{"id":27507,"date":"2022-11-09T06:11:05","date_gmt":"2022-11-09T06:11:05","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=27507"},"modified":"2026-02-16T17:03:02","modified_gmt":"2026-02-16T17:03:02","slug":"expected-value-variance-standard-deviation-covariances-and-correlations-of-portfolio-returns","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/expected-value-variance-standard-deviation-covariances-and-correlations-of-portfolio-returns\/","title":{"rendered":"Expected Value, Variance, Standard Deviation, Covariances, and Correlations of Portfolio Returns"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n\n  \"name\": \"Basics of Derivative Pricing and Valuation (2025 Level I CFA\u00ae Exam \u2013 Derivative \u2013 Module 2)\",\n\n  \"description\": \"This video lesson covers Topic 7 \u2013 Derivatives, Module 2 \u2013 Basics of Derivative Pricing and Valuation. It explores core concepts like arbitrage, replication, and risk neutrality in pricing derivatives. Key highlights include the valuation of forward and futures contracts, factors affecting options pricing, put\u2013call parity, and the binomial model. It also differentiates between European and American options and explains their respective value determinants.\",\n\n  \"uploadDate\": \"2022-06-29T00:00:00+00:00\",\n\n  \"thumbnailUrl\": \"https:\/\/analystprep.com\/path-to-thumbnail\/derivative-pricing-thumbnail.jpg\", \n\n  \"contentUrl\": \"https:\/\/youtu.be\/0Geaej45v7w\",\n\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/0Geaej45v7w\",\n\n  \"duration\": \"PT1H08M27S\",\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<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Assume that we have equally invested in two different companies, ABC and XYZ, with given probabilities and return distributions. The portfolio standard deviation is closest to:\",\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C.\\n\\nFirst, compute covariance:\\nCov(RABC, RXYZ) = 0.15(0.06\u22120.082)(0.04\u22120.04975) + 0.6(0.08\u22120.082)(0.05\u22120.04975) + 0.25(0.10\u22120.082)(0.055\u22120.04975) = 0.0000561.\\n\\nPortfolio variance = 0.5\u00b2 \u00d7 0.01249\u00b2 + 0.5\u00b2 \u00d7 0.0046\u00b2 + 2 \u00d7 0.5 \u00d7 0.5 \u00d7 0.0000561 = 0.00007234.\\n\\nPortfolio standard deviation = \u221a0.00007234 = 0.00851.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"0.0000561\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"0.00007234\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"0.00851\"\n      }\n    ],\n    \"answerCount\": 3\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/PsZrsg3ZUDE\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>A portfolio is a collection of investments a company, mutual fund, or individual investor holds. A portfolio consists 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 individual asset\u2019s expected returns multiplied by its associated weight. Therefore:<\/p>\n<p>$$ E(R_p) = \\sum {W_i R_i} \\text{ where i = 1,2,3 \u2026 n} $$<\/p>\n<p>Where:<\/p>\n<p>\\(W_i\\) = Weights (market value) attached to each asset \\(i\\).<\/p>\n<p>\\(R_i\\) = Returns expected by each each asset \\(i\\).<\/p>\n<p><strong>Example: Portfolio Expected Return<\/strong><\/p>\n<p>Assume that we have a simple portfolio of two mutual funds, one invested in bonds and the other in stocks. Let us 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>$$ \\begin{align*} E(R_p)&amp; = 0.5 \u00d7 0.08 + 0.5 \u00d7 0.06 \\\\ &amp; = 0.07 \\text{ or } 7\\% \\\\ \\end{align*} $$<\/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\"> Practice portfolio return and risk calculations with CFA\u00ae-style problems. <\/a><\/p>\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, then:<\/p>\n<p>$$ \\text{Portfolio variance} = { W }_{ A }^{ 2 } { \\sigma }^{ 2 }\\left( { R }_{ A } \\right) +{ W }_{ B }^{ 2 }{ \\sigma }^{ 2 }\\left( { R }_{ B } \\right) +2 \\left( { W }_{ A } \\right)\u00a0 \\left( { W }_{ B } \\right)\u00a0 Cov\\left( { R }_{ A },{ R }_{ B } \\right) $$<\/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. It is a measure of the riskiness of a portfolio. Thus:<\/p>\n<p>$$ \\text{Standard deviation}=\\sqrt{ { W }_{ A }^{ 2 } \u00a0{ \\sigma }^{ 2 }\\left( { R }_{ A } \\right) +{ W }_{ B }^{ 2 } \u00a0{ \\sigma }^{ 2 }\\left( { R }_{ B } \\right) +2 \u00a0\\left( { W }_{ A } \\right) \u00a0\\left( { W }_{ B } \\right)\u00a0 \\text{cov} \\left( { R }_{ A },{ R }_{ B } \\right)\u00a0 } $$<\/p>\n<p>Where:<\/p>\n<p>\\(\\text{cov} ( { R }_{ A },{ R }_{ B })\\) = Directional relationship between the returns on assets A and B.<\/p>\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 the interest rate and the 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>$$ \\text{cov}[X,Y ] = E [(X &#8211; E[X ])(Y &#8211; E[Y ])] $$<\/p>\n<p>While the abovementioned covariance formula is correct, we use a slightly modified formula to calculate the covariance of returns from a joint probability 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_i\\) and \\(R_j\\) 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.<\/p>\n<ul>\n<li>A positive number indicates co-movement, i.e., the variables tend to move in the <strong>same<\/strong> <strong>direction<\/strong>.<\/li>\n<li>A value of zero indicates <strong>no<\/strong> <strong>relationship<\/strong>.<\/li>\n<li>A negative value shows that the variables move in <strong>opposite directions<\/strong>.<\/li>\n<\/ul>\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>$$ { \\text{Correlation} }\\left( { R }_{ i },{ R }_{ j } \\right) =\\frac { \\text{Covariance}\\left( { R }_{ i },{ R }_{ j } \\right) }{ \\text{Standard deviation}\\left( { R }_{ i } \\right) \u00d7 \\text{Standard deviation}\\left( { R }_{ j } \\right) } $$<\/p>\n<p>Correlation measures the strength of the linear relationship between two variables. While covariance can take on any value between negative infinity and positive infinity, correlation is always a value between -1 and +1.<\/p>\n<ul>\n<li>+1 indicates\u00a0<strong>a perfect linear relationship<\/strong> (i.e., the two variables move in the same direction with equal unit changes).<\/li>\n<li>Zero indicates <strong>no<\/strong>\u00a0<strong>linear relationship<\/strong><strong>\u00a0<\/strong>at all.<\/li>\n<li>-1 indicates a <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.<\/li>\n<\/ul>\n<h4><strong>Example: Calculating the Correlation<\/strong><strong> Coefficient #1<\/strong><\/h4>\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 a 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. The expected value of returns is then 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>$$ { \\text{Correlation} }\\left( { R }_{ i },{ R }_{ j } \\right) =\\frac { \\text{Covariance}\\left( { R }_{ ABC },{ R }_{ XYZ } \\right) }{ \\text{Standard deviation}\\left( { R }_{ ABC } \\right) \u00d7 \\text{Standard deviation}\\left( { R }_{ XYZ } \\right) } $$<\/p>\n<p>Therefore:<\/p>\n<p>$$ \\begin{align*} \\text{Correlation} &amp; =\\cfrac {0.0000561}{(0.01249 \u00d7 0.0046)} \\\\ &amp; = 0.976 \\\\ \\end{align*} $$<\/p>\n<p><em><strong>Interpretation<\/strong><\/em>: 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<h4><strong>Example: Calculating Correlation Coefficient #2<\/strong><\/h4>\n<p>An analyst is analyzing the impact of changes in interest rates introduced by the Central Bank on the country\u2019s inflation rate. He analyzes historical data for five years. The covariance between the interest rate and the inflation rate is -0.00075, while the standard deviation of the interest rate is 5.5%, and the inflation rate is 12%. Calculate and interpret the correlation between interest rate and inflation rate.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$\\begin{align}\\text{Correlation}_{\\text{(Interest rate, Inflation)}}&amp;=\\frac{\\text{Covariance (Interest rate, Inflation)}}{\\text{Standard deviation of interest rate}\\times \\text{Standard deviation of inflation}}\\\\ &amp;=\\frac{-0.00075}{0.055\\times 0.12}=-0.11364\\end{align}$$<\/p>\n<p><em><strong>Interpretation<\/strong><\/em>: A correlation of -0.11364 indicates a negative correlation between the interest rate and the inflation rate.<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Assume that we have equally invested 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% probability 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>Besides, we anticipate that the same probabilities 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>The portfolio standard deviation is <em>closest<\/em> to:<\/p>\n<p>A. 0.0000561.<\/p>\n<p>B. 0.00007234.<\/p>\n<p>C. 0.00851.<\/p>\n<p>The correct answer is <strong>C<\/strong>.<\/p>\n<p><strong>Actual calculation<\/strong>:<\/p>\n<p>$$ \\text{Portfolio variance} = { W }_{ A }^{ 2 } \u00d7 { \\sigma }^{ 2 }\\left( { R }_{ A } \\right) +{ W }_{ B }^{ 2 } \u00d7 { \\sigma }^{ 2 }\\left( { R }_{ B } \\right) +2\u00d7 \\left( { W }_{ A } \\right) \u00d7 \\left( { W }_{ B } \\right) \u00d7 Cov\\left( { R }_{ A },{ R }_{ B } \\right) $$<\/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>$$ \\begin{align*} \\text{Portfolio variance} &amp; = 0.5^2\u00d7 0.01249^2+ 0.5^2\u00d7 0.0046^2+ 2 \u00d7 0.5 \u00d7 0.5\u00d7 0.0000561 \\\\ &amp; = 0.00007234 \\\\ \\end{align*} $$<\/p>\n<p>Therefore, the standard deviation is \\(\\sqrt{0.00007234} = 0.00851\\).<\/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>A portfolio is a collection of investments a company, mutual fund, or individual investor holds. A portfolio consists of assets such as stocks, bonds, or cash equivalents. Financial professionals usually manage a portfolio.<\/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-27507","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 Returns: Expected Value &amp; Variance | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Analyze portfolio returns with metrics like expected value, variance, standard deviation, covariance, and correlation for investment insights.\" \/>\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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