{"id":1241,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1241"},"modified":"2026-03-31T09:48:53","modified_gmt":"2026-03-31T09:48:53","slug":"shortfall-risk-safety-first-criterion-example","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/shortfall-risk-safety-first-criterion-example\/","title":{"rendered":"Shortfall Risk, Safety-first Ratio and Selection of an Optimal Portfolio Using Roy\u2019s Safety-first Criterion"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2021 Level I CFA\u00ae Exam \u2013 Reading 9)\",\n  \"description\": \"This video covers common probability distributions, including defining and distinguishing discrete and continuous random variables, probability and cumulative distribution functions, and the use of normal, binomial, and uniform distributions. It explains key concepts like Z-scores, confidence intervals, Monte Carlo simulation, and their applications in finance for assessing risks and probabilities.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/KbEfz3KiJDo\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/KbEfz3KiJDo\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo\",\n  \"duration\": \"PT1H45M31S\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-131.jpg\",\n  \"caption\": \"Image showing shortfall-risk\",\n  \"width\": 1463,\n  \"height\": 1036,\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  }\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\": \"Which portfolio is preferable for the manager under the safety-first criterion?\",\n    \"text\": \"The returns on a fund are normally distributed. At the end of year t, the fund has a value of $100,000. At the end of year t+1, the fund manager wishes to withdraw $10,000 for further funding and is reluctant to tap into the $100,000. There are two investment options:\\n\\nPortfolio A: Expected return = 14%, Standard deviation = 17%\\nPortfolio B: Expected return = 13%, Standard deviation = 20%\\n\\nWhich portfolio is preferable for the manager?\\n\\nA. Portfolio A\\nB. Portfolio B\\nC. The manager is indifferent between the two portfolios\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A. Portfolio A.\\n\\nUsing the safety-first criterion, the threshold return is 10% (10,000 \/ 100,000). The safety-first ratio for Portfolio A is (14 \u2212 10) \/ 17 = 0.24, while for Portfolio B it is (13 \u2212 10) \/ 20 = 0.15. Portfolio A has the higher safety-first ratio, making it the preferred choice.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\"\n  width=\"611\"\n  height=\"344\"\n  src=\"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo\"\n  title=\"YouTube video player\"\n  frameborder=\"0\"\n  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\"\n  referrerpolicy=\"strict-origin-when-cross-origin\"\n  allowfullscreen>\n<\/iframe>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Shortfall Risk<\/strong><\/h2>\n\n\n\n<p>Shortfall risk refers to the probability that a portfolio will not exceed the minimum (benchmark) return that has been set by an investor. In other words, it is the risk that a portfolio will fall short of the level of return considered acceptable by an investor. As such, shortfall risks are downside risks. While shortfall risk focuses on the downside economic risk, the standard deviation measures the overall volatility of a financial asset.<\/p>\n\n\n\n<!--more-->\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"1463\" height=\"1036\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-131.jpg\" alt=\"Image showing shortfall-risk\" class=\"wp-image-17021\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-131.jpg 1463w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-131-300x212.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-131-768x544.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-131-1024x725.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-131-400x283.jpg 400w\" sizes=\"auto, (max-width: 1463px) 100vw, 1463px\" \/><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\"><strong>Safety-first Ratio<\/strong><\/h2>\n\n\n\n<p>Roy\u2019s safety-first criterion states that the optimal portfolio is the one that minimizes the probability that the portfolio return, denoted by R<sub>P<\/sub>, falls below the threshold level of return, R<sub>L<\/sub>. Writing down this concept in symbols:<\/p>\n\n\n\n<p>the optimal portfolio minimizes \\(P(R_P &lt; R_L)\\).<\/p>\n\n\n\n<p>As such, if returns are normally distributed, the optimal portfolio is the one with the highest safety-first ratio.<\/p>\n\n\n\n<p>$$ \\text{SFRatio}=\\frac { E\\left( { R }_{ p } \\right) -{ R }_{ L } }{ \\sigma _{ p } } $$<\/p>\n\n\n\n<p>The numerator, E(R<sub>P<\/sub>) \u2013 R<sub>L<\/sub>, represents the distance from the mean return to the threshold level,&nbsp; i.e., it measures the excess return over and above the threshold level of return, per unit risk.<\/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: inline-flex; align-items: center; justify-content: center;\n       width: 100%; padding: 12px 20px;\n       border: 2px solid #1a73e8; border-radius: 999px;\n       color: #1a73e8; text-decoration: none;\n       font-size: 15px; font-weight: 600;\n       line-height: 1.2; white-space: nowrap;\">\n      Apply shortfall risk and safety-first concepts through our free trial.\n    <\/a>\n  <\/div>\n<\/div>\n\n\n<h3><strong>Example:\u00a0<\/strong><strong>Safety-first Ratio<\/strong><\/h3>\n<p>An investor sets a minimum threshold of 3%. There are three portfolios from which he is to choose one. The expected return and the standard deviation for each portfolio are as given below:<\/p>\n<p>$$ \\begin{array}{c|c|c|c} {} &amp; \\text{Portfolio A} &amp; \\text{Portfolio B} &amp; \\text{Portfolio C} \\\\ \\hline {\\text E \\left\\{ \\text R \\right\\} } &amp; {5\\%} &amp; {10\\%} &amp; {20\\%} \\\\ \\hline \\text{S.D} &amp; {15\\%} &amp; {20\\%} &amp; {25\\%} \\\\ \\end{array} $$<\/p>\n<p>Determine the optimal portfolio for the investor.<\/p>\n<p>You should compute the safety-first ratio for each of the three portfolios and then compare them.<\/p>\n<p>For portfolio A,<\/p>\n<p>$$ \\text{SFRatio}_{\\text A} =\\cfrac {(5 \u2013 3)}{15} = 0.1333 $$<\/p>\n<p>Similarly, for portfolio B,<\/p>\n<p>$$ \\text{SFRatio}_{\\text B} =\\cfrac {(10 \u2013 3)}{20} = 0.35 $$<\/p>\n<p>Lastly,<\/p>\n<p>$$ \\text{SFRatio}_{\\text C}=\\cfrac {(20 \u2013 3)}{25} = 0.68 $$<\/p>\n<p>The optimal portfolio should minimize the safety-first ratio. Comparing the three ratios, it\u2019s easy to notice that the safety-first ratio for the portfolio C is the highest. Therefore, the investor should choose portfolio C.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>The returns on a fund are normally distributed. At the end of year t, the fund has a value of $100,000. At the end of year t+1, the fund manager wishes to withdraw $10,000 for further funding, but is reluctant to tap into the $100,000 dollars. There are two investment options:<\/p>\n<p>$$ \\begin{array}{c|c|c} {} &amp; \\text{Portfolio A} &amp; \\text {Portfolio B} \\\\ \\hline {\\text{Expected return} } &amp; {14\\%} &amp; {13\\%} \\\\ \\hline \\text{Standard deviation} &amp; {17\\%} &amp; {20\\%} \\\\ \\end{array} $$<\/p>\n<p>Which portfolio is preferable for the manager?<\/p>\n<p>A. Portfolio A<\/p>\n<p>B. Portfolio B<\/p>\n<p>C. The manager is indifferent between the two portfolios<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is A.<\/p>\n<p>First, you should calculate the threshold return from the information given. Since there should be no tapping into the fund, the threshold return is \\(\\cfrac {10,000}{100,000} = 10\\% \\text { or } 0.1\\).<\/p>\n<p>You should then calculate the safety-first ratio for each portfolio:<\/p>\n<p>$$ \\text{SFRatio}_{\\text A} =\\cfrac {(14 \u2013 10)}{17} = 0.24 $$<\/p>\n<p>$$ \\text{SFRatio}_{\\text B} =\\cfrac {(13 \u2013 10)}{20} = 0.15 $$<\/p>\n<p>Portfolio A has the highest safety-first ratio. Hence, it\u2019s the most desirable.<\/p>\n<p>Note that you can also go a step further and calculate \\(P(R_P &lt; R_L)\\). To do this, you would have to negate each safety-first ratio and then find the CDF of the standard normal distribution for the resulting value. That is,<\/p>\n<p>$$ \\begin{align*}<br \/>P(R_P &lt; R_L) &amp; = F(-\\text{SFRatio}) \\\\<br \/>F(-0.24)&amp; = 1 \u2013 0.5948 = 0.4052 \\\\<br \/>F(-0.15) &amp; = 1 \u2013 0.5596 = 0.4404 \\\\<br \/>\\end{align*} $$<\/p>\n<p>$$ (-\\text{SFRatio is the z-value}) $$<\/p>\n<p>Interpretation: for portfolio A, there is approximately a 40% probability of obtaining a return below the threshold return. For portfolio B, this probability rises to 44%. Therefore, we choose the option for which the chance of not exceeding the benchmark return is lowest \u2013 portfolio A.<\/p>\n<\/blockquote>\n<p><em>Reading 9 LOS 9m:<\/em><\/p>\n<p><em>Define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy&#8217;s safety-first criterion<\/em><\/p>\n<div class=\"notes_inv\">\n<hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>\n\n\n<div style=\"text-align: center; margin: 40px 0;\">\n  <a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 20px; border-radius: 999px; background-color: #1a73e8; color: #ffffff; text-decoration: none; font-weight: 600;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"font-size: 15px; margin-top: 12px; color: #555;\">\n    Practice shortfall risk, Roy\u2019s safety-first ratio, and portfolio risk evaluation with CFA Level I questions.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Shortfall Risk Shortfall risk refers to the probability that a portfolio will not exceed the minimum (benchmark) return that has been set by an investor. In other words, it is the risk that a portfolio will fall short of the&#8230;<\/p>\n","protected":false},"author":2,"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-1241","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>Shortfall Risk &amp; Safety-First Ratio | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn about shortfall risk and Roy&#039;s safety-first ratio, key measures to evaluate the probability of falling below a portfolio&#039;s benchmark return.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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