\% \u00d7 6.5\\%) + (60\\% \u00d7 3\\%) = 4.40\\%\\)<\/p>\n<p><em><strong>Note<\/strong><\/em>: For the rest of the calculation, your curriculum sometimes ditches the percentage signs so that 4.40% becomes 4.40.<\/p>\n<p>The deviations of returns at economic growth of &lt; 4% \\(= (2.3 &#8211; 5.72) \u00d7 (6.5-\u00a0 4.40) = -7.182\\)<\/p>\n<p>The deviations of returns at economic growth of &gt;4% \\(= (8 -5.72) \u00d7 (3-4.40) = -3.192\\)<\/p>\n<p>The covariance of returns between stock A and stock B is computed as follows:<\/p>\n<p>$$\\text{Cov}(\\text R_{\\text{A},\\text{B}}) = (-7.182 \u00d7 \u00a00.40) + (-3.192 \u00d7 0.60) = -4.788$$<\/p>\n<p><em><strong>Interpretation<\/strong><\/em>: Since covariance is negative, the two returns show some co-movement in opposite signs.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>The following table represents the estimated returns for two motor vehicle production brands \u2013 TY and Ford, in 3 industrial environments: strong (50% probability), average (30% probability), and weak (20% probability).<\/p>\n<p>$$ \\begin{array}{c|c|c|c} {} &amp; \\textbf{TY Returns +6%} &amp; \\text{TY Returns +3%} &amp; \\textbf{TY Returns -1%} \\\\ \\hline {\\text{Ford Sales }+10\\%} &amp; \\text{Strong (0.5)} &amp; {} &amp; {} \\\\ \\hline {\\text{Ford Sales }+4\\%} &amp; {} &amp; \\text{Average (0.3)} &amp; {} \\\\ \\hline {\\text{Ford Sales }-4\\%} &amp; {} &amp; {} &amp; \\text{Weak (0.2)} \\\\ \\end{array} $$<\/p>\n<p>Given the above joint probability function, the covariance between TY and Ford returns is <em>closest<\/em> to:<\/p>\n<p>A. 0.054.<\/p>\n<p>B. 0.1542.<\/p>\n<p>C. 0.1442.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>C<\/strong>.<\/p>\n<p>First, we must start by calculating the expected return for each brand:<\/p>\n<p>$$ \\text{Expected return for TY} = (0.5 \u00d7 6\\%) + (0.3 \u00d7 3\\%) + (0.2 \u00d7 (-1\\%)) = 3\\% + 0.9\\% \u2013 0.2\\% = 3.7\\% $$<\/p>\n<p>$$ \\text{Expected return for Ford} = (0.5 \u00d7 10\\%) + (0.3 \u00d7 4\\%) + (0.2 \u00d7 (-4\\%)) = 5\\% + 1.2\\% \u2013 0.8\\% = 5.4\\% $$<\/p>\n<p>Next, we can now compute the covariance:<\/p>\n<p>$$ \\begin{align*}<br \/>\\text{Covariance} &amp; = 0.5(6\\% \u2013 3.7\\%)(10\\% \u2013 5.4\\%) \\\\<br \/>&amp; + 0.3(3\\% \u2013 3.7\\%)(4\\% \u2013 5.4\\%) \\\\<br \/>&amp; + 0.2(-1\\% \u2013 3.7\\%)(-4\\% \u2013 5.4\\%) \\\\<br \/>&amp; = 5.29\\% + 0.294\\% + 8.836\\% \\\\<br \/>&amp; = 0.1442 \\\\<br \/>\\end{align*} $$<\/p>\n<p><em><strong>Interpretation<\/strong><\/em>: The covariance is positive. This means that the returns for the two brands show some co-movement in the same direction.<\/p>\n<p><em><strong>Note<\/strong><\/em>: This would most likely be the case in real life because the companies are in the same industry, and therefore, the systematic risks affecting them are quite similar.<\/p>\n<\/blockquote>\n\n\n<!-- BOTTOM CTA \u2013 Refined Version -->\n<div style=\"text-align:center; background-color:#f4f6f9; padding:35px 20px; border-radius:12px; margin-top:40px;\">\n\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:inline-block;\n       padding:14px 34px;\n       background-color:#3b6fd8;\n       color:#ffffff;\n       border-radius:50px;\n       font-size:16px;\n       font-weight:600;\n       text-decoration:none;\n       margin-bottom:18px;\n     \">\n     Start Free Trial\n  <\/a>\n\n  <p style=\"max-width:700px; margin:0 auto; font-size:16px; line-height:1.6; color:#333;\">\n    Build confidence with joint probability tables and portfolio covariance calculations using exam-style practice questions, detailed solutions, and timed quizzes inside AnalystPrep\u2019s free trial.\n  <\/p>\n\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Covariance between variables can be calculated in two ways. One method is the historical sample covariance between two random variables \\(X_i\\)&nbsp;and \\(Y_i\\). It is based on a sample of past data of size \\(n\\) and is given by: $$\\text{Cov}_{X_i,Y_i}=\\frac{\\sum_{i=1}^{n}{(X_i -\\bar{X})(Y_i&#8230;<\/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":[2],"tags":[],"class_list":["post-27515","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Covariance of Portfolio Returns | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn how to calculate portfolio return covariance using joint probability distributions and historical sample data.\" \/>\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\/quantitative-methods\/covariance-of-portfolio-returns-given-a-joint-probability-distribution\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Covariance of Portfolio Returns | CFA Level 1 - 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