{"id":45264,"date":"2023-07-03T18:35:31","date_gmt":"2023-07-03T18:35:31","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=45264"},"modified":"2026-02-07T09:55:17","modified_gmt":"2026-02-07T09:55:17","slug":"covariance-and-correlation","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/covariance-and-correlation\/","title":{"rendered":"Covariance and Correlation"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Statistical Measures and Asset Returns (2025 CFA\u00ae Level I Exam \u2013 Quantitative Methods \u2013 Learning Module 3)\",\n  \"description\": \"CFA\u00ae Level I Quantitative Methods lesson covering Statistical Measures and Asset Returns. This session explains measures of central tendency, dispersion, skewness, kurtosis, and correlation. Learn how to calculate, interpret, and evaluate statistical measures in investment contexts, and apply them to CFA-style exam problems for the 2025 exam cycle.\",\n  \"uploadDate\": \"2024-02-16\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/PYgG65-xki8\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=PYgG65-xki8\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/PYgG65-xki8\",\n  \"duration\": \"PT48M36S\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Calculating variance given correlation and covariance\",\n    \"text\": \"The correlation coefficient between X and Y is 0.7, and the covariance is 29. If the variance of Y is 25, the variance of X is closest to:\\n\\nA. 8.29.\\n\\nB. 29.00.\\n\\nC. 68.65.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"68.65.\",\n      \"explanation\": \"Using the correlation formula rXY = sXY \/ (sX \u00d7 sY), we have 0.7 = 29 \/ (sX \u00d7 5). Solving gives sX \u2248 8.2857. Squaring this value gives the variance of X: (8.2857)\u00b2 \u2248 68.65.\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1\/images\/correlation\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Img_23.png\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Img_23.png\",\n  \"caption\": \"Correlation between asset returns\",\n  \"width\": 1590,\n  \"height\": 1468,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"url\": \"https:\/\/analystprep.com\/\"\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/PYgG65-xki8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Covariance<\/h2>\n<p>Covariance is a measure of how two variables move together. The\u00a0sample covariance\u00a0of\u00a0<em>X<\/em>\u00a0and\u00a0<em>Y<\/em>\u00a0is calculated as follows:<\/p>\n<p>$${s}_{XY}=\\frac{\\sum_{i=1}^{N}\\left(X_i-\\bar{X}\\right)\\left(Y_i-\\bar{Y}\\right)}{n-1}$$<\/p>\n<p>The formula above implies that the sample covariance is the mean of the product of the deviations in the two random variables and from their sample means.<\/p>\n<p>If the covariance between two random variables is positive, it means they move in the same direction. When one is below its mean, the other is also below its mean, and vice versa.<\/p>\n<p>A major drawback of covariance is that it is difficult to interpret since its value can vary from negative infinity to positive infinity.<\/p>\n<p><a style=\"display: block; margin: 20px 0 28px; padding: 14px 18px; border: 2px solid #2563eb; border-radius: 12px; text-align: center; color: #2563eb; text-decoration: none; font-weight: 500; font-size: 15px; background-color: #ffffff;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Turn covariance and correlation into scoring points with a free trial. <\/a><\/p>\n<h2>Correlation<\/h2>\n<p>Correlation is a standardized measure of the linear relationship between two variables. It takes the covariance and divides it by the product of the standard deviations of both variables. As a result, its value ranges between -1 and +1 and is easier to interpret.<\/p>\n<p>The sample correlation coefficient is calculated as follows:<\/p>\n<p>$$r_{xy}=\\frac{s_{XY}}{s_X\\times s_Y}$$<\/p>\n<p>Where:<\/p>\n<p>\\(s_{X Y}\\) = Covariance between variable X and Y.<\/p>\n<p>\\(s_{X}\\) = Standard deviation of variable X.<\/p>\n<p>\\(s_{Y}\\) = Standard deviation of variable Y.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1590\" height=\"1468\" class=\"aligncenter size-full wp-image-31069\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Img_23.png\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Img_23.png 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Img_23-300x277.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Img_23-1024x945.png 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Img_23-768x709.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Img_23-1536x1418.png 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Img_23-400x369.png 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<h3><strong>Properties of Correlation<\/strong><\/h3>\n<ul>\n<li>Correlation ranges between -1 to +1 for two random variables, <em>X <\/em>and <em>Y<\/em>.<\/li>\n<li>A correlation of 0 (uncorrelated variables) indicates that no linear (straight-line) relationship exists between the variables.<\/li>\n<li>A positive correlation close to +1 indicates a strong positive linear relationship.\n<ul>\n<li>A correlation of 1 indicates a perfect linear relationship.<\/li>\n<\/ul>\n<\/li>\n<li>A negative correlation close to -1 indicates a strong negative linear relationship.\n<ul>\n<li>A correlation of -1 indicates a perfect inverse linear relationship.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3><strong>Limitations of Correlation Analysis<\/strong><\/h3>\n<ul>\n<li>Two variables can have a very low correlation despite having a strong <strong><em>nonlinear <\/em><\/strong>relationship.<\/li>\n<li>Correlation can be an unreliable measure when outliers are present in the data.<\/li>\n<li>Correlation does not imply causation. This implies that the correlation may be spurious. A spurious correlation refers to:\n<ul>\n<li>Correlation between two variables due to chance relationships in a particular dataset.<\/li>\n<li>Correlation arising between variables when they are divided by a third variable.<\/li>\n<li>Correlation between two variables arising from their relation to a third variable.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>The correlation coefficient between X and Y is 0.7, and the covariance is 29. If the variance of Y is 25, the variance of X is <em>closest <\/em>to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>8.29.<\/li>\n<li>29.00<\/li>\n<li>68.65.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>The correct answer is<\/strong> <strong>C<\/strong>.<\/p>\n<p>$$\\begin{align} r_{X Y} &amp;=\\frac{s_{X Y}}{s_{X} \\times s_{Y}}\\\\ \\Rightarrow 0.7 &amp;=\\frac{29}{s_{X} \\times 5} \\\\ \\therefore s_{X}&amp;=8.2857\\\\ \\\\ \\text{Variance} &amp;=8.2857^2=68.65 \\end{align}$$<\/p>\n<\/blockquote>\n<div style=\"text-align: center; margin-top: 32px;\"><a style=\"display: inline-block; padding: 16px 40px; background-color: #2563eb; color: #ffffff; text-decoration: none; border-radius: 14px; font-weight: bold; font-size: 17px;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a><\/div>","protected":false},"excerpt":{"rendered":"<p>Covariance Covariance is a measure of how two variables move together. The\u00a0sample covariance\u00a0of\u00a0X\u00a0and\u00a0Y\u00a0is calculated as follows: $${s}_{XY}=\\frac{\\sum_{i=1}^{N}\\left(X_i-\\bar{X}\\right)\\left(Y_i-\\bar{Y}\\right)}{n-1}$$ The formula above implies that the sample covariance is the mean of the product of the deviations in the two random variables and&#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-45264","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 and Correlation Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how covariance and correlation 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