{"id":8178,"date":"2020-08-13T19:10:16","date_gmt":"2020-08-13T19:10:16","guid":{"rendered":"https:\/\/analystprep.com\/blog\/?p=8178"},"modified":"2026-01-07T11:15:15","modified_gmt":"2026-01-07T11:15:15","slug":"sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams","status":"publish","type":"post","link":"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/","title":{"rendered":"Sharpe Ratio, Treynor Ratio and Jensen&#8217;s Alpha (Calculations for CFA\u00ae and FRM\u00ae Exams)"},"content":{"rendered":"\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 difference between Sharpe and Treynor ratios?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"The Sharpe Ratio evaluates returns per unit of total risk (standard deviation), while the Treynor Ratio focuses only on systematic risk (beta). Use Sharpe when assessing the absolute risk-adjusted return of a portfolio, and Treynor when comparing diversified portfolios relative to market risk.\"\n      }\n    },\n    {\n      \"@type\": \"Question\",\n      \"name\": \"How is Jensen\u2019s Alpha calculated in CFA exams?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"Jensen\u2019s Alpha is calculated by subtracting the expected return (based on CAPM) from the actual return of a portfolio. Formula: Alpha = R_P - [R_F + \u03b2(R_M - R_F)], where R_P is the portfolio return, R_F is the risk-free rate, R_M is the market return, and \u03b2 is the portfolio\u2019s beta.\"\n      }\n    },\n    {\n      \"@type\": \"Question\",\n      \"name\": \"When should you use Sharpe vs Treynor?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"Use Sharpe when evaluating portfolios that may include unsystematic risk (like actively managed funds). Use Treynor when comparing well-diversified portfolios where unsystematic risk is negligible. In short: Sharpe = total risk, Treynor = market risk only.\"\n      }\n    },\n    {\n      \"@type\": \"Question\",\n      \"name\": \"Are Sharpe and Treynor ratios tested in CFA Level 1 or 2?\",\n      \"acceptedAnswer\": {\n        \"@type\": \"Answer\",\n        \"text\": \"Both Sharpe and Treynor ratios are introduced in CFA Level 1, particularly under Portfolio Management. However, deeper application and comparative analysis (including Jensen\u2019s Alpha) are often emphasized more in CFA Level 2.\"\n      }\n    }\n  ]\n}\n<\/script>\n\n\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@graph\": [\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-systematic-vs-unsystematic-risk-484x284.png\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-systematic-vs-unsystematic-risk-484x284.png\",\n      \"caption\": \"systematic-vs-unsystematic-risk\",\n      \"width\": 484,\n      \"height\": 284,\n      \"fileFormat\": \"image\/png\",\n      \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n      \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n      \"creditText\": \"AnalystPrep Design Team\",\n      \"creator\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\",\n        \"url\": \"https:\/\/analystprep.com\"\n      }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-capital-market-line-cml-484x332.png\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-capital-market-line-cml-484x332.png\",\n      \"caption\": \"capital-market-line\",\n      \"width\": 484,\n      \"height\": 332,\n      \"fileFormat\": \"image\/png\",\n      \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n      \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n      \"creditText\": \"AnalystPrep Design Team\",\n      \"creator\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\",\n        \"url\": \"https:\/\/analystprep.com\"\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 Sharpe and Treynor ratios?\",\n    \"text\": \"What is the difference between Sharpe and Treynor ratios?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2020-08-13T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The Sharpe Ratio evaluates returns per unit of total risk (standard deviation), while the Treynor Ratio focuses only on systematic risk (beta). Use Sharpe when assessing absolute risk-adjusted returns and Treynor when comparing diversified portfolios relative to market risk.\",\n      \"dateCreated\": \"2020-08-13T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#faq-sharpe-vs-treynor-difference\",\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 is Jensen\u2019s Alpha calculated in CFA exams?\",\n    \"text\": \"How is Jensen\u2019s Alpha calculated in CFA exams?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2020-08-13T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Jensen\u2019s Alpha is calculated by subtracting the CAPM-based expected return from the actual portfolio return: \u03b1p = Rp \u2013 [Rf + \u03b2p (E(Rm) \u2013 Rf)], where Rp is portfolio return, Rf is risk-free rate, E(Rm) is expected market return, and \u03b2p is portfolio beta.\",\n      \"dateCreated\": \"2020-08-13T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#faq-jensen-alpha-calculation\",\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\": \"When should you use Sharpe vs Treynor?\",\n    \"text\": \"When should you use Sharpe vs Treynor?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2020-08-13T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Use the Sharpe Ratio when evaluating portfolios that may still contain unsystematic risk, such as actively managed funds. Use the Treynor Ratio when comparing well-diversified portfolios where unsystematic risk is negligible and you want to focus on market risk only.\",\n      \"dateCreated\": \"2020-08-13T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#faq-when-use-sharpe-vs-treynor\",\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\": \"Are Sharpe and Treynor ratios tested in CFA Level 1 or 2?\",\n    \"text\": \"Are Sharpe and Treynor ratios tested in CFA Level 1 or 2?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2020-08-13T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Both Sharpe and Treynor ratios are introduced in CFA Level 1 under Portfolio Management, while deeper applications and comparisons, including Jensen\u2019s Alpha, tend to be emphasized more in CFA Level 2.\",\n      \"dateCreated\": \"2020-08-13T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#faq-which-level-tests-sharpe-treynor\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\n}\n<\/script>\n\n\n\n<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 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Portfolio_Performance_Measures\" >Portfolio Performance Measures<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Sharpe_Ratio\" >Sharpe Ratio<\/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'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Example_Calculating_the_Sharpe_Ratio\" >Example: Calculating the Sharpe Ratio\u00a0\u00a0\u00a0\u00a0\u00a0<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Type_of_Risks\" >Type of Risks<\/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'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Example_Calculating_the_Beta_of_a_Security\" >Example: Calculating the Beta of a Security<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Treynor_Ratio\" >Treynor Ratio<\/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'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Example_Calculating_Treynors_Ratio\" >Example: Calculating Treynor&#8217;s Ratio<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Jensens_Alpha\" >Jensen&#8217;s Alpha<\/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'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Example_Calculating_Jensens_Alpha\" >Example:\u00a0Calculating Jensen&#8217;s Alpha<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/analystprep.com\/blog\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Formula_Summary_Card\" >Formula Summary Card<\/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\/sharpe-ratio-treynor-ratio-and-jensens-alpha-calculations-for-cfa-and-frm-exams\/#Frequently_Asked_Questions\" >Frequently Asked Questions<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Portfolio_Performance_Measures\"><\/span>Portfolio Performance Measures<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Portfolio management involves a trade-off between risk and return. Most amateur investors mistakenly focus only on the return aspect and lose sight of the risk taken to achieve the return. The portfolio performance measures are intended as metrics to compare different portfolios quickly. Like any financial ratio, these are not intended to be a one-stop answer to decide on a portfolio and should instead be used in conjunction with other data points.<\/p>\n\n\n\n<p>The measure themselves are quite different from one another in the way that they measure the trade-off and how they define risk.<\/p>\n\n\n\n<div style=\"margin: 20px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     class=\"ap-cta\"\n     data-cta-text=\"Ready to test these performance-metrics yourself\"\n     data-cta-type=\"button\"\n     data-cta-location=\"top_content\"\n     data-page-type=\"blog\"\n     style=\"\n       display: inline-block;\n       padding: 10px 16px;\n       font-size: 14px;\n       font-weight: 600;\n       color: #0b5ed7;\n       border: 2px solid #0b5ed7;\n       border-radius: 6px;\n       text-decoration: none;\n       background-color: transparent;\n     \">\n    Ready to test these performance\u2011metrics yourself? Try AnalystPrep\u2019s free trial now.\n  <\/a>\n<\/div>\n\n\n\n<p><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Sharpe_Ratio\"><\/span>Sharpe Ratio<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Sharpe Ratio is defined as the portfolio risk premium divided by the portfolio risk:<\/p>\n<p>$$ \\begin{align} \\text{Sharpe ratio} &amp;= \\frac{\\text{Expected Return on the portfolio} &#8211; \\text{Risk-free rate of interest}}{\\text{Standard deviation of the portfolio}} \\\\ &amp;= \\frac{E(R_p) &#8211; R_f}{\\sigma_p} \\end{align} $$<\/p>\n<p>The Sharpe ratio, or reward-to-variability ratio, is the slope of the capital allocation line (CAL). The greater the slope (higher number) the better the asset.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"206\" class=\"size-medium wp-image-8180 aligncenter\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-capital-market-line-cml-300x206.png\" alt=\"cfa-frm-capital-market-line-cml\" srcset=\"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-capital-market-line-cml-300x206.png 300w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-capital-market-line-cml-768x527.png 768w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-capital-market-line-cml-400x274.png 400w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-capital-market-line-cml-484x332.png 484w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-capital-market-line-cml-570x391.png 570w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-capital-market-line-cml.png 974w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Note that the risk being used is the total risk of the portfolio, not its systematic risk, which is a limitation of the measure. The portfolio with the highest Sharpe ratio has the best performance, but the Sharpe ratio by itself is not informative. In order to rank portfolios, the Sharpe ratio for each portfolio must be computed.<\/p>\n<p>A further limitation occurs when the numerators are negative. In this instance, the Sharpe ratio will be less negative for a riskier portfolio, resulting in incorrect rankings.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Example_Calculating_the_Sharpe_Ratio\"><\/span><b>Example: Calculating the Sharpe Ratio\u00a0\u00a0\u00a0\u00a0\u00a0 <\/b><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>A client has three portfolio choices, each with the following characteristics:<\/p>\n<p><strong>Portfolio Statistics<\/strong><\/p>\n<p>$$\\begin{array}{l|c|c|c}<br \/>&amp; \\textbf{Expected Return} &amp; \\textbf{Volatility} &amp; \\textbf{Beta} \\\\<br \/>\\hline<br \/>\\text{Portfolio A} &amp; 15\\% &amp; 12\\% &amp; 10\\% \\\\<br \/>\\hline<br \/>\\text{Portfolio B} &amp; 18\\% &amp; 14\\% &amp; 11\\% \\\\<br \/>\\hline<br \/>\\text{Portfolio C} &amp; 12\\% &amp; 9\\% &amp; 5\\% \\\\<br \/>\\end{array}$$<\/p>\n<p>The efficient market portfolio has an expected return of 20% and a standard deviation of 12%, and the risk-free rate of interest is 5%.<\/p>\n<p>Based on the Sharpe ratio for each portfolio, the client should choose:<\/p>\n<ol type=\"A\">\n<li>Portfolio A<\/li>\n<li>Portfolio B<\/li>\n<li>Portfolio C<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>\\( \\text{Sharpe ratio} =\u00a0\\frac{\\text{Return on the portfolio} &#8211; \\text{Risk-free rate of interest}}{\\text{Standard deviation of the portfolio}} = \\frac{R_p &#8211; R_f}{\u03c3_p} \\)<\/p>\n<p>\\(\\text{Portfolio A&#8217;s Sharpe Ratio} = \\frac{15\\%\u22125\\%}{12\\%} =\u00a00.83\\)<\/p>\n<p>\\(\\text{Portfolio B&#8217;s\u00a0Sharpe Ratio} = \\frac{18\\%\u22125\\%}{14\\%} = 0.93\\)<\/p>\n<p>\\(\\text{Portfolio C\u00a0Sharpe Ratio} = \\frac{12\\%\u22125\\%}{9\\%} = 0.77\\)<\/p>\n<p>The client should choose portfolio B as it gives the highest Sharpe ratio.<\/p>\n<p>\u00a0<\/p>\n\n\n\n<h2><span class=\"ez-toc-section\" id=\"Type_of_Risks\"><\/span>Type of Risks<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Sharpe Ratio defines the risk in terms of standard deviation, which is a measure of total risk. Hence, it includes both systematic as well as unsystematic risk. The next measures that we look at\u00a0\u2013 Treynor Ratio and Jensen&#8217;s Alpha\u00a0\u2013 define the risk in a narrower way. In order to understand the applicability of the measure, we first need to understand the different types of risks.<\/p>\n<p>CAPM suggests that investors should hold the market portfolio and a risk-free asset. The true market portfolio consists of a large number of securities, and it may not be practical for an investor to own them all. Much of the non-systematic risk can be diversified away by holding 30 or more individual securities. However, these securities should be randomly selected from multiple asset classes. An index may serve as the best method of creating diversification.<\/p>\n<p>It is important to note that only non-systematic risk can be eliminated through the addition of different securities into the portfolio. Systematic risk\u00a0\u2013 the\u00a0risk\u00a0inherent to the entire market \u2013 cannot be diversified away.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"176\" class=\"size-medium wp-image-8185 aligncenter\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-systematic-vs-unsystematic-risk-300x176.png\" alt=\"cfa-frm-systematic-vs-unsystematic-risk\" srcset=\"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-systematic-vs-unsystematic-risk-300x176.png 300w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-systematic-vs-unsystematic-risk-400x234.png 400w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-systematic-vs-unsystematic-risk-484x284.png 484w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-systematic-vs-unsystematic-risk-570x334.png 570w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/cfa-frm-systematic-vs-unsystematic-risk.png 664w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The systematic risk of a portfolio is denoted by Beta:<\/p>\n<p>$$\\begin{align} \\beta_i &amp;= \\frac{\\text{Covariance\u00a0 between the security and the market returns}}{\\text{Variance of the market return}}\\\\&amp; = \\frac{Cov(R_i, R_m)}{\\sigma_m^2} \\end{align}$$<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Example_Calculating_the_Beta_of_a_Security\"><\/span>Example: Calculating the Beta of a Security<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Assume the risk-free rate is 2%, security has a correlation of 0.8 with the market index, and a standard deviation of 16%, while the standard deviation of the market is 12%. If the market expected return is 8%, the beta of the security is<em> closest to<\/em>:<\/p>\n<ol type=\"A\">\n<li>0.128<\/li>\n<li>1.067<\/li>\n<li>1.333<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is B.<\/p>\n<p>Recall that for a two-asset portfolio,\u00a0<\/p>\n<p>$$\\text{Cov}(X, Y) = \\rho_{X,Y} \\times \\sigma_X \\times \\sigma_Y$$<\/p>\n<p>and since\u00a0<\/p>\n<p>$$\\beta_i= \\frac{Cov(R_i, R_m)}{\\sigma_m^2} $$<\/p>\n<p>then,<\/p>\n<p>\\(\\beta_i= \\frac{0.8 \u00d7 0.16\\times 0.12}{0.12^2}\u00a0 = 1.07\\)<\/p>\n<p><strong data-start=\"126\" data-end=\"176\">Ready to take your FRM prep to the next level?<\/strong> <a class=\"cursor-pointer\" href=\"https:\/\/analystprep.com\/shop\/frm-part-1-and-part-2-complete-course-by-analystprep\/\" target=\"_new\" rel=\"noopener\" data-start=\"177\" data-end=\"304\">Explore AnalystPrep\u2019s Complete FRM Course<\/a> and start mastering every concept with confidence.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Treynor_Ratio\"><\/span>Treynor Ratio<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Treynor ratio is an extension of the Sharpe ratio that, instead of using total risk, uses beta or systematic risk in the denominator. As such, this is better suited to investors who hold diversified portfolios.<\/p>\n<p>$$ \\text{Treynor ratio} = \\frac{\\text{Expected return on the portfolio} &#8211; \\text{Risk-free rate}}{\\text{Beta of the portfolio}} = \\frac{E(R_p) &#8211; R_f}{\\beta_p} $$<\/p>\n<p>As with the Sharpe ratio, the Treynor ratio requires positive numerators to give meaningful comparative results and, the Treynor ratio does not work for negative beta assets. Also, while both the Sharpe and Treynor ratios can rank portfolios, they do not provide information on whether the portfolios are better than the market portfolio or information about the degree of superiority of a higher ratio portfolio over a lower ratio portfolio.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Example_Calculating_Treynors_Ratio\"><\/span><strong>Example: Calculating Treynor&#8217;s Ratio<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>A portfolio manager earned an average annual return of 12%. The beta of the portfolio is 0.9, and the volatility of returns is 25%. The average annual return for the market index was 14%, and the standard deviation of the market returns is 30%. The risk-free rate is 5%. Calculate the Treynor measure for the portfolio.<\/p>\n<ol type=\"A\">\n<li>10.0%<\/li>\n<li>5.6%<\/li>\n<li>7.8%<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>C.<\/strong><\/p>\n<p>Recall that,\u00a0<\/p>\n<p>$$ \\begin{align} \\text{Treynor Ratio} &amp;= \\frac{E(R_p) &#8211; R_f}{\\beta_p} \\\\ &amp;= \\frac{12\\% &#8211; 5\\%}{9\\%} = 7.8\\% \\end{align} $$<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Jensens_Alpha\"><\/span>Jensen&#8217;s Alpha<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Jensen&#8217;s Alpha is based on systematic risk. The daily returns of the portfolio are regressed against the daily returns of the market in order to compute a measure of this systematic risk in the same manner as the CAPM. The difference between the actual return of the portfolio and the calculated or modeled risk-adjusted return is a measure of performance relative to the market.<\/p>\n<p>$$ \\text{Jensen\u2019s alpha} = \\alpha_p = R_p \u2013 [R_f + \\beta_p (E(R_m)\u2013 R_f)]$$<\/p>\n<p>If\u00a0\\(\u03b1_p\\) is positive, the portfolio has outperformed the market, whereas a negative value indicates underperformance. The values of Alpha can also be used to rank portfolios\u00a0or the managers of those portfolios, with the Alpha being a representation of the maximum an investor should pay for the active management of that portfolio.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Example_Calculating_Jensens_Alpha\"><\/span>Example:\u00a0<b>Calculating Jensen&#8217;s Alpha<\/b><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Two portfolios have the following characteristics:<\/p>\n<p>$$\\begin{array}{c|c|c} \\textbf{Portfolio} &amp; \\textbf{Return} &amp; \\textbf{Beta} \\\\ \\hline A &amp; 8\\% &amp; 0.7 \\\\ \\hline B &amp; 7\\% &amp; 1.1 \\\\ \\end{array}$$<\/p>\n<p>Given a market return of 10% and a risk-free rate of 4%, calculate Jensen&#8217;s Alpha for both portfolios and comment on which portfolio has performed better.<\/p>\n<ol type=\"A\">\n<li>-0.2%\u00a0 and -3.6% respectively; Portfolio A has performed better than Portfolio B.<\/li>\n<li>-0.2% and -3.6% respectively; Portfolio B has performed better than Portfolio A.<\/li>\n<li>0.2% and 3.6% respectively; Portfolio B has performed better than Portfolio A.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>$$ \\text{Jensen\u2019s alpha } ({ \\alpha }_{ \\text{p} })=R_p \u2013 [R_f + \\beta_p (E(R_m)\u2013 R_f)] $$<\/p>\n<p>\\(\\text{Jensen\u2019s Alpha for Portfolio A} = 0.08 &#8211; [0.04 + 0.7(0.1 &#8211; 0.04)] = -0.002\\)<\/p>\n<p>\\(\\text{Jensen\u2019s Alpha for Portfolio B} = 0.07\u00a0 &#8211; [0.04 + 1.1(0.1 &#8211; 0.04)] = -0.036\\)<\/p>\n<p>Jensen&#8217;s Alpha is -0.2% and -3.6% for portfolios A\u00a0 and B, respectively. A higher Jensen&#8217;s Alpha (-0.2% in this case) indicates that a portfolio has performed better. Also note that both portfolio managers have been unable to create Alpha, but the manager of portfolio A has not been as bad as portfolio B&#8217;s manager.<\/p>\n<h3 data-start=\"2557\" data-end=\"2614\"><span class=\"ez-toc-section\" id=\"Formula_Summary_Card\"><\/span>Formula Summary Card<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n<p><strong>Portfolio Performance Metrics<\/strong><\/p>\n$$\n\\begin{array}{l|l|l|l}\n\\textbf{Metric} &#038; \\textbf{What It} &#038; \\textbf{Formula} &#038; \\textbf{When to Use} \\\\\n&#038; \\textbf{Measures} &#038; &#038; \\\\\n\\hline\n\\text{Sharpe Ratio} &#038; \\text{Excess return per} &#038; (R_p &#8211; R_f)\/\\sigma_p &#038; \\text{When comparing} \\\\\n&#038; \\text{unit of total} &#038; &#038; \\text{portfolios or} \\\\\n&#038; \\text{risk (volatility)} &#038; &#038; \\text{investments using} \\\\\n&#038; &#038; &#038; \\text{total risk} \\\\\n&#038; &#038; &#038; \\text{(standard deviation)} \\\\\n\\hline\n\\text{Treynor Ratio} &#038; \\text{Excess return per} &#038; (R_p &#8211; R_f)\/\\beta_p &#038; \\text{When portfolios} \\\\\n&#038; \\text{unit of} &#038; &#038; \\text{are well-} \\\\\n&#038; \\text{systematic risk (beta)} &#038; &#038; \\text{diversified and} \\\\\n&#038; &#038; &#038; \\text{you want to} \\\\\n&#038; &#038; &#038; \\text{isolate market risk} \\\\\n\\hline\n\\text{Jensen&#8217;s Alpha} &#038; \\text{Abnormal return} &#038; R_p &#8211; [R_f + &#038; \\text{To measure manager} \\\\\n&#038; \\text{above CAPM-} &#038; \\beta_p (E(R_m) &#8211; R_f)] &#038; \\text{performance relative} \\\\\n&#038; \\text{predicted return} &#038; &#038; \\text{to expected market} \\\\\n&#038; &#038; &#038; \\text{return} \\\\\n\\end{array}\n$$\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Frequently_Asked_Questions\"><\/span><strong>Frequently Asked Questions<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p><strong>1. What is the difference between Sharpe and Treynor ratios?<\/strong><br>The Sharpe Ratio evaluates returns per unit of total risk (standard deviation), while the Treynor Ratio focuses only on systematic risk (beta). Use Sharpe when assessing the absolute risk-adjusted return of a portfolio, and Treynor when comparing diversified portfolios relative to market risk.<\/p>\n\n\n\n<p><strong>2. How is Jensen\u2019s Alpha calculated in CFA exams?<\/strong><br>Jensen\u2019s Alpha is calculated by subtracting the expected return (based on CAPM) from the actual return of a portfolio. Formula: $$ \\text{Jensen\u2019s alpha} = \\alpha_p = R_p \u2013 [R_f + \\beta_p (E(R_m)\u2013 R_f)]$$<\/p>\n\n\n\n<p>Where \\(R_p \\)\u200b is the portfolio return, \\(R_f\\)\u200b is the risk-free rate, \\(E(R_m\\)\u200b is the expected market return, and  \\(\\beta_p\\) is the portfolio\u2019s beta.<\/p>\n\n\n\n<p><strong>3. When should you use Sharpe vs Treynor?<\/strong><br>Use Sharpe when evaluating portfolios that may include unsystematic risk (like actively managed funds). Use Treynor when comparing well-diversified portfolios where unsystematic risk is negligible. In short: Sharpe = total risk, Treynor = market risk only.<\/p>\n\n\n\n<p><strong>4. Are Sharpe and Treynor ratios tested in CFA Level 1 or 2?<\/strong><br>Both Sharpe and Treynor ratios are introduced in <strong>CFA Level 1<\/strong>, particularly under Portfolio Management. However, deeper application and comparative analysis (including Jensen\u2019s Alpha) are often emphasized more in <strong>CFA Level 2<\/strong>.<\/p>\n\n\n\n<div style=\"margin: 40px 0; padding: 30px; text-align: center; background-color: #f5f8fc; border-radius: 10px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     class=\"ap-cta\"\n     data-cta-text=\"Start Free Trial\"\n     data-cta-type=\"button\"\n     data-cta-location=\"bottom_content\"\n     data-page-type=\"blog\"\n     style=\"\n       display: inline-block;\n       padding: 14px 26px;\n       font-size: 18px;\n       font-weight: 700;\n       color: #ffffff;\n       background-color: #0b5ed7;\n       border-radius: 8px;\n       text-decoration: none;\n     \">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"margin-top: 12px; font-size: 15px; color: #333;\">\n    Practice Sharpe, Treynor &#038; Jensen\u202fAlpha calculations with full questions and solutions.\n  <\/p>\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Portfolio Performance Measures Portfolio management involves a trade-off between risk and return. Most amateur investors mistakenly focus only on the return aspect and lose sight of the risk taken to achieve the return. The portfolio performance measures are intended as&#8230;<\/p>\n","protected":false},"author":2,"featured_media":8311,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[70,71],"tags":[],"class_list":["post-8178","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\/8178","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=8178"}],"version-history":[{"count":42,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/posts\/8178\/revisions"}],"predecessor-version":[{"id":13623,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/posts\/8178\/revisions\/13623"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/media\/8311"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/media?parent=8178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/categories?post=8178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/tags?post=8178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}