{"id":575,"date":"2019-10-10T13:27:00","date_gmt":"2019-10-10T13:27:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=575"},"modified":"2026-02-17T18:58:44","modified_gmt":"2026-02-17T18:58:44","slug":"portfolio-expected-return-variance","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/portfolio-expected-return-variance\/","title":{"rendered":"Portfolio Returns"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Calculating portfolio standard deviation with two equally weighted assets\",\n    \"text\": \"Assume an equal investment in two companies, ABC and XYZ. For ABC, the probabilities of returns are 15% for 6%, 60% for 8%, and 25% for 10%, with an expected return of 8.2% and a standard deviation of 1.249%. For XYZ, the same probabilities correspond to returns of 4%, 5%, and 5.5%, with an expected return of 4.975% and a standard deviation of 0.46%.\\n\\nCalculate the portfolio standard deviation.\\n\\nA. 0.0000561\\n\\nB. 0.0000\\n\\nC. 0.00851\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"C. 0.00851.\\n\\nUsing equal weights of 0.5 for each asset, the portfolio variance equals the weighted variances of each asset plus twice the weighted covariance between them. The covariance between ABC and XYZ is 0.0000561, leading to a portfolio variance of 0.00007234. Taking the square root yields a portfolio standard deviation of 0.00851.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"A. 0.0000561\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"B. 0.0000\"\n      }\n    ]\n  }\n}\n<\/script>\n\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n\n  \"name\": \"Probability Concepts (2021 Level I CFA\u00ae Exam \u2013 Reading 8)\",\n\n  \"description\": \"This video lesson explores Topic 1 \u2013 Quantitative Methods, Module 2, delving into probability concepts. Key highlights include defining random variables, understanding mutual exclusivity, conditional vs. unconditional probabilities, Bayes' formula, expected values, covariance, correlation, probability distributions, and advanced counting techniques like factorials, permutations, and combinations.\",\n\n  \"uploadDate\": \"2021-10-01T00:00:00+00:00\",\n\n  \"thumbnailUrl\": \"https:\/\/analystprep.com\/path-to-thumbnail\/probability-concepts-thumbnail.jpg\",\n\n  \"contentUrl\": \"https:\/\/youtu.be\/hu47ZbsskEw\",\n\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw\",\n\n  \"duration\": \"PT48M33S\",\n\n  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/path-to-logo\/logo.jpg\",\n      \"width\": 600,\n      \"height\": 60\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw?si=_uXr4yALZfkWvyTr\" 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>A portfolio is basically a collection of investments held by a company, mutual fund, or even an individual investor, consisting of assets such as stocks, bonds, or cash equivalents. Financial professionals usually manage a portfolio.<\/p>\n<p><!--more--><\/p>\n<h2><strong>Portfolio Expected Return<\/strong><\/h2>\n<p>Portfolio expected return is the sum of each of the individual asset\u2019s expected returns multiplied by its associated weight. Therefore:<\/p>\n<p>E(R<sub>p<\/sub>) = \u03a3W<sub>i<\/sub> R<sub>i<\/sub> where i = 1,2,3 \u2026 n<\/p>\n<p>Where W<sub>i <\/sub>represents the weight attached to the asset, i, and R<sub>i <\/sub>is the asset\u2019s return.<\/p>\n<p>The weight attached to an asset = market value of an asset\/market value of a portfolio<\/p>\n<p><strong>Example<\/strong><\/p>\n<p>Assume that we have a simple portfolio of two mutual funds, one invested in bonds and the other invested in stocks. Let\u2019s further assume that we expect a stock return of 8% and a bond return of 6% and our allocation is equal in both funds. Then:<\/p>\n<p>$$ \\begin{align*} E(R_p)&amp; = 0.5 * 0.08 + 0.5 * 0.06 \\\\ &amp; = 0.07 \\text{ or } 7\\% \\\\ \\end{align*} $$<\/p>\n<div style=\"margin:18px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" \n     target=\"_blank\" \n     rel=\"noopener noreferrer\"\n     style=\"\n        display:block;\n        text-align:center;\n        padding:14px 18px;\n        border:2px solid #2F5BFF;\n        border-radius:18px;\n        color:#2F5BFF;\n        font-weight:600;\n        font-size:16px;\n        text-decoration:none;\n        background-color:#f9faff;\n     \">\n     Practice CFA questions on portfolio return and variance.\n  <\/a>\n<\/div>\n\n<h2><strong>Portfolio Variance<\/strong><\/h2>\n<p>The variance of a portfolio\u2019s return is always a function of the individual assets as well as the covariance between each of them. If we have two assets, A and B,<\/p>\n<p>$$ \\text{Portfolio variance} = { W }_{ A }^{ 2 }\\ast { \\sigma }^{ 2 }\\left( { R }_{ A } \\right) +{ W }_{ B }^{ 2 }\\ast { \\sigma }^{ 2 }\\left( { R }_{ B } \\right) +2\\ast \\left( { W }_{ A } \\right) \\ast \\left( { W }_{ B } \\right) \\ast Cov\\left( { R }_{ A },{ R }_{ B } \\right) $$<\/p>\n<p>Portfolio variance is a measure of risk. More variance translates to more risk. Investors usually reduce the portfolio variance by choosing assets that have low or negative covariance, e.g. stocks and bonds.<\/p>\n<h2><strong>Portfolio Standard Deviation<\/strong><\/h2>\n<p>This is simply the square root of the portfolio variance. Therefore:<\/p>\n<p>$$ S.D={ \\left\\{ { W }_{ A }^{ 2 }\\ast { \\sigma }^{ 2 }\\left( { R }_{ A } \\right) +{ W }_{ B }^{ 2 }\\ast { \\sigma }^{ 2 }\\left( { R }_{ B } \\right) +2\\ast \\left( { W }_{ A } \\right) \\ast \\left( { W }_{ B } \\right) \\ast Cov\\left( { R }_{ A },{ R }_{ B } \\right) \\right\\} }^{ \\frac { 1 }{ 2 } } $$<\/p>\n<p>It is also a measure of the riskiness of a portfolio.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Assume that we have invested equally in two different companies; ABC and XYZ. 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% <a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/introduction-probability-defining-basic-terms\/\">probability <\/a>that they will be 10%. In addition, 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. The expected value of returns is then 4.975 and the standard deviation is 0.46%.<\/p>\n<p>Calculate the portfolio standard deviation:<\/p>\n<p>A. 0.0000561<\/p>\n<p>B. 0.0000<\/p>\n<p>C. 0.00851<\/p>\n<p>The correct answer is C.<\/p>\n<p>Working<\/p>\n<p>$$ \\text{Portfolio variance} = { W }_{ A }^{ 2 }\\ast { \\sigma }^{ 2 }\\left( { R }_{ A } \\right) +{ W }_{ B }^{ 2 }\\ast { \\sigma }^{ 2 }\\left( { R }_{ B } \\right) +2\\ast \\left( { W }_{ A } \\right) \\ast \\left( { W }_{ B } \\right) \\ast Cov\\left( { R }_{ A },{ R }_{ B } \\right) $$<\/p>\n<p>First, we must calculate the covariance between the two stocks:<\/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>Since we already have the weight and the standard deviation of each asset, we can proceed and calculate the portfolio variance:<\/p>\n<p>$$ \\begin{align*} &amp; = 0.5^2* 0.01249^2+ 0.5^2* 0.0046^2+ 2 * 0.5 * 0.5 * 0.0000561 \\\\ &amp; = 0.00007234 \\\\ \\end{align*} $$<\/p>\n<p>Therefore, the standard deviation is \\(0.00007234^{\\frac {1}{2}} = 0.00851\\)<\/p>\n<\/blockquote>\n<p><em>Reading 8 LOS 8l<\/em><\/p>\n<p><em>Calculate and interpret the expected value, variance, and standard deviation of a random variable and returns on a portfolio.<\/em><\/p>\n<div class=\"notes_inv\"><hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum-quantitative-methods\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>\n<p>\u00a0<\/p>\n<div style=\"text-align:center; margin:40px 0 20px;\">\n\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" \n     target=\"_blank\" \n     rel=\"noopener noreferrer\"\n     style=\"\n        display:inline-block;\n        background:#3E73D9;\n        color:#ffffff;\n        padding:16px 34px;\n        border-radius:50px;\n        font-weight:700;\n        font-size:18px;\n        text-decoration:none;\n     \">\n     Start Free Trial\n  <\/a>\n\n  <p style=\"\n        margin-top:18px;\n        font-size:16px;\n        line-height:1.6;\n        max-width:700px;\n        margin-left:auto;\n        margin-right:auto;\n        color:#333333;\n     \">\n     Strengthen your CFA Level I Quantitative Methods preparation with exam-style questions, QBank drills, and full mock exams designed to help you master portfolio risk, covariance, and diversification concepts under exam conditions.\n  <\/p>\n\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>A portfolio is basically a collection of investments held by a company, mutual fund, or even an individual investor, consisting of assets such as stocks, bonds, or cash equivalents. Financial professionals usually manage a portfolio.<\/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-575","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>Portfolio Expected Return &amp; Variance | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Calculate portfolio expected return and variance, analyzing a mix of assets like stocks and bonds to evaluate investment performance and risk.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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