{"id":46779,"date":"2023-09-22T17:50:50","date_gmt":"2023-09-22T17:50:50","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=46779"},"modified":"2026-02-13T19:27:31","modified_gmt":"2026-02-13T19:27:31","slug":"bonds-percentage-price-change-using-curve-based-duration-and-convexity","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/bonds-percentage-price-change-using-curve-based-duration-and-convexity\/","title":{"rendered":"Bond\u2019s Percentage Price Change Using Curve-based Duration and Convexity"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Curve-Based and Empirical Fixed Income Risk Measures (2025 CFA\u00ae Level I Exam \u2013 Fixed Income \u2013 Learning Module 13)\",\n  \"description\": \"This CFA\u00ae Level I Fixed Income lesson explains curve-based and empirical measures of interest rate risk. The video covers modified duration, money duration, and price value of a basis point (PVBP), along with how a bond\u2019s maturity, coupon rate, and yield level affect interest rate risk. Designed for exam preparation, the session focuses on calculating, interpreting, and applying these risk measures to analyze fixed-income securities and portfolios.\",\n  \"uploadDate\": \"2023-12-05\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/eQ0BoYEhni0\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=eQ0BoYEhni0\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/eQ0BoYEhni0\",\n  \"duration\": \"PT31M09S\"\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\": \"What are the percentage changes in a bond\u2019s full price for \u00b1100 bp shifts in the benchmark government par curve, given effective duration and effective convexity?\",\n    \"text\": \"The effective duration and effective convexity of a bond are 4.816 and 26.723, respectively. The percentage changes in the bond\u2019s full price for \u00b1100 bp shifts in the benchmark government par curve are closest to?\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is \u22124.68% for a +100 bp (upward) shift and +4.95% for a \u2212100 bp (downward) shift. Using %\u0394PVFull \u2248 (\u2212EffDur \u00d7 \u0394Curve) + 0.5 \u00d7 EffCon \u00d7 (\u0394Curve)^2 with \u0394Curve = +0.01: %\u0394PVFull = (\u22124.816 \u00d7 0.01) + 0.5 \u00d7 26.723 \u00d7 (0.01)^2 = \u22120.04682 \u2248 \u22124.68%. With \u0394Curve = \u22120.01: %\u0394PVFull = (\u22124.816 \u00d7 \u22120.01) + 0.5 \u00d7 26.723 \u00d7 (\u22120.01)^2 = 0.049492 \u2248 +4.95%.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p>\n  <iframe loading=\"lazy\"\n    src=\"\/\/www.youtube.com\/embed\/eQ0BoYEhni0\"\n    width=\"611\"\n    height=\"343\"\n    allowfullscreen=\"allowfullscreen\">\n  <\/iframe>\n<\/p>\n\n\n\n<p>Effective duration and effective convexity are curve-based metrics that are crucial for assessing the interest rate risk of complex instruments, such as those with embedded contingency provisions. These metrics are typically determined from bond prices derived using an option valuation model, given specific changes in the underlying benchmark government yield curve. Effective duration and effective convexity can be used to estimate the percentage change in a bond\u2019s full price for a given shift in the benchmark yield curve (?Curve). It is calculated as:<\/p>\n\n\n\n<p>\\[\\%\\Delta PV^{Full} \\approx ( &#8211; EffDur \\times \\Delta Curve) + \\left\\lbrack \\frac{1}{2} \\times EffCon \\times (\\Delta Curve)^{2} \\right\\rbrack\\]<\/p>\n\n\n\n<!-- TOP CTA \u2013 Full Width Outline Button -->\n<div style=\"margin:24px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:block;\n       width:100%;\n       padding:13px 0;\n       border:2px solid #3b6fd8;\n       border-radius:50px;\n       font-size:17px;\n       font-weight:500;\n       text-align:center;\n       text-decoration:none;\n       color:#3b6fd8;\n       background-color:#f4f6f9;\n       box-sizing:border-box;\n     \">\n     Practice duration and convexity questions with free trial access.\n  <\/a>\n<\/div>\n\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Example: Bond Price Impact from Par Curve Shifts<\/strong><\/h4>\n\n\n\n<p>Consider the effective duration and effective convexity data for Bond X and Bond Y. We will examine the effects of a 200bps shift in the benchmark government par curve.<\/p>\n\n\n\n<p>$$\\begin{array}{c|c|c} \\text{Bond} &amp; \\text{EffDur} &amp; \\text{EffCon} \\\\ \\hline \\text{Bond } X &amp; 7.425 &amp; -295.0 \\\\ \\hline \\text{Bond } Y &amp; 6.891 &amp; -278.310 \\\\&nbsp; \\end{array}$$<\/p>\n\n\n\n<p><strong><u>Upward Shift by 200 bps:<\/u><\/strong><\/p>\n\n\n\n<p>\\[\\%\\Delta PV^{Full} \\approx ( &#8211; EffDur \\times \\Delta Curve) + \\left\\lbrack \\frac{1}{2} \\times EffCon \\times (\\Delta Curve)^{2} \\right\\rbrack\\]<\/p>\n\n\n\n<p>Where \\(\\Delta Curve = 0.02\\),<\/p>\n\n\n\n<p>Bond X: \\(\\%\\Delta PV^{\\text{Full}} = ( &#8211; 7.425 \\times 0.02) + \\left( 0.5 \\times &#8211; 295.0 \\times (0.02)^{2} \\right) = &#8211; 20.75\\%\\)<\/p>\n\n\n\n<p>Bond Y: \\(\\%\\Delta PV^{Full} = ( &#8211; 6.891 \\times 0.02) + \\left( 0.5 \\times &#8211; 278.310 \\times (0.02)^{2} \\right) = &#8211; 19.35\\%\\)<\/p>\n\n\n\n<p>The significant decline in bond prices in response to increasing interest rates can be attributed to the high effective durations of Bond X and Bond Y, indicating their increased vulnerability to changes in interest rates. The situation is compounded by the negative convexities, which amplify the declines in bond prices. Investors who hold these bonds are, therefore, at risk of incurring losses in the event of rising interest rates. The negative convexity also indicates that bond prices may not experience significant increases even if interest rates subsequently decrease, as we will see below.<\/p>\n\n\n\n<p><strong><u>Downward Shift by 200 bps:<\/u><\/strong><\/p>\n\n\n\n<p>\\(\\Delta Curve = &#8211; 0.02\\).<\/p>\n\n\n\n<p>Bond X: \\(\\%\\Delta PV^{\\text{Full}} = ( &#8211; 7.425 \\times &#8211; 0.02) + \\left( 0.5 \\times &#8211; 295.0 \\times ( &#8211; 0.02)^{2} \\right) = 8.95\\%\\)<\/p>\n\n\n\n<p>Bond Y: \\(\\%\\Delta PV^{Full} = ( &#8211; 6.891 \\times &#8211; 0.02) + \\left( 0.5 \\times &#8211; 278.310 \\times ( &#8211; 0.02)^{2} \\right) = 8.22\\%\\)<\/p>\n\n\n\n<p>Despite declining interest rates, the price appreciation of both bonds is moderate. This seemingly counterintuitive behavior can be attributed back to the negative convexities of the bonds. In rate-declining scenarios, one would anticipate bond prices to rise more significantly. However, the negative convexity lowers this increase, causing the price appreciation to be less than would be predicted based solely on duration.<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<h3 class=\"wp-block-heading\"><strong>Question<\/strong><\/h3>\n\n\n\n<p>The effective duration and effective convexity of a bond are 4.816 and 26.723, respectively, The percentage changes in the bond\u2019s full price for \u00b1100bp shifts in the benchmark government par curve are closest to?<\/p>\n\n\n\n<ol style=\"list-style-type:upper-alpha\" class=\"wp-block-list\">\n<li>-4.68% and 4.95%<\/li>\n\n\n\n<li>-4.47% and 4.71%<\/li>\n\n\n\n<li>-4.816% and 26.72%<\/li>\n<\/ol>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<p><strong>The correct answer is A.<\/strong><\/p>\n\n\n\n<p>\\(\\%\\Delta PV^{Full} \\approx ( &#8211; EffDur \\times \\Delta Curve) + \\left\\lbrack \\frac{1}{2} \\times EffCon \\times (\\Delta Curve)^{2} \\right\\rbrack\\)<\/p>\n\n\n\n<p><strong><u>Upward shift:<\/u><\/strong><\/p>\n\n\n\n<p>\\(\\Delta Curve = 0.01\\)<\/p>\n\n\n\n<p>\\(\\%\\Delta PV^{\\text{Full}} = ( &#8211; 4.816 \\times 0.01) + \\left( 0.5 \\times 26.723 \\times (0.01)^{2} \\right) = &#8211; 4.68\\%\\)<\/p>\n\n\n\n<p><strong><u>Downward shift:<\/u><\/strong><\/p>\n\n\n\n<p>\\(\\Delta Curve = &#8211; 0.01\\)<\/p>\n\n\n\n<p>\\(\\%\\Delta PV^{\\text{Full}} = ( &#8211; 4.816 \\times &#8211; 0.01) + \\left( 0.5 \\times 26.723 \\times ( &#8211; 0.01)^{2} \\right) = 4.95\\%\\)<\/p>\n<\/blockquote>\n\n\n\n<div class=\"wp-block-group is-nowrap is-layout-flex wp-container-core-group-is-layout-ad2f72ca wp-block-group-is-layout-flex\"><\/div>\n\n\n\n<!-- BOTTOM CTA \u2013 Refined Version -->\n<div style=\"text-align:center; background-color:#f4f6f9; padding:32px 20px; border-radius:12px; margin-top:40px;\">\n\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:inline-block;\n       padding:13px 30px;\n       background-color:#3b6fd8;\n       color:#ffffff;\n       border-radius:50px;\n       font-size:15.5px;\n       font-weight:600;\n       text-decoration:none;\n       margin-bottom:16px;\n     \">\n     Start Free Trial\n  <\/a>\n\n  <p style=\"max-width:680px; margin:0 auto; font-size:15.5px; line-height:1.6; color:#333;\">\n    Strengthen your CFA Level I fixed income skills with exam-style curve based duration and convexity problems, structured explanations, and timed practice designed to improve calculation speed and accuracy.\n  <\/p>\n\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Effective duration and effective convexity are curve-based metrics that are crucial for assessing the interest rate risk of complex instruments, such as those with embedded contingency provisions. These metrics are typically determined from bond prices derived using an option valuation&#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-46779","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>Bond Price Change Using Duration &amp; Convexity | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how effective duration and convexity help estimate percentage price changes in bonds due to interest rate shifts.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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