{"id":46755,"date":"2023-09-22T17:03:52","date_gmt":"2023-09-22T17:03:52","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=46755"},"modified":"2026-04-06T13:04:42","modified_gmt":"2026-04-06T13:04:42","slug":"convexity-and-convexity-adjustment","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/convexity-and-convexity-adjustment\/","title":{"rendered":"Convexity and Convexity Adjustment"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"@id\": \"https:\/\/analystprep.com\/#video-VEg8_rxtZuI\",\n  \"name\": \"Yield-Based Bond Convexity and Portfolio Properties (2025 CFA\u00ae Level I Exam \u2013 Fixed Income \u2013 Learning Module 12)\",\n  \"description\": \"Topic 7 \u2013 Fixed Income. Learning Module 12 \u2013 Yield-Based Bond Convexity and Portfolio Properties. This lesson explains yield-based measures of interest rate risk, including modified duration, money duration, and price value of a basis point (PVBP). It also examines how a bond\u2019s maturity, coupon rate, and yield level affect interest rate risk, with exam-focused explanations aligned to the CFA Level I Fixed Income curriculum.\",\n  \"uploadDate\": \"2023-12-03T00:00:00+00:00\",\n  \"thumbnailUrl\": [\n    \"https:\/\/img.youtube.com\/vi\/VEg8_rxtZuI\/maxresdefault.jpg\",\n    \"https:\/\/img.youtube.com\/vi\/VEg8_rxtZuI\/hqdefault.jpg\"\n  ],\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=VEg8_rxtZuI\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/VEg8_rxtZuI\",\n  \"duration\": \"PT29M44S\",\n  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/default-logo.jpg\",\n      \"width\": 600,\n      \"height\": 60\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\": \"Which of the following factors most likely increases the convexity of a bond?\",\n    \"text\": \"Which of the following factors most likely increases the convexity of a bond?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. Lower yield-to-maturity (YTM) increases a bond\u2019s convexity. Bonds exhibit greater curvature in the price\u2013yield relationship at lower yields. Higher coupon rates and shorter maturities reduce convexity rather than increase it.\",\n      \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/convexity-and-convexity-adjustment\/\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\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\/VEg8_rxtZuI\"\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<p><\/p>\n\n\n\n<p>Duration provides a linear approximation of the change in a bond&#8217;s price with respect to changes in yield. On the other hand, convexity measures the non-linear, second-order effect of yield changes on a bond&#8217;s price. It captures the curvature of the price-yield relationship.<\/p>\n\n\n\n<p>While duration estimates price changes linearly, the true bond price-yield relationship is convex. Convexity becomes particularly crucial when considering significant yield changes and for bonds with longer maturities.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Calculating Convexity<\/strong><\/h3>\n\n\n\n<p>Convexity can be calculated using the formula:<\/p>\n\n\n\n<p>\\[\\%\\Delta P_{VFull} \\approx ( &#8211; \\text{AnnModDur} \\times \\Delta\\text{Yield}) + \\left\\lbrack \\frac{1}{2} \\times \\text{AnnConvexity} \\times (\\Delta\\text{Yield})^{2} \\right\\rbrack\\]<\/p>\n\n\n\n<p>The first term captures the effect from modified duration. The second term represents the convexity adjustment.<\/p>\n\n\n\n<p>Convexity can also be approximated using the following formula:<\/p>\n\n\n\n<p>\\[ApproxCon\\ = \\frac{\\left( PV_{-} \\right) + \\left( PV_{+} \\right) &#8211; \\left\\lbrack 2 \\times \\left( PV_{0} \\right) \\right\\rbrack}{(\\Delta\\text{Yield})^{2} \\times \\left( PV_{0} \\right)}\\]<\/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 convexity and bond risk practice\n  <\/a>\n<\/div>\n\n\n<h3><strong>Factors Affecting Convexity<\/strong><\/h3>\n<ol type=\"i\">\n<li>Maturity: Longer maturity increases convexity.<\/li>\n<li>Coupon rate: Lower coupon rate increases convexity.<\/li>\n<li>YTM: Lower YTM increases convexity.<\/li>\n<li>Cash Flow Dispersion: For two bonds with the same duration, the one with more dispersed cash flows will have greater convexity.<\/li>\n<\/ol>\n<h3><strong>Benefits of Convexity<\/strong><\/h3>\n<p>Bonds with greater convexity perform better in both rising and falling yield scenarios, making them less risky for investors. This assumes that the difference in convexity is not reflected in the bond&#8217;s price. For large yield changes, a bond&#8217;s price will rise more with a decrease in yield and fall less with an increase in yield if it has higher convexity.<\/p>\n<h4><strong>Example: Calculating Convexity<\/strong><\/h4>\n<p>Consider a bond that has a term to maturity of 3 years, an annual coupon rate of 2%, a yield-to-maturity (YTM) of 2%, and is priced at 100 per 100 par value.<\/p>\n<ol type=\"i\">\n<li>Calculate the modified duration and convexity for the bond at issuance.<\/li>\n<li>Calculate ApproxModDur and ApproxCon for the bond using a 10 bp increase and decrease in the yield-to-maturity.<\/li>\n<\/ol>\n<p><strong>Calculating Modified Duration and Convexity<\/strong><\/p>\n<p>\\[ \\begin{array}{c|c|c|c|c|c|c} \\textbf{Period} &amp; \\textbf{Time to Receipt} &amp; \\textbf{Cashflow Amount} &amp; \\textbf{Present Value} &amp; \\textbf{Weights} &amp; \\textbf{Time to Receipt*Weight} &amp; \\textbf{Convexity of Cashflows} \\\\ \\hline 1 &amp; 1.0000 &amp; 2 &amp; 1.9608 &amp; 0.01960 &amp; 0.02 &amp; 0.04 \\\\ \\hline 2 &amp; 2.0000 &amp; 2 &amp; 1.9223 &amp; 0.01922 &amp; 0.04 &amp; 0.11 \\\\ \\hline 3 &amp; 3.0000 &amp; 102 &amp; 96.1169 &amp; 0.96118 &amp; 2.88 &amp; 11.31 \\\\ \\hline \\textbf{Total} &amp; &amp; &amp; \\textbf{100.0000} &amp; \\textbf{1.0000} &amp; \\textbf{2.94} &amp; \\textbf{11.46} \\\\ \\end{array} \\]<\/p>\n<p>Annualized Macaulay Duration = 2.94<\/p>\n<p>Annualized convexity = 11.46<\/p>\n<p>Convexity for each period has been calculated as:<\/p>\n<p><span class=\"math display\">\\[\\text{Convexity}= \\text{Time to receipt of cashflows} \\times \\left(\\text{Time to receipt of cashflows} + 1\\right) \\times \\text{Weight} \\times \\left( 1 + \\frac{YTM}{m} \\right)^{- m}\\]<\/span><\/p>\n<p>Where m is the periodicity.<\/p>\n<h3><strong>Calculating ApproxModDur and ApproxCon<\/strong><\/h3>\n<p><span class=\"math display\">\\[ApproxCon = \\frac{\\left( PV_{-} \\right) + \\left( PV_{+} \\right) &#8211; \\left\\lbrack 2 \\times \\left( PV_{0} \\right) \\right\\rbrack}{(\\Delta\\text{Yield})^{2} \\times \\left( PV_{0} \\right)}\\]<\/span><\/p>\n<p><span class=\"math display\">\\[{PV}_{0} = \\frac{2}{1.02} + \\frac{2}{{1.02}^{2}} + \\frac{102}{{1.02}^{3}} = 100\\]<\/span><\/p>\n<p><span class=\"math display\">\\[{PV}_{-} = \\frac{2}{1.019} + \\frac{2}{{1.019}^{2}} + \\frac{102}{{1.019}^{3}} = 100.288951\\]<\/span><\/p>\n<p><span class=\"math display\">\\[{PV}_{+} = \\frac{2}{1.021} + \\frac{2}{{1.021}^{2}} + \\frac{102}{{1.021}^{3}} = 99.71217249\\]<\/span><\/p>\n<p><span class=\"math display\">\\[\\text{ApproxCon} = \\frac{100.288951\\ + 99.71217249\\ &#8211; \\lbrack 2 \\times 100\\rbrack}{(0.001)^{2} \\times (100)} = 11.2349\\]<\/span><\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Which of the following factors <em>most likely<\/em> increases the convexity of a bond?<\/p>\n<ol style=\"list-style-type: upper-alpha; text-align: left;\">\n<li>Higher coupon rate<\/li>\n<li>Shorter maturity<\/li>\n<li>Lower yield-to-maturity (YTM)<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>The correct answer is C:<\/strong><\/p>\n<p>Lower YTM increases convexity.<\/p>\n<p><strong>A is incorrect:<\/strong> Lower coupon rates, not higher, increase convexity.<\/p>\n<p><strong>B is incorrect:<\/strong> Longer maturities, not shorter, increase convexity.<\/p>\n<\/blockquote>\n\n\n<div style=\"background:#f2f4f8; border-radius:16px; padding:32px 20px; text-align:center; margin:40px 0;\">\n  \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 convexity, price sensitivity, and interest rate risk with structured CFA Level I practice.\n  <\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Duration provides a linear approximation of the change in a bond&#8217;s price with respect to changes in yield. On the other hand, convexity measures the non-linear, second-order effect of yield changes on a bond&#8217;s price. It captures the curvature of&#8230;<\/p>\n","protected":false},"author":12,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-46755","post","type-post","status-publish","format-standard","hentry","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>Convexity and Convexity Adjustment | CFA Level I<\/title>\n<meta name=\"description\" content=\"Learn how bond convexity measures price sensitivity to interest rate changes, including the convexity formula and how convexity adjustment is applied.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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