{"id":29825,"date":"2021-09-09T08:09:03","date_gmt":"2021-09-09T08:09:03","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29825"},"modified":"2026-03-05T19:01:49","modified_gmt":"2026-03-05T19:01:49","slug":"shortfall-risk-safety-first-ratio-and-identification-of-an-optimal-portfolio-using-roys-safety-first-criterion","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/shortfall-risk-safety-first-ratio-and-identification-of-an-optimal-portfolio-using-roys-safety-first-criterion\/","title":{"rendered":"Shortfall Risk, Safety-first Ratio, and Identification 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\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Which portfolio is preferable using the safety-first criterion?\",\n    \"text\": \"A fund currently valued at $100,000 must generate at least $10,000 next year without reducing the principal. Portfolio A has an expected return of 14% and a standard deviation of 17%. Portfolio B has an expected return of 13% and a standard deviation of 20%. Which portfolio is preferable for the manager?\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A. The threshold return is 10% (10,000 \/ 100,000). Using the safety-first ratio: SFR(A) = (14 \u2212 10) \/ 17 = 0.24 and SFR(B) = (13 \u2212 10) \/ 20 = 0.15. Since Portfolio A has the higher safety-first ratio, it minimizes the probability of falling below the required return, making it the preferred portfolio.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"Portfolio A.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"Portfolio B.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"The manager is indifferent to the two portfolios.\"\n      }\n    ]\n  }\n}\n<\/script>\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 using the safety-first criterion?\",\n    \"text\": \"A fund currently valued at $100,000 must generate at least $10,000 next year without reducing the principal. Portfolio A has an expected return of 14% and a standard deviation of 17%. Portfolio B has an expected return of 13% and a standard deviation of 20%. Which portfolio is preferable for the manager?\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A. The threshold return is 10% (10,000 \/ 100,000). Using the safety-first ratio: SFR(A) = (14 \u2212 10) \/ 17 = 0.24 and SFR(B) = (13 \u2212 10) \/ 20 = 0.15. Since Portfolio A has the higher safety-first ratio, it minimizes the probability of falling below the required return, making it the preferred portfolio.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"Portfolio A.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"Portfolio B.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"The manager is indifferent to the two portfolios.\"\n      }\n    ]\n  }\n}\n<\/script>\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\": \"Shortfall Risk\",\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\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/TXO9ODcLWiU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Shortfall risk refers to the probability that a portfolio will not exceed the minimum (benchmark) return that an investor has set. In other words, it is the risk that a portfolio will fall short of the level of return an investor considers acceptable. As such, shortfall risks are downside risks. While a shortfall risk focuses on the downside economic risk, the standard deviation measures the overall volatility of a financial asset.<\/p>\n<p><!--more--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-17021\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-131.jpg\" alt=\"shortfall-risk\" width=\"1463\" height=\"1036\" 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\" \/><\/p>\n<h2><strong>Safety-first Ratio<\/strong><\/h2>\n<p>Roy\u2019s safety-first criterion states that the optimal portfolio is the one that minimizes the probability that a portfolio return, denoted by \\(R_P\\), may fall below the threshold level of return, \\(R_L\\).&nbsp;The optimal portfolio minimizes \\(P(R_P &lt; R_L)\\).<\/p>\n<p>As such, if returns are distributed normally, the optimal portfolio is the one with the highest safety-first ratio.<\/p>\n<p>$$ \\text{SFRatio}=\\frac { E\\left( { R }_{ p } \\right) -{ R }_{ L } }{ \\sigma _{ p } } $$<\/p>\n<p>The numerator, \\(E(R_P)\\) \u2013 \\(R_L\\), represents the distance from the mean return to the threshold level, i.e., it measures the excess return over and above the threshold level of return per unit risk.<\/p>\n<h4><strong>Example:&nbsp;<\/strong><strong>Safety-first Ratio<\/strong><\/h4>\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>$$ \\begin{array}{c|c|c|c} {} &amp; \\textbf{Portfolio A} &amp; \\textbf{Portfolio B} &amp; \\textbf{Portfolio C} \\\\ \\hline {\\text {Expected return}} &amp; {5\\%} &amp; {10\\%} &amp; {20\\%} \\\\ \\hline \\text{Standard deviation} &amp; {15\\%} &amp; {20\\%} &amp; {25\\%} \\\\ \\end{array} $$<\/p>\n<p>What is the optimal portfolio for the investor?<\/p>\n<p><strong>Solution<\/strong><\/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 is easy to notice that the safety-first ratio for 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 distributed normally. 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 to the two portfolios.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/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. This is the reason it is the most desirable.<\/p>\n<p><em><strong>Note to candidates<\/strong><\/em>: 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>$$ \\begin{align*}<br \/>\nP(R_P &lt; R_L) &amp; = F(-\\text{SFRatio}) \\\\<br \/>\nF(-0.24)&amp; = 1 \u2013 0.5948 = 0.4052 \\\\<br \/>\nF(-0.15) &amp; = 1 \u2013 0.5596 = 0.4404 \\\\<br \/>\n\\end{align*} $$<\/p>\n<p>$$ (-\\text{Where SFRatio is the z-value}) $$<\/p>\n<p><em><strong>Interpretation<\/strong><\/em>: 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><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Shortfall risk refers to the probability that a portfolio will not exceed the minimum (benchmark) return that an investor has set. In other words, it is the risk that a portfolio will fall short of the level of return an&#8230;<\/p>\n","protected":false},"author":10,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-29825","post","type-post","status-publish","format-standard","hentry","category-uncategorized","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=\"Understand shortfall risk, the safety-first ratio, and how Roy\u2019s safety-first criterion helps identify an optimal portfolio.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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