{"id":25182,"date":"2021-08-12T12:58:53","date_gmt":"2021-08-12T12:58:53","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=25182"},"modified":"2026-04-28T09:24:00","modified_gmt":"2026-04-28T09:24:00","slug":"downside-deviation","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/downside-deviation\/","title":{"rendered":"Downside Deviation"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Organizing, Visualizing, and Describing Data \u2013 Part II: Measures of Central Tendency (Level I CFA)\",\n  \"description\": \"Explore key concepts in data analysis with this session on measures of central tendency, dispersion, skewness, kurtosis, and correlation. Learn to calculate, interpret, and visualize data effectively for investment and analysis problems. Perfect for mastering quantitative techniques!\",\n  \"uploadDate\": \"2021-10-08T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/M0gKgPztSoM\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/M0gKgPztSoM\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/M0gKgPztSoM\",\n  \"duration\": \"PT41M38S\"\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\": \"If a security has a mean expected return of 10% and a standard deviation of 5%, its coefficient of variation is closest to:\",\n    \"text\": \"If a security has a mean expected return of 10% and a standard deviation of 5%, its coefficient of variation is closest to:\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. The coefficient of variation (CV) is calculated as standard deviation divided by mean return: CV = 0.05 \/ 0.10 = 0.5, which is closest to 0.500.\",\n      \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/downside-deviation\/\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\n}\n<\/script>\n\n\n\n<p>When trying to estimate downside risk (i.e., returns below the mean), we can use the following measures:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Semi-variance<\/strong>: The average squared deviation below the mean.<\/li>\n\n\n\n<li><strong>Semi-deviation<\/strong> (also known as semi-standard deviation): The positive square root of semi-variance.<\/li>\n\n\n\n<li><strong>Target semi-variance<\/strong>: The sum of the squared deviations from a specific target return.<\/li>\n\n\n\n<li><strong>Target semi-deviation<\/strong>: The square root of target semi-variance.<\/li>\n<\/ul>\n\n\n\n<!--more-->\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Sample Target Semi-deviation<\/strong><\/h2>\n\n\n\n<p>The target semi deviation, \\(s_{\\text {Target }}\\), is calculated as follows:<\/p>\n\n\n\n<p>$$s_{\\text {Target }}=\\sqrt{ \\sum_{\\text {for all } X_{i} \\leq B}^{n} \\frac{\\left(X_{i}-B\\right)^{2}}{n-1}}$$<\/p>\n\n\n\n<p>Where \\(B\\) is the target and \\(n\\)&nbsp;is the total number of sample observations.<\/p>\n\n\n\n<div style=\"text-align:center;margin:25px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\"\n     style=\"display:inline-block;padding:12px 20px;border:2px solid #2f5cff;border-radius:999px;\n            color:#2f5cff;text-decoration:none;background:#f7f9fc;white-space:nowrap;\">\n     Understand downside risk with our free trial.\n  <\/a>\n<\/div>\n\n\n<h4>\u00a0<\/h4>\n<p>Example<strong>: <\/strong>Target Downside Deviation<\/p>\n<p>Yearly returns of an equity mutual fund are provided as follows.<\/p>\n<p>$$<br \/>\\begin{array}{c|c}<br \/>\\textbf { Month } &amp; \\textbf { Return % } \\\\<br \/>\\hline 2010 &amp; 36 \\% \\\\<br \/>\\hline 2011 &amp; 29 \\% \\\\<br \/>\\hline 2012 &amp; 10 \\% \\\\<br \/>\\hline 2013 &amp; 52 \\% \\\\<br \/>\\hline 2014 &amp; 41 \\% \\\\<br \/>\\hline 2015 &amp; 16 \\% \\\\<br \/>\\hline 2016 &amp; 10 \\% \\\\<br \/>\\hline 2017 &amp; 23 \\% \\\\<br \/>\\hline 2018 &amp; -10 \\% \\\\<br \/>\\hline 2019 &amp; -19 \\% \\\\<br \/>\\hline 2020 &amp; 2 \\% \\\\<br \/>\\end{array}<br \/>$$<\/p>\n<p>What is the target downside deviation if the target return is 20%?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$<br \/>\\begin{array}{c|c|c|c|c}<br \/>\\textbf { Month } &amp; \\begin{array}{c}<br \/>\\textbf { Return } \\\\<br \/>\\%<br \/>\\end{array} &amp; \\begin{array}{c}<br \/>\\textbf { Deviation } \\\\<br \/>\\textbf { from the 20% } \\\\<br \/>\\textbf { target }<br \/>\\end{array} &amp; \\begin{array}{c}<br \/>\\textbf { Deviation } \\\\<br \/>\\textbf { below the } \\\\<br \/>\\textbf { target }<br \/>\\end{array} &amp; \\begin{array}{c}<br \/>\\textbf { Squared } \\\\<br \/>\\textbf { deviations } \\\\<br \/>\\textbf { below the } \\\\<br \/>\\textbf { target }<br \/>\\end{array} \\\\<br \/>\\hline 2010 &amp; 36.00 &amp; 16.00 &amp; &#8211; &amp; &#8211; \\\\<br \/>\\hline 2011 &amp; 29.00 &amp; 9.00 &amp; &#8211; &amp; &#8211; \\\\<br \/>\\hline 2012 &amp; 10.00 &amp; (10.00) &amp; (10.00) &amp; 100 \\\\<br \/>\\hline 2013 &amp; 52.00 &amp; 32.00 &amp; &#8211; &amp; \\\\<br \/>\\hline 2014 &amp; 41.00 &amp; 21.00 &amp; &#8211; &amp; \\\\<br \/>\\hline 2015 &amp; 16.00 &amp; (4.00) &amp; (4.00) &amp; 16 \\\\<br \/>\\hline 2016 &amp; 10.00 &amp; (10.00) &amp; (10.00) &amp; 100 \\\\<br \/>\\hline 2017 &amp; 23.00 &amp; 3.00 &amp; &#8211; &amp; \\\\<br \/>\\hline 2018 &amp; (10.00) &amp; (30.00) &amp; (30.00) &amp; 900 \\\\<br \/>\\hline 2019 &amp; (19.00) &amp; (39.00) &amp; (39.00) &amp; 1,521 \\\\<br \/>\\hline 2020 &amp; 2.00 &amp; (18.00) &amp; (18.00) &amp; 324 \\\\<br \/>\\hline {\\text { Sum }} &amp; {}&amp;{}&amp;{}&amp;{\\textbf{2,961}}\\\\<br \/>\\end{array}<br \/>$$<\/p>\n<p>Here \\(n = 11 \u2013 1 = 10\\) so that:<\/p>\n<p>$$\\text{Target semi-deviation} = \\left(\\frac{2961 }{10}\\right)^{0.5} = 17.21\\%$$<\/p>\n<h2><strong>Coefficient of Variation<\/strong><\/h2>\n<p>The coefficient of variation, \\(CV\\), is a measure of spread that describes the amount of variability of data relative to its mean. It has <strong>no units,<\/strong> so we can use it as an alternative to the standard deviation to compare the variability of data sets that have different means. The coefficient of variation is given by:<\/p>\n<p><span style=\"color: initial; font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: revert;\">$$ \\text{CV} = \\cfrac {S}{\\bar{X}} $$<\/span><\/p>\n<p>Where:<\/p>\n<p>\\(S\\) = The standard deviation of a sample.<\/p>\n<p>\\(\\bar{X}\\) = The mean of the sample.<\/p>\n<p><em><strong>Note<\/strong><\/em><strong>: <\/strong>The formula can be replaced with \\(\\frac{\u03c3}{\u03bc}\\) when dealing with a population.<\/p>\n<p><strong>Procedure to Follow While Calculating the Coefficient of Variation<\/strong>:<\/p>\n<ol>\n<li>Compute the mean of the data.<\/li>\n<li>Calculate the sample standard deviation of the data set, \\(S\\).<\/li>\n<li>Find the ratio of \\(S\\) to the mean, \\(x\u0304\\).<\/li>\n<\/ol>\n<h4>Example<strong>: <\/strong>Coefficient of Variation<\/h4>\n<p>What is the relative variability for the samples 40, 46, 34, 35, and 45 of a population?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>Step 1:<\/strong> Calculate the mean.<\/p>\n<p>$$ \\text{Mean} =\\cfrac {(40 + 46 + 34 + 35 + 45)}{5} =\\cfrac {200}{5} = 40 $$<\/p>\n<p><strong>Step 2:<\/strong> Calculate the sample standard deviation. (Start with the variance, \\(S^2\\).)<\/p>\n<p>$$ \\begin{align*} S^2 &amp; =\\cfrac {{(40 \u2013 40)^2 + &#8230; + (45 \u2013 40)^2 }}{4} \\\\ &amp;=\\cfrac {122}{4} \\\\ &amp; = 30.5 \\\\ \\end{align*} $$<\/p>\n<p><em><strong>Note<\/strong><\/em><strong>: <\/strong>Since it is the sample standard deviation (not the population standard deviation), we use \\(n &#8211; 1\\) as the denominator.<\/p>\n<p>Therefore,<\/p>\n<p>$$ S = \\sqrt{30.5} = 5.52268 $$<\/p>\n<p><strong>Step 3<\/strong>: Calculate the ratio.<\/p>\n<p>$$ \\text{Ratio} =\\cfrac {5.52268}{40} = 0.13806 \\text{ or } 13.81\\% $$<\/p>\n<h3><strong>Interpreting the Coefficient of Variation<\/strong><\/h3>\n<p>In finance, the coefficient of variation is used to measure the <strong>risk per unit of return<\/strong>. For example, imagine that the mean monthly return on a T-Bill is 0.5% with a standard deviation of 0.58%. Suppose we have another investment, say, Y, with a 1.5% mean monthly return and standard deviation of 6%, then,<\/p>\n<p>$$ \\text{CV}_{\\text T-\\text {Bill}} =\\cfrac {0.58}{0.5} = 1.16 $$<\/p>\n<p>$$ \\text{CV}_\\text{Y} =\\cfrac {6}{1.5} = 4 $$<\/p>\n<p><em><strong>Interpretation<\/strong><\/em>: The dispersion per unit monthly return of T-Bills is less than that of Y. Therefore, investment Y is riskier than an investment on T-Bills.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>If a security has a mean expected return of 10% and a standard deviation of 5%, its coefficient of variation is <em>closest<\/em> to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>0.005.<\/li>\n<li>0.500.<\/li>\n<li>2.000.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>$$ \\text{CV} = \\cfrac {S}{\\text x\u0304} = \\cfrac {0.05}{0.10} = 0.5$$<\/p>\n<p>Where:<\/p>\n<p>\\(S\\) = The standard deviation of the sample.<\/p>\n<p>\\(x\u0304\\) = The mean of the sample.<\/p>\n<p><strong>A is incorrect<\/strong>. It assumes the following calculation.<\/p>\n<p>$$CV=\\frac{0.05}{10}=0.005$$<\/p>\n<p><strong>C is incorrect.<\/strong> It assumes the following calculation.<\/p>\n<p>$$CV=\\frac{10}{5}=2$$<\/p>\n<\/blockquote>\n\n\n<h2>Build Confidence With CFA Quant Risk Measures<\/h2>\n\n<p>Downside deviation, coefficient of variation, and risk-adjusted return concepts are common CFA Level I test areas.<\/p>\n\n<p>Strengthen your understanding with guided lessons and exam-style practice inside AnalystPrep\u2019s <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\">Free Trial<\/a>.<\/p>\n\n\n\n<div style=\"text-align:center;margin:40px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\"\n     style=\"display:inline-block;padding:14px 26px;background:#4a76d1;color:#fff;border-radius:999px;text-decoration:none;\">\n     Start Free Trial \u2192\n  <\/a>\n  <p style=\"margin-top:10px;\">\n    Learn semi-deviation, downside measures, and risk analysis with clear study tools.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>When trying to estimate downside risk (i.e., returns below the mean), we can use the following measures:<\/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-25182","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Downside Deviation Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Understand downside deviation, semi-variance, and other measures used to assess downside risk and returns below the mean. 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