{"id":8212,"date":"2020-08-19T05:43:48","date_gmt":"2020-08-19T05:43:48","guid":{"rendered":"https:\/\/analystprep.com\/blog\/?p=8212"},"modified":"2026-03-30T12:21:03","modified_gmt":"2026-03-30T12:21:03","slug":"covariance-and-correlation-calculations-for-cfa-and-frm-exams","status":"publish","type":"post","link":"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/","title":{"rendered":"Covariance and Correlation (Calculations for CFA\u00ae and FRM\u00ae Exams)"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_80 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#Example_1_Calculating_the_Covariance_of_a_Portfolio_of_Two_Assets\" >Example 1: Calculating the Covariance of a Portfolio of Two Assets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#Solution\" >Solution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#Example_2_Covariance\" >Example 2: Covariance<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#Solution-2\" >Solution<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#Correlation\" >Correlation<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#Example_Calculating_the_Covariance\" >Example: Calculating the Covariance<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#How_does_Correlation_Impact_Portfolio_Risk\" >How does Correlation Impact Portfolio Risk?<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#Ready_to_Apply_These_Formulas\" >Ready to Apply These Formulas?<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#FAQs_on_Covariance_and_Correlation\" >FAQs on Covariance and Correlation<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#1_What_is_the_covariance_formula_for_CFA_Level_1\" >1. What is the covariance formula for CFA Level 1?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#2_Whats_the_difference_between_covariance_and_correlation\" >2. What\u2019s the difference between covariance and correlation?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#3_How_do_I_convert_covariance_to_correlation\" >3. How do I convert covariance to correlation?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#4_Why_is_a_correlation_of_zero_not_the_same_as_independence\" >4. Why is a correlation of zero not the same as independence?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#5_How_does_correlation_affect_portfolio_risk\" >5. How does correlation affect portfolio risk?<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"FAQPage\",\n  \"mainEntity\": [\n    {\n      \"@type\": \"Question\",\n      \"name\": \"What is the covariance formula for the CFA Level I exam?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"The covariance between two random variables X and Y is given by Cov(X,Y) = E[(X \u2013 E[X])(Y \u2013 E[Y])], or equivalently Cov(X,Y) = E[XY] \u2013 E[X]E[Y].\"\n      }\n    },\n    {\n      \"@type\": \"Question\",\n      \"name\": \"What\u2019s the difference between covariance and correlation?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"Covariance measures how two variables move together but is expressed in the units of the variables and can range from negative infinity to positive infinity. Correlation standardises this by dividing covariance by the product of the two variables\u2019 standard deviations, giving a value between \u20131 and +1.\"\n      }\n    },\n    {\n      \"@type\": \"Question\",\n      \"name\": \"How do I convert covariance to correlation?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"You convert covariance to correlation by dividing the covariance between X and Y by the product of their standard deviations: Corr(X,Y) = Cov(X,Y) \/ (\u03c3_X \u00d7 \u03c3_Y).\"\n      }\n    },\n    {\n      \"@type\": \"Question\",\n      \"name\": \"Why is a correlation of zero not the same as independence?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"A correlation of zero means there is no linear relationship between the two variables, but they may still have a non-linear relationship (so are not necessarily independent). Independence is a stronger condition than zero correlation.\"\n      }\n    },\n    {\n      \"@type\": \"Question\",\n      \"name\": \"How does correlation affect portfolio risk?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"Correlation between assets affects diversification: if assets are perfectly positively correlated (correlation = +1), there is no diversification benefit; if correlation = \u20131, maximum diversification benefit. Lower correlation between portfolio components tends to reduce overall volatility.\"\n      }\n    }\n  ]\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\": \"What is the covariance formula for CFA Level I?\",\n    \"text\": \"What is the covariance formula for CFA Level I?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2024-09-01T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"For CFA Level I, covariance between two variables X and Y is defined as:\\nPopulation covariance: Cov(X,Y) = \u03a3[(Xi \u2212 \u03bcX)(Yi \u2212 \u03bcY)] \/ N.\\nSample covariance: Cov(X,Y) = \u03a3[(Xi \u2212 X\u0304)(Yi \u2212 \u0232)] \/ (n \u2212 1).\\nIt measures how two variables move together relative to their means.\",\n      \"dateCreated\": \"2024-09-01T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#faq-covariance-formula\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\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\": \"What is the difference between covariance and correlation?\",\n    \"text\": \"What is the difference between covariance and correlation?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2024-09-01T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Covariance is an unscaled measure of how two variables move together and can take any value from negative to positive infinity. Correlation standardizes covariance by dividing by the product of the variables\u2019 standard deviations: \u03c1XY = Cov(X,Y) \/ (\u03c3X\u03c3Y). As a result, correlation is unitless and always lies between \u22121 and +1, making it easier to interpret.\",\n      \"dateCreated\": \"2024-09-01T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#faq-covariance-vs-correlation\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\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\": \"How do I convert covariance to correlation?\",\n    \"text\": \"How do I convert covariance to correlation?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2024-09-01T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"To convert covariance to correlation, divide by the product of the standard deviations of the two variables: \u03c1XY = Cov(X,Y) \/ (\u03c3X\u03c3Y). This rescales the measure so that correlation is bounded between \u22121 and +1, indicating the strength and direction of the linear relationship.\",\n      \"dateCreated\": \"2024-09-01T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#faq-convert-cov-to-corr\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\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\": \"Why is a correlation of zero not the same as independence?\",\n    \"text\": \"Why is a correlation of zero not the same as independence?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2024-09-01T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"A correlation of zero means there is no linear relationship between two variables, but they can still be related in a non-linear way. True independence means the joint behavior of the variables factorizes and all forms of dependence vanish. Independence implies zero correlation, but zero correlation does not necessarily imply independence.\",\n      \"dateCreated\": \"2024-09-01T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#faq-zero-corr-vs-independence\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\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\": \"How does correlation affect portfolio risk?\",\n    \"text\": \"How does correlation affect portfolio risk?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2024-09-01T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Correlation drives the diversification benefit in a portfolio. When asset returns are highly positively correlated, portfolio variance is high because assets tend to move together. As correlation decreases toward zero, combining assets reduces overall risk. With negative correlation, especially near \u22121, portfolio risk can be reduced dramatically, and in the case of perfect negative correlation, certain combinations can even eliminate variance.\",\n      \"dateCreated\": \"2024-09-01T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/blog\/covariance-and-correlation-calculations-for-cfa-and-frm-exams\/#faq-correlation-portfolio-risk\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\n}\n}\n<\/script>\n\n\n\n<p>The 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 the variables X and Y, where we can let:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>X = interest rate<\/li>\n\n\n\n<li>Y = inflation<\/li>\n<\/ul>\n\n\n\n<p>The general formula used to calculate the covariance between two random variables, X and Y, is:<\/p>\n\n\n\n<p>$$COV\\ [X,\\ Y] =E[(X-E[X]) (Y-E[Y])]$$<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<p>\\(X\\) = a random variable such as lending interest rates.<\/p>\n\n\n\n<p>\\(Y\\) = another random variable, such as inflation rates.<\/p>\n\n\n\n<p>\\(E[X]\\) = The expected value (mean) of X.<\/p>\n\n\n\n<p>\\(E[Y]\\) = The expected value (mean) of Y.<\/p>\n\n\n\n<p>The covariance between two random variables can be <strong>positive<\/strong>, <strong>negative,<\/strong> or <strong>zero<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A <strong>positive<\/strong> number indicates co-movement (i.e., the variables tend to move in the <strong>same direction<\/strong>);<\/li>\n\n\n\n<li>A value of <strong>zero<\/strong> indicates <strong>no relationship;<\/strong> and<\/li>\n\n\n\n<li>A <strong>negative<\/strong> value shows that the variables move in <strong>opposite directions<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>Moreover, if variables are independent, their covariance is zero, i.e.,<\/p>\n\n\n\n<p>$$COV\\left(X,Y\\right)=E\\left(XY\\right)-E\\left(X\\right)E\\left(Y\\right)=0$$<\/p>\n\n\n\n<p>Covariances can be represented in a tabular format in a covariance matrix as follows:<\/p>\n\n\n\n<p>$$ \\begin{array}{c|ccc} \\text{Asset} &amp; A &amp; B &amp; C \\\\ \\hline A &amp; \\text{Cov}(A, A) &amp; \\text{Cov}(A, B) &amp; \\text{Cov}(A, C) \\\\ B &amp; \\text{Cov}(B, A) &amp; \\text{Cov}(B, B) &amp; \\text{Cov}(B, C) \\\\ C &amp; \\text{Cov}(C, A) &amp; \\text{Cov}(C, B) &amp; \\text{Cov}(C, C) \\\\ \\end{array} $$<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The off-diagonal terms represent variances since, for example, \\(\\text{Cov(C, C) = Var(C)}\\)<\/li>\n\n\n\n<li>A two-asset portfolio would have a similar 2 \u00d7 2 matrix.<\/li>\n\n\n\n<li>A correlation matrix can also be created to represent the correlations between various assets in a large portfolio.<\/li>\n<\/ul>\n\n\n\n<div style=\"margin:20px 0\">\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\"\nstyle=\"display:block;width:100%;text-align:center;padding:10px;border:2px solid #2f5bea;border-radius:40px;font-size:16px;color:#2f5bea;text-decoration:none\">\nPractice covariance and correlation questions with our free trial.\n<\/a>\n<\/div>\n\n\n<h4><span class=\"ez-toc-section\" id=\"Example_1_Calculating_the_Covariance_of_a_Portfolio_of_Two_Assets\"><\/span>Example 1: Calculating the Covariance of a Portfolio of Two Assets<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>A portfolio comprises two stocks \u2013 1 and 2. The returns for the last 5 years are as follow:<\/p>\n<p style=\"padding-left: 30px;\">Stock 1: 5%; 4.5%; 4.8%; 5.5%; 6%.<\/p>\n<p style=\"padding-left: 30px;\">Stock 2: 6%; 6.2%; 5.7%; 6.1%; 6.5%.<\/p>\n<p>Compute the covariance.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Solution\"><\/span>Solution<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Step 1<\/strong>: We calculate the weighted sum of each stock to get the expected return on Stock 1 and Stock 2<\/p>\n<p>$$\\begin{align} E(R_1)&amp; = \\frac{5\\%+4.5\\%+4.8\\%+5.5\\%+ 6\\%}{5} = 5.2\\% \\\\ E(R_2) &amp;= \\frac{6\\%+ 6.2\\%+ 5.7\\%+ 6.1\\%+ 6.5\\%}{5} = 6.1\\% \\end{align}$$<\/p>\n<p><strong>Step 2:<\/strong> We subtract each year&#8217;s return from the expected return to obtain [R1-E(R1)] and [R2-E(R2)] as follows:<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} \\textbf{Year} &amp; R_1 &amp; R_2 &amp; R_1 &#8211; \\mathbb{E}(R_1) &amp; R_2 &#8211; \\mathbb{E}(R_2) \\\\ \\hline 1 &amp; 5\\% &amp; 6\\% &amp; -0.2\\% &amp; -0.1\\% \\\\ 2 &amp; 4.5\\% &amp; 6.2\\% &amp; -0.7\\% &amp; 0.1\\% \\\\ 3 &amp; 4.8\\% &amp; 5.7\\% &amp; -0.4\\% &amp; -0.4\\% \\\\ 4 &amp; 5.5\\% &amp; 6.1\\% &amp; 0.3\\% &amp; 0\\% \\\\ 5 &amp; 6\\% &amp; 6.5\\% &amp; 0.8\\% &amp; 0.4\\% \\\\ \\end{array} $$<\/p>\n<p>Step 3: We multiply the values obtained in Step 2 and we divide by the number of observations to get a <em>mean<\/em> observation.<\/p>\n<p>$$\\begin{align}<br \/>\\text{Cov}(R_1, R_2) &amp;= \\mathbb{E}[(R_1 &#8211; \\mathbb{E}[R_1])(R_2 &#8211; \\mathbb{E}[R_2])] \\\\<br \/>&amp;= \\frac{<br \/>\\begin{array}{l}<br \/>(-0.002 \\times -0.001) + (-0.007 \\times 0.001) + \\\\<br \/>(-0.004 \\times -0.004) + (0.003 \\times 0) + (0.008 \\times 0.004)<br \/>\\end{array}<br \/>}{5} \\\\<br \/>&amp;= \\frac{0.000002 + (-0.000007) + 0.000016 + 0 + 0.000032}{5} \\\\<br \/>&amp;= \\frac{0.000043}{5} = 0.0000086<br \/>\\end{align}$$<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Example_2_Covariance\"><\/span>Example 2: Covariance<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Consider a set of a well-diversified portfolios X and Y. Suppose that the mean returns for X and Y are 6.12 and 7.04, respectively, what is the covariance between these portfolios?<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Solution-2\"><\/span><strong>Solution<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Recall,<\/p>\n<p>$$COV\\left(X,Y\\right)=E\\left(XY\\right)-E\\left(X\\right)E\\left(Y\\right)\\)$$<\/p>\n<p>For independent variables,<\/p>\n<p>$$E\\left(XY\\right)=E\\left(X\\right)E\\left(Y\\right)=6.12\\times7.04=43.0848$$<\/p>\n<p>Which implies that,<\/p>\n<p>$$COV\\left(X,Y\\right)=E\\left(XY\\right)-E\\left(X\\right)E\\left(Y\\right)=43.0848-6.12\\left(7.04\\right)=0$$<\/p>\n<h1 style=\"margin-top: 12pt; margin-bottom: 0pt; text-align: justify; page-break-inside: avoid; page-break-after: avoid; line-height: 115%; font-size: 20pt;\"><span class=\"ez-toc-section\" id=\"Correlation\"><\/span>Correlation<span class=\"ez-toc-section-end\"><\/span><\/h1>\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>$$\\text{Correlation}\\ (X_1,X_2\\ )=\\frac{Cov(X_1,X_2\\ )}{\\text{Standard deviation}\\ (X_1\\ )\\times \\text{Standard deviation}\\ (X_2\\ )}$$<\/p>\n<p>Correlation measures the strength of the linear relationship between two variables. While the covariance can take on any value between negative infinity and positive infinity, the correlation is always a value between -1 and +1.<\/p>\n<p>A correlation of -1 indicates a perfect inverse relationship (i.e. a unit change in one means that the other will have a unit change in the opposite direction). Secondly, a correlation of +1 indicates a perfect linear relationship (i.e. the two variables move in the same direction with the unit changes being equal). Finally, a correlation of zero implies that there is no linear relationship between the variables.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Example_Calculating_the_Covariance\"><\/span>Example: Calculating the Covariance<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>We anticipate that there is 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 that they will be 10%. We already know 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. The expected value of returns is then 4.975 and the standard deviation is 0.46%.<\/p>\n<p>$$ \\begin{align}<br \/>\\text{cov}(\\text{R}_{ \\text{ABC}},\\text{R}_{ \\text{XYZ}}) &amp; = 0.15(0.06 \u2013 0.082)(0.04 \u2013 0.04975) \\\\<br \/>&amp; + 0.6(0.08 \u2013 0.082)(0.05 \u2013 0.04975) \\\\<br \/>&amp; + 0.25(0.10 \u2013 0.082)(0.055 \u2013 0.04975) \\\\<br \/>&amp; = 0.0000561 \\\\<br \/>\\end{align} $$<\/p>\n<p>Recall that,<\/p>\n<p>$$\\text{Corr}(\\text{R}_{ \\text{ABC}},\\text{R}_{ \\text{XYZ}})=\\frac{\\text{Cov}(\\text{R}_{ \\text{ABC}},\\text{R}_{ \\text{XYZ}}\\ )}{(\\text{Standard deviation}\\ (\\text{R}_{ \\text{ABC}}\\ )\\times \\text{Standard deviation} (\\text{R}_{ \\text{XYZ}}\\ ))}$$<\/p>\n<p>Thus,<\/p>\n<p>$$\\text{Correlation}=0.0000561\\left(0.01249\\times0.0046\\right)=0.976$$<\/p>\n<p><strong>Interpretation<\/strong>: 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<p><blockquote class=\"wp-embedded-content\" data-secret=\"4LnQjQMOb6\"><a href=\"https:\/\/analystprep.com\/shop\/all-3-levels-of-the-cfa-exam-complete-course-by-analystprep\/\">All 3 Levels of the CFA Exam &#8211; Complete Course offered by AnalystPrep<\/a><\/blockquote><iframe loading=\"lazy\" class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; visibility: hidden;\" title=\"&#8220;All 3 Levels of the CFA Exam &#8211; Complete Course offered by AnalystPrep&#8221; &#8212; AnalystPrep\" src=\"https:\/\/analystprep.com\/shop\/all-3-levels-of-the-cfa-exam-complete-course-by-analystprep\/embed\/#?secret=gmDahvQ32p#?secret=4LnQjQMOb6\" data-secret=\"4LnQjQMOb6\" width=\"600\" height=\"338\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe><\/p>\n<h1 style=\"margin-top: 12pt; margin-bottom: 0pt; text-align: justify; page-break-inside: avoid; page-break-after: avoid; line-height: 115%; font-size: 20pt;\"><span class=\"ez-toc-section\" id=\"How_does_Correlation_Impact_Portfolio_Risk\"><\/span>How does Correlation Impact Portfolio Risk?<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>As mentioned earlier, correlation ranges from -1 to +1<\/p>\n<ul type=\"disc\">\n<li>+1 = returns are perfectly positively correlated.<\/li>\n<li>0 = returns of two assets are not correlated.<\/li>\n<li>-1 = returns are perfectly negatively correlated.<\/li>\n<li>What happens to portfolio risk (in a portfolio of two risky assets) when the two assets are perfectly correlated?<\/li>\n<li>Risk is unaffected; no benefit of diversification<\/li>\n<\/ul>\n<p>In conclusion, using negatively correlated investments to form a portfolio helps to reduce the overall volatility of the portfolio.<\/p>\n<h3 data-start=\"155\" data-end=\"192\"><span class=\"ez-toc-section\" id=\"Ready_to_Apply_These_Formulas\"><\/span>Ready to Apply These Formulas?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p data-start=\"194\" data-end=\"497\">Understanding the formulas is one thing\u2014mastering them under exam pressure is another. Put your knowledge to the test with AnalystPrep\u2019s CFA Level 1 practice questions. You\u2019ll get access to hundreds of quant-focused problems, detailed solutions, and performance tracking tools to help you study smarter.<\/p>\n<p><blockquote class=\"wp-embedded-content\" data-secret=\"j2C34ypFXl\"><a href=\"https:\/\/analystprep.com\/cfa-level-1-practice-questions\/\">CFA Level I Practice Questions<\/a><\/blockquote><iframe loading=\"lazy\" class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; visibility: hidden;\" title=\"&#8220;CFA Level I Practice Questions&#8221; &#8212; AnalystPrep\" src=\"https:\/\/analystprep.com\/cfa-level-1-practice-questions\/embed\/#?secret=H6XjUk6EI4#?secret=j2C34ypFXl\" data-secret=\"j2C34ypFXl\" width=\"600\" height=\"338\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe><\/p>\n<h1><span class=\"ez-toc-section\" id=\"FAQs_on_Covariance_and_Correlation\"><\/span>FAQs on Covariance and Correlation<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h3><span class=\"ez-toc-section\" id=\"1_What_is_the_covariance_formula_for_CFA_Level_1\"><\/span>1. What is the covariance formula for CFA Level 1?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The formula for covariance between two random variables \\( X \\) and \\( Y \\) is:<\/p>\n<p>\\[ \\text{Cov}(X, Y) = \\mathbb{E}\\left[(X &#8211; \\mathbb{E}[X])(Y &#8211; \\mathbb{E}[Y])\\right] \\]<\/p>\n<p>Alternatively:<\/p>\n<p>\\[ \\text{Cov}(X, Y) = \\mathbb{E}[XY] &#8211; \\mathbb{E}[X]\\mathbb{E}[Y] \\]<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\( X, Y \\) = Random variables (e.g., returns, interest rates)<\/li>\n<li>\\( \\mathbb{E}[X], \\mathbb{E}[Y] \\) = Expected values (means) of \\( X \\) and \\( Y \\)<\/li>\n<li>\\( \\mathbb{E}[XY] \\) = Expected value of the product \\( XY \\)<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"2_Whats_the_difference_between_covariance_and_correlation\"><\/span>2. What\u2019s the difference between covariance and correlation?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Covariance indicates the direction of co-movement between two variables but is affected by the units of measurement. Correlation standardizes this to a scale from -1 to +1.<\/p>\n<p>\\[ \\rho(X, Y) = \\frac{\\text{Cov}(X, Y)}{\\sigma_X \\cdot \\sigma_Y} \\]<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\( \\rho(X, Y) \\): Correlation between \\( X \\) and \\( Y \\)<\/li>\n<li>\\( \\text{Cov}(X, Y) \\): Covariance between \\( X \\) and \\( Y \\)<\/li>\n<li>\\( \\sigma_X, \\sigma_Y \\): Standard deviations of \\( X \\) and \\( Y \\), respectively<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"3_How_do_I_convert_covariance_to_correlation\"><\/span>3. How do I convert covariance to correlation?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Use the correlation formula to convert covariance to a dimensionless metric:<\/p>\n<p>\\[ \\text{Correlation} = \\frac{\\text{Cov}(X, Y)}{\\sigma_X \\cdot \\sigma_Y} \\]<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\( \\text{Cov}(X, Y) \\): Covariance between the two variables<\/li>\n<li>\\( \\sigma_X \\): Standard deviation of variable \\( X \\)<\/li>\n<li>\\( \\sigma_Y \\): Standard deviation of variable \\( Y \\)<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"4_Why_is_a_correlation_of_zero_not_the_same_as_independence\"><\/span>4. Why is a correlation of zero not the same as independence?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A correlation of zero means there&#8217;s no <em>linear<\/em> relationship between variables, but they may still be dependent in other ways. Independence is a much stronger condition.<\/p>\n<p>$$\\begin{align}\\rho(X, Y) = 0 &amp;\\Rightarrow \\text{No linear relationship} \\\\ \\text{Cov}(X, Y) = 0 &amp; \\Rightarrow \\text{Independence} \\end{align}$$<\/p>\n<p>However, the converse is not always true: zero covariance does <em>not<\/em> imply independence unless the variables are jointly normally distributed.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5_How_does_correlation_affect_portfolio_risk\"><\/span>5. How does correlation affect portfolio risk?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The correlation between assets determines the potential for diversification in a portfolio. The lower the correlation, the greater the risk-reduction benefit.<\/p>\n<ul>\n<li>\\( \\rho = +1 \\): Perfect positive correlation \u2013 no diversification benefit<\/li>\n<li>\\( \\rho = 0 \\): No linear relationship \u2013 some diversification benefit<\/li>\n<li>\\( \\rho = -1 \\): Perfect negative correlation \u2013 maximum diversification benefit<\/li>\n<\/ul>\n<p>In general, combining negatively or weakly correlated assets helps reduce portfolio volatility.<\/p>\n\n\n<div style=\"text-align:center;margin:40px 0\">\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\"\nstyle=\"display:inline-block;padding:10px 26px;background:#3f78d7;color:#fff;border-radius:40px;font-size:16px;text-decoration:none\">\nStart Free Trial \u2192\n<\/a>\n<p style=\"margin-top:10px;max-width:600px;margin-left:auto;margin-right:auto;font-size:14px\">\nSolve CFA and FRM-style questions and master covariance and correlation calculations.\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The 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 the variables X and Y, where we can let: The general formula used to&#8230;<\/p>\n","protected":false},"author":2,"featured_media":8308,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[70,71],"tags":[],"class_list":["post-8212","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-cfa","category-frm","blog-post","animate"],"acf":[],"post_mailing_queue_ids":[],"_links":{"self":[{"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/posts\/8212","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/comments?post=8212"}],"version-history":[{"count":36,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/posts\/8212\/revisions"}],"predecessor-version":[{"id":14359,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/posts\/8212\/revisions\/14359"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/media\/8308"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/media?parent=8212"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/categories?post=8212"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/tags?post=8212"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}