{"id":561,"date":"2019-10-10T13:27:00","date_gmt":"2019-10-10T13:27:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=561"},"modified":"2026-03-27T13:12:02","modified_gmt":"2026-03-27T13:12:02","slug":"covariance-and-correlation-example","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/covariance-and-correlation-example\/","title":{"rendered":"Covariance and Correlation"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Probability Concepts (2021 Level I CFA\u00ae Exam \u2013 Reading 8)\",\n  \"description\": \"This video lesson explores probability concepts, including definitions, rules (multiplication, addition, total probability), conditional expectations, Bayes' formula, and counting problems. It explains key formulas for permutations, combinations, and labeling, concluding with applications in portfolio analysis and investment decision-making.\",\n  \"uploadDate\": \"2020-10-01T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/hu47ZbsskEw\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=hu47ZbsskEw\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw\",\n  \"duration\": \"PT48M33S\",\n  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/wp-content\/uploads\/2023\/01\/analystprep-logo.png\",\n      \"width\": 250,\n      \"height\": 50\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\"\n  width=\"611\"\n  height=\"344\"\n  src=\"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw?rel=0&#038;modestbranding=1&#038;playsinline=1&#038;vq=hd1080\"\n  title=\"YouTube video player\"\n  frameborder=\"0\"\n  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\"\n  allowfullscreen>\n<\/iframe>\n\n\n\n<h2><strong>Covariance<\/strong><\/h2>\n<p>Covariance is a measure of the degree of co-movement between two random variables. For instance, we could be interested in the degree of co-movement between interest rate and inflation rate. The general formula used to calculate the covariance between two random variables, X and Y, is:<\/p>\n<p><!--more--><\/p>\n<p>$$ \\ text {cov} [X,Y]\u00a0 =\u00a0 E [(X &#8211; E[X ])(Y &#8211; E[Y])] $$<\/p>\n<p>While the formula for covariance given above is correct, we use a slightly modified formula to calculate the covariance of returns from a <a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-applications-multiplication-addition-total\/\">joint probability<\/a> model. It is based on the probability-weighted average of the cross-products of the random variables\u2019 deviations from their expected values for each possible outcome. Therefore, if we have two assets, I and J, with returns R<sub>i <\/sub>and R<sub>j<\/sub> respectively, then:<\/p>\n<p>$$ { \\sigma }_{ { R }_{ i },{ R }_{ j } }=\\sum _{ i=1 }^{ n }{ P\\left( { R }_{ i } \\right) \\left[ { R }_{ i }-E\\left( { R }_{ i } \\right) \\right] \\left[ { R }_{ j }-E\\left( { R }_{ j } \\right) \\right] } $$<\/p>\n<p>The covariance between two random variables can be positive, negative, or zero. A positive number indicates co-movement (i.e. the variables tend to move in the <strong>same<\/strong> <strong>direction<\/strong>); a value of zero indicates <strong>no<\/strong> <strong>relationship<\/strong>, and a negative value shows that the variables move in <strong>opposite directions<\/strong>.<\/p>\n<div style=\"text-align:center; background:#f3f5f9; padding:20px 16px; margin:24px 0;\">\n  <div style=\"max-width:760px; margin:0 auto;\">\n    <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n       style=\"display:inline-flex; align-items:center; justify-content:center; width:100%; padding:12px 32px; border:2px solid #2f6fdd; border-radius:999px; color:#2f6fdd; text-decoration:none; font-size:15.5px; font-weight:500; line-height:1; text-align:center;\">\n      Practice covariance and correlation questions with a Free Trial.\n    <\/a>\n  <\/div>\n<\/div>\n\n<h2><strong>Correlation<\/strong><\/h2>\n<p>Correlation is the ratio of the covariance between two random variables and the product of their two standard deviations i.e.<\/p>\n<p>$$ { \\text{Correlation} }\\left( { R }_{ i },{ R }_{ j } \\right) =\\frac { \\text{Covariance}\\left( { R }_{ i },{ R }_{ j } \\right) }{ \\text{Standard deviation}\\left( { R }_{ i } \\right) \\ast \\text{Standard deviation}\\left( { R }_{ j } \\right) } $$<\/p>\n<p>It measures the strength of the <strong>linear relationship<\/strong> between two variables. While the covariance can take on any value between negative infinity and positive infinity, the correlation is <strong>always a value between -1 and +1<\/strong>.<\/p>\n<p>You should note the following:<\/p>\n<p>First, -1 indicates a\u00a0<strong>perfect inverse relationship<\/strong> (i.e., a unit change in one means that the other will have a unit change in the opposite direction). Secondly, +1 indicates\u00a0<strong>a perfect linear relationship<\/strong> (i.e., the two variables move in the same direction with equal unit changes). Finally, if there is<strong>\u00a0<\/strong><strong>no<\/strong>\u00a0<strong>linear relationship<\/strong><strong>\u00a0<\/strong>at all, then the correlation will be zero.<\/p>\n<h3><strong>Example<\/strong><\/h3>\n<p>We anticipate a 15% chance that next year\u2019s stock returns for ABC Corp will be 6%, a 60% probability that they will be 8%, and a 25% probability of 10% return. We already know that the expected value of returns is 8.2%, and the standard deviation is 1.249%.<\/p>\n<p>We also anticipate that the same probabilities and states are associated with a 4% return for XYZ Corp, a 5% return, and a 5.5% return. Then the expected value of returns is 4.975, and the standard deviation is 0.46%.<\/p>\n<p>Suppose we wish to calculate the covariance and the correlation between ABC and XYZ returns. Then:<\/p>\n<p>$$ \\begin{align*} \\text{Covariance}, \\text{cov}(\\text R_{\\text{ABC}},\\text R_{\\text{XYZ}}) &amp; = 0.15(0.06 \u2013 0.082)(0.04 \u2013 0.04975) \\\\ &amp; + 0.6(0.08 \u2013 0.082)(0.05 \u2013 0.04975) \\\\ &amp; + 0.25(0.10 \u2013 0.082)(0.055 \u2013 0.04975) \\\\ &amp; = 0.0000561 \\\\ \\end{align*} $$<\/p>\n<p>$$ { \\text{Correlation} }\\left( { R }_{ i },{ R }_{ j } \\right) =\\frac { \\text{Covariance}\\left( { R }_{ i },{ R }_{ j } \\right) }{ \\text{Standard deviation}\\left( { R }_{ i } \\right) \\ast \\text{Standard deviation}\\left( { R }_{ j } \\right) } $$<\/p>\n<p>Therefore:<\/p>\n<p>$$ \\begin{align*} \\text{Correlation} &amp; =\\cfrac {0.0000561}{(0.01249 * 0.0046)} \\\\ &amp; = 0.976 \\\\ \\end{align*} $$<\/p>\n<p>Interpretation: the correlation between the returns of the two companies is very strong (almost +1) and the returns move linearly in the same direction.<\/p>\n<div class=\"notes_inv\"><hr \/>\n<p>\u00a0<\/p>\n<\/div>\n<p>\u00a0<\/p>\n<div style=\"text-align:center; background:#f3f5f9; padding:44px 20px 24px; margin:40px 0;\">\n  \n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-flex; align-items:center; justify-content:center; background:#4274d8; color:#ffffff; text-decoration:none; padding:14px 36px; border-radius:999px; font-size:16px; font-weight:700; line-height:1; text-align:center;\">\n    Start Free Trial\n  <\/a>\n\n  <p style=\"max-width:640px; margin:14px auto 0; font-size:15.5px; line-height:1.5; color:#1f2937;\">\n    Strengthen covariance and correlation analysis with\n    exam-focused CFA Level I practice.\n  <\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Covariance Covariance is a measure of the degree of co-movement between two random variables. For instance, we could be interested in the degree of co-movement between interest rate and inflation rate. The general formula used to calculate the covariance between&#8230;<\/p>\n","protected":false},"author":2,"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-561","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 &amp; Correlation Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Covariance shows the degree of co-movement between variables, while correlation measures their linear relationship. 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