{"id":17607,"date":"2021-07-12T14:34:15","date_gmt":"2021-07-12T14:34:15","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17607"},"modified":"2026-03-26T06:23:20","modified_gmt":"2026-03-26T06:23:20","slug":"compare-effective-convexities-of-callable-putable-and-straight-bonds","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/compare-effective-convexities-of-callable-putable-and-straight-bonds\/","title":{"rendered":"Effective Convexities"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"QAPage\",\r\n  \"mainEntity\": {\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Which statement is least accurate regarding the effective convexity of a callable bond?\",\r\n    \"text\": \"Which of the following statements is least accurate: The effective convexity of a callable bond:\\n\\nA. Becomes negative when the call option is near the money.\\n\\nB. Is always positive.\\n\\nC. Is similar to that of a straight bond when interest rates are high.\",\r\n    \"answerCount\": 1,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is B.\\n\\nThe effective convexity of a callable bond is not always positive. When the embedded call option is near the money, the bond\u2019s price appreciation is capped as interest rates fall, causing the convexity to become negative.\\n\\nOption A is correct because callable bonds exhibit negative convexity when the call option is near the money. Option C is also correct because when interest rates are high, the likelihood of the bond being called is low, and the callable bond behaves similarly to a straight bond with positive convexity.\"\r\n    }\r\n  }\r\n}\r\n<\/script><\/p>\r\n<p>&nbsp;<\/p>\r\n\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/5zZuTYmjxW8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p><em><strong>Effective convexity<\/strong><\/em> is the sensitivity of duration to changes in interest rates.<\/p>\r\n<p>$$ \\text{Effective convexity} =\\cfrac {P_{i-}+P_{i+}-2 \\times P_o}{P_0 (\\Delta \\text{Curve})^2} $$<\/p>\r\n<p>Both callable and straight bonds experience similar positive convexity when interest rates are high. However, the effective convexity of a callable bond turns negative when the call option is near the money. This is because the upside for a callable bond is much smaller than the downside.<\/p>\r\n<p>On the other hand, putable and straight bonds have similar positive convexity when interest rates are low. It is worth noting that putable bonds have positive convexity throughout. Besides, the convexity of a putable bond becomes greater than that of a straight bond when the put is near the money.<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Which of the following statements is <em>least<\/em> accurate: The effective convexity of a callable bond:<\/p>\r\n<ol type=\"A\">\r\n\t<li>Becomes negative when the call option is near the money.<\/li>\r\n\t<li>Is always positive.<\/li>\r\n\t<li>Is similar to that of a straight when interest rates are high.<\/li>\r\n<\/ol>\r\n<h4><strong>Solution<\/strong><\/h4>\r\n<p><strong>The correct answer is B.<\/strong><\/p>\r\n<p>The effective convexity for a callable bond is not always positive as it becomes negative when the embedded option is near the money<\/p>\r\n<p><strong>A is incorrect.<\/strong>\u00a0The effective convexity of a callable bond turns negative when the call option is near the money because the price response of a callable bond to lower interest rates is capped by the call option. In other words, the price of a callable bond has limited upside potential<\/p>\r\n<p><strong>C is incorrect.<\/strong>\u00a0Both callable and straight bonds experience similar positive convexity when interest rates are high.<\/p>\r\n<\/blockquote>\r\n<p>Reading 30: Valuation and Analysis of Bonds with Embedded Options<\/p>\r\n<p><em>LOS 30 (l) Compare effective convexities of callable, putable, and straight bonds.<\/em><\/p>\r\n\n            <div \n                class=\"elfsight-widget-pricing-table elfsight-widget\" \n                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\n                data-elfsight-pricing-table-version=\"2.6.1\"\n                data-elfsight-widget-id=\"elfsight-pricing-table-3\">\n            <\/div>\n            \r\n\r\n<div style=\"text-align: center; margin: 40px 0;\"><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\"> Start Free Trial \u2192 <\/a>\r\n<p style=\"font-size: 15px; margin-top: 12px; color: #555;\">Practice comparing callable, putable, and straight bond convexities with exam-style CFA Level II questions on AnalystPrep.<\/p>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>&nbsp; Effective convexity is the sensitivity of duration to changes in interest rates. $$ \\text{Effective convexity} =\\cfrac {P_{i-}+P_{i+}-2 \\times P_o}{P_0 (\\Delta \\text{Curve})^2} $$ Both callable and straight bonds experience similar positive convexity when interest rates are high. However, the effective&#8230;<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,472],"tags":[],"class_list":["post-17607","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-fixed-income","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>Effective Convexities - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"Know effective convexity in bond valuation, including its calculation formula and how it differs among callable, putable, and straight bonds.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/compare-effective-convexities-of-callable-putable-and-straight-bonds\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Effective Convexities - 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