{"id":45440,"date":"2023-08-06T10:59:31","date_gmt":"2023-08-06T10:59:31","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=45440"},"modified":"2026-03-27T16:06:51","modified_gmt":"2026-03-27T16:06:51","slug":"portfolio-expected-return-and-variance-of-return","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/portfolio-expected-return-and-variance-of-return\/","title":{"rendered":"Portfolio Expected Return and Variance of Return"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Portfolio Mathematics (2025 CFA\u00ae Level I Exam \u2013 Quantitative Methods \u2013 Learning Module 5)\",\n  \"description\": \"This video covers Portfolio Mathematics for the CFA Level I Quantitative Methods curriculum. Topics include calculating and interpreting the expected value, variance, standard deviation, covariance, and correlation of portfolio returns. It also explains how to compute covariance and correlation using a joint probability function for returns, and introduces shortfall risk, the safety-first ratio, and portfolio selection using Roy\u2019s safety-first criterion.\",\n  \"uploadDate\": \"2024-08-03T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/bQUXvapTZrg\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/bQUXvapTZrg\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/bQUXvapTZrg\",\n  \"duration\": \"PT32M42S\"\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\": \"Assume that we have investments in two companies, ABC and XYZ. For ABC, there\u2019s a 15% chance of a 6% return, a 60% chance of an 8% return, and a 25% chance of a 10% return. The expected return for ABC is 8.2%, and the standard deviation is 1.249%. For XYZ, there are similar probabilities of 4%, 5%, and 5.5% returns. The expected return for XYZ is 4.975%, and the standard deviation is 0.46%. Assuming equal weights, the portfolio standard deviation is closest to:\",\n    \"text\": \"Assume that we have investments in two companies, ABC and XYZ. For ABC, there\u2019s a 15% chance of a 6% return, a 60% chance of an 8% return, and a 25% chance of a 10% return. The expected return for ABC is 8.2%, and the standard deviation is 1.249%. For XYZ, there are similar probabilities of 4%, 5%, and 5.5% returns. The expected return for XYZ is 4.975%, and the standard deviation is 0.46%. Assuming equal weights, the portfolio standard deviation is closest to:\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. First compute the covariance: Cov(R_ABC, R_XYZ) = 0.15(0.06\u22120.082)(0.04\u22120.04975) + 0.60(0.08\u22120.082)(0.05\u22120.04975) + 0.25(0.10\u22120.082)(0.055\u22120.04975) = 0.0000561. With equal weights (0.5 each), portfolio variance = (0.5^2)(0.01249^2) + (0.5^2)(0.0046^2) + 2(0.5)(0.5)(0.0000561) = 0.00007234. Therefore, portfolio standard deviation = sqrt(0.00007234) = 0.00851.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/bQUXvapTZrg?si=NiNwTyzaXr1_zmY1\" 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 a collection of investments a company, mutual fund, or individual investor holds. It consists of assets such as stocks, bonds, or cash equivalents. Financial professionals usually manage a portfolio.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Portfolio Expected Return<\/h3>\n\n\n\n<p>To calculate the portfolio&#8217;s expected return, you take the expected returns of each security in the portfolio. Then, you multiply each security&#8217;s expected return by its proportion in the portfolio and add them up. The formula below helps you find the portfolio&#8217;s expected return:<\/p>\n\n\n\n<p>$$ E(R_p)= w_1 E(R_1)+w_2 E(R_2)+\\cdots w_n E(R_n) $$<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<p>\\(w_1,w_2,\\dots,w_n\\) = Weights (market value of asset\/market value of the portfolio) attached to assets \\(1,2,\\dots,n\\).<br>\\(R_1,R_2,\\dots,R_n\\) = Expected returns for assets \\(1,2,\\dots,n\\).<\/p>\n\n\n\n<div style=\"text-align: center; margin: 24px 0;\">\n  <div style=\"max-width: 680px; margin: 0 auto;\">\n    <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n       style=\"display: flex; align-items: center; justify-content: center;\n       width: 100%; padding: 8px 16px;\n       border: 2px solid #2f6fed; border-radius: 999px;\n       color: #2f6fed; text-decoration: none;\n       font-size: 15px; font-weight: 500;\n       line-height: 1.2; white-space: nowrap;\">\n      Practice portfolio return and variance calculations with our free trial.\n    <\/a>\n  <\/div>\n<\/div>\n\n\n<\/p>\n<p><strong>Example: Portfolio Expected Return<\/strong><\/p>\n<p>Assume we have a simple portfolio of two mutual funds, one invested in bonds and the other invested 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. The expected return would be calculated as follows:<\/p>\n<p>$$ E(Rp)=(0.5 \\times 0.08)+(0.5 \\times 0.06)=0.07 \\text{ or } 7\\% $$<\/p>\n<h3>Portfolio Variance<\/h3>\n<p>The variance of a portfolio&#8217;s return is a function of the individual asset covariances and the covariance between each of them.<\/p>\n<p>Consider a portfolio with three assets: A, B, and C. The portfolio variance is given by:<\/p>\n<p>$$ \\begin{align*}<br \/>\\text{Portfolio} &amp; \\text{ Variance} \\\\ &amp; = W_A^2 \\sigma^2 (R_A)+W_B^2 \\sigma^2 (R_B)+W_C^2 \\sigma^2 (R_C)+2(W_A) (W_B)Cov(R_A,R_B ) \\\\ &amp; +2 (W_A)(W_C)Cov(R_A,R_C )+2 (W_B)(W_C)Cov(R_B,R_C) \\end{align*} $$<\/p>\n<p>If we have two assets, \\(A\\) and \\(B\\), then:<\/p>\n<p>$$ \\text{Portfolio Variance}= W_A^2 \\sigma^2 (R_A)+W_B^2 \\sigma^2 (R_B)+2 (W_A)(W_B)Cov(R_A,R_B) $$<\/p>\n<p>Where:<\/p>\n<p>\\(W_A\\) = Weight of assets \\(A\\) in the portfolio.<\/p>\n<p>\\(W_B\\) = Weight of assets \\(B\\) in the portfolio.<\/p>\n<p>\\(\\sigma^2 (R_A)\\) = Variance of the returns on assets \\(A\\).<\/p>\n<p>\\(\\sigma^2 (R_B)\\) = Variance of the returns on assets \\(B\\).<\/p>\n<p>Portfolio variance is a measure of risk. The higher the variance, the higher the risk. Investors usually reduce the portfolio variance by choosing assets with low or negative covariance, e.g., stocks and bonds.<\/p>\n<h4>Portfolio Standard Deviation<\/h4>\n<p>Portfolio standard deviation is simply the square root of the portfolio variance. It measures a portfolio&#8217;s riskiness.<\/p>\n<p>Considering a portfolio with two assets, A and B, the portfolio standard deviation is given by:<\/p>\n<p>$$ \\text{Standard deviation}= \\sqrt{ (W_A^2 \\sigma^2 (R_A)+W_B^2 \\sigma^2 (R_B)+2(W_A)(W_B)Cov(R_A,R_B) } $$<\/p>\n<h3>Covariance<\/h3>\n<p>Covariance is a measure of the degree of co-movement between two random variables. The general formula used to calculate the covariance between two random variables, \\(X\\) and \\(Y\\) is:<\/p>\n<p>$$ Cov(X,Y)=\\sigma(X,Y)=E[(X-E[X])(Y-E[Y]) $$<\/p>\n<p>Where:<\/p>\n<p>\\(Cov(X, Y)\\) = Covariance of \\(X\\) and \\(Y\\).<\/p>\n<p>\\(E[X]\\) = Expected value of the random variable X.<\/p>\n<p>\\(E[Y]\\) = Expected values of the random variable Y.<\/p>\n<p>This formula calculates the population covariance. It does this by taking the probability-weighted average of the cross-products of the random variables&#8217; deviations from their expected values for every possible outcome.<\/p>\n<h3>Sample Covariance<\/h3>\n<p>The sample covariance between two variables, \\(X\\) and \\(Y\\), based on a sample data of size \\(n\\) is:<\/p>\n<p>$$ Cov(X,Y)=\\sum_{i=1}^n {\\frac { (X_i-\\bar X)(Y_i-\\bar Y)}{n-1}} $$<\/p>\n<p>Where:<\/p>\n<p>\\(\\bar X\\)= Sample mean of \\(X\\).<\/p>\n<p>\\(\\bar Y\\)= Sample mean of \\(Y\\).<\/p>\n<p>\\(X_i\\) and \\(Y_i\\) = i-th data points of \\(X\\) and \\(Y\\), respectively.<\/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. The variables tend to move in the same direction.<\/li>\n<li>A value of zero indicates no relationship.<\/li>\n<li>A negative value shows that the variables move in opposite directions.<\/li>\n<\/ul>\n<h3>Covariance Matrix<\/h3>\n<p>A covariance matrix displays a complete list of covariances between assets needed to calculate the portfolio variance. Consider a portfolio with three assets: A, B, and C. The covariance matrix is as follows:<\/p>\n<p>$$ \\begin{array}{c|c|c|c}<br \/>\\textbf{Asset} &amp; \\bf A &amp; \\bf B &amp; \\bf C \\\\ \\hline<br \/>A &amp; \\bf{Cov(R_A,R_A)} &amp; Cov(R_A,R_B) &amp; Cov(R_A,R_C) \\\\ \\hline<br \/>B &amp; Cov(R_B,R_A) &amp; \\bf{Cov(R_B,R_B)} &amp; Cov(R_B,R_C) \\\\ \\hline<br \/>C &amp; Cov(R_C,R_A) &amp; Cov(R_C,R_B) &amp; \\bf{Cov(R_C,R_C)}<br \/>\\end{array} $$<\/p>\n<p>The off-diagonal (bolded) terms represent variances since, for example:<\/p>\n<p>$$ Cov(R_A,R_A )=\\rho(A,A) \\sigma_A \\sigma_A =1?\\sigma_A^2=\\sigma_A^2 $$<\/p>\n<p>As such, the table above transforms:<\/p>\n<p>$$ \\begin{array}{c|c|c|c}<br \/>\\textbf{Asset} &amp; \\bf A &amp; \\bf B &amp; \\bf C \\\\ \\hline<br \/>A &amp; \\sigma_A^2 &amp; Cov(R_A,R_B) &amp; Cov(R_A,R_C) \\\\ \\hline<br \/>B &amp; Cov(R_B,R_A) &amp; \\sigma_B^2 &amp; Cov(R_B,R_C) \\\\ \\hline<br \/>C &amp; Cov(R_C,R_A) &amp; Cov(R_C,R_B) &amp; \\sigma_C^2<br \/>\\end{array} $$<\/p>\n<p>Intuitively, a three-asset portfolio would have \\(3 \\times 3 = 9\\) entries of covariances. However, we do not count the off-diagonal terms since they contain the individual variances of the assets. As such, we have \\(6 (= 9 &#8211; 3)\\) covariances.<\/p>\n<p>Note that:<\/p>\n<p>$$ \\begin{align*}<br \/>Cov(R_B,R_A)&amp;=Cov(R_A,R_B) \\\\<br \/>Cov(R_A,R_C) &amp;=Cov(R_A,R_C) \\\\<br \/>Cov(R_C,R_B) &amp; =Cov(R_B,R_C)<br \/>\\end{align*} $$<\/p>\n<p>Therefore, there are \\(\\frac {6}{2}=3\\) distinct covariance terms in the above covariance matrix.<\/p>\n<p>In general, if we have \\(n\\) securities in a portfolio, there are \\(\\frac {n(n-1)}{2}\\) distinct covariances and \\(n\\) variances to estimate.<\/p>\n<h3>Correlation<\/h3>\n<p>Correlation is the covariance ratio between two random variables and the product of their two standard deviations. The correlation formula for random variables \\(X\\) and \\(Y\\) is:<\/p>\n<p>$$ \\begin{align*}<br \/>\\text{Correlation (X,Y)}&amp; =Corr(X,Y)=\\rho(X,Y)\\\\ &amp;=\\frac {Cov(X,Y)}{\\text{Standard deviation(X)} \\times \\text{Standard deviation(Y)} } \\\\<br \/>&amp; =\\frac { Cov(X,Y)}{\\sigma_X \\sigma_Y} \\end{align*} $$<\/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 between -1 and +1:<\/p>\n<ul>\n<li>+1 indicates a perfect linear relationship (i.e., the two variables move in the same direction with equal unit changes).<\/li>\n<li>Zero indicates no linear relationship at all.<\/li>\n<li>-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.<\/li>\n<\/ul>\n<p><strong>Example: Calculating Correlation Coefficient from the Covariance Matrix \\(\\#1\\)<\/strong><\/p>\n<p>Harrison is a portfolio manager who oversees three assets: A, B, and C. The covariance matrix of these assets is shown below:<\/p>\n<p>$$ \\begin{array}{c|c|c|c}<br \/>\\textbf{Asset} &amp; \\bf A &amp; \\bf B &amp; \\bf C \\\\ \\hline<br \/>A &amp; 0.04 &amp; 0.02 &amp; 0.01 \\\\ \\hline<br \/>B &amp; 0.02 &amp; 0.05 &amp; 0.015 \\\\ \\hline<br \/>C &amp; 0.01 &amp; 0.015 &amp; 0.09<br \/>\\end{array} $$<\/p>\n<p>Using this information, what is the correlation coefficient between assets \\(B\\) and \\(C\\)?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Note:<\/p>\n<p>$$ \\begin{align*}<br \/>\\text{Correlation } (B,C)&amp;=\\frac {Cov(B,C)}{\\sigma_B \\sigma_C } \\\\<br \/>&amp;=\\frac {0.015}{\\sqrt{0.05 \\times 0.09}}=0.224 \\end{align*} $$<\/p>\n<p><strong>Example: Calculating the Correlation Coefficient \\(\\#2\\)<\/strong><\/p>\n<p>We expect a 15% chance that ABC Corp&#8217;s stock returns for the next year will be 6%. There&#8217;s a 60% probability that they will be 8% and a 25% probability of a 10% return. The expected return 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%, 5%, and 5.5% return for XYZ Corp. The expected value of returns is then 4.975%, and the standard deviation is 0.46%.<\/p>\n<p>To calculate the covariance and the correlation between ABC and XYZ returns, then:<\/p>\n<p>$$ \\begin{align*} Cov(R_{ABC},R_{XYZ} )&amp; =0.15(0.06-0.082)(0.04-0.04975) \\\\ &amp; +0.6(0.08-0.082)(0.05-0.04975)\\\\ &amp; +0.25(0.10-0.082)(0.055-0.04975)\\\\ &amp;=0.0000561 \\end{align*} $$<\/p>\n<p>$$ \\begin{align*}<br \/>&amp; \\text{Correlation}(Ri,Rj) \\\\ &amp; =\\frac {\\text{Covariance}(R_{ABC},R_{XYZ})}{\\text{Standard deviation(RABC)} \\times \\text{Standard deviation(RXYZ)}} \\end{align*} $$<\/p>\n<p>Therefore:<\/p>\n<p>$$ \\text{Correlation}=\\frac {0.0000561}{(0.01249 \\times 0.0046)}=0.976 $$<\/p>\n<p>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><strong>Example: Calculating Correlation Coefficient \\(\\#3\\)<\/strong><\/p>\n<p>An analyst studied five years of historical data to examine how changes in Central Bank interest rates affect the country&#8217;s inflation rate. The covariance between the interest rate and inflation rate is -0.00075. The standard deviation of the interest rate is 5.5%, and the inflation rate is 12%. Now, let&#8217;s calculate and interpret the correlation between these two variables.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\begin{align*}<br \/>&amp;\\text{Correlation}_{\\text{Interest rate, Inflation}}\\\\&amp;=\\frac { \\text{Covariance}_{\\text{Interest Rate, Inflation}}}{\\text{Standard deviation}_{\\text{Interest rate}} \\times {\\text{Standard deviation}}_{\\text{Inflation}}} \\\\<br \/>&amp; \\text{Correlation}_{\\text{Interest rate, Inflation}} = \\frac {-0.00075}{(0.055 \\times 0.12)}=-0.11364 \\end{align*} $$<\/p>\n<p>A correlation of -0.11364 indicates a negative correlation between the interest rate and the inflation rate.<\/p>\n<p>Note that if we consider, say, assets \\(A\\) and \\(B\\), then:<\/p>\n<p>$$ \\begin{align*} Corr(A,B) &amp; =\\rho(A,B)=\\frac {Cov(A,B)}{\\sigma_A \\sigma_B } \\\\<br \/>\\Rightarrow Cov(A,B) &amp; =\\sigma_A \\sigma_B \\rho(A,B) \\end{align*} $$<\/p>\n<p>Consequently, in the formula for calculating portfolio variance, consisting of two assets, A and B, we substitute for \\(Cov(A, B)\\) so that:<\/p>\n<p>$$ \\text{Portfolio Variance}= W_A^2 \\sigma^2 (R_A)+W_B^2 \\sigma^2 (R_B)+2(W_A)(W_B) \\sigma_A \\sigma_B \\rho(A,B) $$<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Assume that we have investments in two companies, ABC and XYZ. For ABC, there&#8217;s a 15% chance of a 6% return, a 60% chance of an 8% return, and a 25% chance of a 10% return. The expected return for ABC is 8.2%, and the standard deviation is 1.249%. For XYZ, there are similar probabilities of 4%, 5%, and 5.5% returns. The expected return for XYZ is 4.975%, and the standard deviation is 0.46%.<\/p>\n<p>Assuming equal weights, the portfolio standard deviation is <em>closest to<\/em>:<\/p>\n<ol type=\"A\">\n<li>0.0000561.<\/li>\n<li>0.00007234.<\/li>\n<li>0.00851.<\/li>\n<\/ol>\n<p><strong>The correct answer is C<\/strong>.<\/p>\n<p>$$ \\text{Portfolio Variance}= W_A^2 \\sigma^2 (R_A)+W_B^2 \\sigma^2 (R_B)+2(W_A)(W_B)Cov(R_A,R_B) $$<\/p>\n<p>First, we must calculate the covariance between the two stocks:<\/p>\n<p>$$ \\begin{align*} Cov(R_{ABC},R_{XYZ}) &amp; =0.15(0.06-0.082)(0.04-0.04975) \\\\ &amp; +0.6(0.08-0.082)(0.05-0.04975) \\\\ &amp; +0.25(0.10-0.082)(0.055-0.04975) \\\\ &amp; =0.0000561 \\end{align*} $$<\/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>$$ \\begin{align*} \\text{Portfolio variance} &amp; =0.5^2 \\times 0.01249^2 + 0.5^2 \\times 0.0046^2 \\\\ &amp; +2 \\times 0.5 \\times 0.5 \\times 0.0000561 \\\\ &amp; =0.00007234 \\end{align*} $$<\/p>\n<p>Therefore, the standard deviation is:<\/p>\n<p>$$ \\sqrt{0.00007234}=0.00851 $$<\/p>\n<\/blockquote>\n\n\n<div style=\"text-align: center; margin: 40px 0 10px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display: inline-flex; align-items: center; justify-content: center;\n     padding: 10px 24px; border-radius: 999px;\n     font-size: 15px; font-weight: 600;\n     background-color: #4a72c9; color: #ffffff;\n     text-decoration: none; line-height: 1.2; white-space: nowrap;\">\n    Start Free Trial \u2192\n  <\/a>\n\n  <p style=\"margin: 14px auto 0; max-width: 680px; font-size: 15px; line-height: 1.6; color: #333333;\">\n    Strengthen your understanding of portfolio expected return, variance, and diversification with structured CFA Level I practice and step-by-step explanations.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>A portfolio is a collection of investments a company, mutual fund, or individual investor holds. It consists of assets such as stocks, bonds, or cash equivalents. Financial professionals usually manage a portfolio. Portfolio Expected Return To calculate the portfolio&#8217;s expected&#8230;<\/p>\n","protected":false},"author":7,"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-45440","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 I<\/title>\n<meta name=\"description\" content=\"Learn how to calculate portfolio expected return and variance, including covariance, correlation, and multi-asset variance formulas.\" \/>\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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