{"id":5963,"date":"2019-09-01T11:00:00","date_gmt":"2019-09-01T11:00:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=5963"},"modified":"2026-04-03T16:31:11","modified_gmt":"2026-04-03T16:31:11","slug":"calculate-and-interpret-the-sharpe-ratio-treynor-ratio-m2-and-jensens-alpha","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/portfolio-management\/calculate-and-interpret-the-sharpe-ratio-treynor-ratio-m2-and-jensens-alpha\/","title":{"rendered":"Sharpe Ratio, Treynor Ratio, M2, and Jensen\u2019s Alpha"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"The Time Value of Money (2021 Level I CFA\u00ae Exam \u2013 Reading 6)\",\n  \"description\": \"This video covers the fundamentals of time value of money, including interest rates, compounding, present and future value calculations, annuities, and cash flow analysis using financial calculators. It provides practical examples and techniques for understanding time value concepts in financial decision-making.\",\n  \"uploadDate\": \"2019-12-16T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/qk19_k31K-4\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/qk19_k31K-4\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/qk19_k31K-4\",\n  \"duration\": \"PT22M42S\"\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\": \"Based on the Sharpe ratio for each portfolio, which portfolio should the client choose?\",\n    \"text\": \"Based on the Sharpe ratio for each portfolio, which portfolio should the client choose?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2025-07-01T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Portfolio B should be chosen because it has the highest Sharpe ratio. The Sharpe ratio measures excess return per unit of total risk, and Portfolio B\u2019s ratio is higher than that of Portfolios A and C, indicating superior risk-adjusted performance.\",\n      \"dateCreated\": \"2025-07-01T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/portfolio-management\/calculate-and-interpret-the-sharpe-ratio-treynor-ratio-m2-and-jensens-alpha\/\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/OCZddCJ_HwM?si=7iNG32cDdLz_eeml\" 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><br>The following are the four ratios commonly used in performance evaluation.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Sharpe Ratio<\/strong><\/h3>\n\n\n\n<p>The Sharpe Ratio is the portfolio risk premium divided by the portfolio risk.<\/p>\n\n\n\n<p>$$ \\text{Sharpe ratio} = \\frac{ R_p \u2013 R_f } { \\sigma_p } $$<\/p>\n\n\n\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. 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 is not informative. In order to rank portfolios, the Sharpe ratio for each portfolio must be computed.<\/p>\n\n\n\n<p>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\n\n\n<div style=\"width:100%; margin:30px 0; box-sizing:border-box;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:block;\n            width:100%;\n            max-width:100%;\n            box-sizing:border-box;\n            padding:16px 20px;\n            border:2px solid #2f6fed;\n            border-radius:999px;\n            background:#f2f4f8;\n            color:#2f6fed;\n            font-size:16px;\n            font-weight:500;\n            text-decoration:none;\n            text-align:center;\n            line-height:1.2;\">\n    Access our CFA free trial for performance metric practice\n  <\/a>\n<\/div>\n\n\n<p>\n\n\n\n<\/p>\n<p>\n\n\n\n<p><!-- \/wp:post-content --> <!-- wp:heading {\"level\":3} --><\/p>\n<h3><strong>Treynor Ratio<\/strong><\/h3>\n<p><!-- \/wp:heading --> <!-- wp:paragraph --><\/p>\n<p>The Treynor ratio is an extension of the Sharpe ratio. Instead of using total risk, Treynor uses beta or systematic risk in the denominator.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$ \\text{Treynor ratio} = \\frac{ R_p \u2013 R_f } { \\beta _p } $$<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>As with the Sharpe ratio, the Treynor ratio requires positive numerators to give meaningful comparative results. Apart from this, 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. Similarly, they do not offer information about the degree of superiority of a higher ratio portfolio over a lower ratio portfolio.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:heading {\"level\":3} --><\/p>\n<h3><strong>M-Squared (M\u00b2) Ratio<\/strong><\/h3>\n<p><!-- \/wp:heading --> <!-- wp:paragraph --><\/p>\n<p>The concept behind the M\u00b2 ratio is to create a portfolio P&#8217; that mimics the risk of the market portfolio by altering the weights of the actual portfolio P and the risk-free asset until portfolio P&#8217; has the same total risk as the market. The return on the mimicking portfolio P&#8217; is determined and compared with the market return.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>The weight in portfolio P (w<sub>p<\/sub>), which sets the portfolio risk equal to the market risk, can be written as:<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$ w_p = \\frac{ \\sigma_m } { \\sigma_p } $$<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>With the balance (1 &#8211; w<sub>p<\/sub>) invested in the risk-free asset.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>The return for the mimicking portfolio P&#8217; is as follows:<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$ R_{ p\u2019} = w_p R_p + (1 \u2013 w_p ) R_f $$<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>Which we can reformulate as:<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$ R_{ p\u2019} = \\frac{ \\sigma_m } { \\sigma_p } \u00d7 R_p + (1 \u2013 \\frac{ \\sigma_m } { \\sigma_p } )\u00d7R_f $$<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>Therefore,<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$ R_{ p\u2019} = R_f + \\sigma_m \\frac{ [R_p \u2013 R_f ] } { \\sigma_p } $$<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>The difference in return between the mimicking portfolio and the market return is M\u00b2 which is expressed as:<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$ M^2 = \\left[R_p \u2013 R_f \\right] \\frac{ \\sigma_m } { \\sigma_p }+ R_f=SR\\times\\sigma_m+R_f $$<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>A portfolio that matches the market&#8217;s return will have an M\u00b2 value equal to zero, while a portfolio that outperforms will have a positive value. By using the M\u00b2 measure, it is possible to rank portfolios and also determine which portfolios beat the market on a risk-adjusted basis.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:heading {\"level\":3} --><\/p>\n<h3><strong>Jensen&#8217;s Alpha<\/strong><\/h3>\n<p><!-- \/wp:heading --> <!-- wp:paragraph --><\/p>\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. Essentially, this is done 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 gauge of performance relative to the market.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$ \\text{Jensen\u2019s alpha} = \\alpha_p = R_p \u2013 [R_f + \\beta_p (R_m\u2013 R_f)] $$<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>If \\alpha<sub>p<\/sub> is positive, the portfolio has outperformed the market, while a negative value indicates underperformance. The alpha values can also be used to rank portfolios or the managers of those portfolios, with the alpha being a representation of the maximum amount an investor should pay for the active management of that portfolio.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:html --><\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p><!-- \/wp:html --> <!-- wp:paragraph --><\/p>\n<p>A client has three portfolio choices, each with the following characteristics:<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$ \\begin{array}{l|r|r|r} \\textbf{} &amp; \\textbf{Expected Return} &amp; \\textbf{Volatility} &amp; \\textbf{Beta} \\\\ \\hline \\text{Portfolio A} &amp; 15\\% &amp; 12\\% &amp; 10\\% \\\\ \\text{Portfolio B} &amp; 18\\% &amp; 14\\% &amp; 11\\% \\\\ \\text{Portfolio C} &amp; 12\\% &amp; 9\\% &amp; 5\\% \\\\ \\end{array} $$<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>The efficient market portfolio has an expected return of 20%, a standard deviation of 12%, and a risk-free interest rate of 5%.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>Based on the Sharpe ratio for each portfolio, the client should choose:<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:list {\"ordered\":true} --><\/p>\n<ol>\n<li>Portfolio A.<\/li>\n<li>Portfolio B.<\/li>\n<li>Portfolio C.<\/li>\n<\/ol>\n<p><!-- \/wp:list --> <!-- wp:paragraph --><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>The correct answer is portfolio <strong>B<\/strong>.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$ \\text{Sharpe ratio} = \\frac{ R_p \u2013 R_f } { \\sigma_p } $$<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>The portfolio with the highest Sharpe ratio has the best performance.<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:paragraph --><\/p>\n<p>$$ \\begin{array}{l|r|r} \\textbf{} &amp; \\textbf{Calculation} &amp; \\textbf{Sharpe Measure} \\\\ \\hline \\text{Portfolio A} &amp; (15\\%-5\\%)\/12\\% &amp; 0.83 \\\\ \\text{Portfolio B} &amp; (18\\%-5\\%)\/14\\% &amp; 0.93 \\\\ \\text{Portfolio C} &amp; (12\\%-5\\%)\/9\\% &amp; 0.77 \\\\ \\end{array} $$<\/p>\n<p><!-- \/wp:paragraph --> <!-- wp:html --> <strong>Note<\/strong>: The Sharpe ratio uses total risk, not just the systematic risk of a portfolio (as represented by beta). Further, note that the information about the efficient market portfolio is useless in this case.<\/p>\n<\/blockquote>\n<p><!-- \/wp:html --><\/p>\n\n<!-- wp:html -->\n<div style=\"background:#f2f4f8; border-radius:16px; padding:32px 20px; text-align:center; margin:40px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-block;\n            background:#4a74c9;\n            color:#ffffff;\n            font-size:16px;\n            font-weight:600;\n            text-decoration:none;\n            padding:14px 34px;\n            border-radius:999px;\n            margin-bottom:16px;\">\n    Start Free Trial\n  <\/a>\n\n  <p style=\"margin:0 auto; max-width:520px; font-size:16px; line-height:1.6; color:#1f2937;\">\n    Strengthen your understanding of Sharpe ratio, Treynor ratio, M\u00b2, and Jensen\u2019s alpha with structured CFA Level I practice.\n  <\/p>\n<\/div>\n<!-- \/wp:html -->","protected":false},"excerpt":{"rendered":"<p>The following are the four ratios commonly used in performance evaluation. Sharpe Ratio The Sharpe Ratio is the portfolio risk premium divided by the portfolio risk. $$ \\text{Sharpe ratio} = \\frac{ R_p \u2013 R_f } { \\sigma_p } $$ The&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[],"class_list":["post-5963","post","type-post","status-publish","format-standard","hentry","category-portfolio-management","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>Sharpe Ratio, Treynor Ratio, M2, and Jensen\u2019s Alpha<\/title>\n<meta name=\"description\" content=\"Learn about the key ratios used in performance evaluation, including the Sharpe Ratio, Treynor Ratio, M-Squared Ratio, and Jensen&#039;s Alpha.\" \/>\n<meta name=\"robots\" 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