{"id":3464,"date":"2019-09-06T12:00:00","date_gmt":"2019-09-06T12:00:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=3464"},"modified":"2026-03-31T12:46:05","modified_gmt":"2026-03-31T12:46:05","slug":"percentage-price-change-bond-duration-convexity","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/percentage-price-change-bond-duration-convexity\/","title":{"rendered":"Duration and Convexity Effect on the Price Change of a Bond"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Understanding Fixed-Income Risk and Return (2025 Level I CFA\u00ae Exam \u2013 Fixed Income \u2013 Module 5)\",\n  \"description\": \"This video covers fixed-income risk and return, including calculations and interpretations of bond return sources, durations (Macaulay, modified, and effective), portfolio duration, and convexity. It explores interest rate risk, yield curve sensitivity, price changes, term structure volatility, holding period returns, credit spread effects, and empirical vs. analytical duration.\",\n  \"uploadDate\": \"2022-06-01T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/ys7hMfL_EIs\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/ys7hMfL_EIs\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/ys7hMfL_EIs\",\n  \"duration\": \"PT1H1S\"\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 is the estimated convexity-adjusted percentage price change resulting from a 100 bps decrease in yield-to-maturity?\",\n    \"text\": \"An investment bank holds a considerable position in a 7% annual coupon paying bond. The bond\u2019s yield-to-maturity is 8%. The settlement date is 83 days into the 360-day year. The approximate modified duration is 9 years and approximate convexity is 105. What is the estimated convexity-adjusted percentage price change resulting from a 100 bps decrease in the yield-to-maturity?\",\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. Using duration and convexity, the estimated price change is: %\u0394PV \u2248 (\u22129.00 \u00d7 \u22120.0100) + (1\/2 \u00d7 105.00 \u00d7 (\u22120.01)^2) \u2248 0.09 + 0.0053 \u2248 0.0953, or 9.53%. Duration alone underestimates the gain, while convexity adds approximately 53 basis points.\",\n      \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/percentage-price-change-bond-duration-convexity\/\",\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\/ys7hMfL_EIs\" \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>The change in the price of a bond can be summarized as follow:<\/p>\n\n\n\n<p>$$\\text{Change in price} = \\text{Duration effect} + \\text{Convexity effect} $$<\/p>\n\n\n\n<p>$$\u2248(\\text{-AnnModDur}\u00d7\u0394Yield)+(\\frac{1}{2}\u00d7\\text{AnnConvexity}\u00d7(\u0394Yield)^2)$$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Example: Change in Price of the Bond when Interest Rate Falls<\/strong><\/h4>\n\n\n\n<p>Suppose the yield-to-maturity is expected to fall by 10 bps tomorrow, from 2.95% to 2.85%. A bond has an annual (modified) duration of 24.500 and annual convexity of 775.0. What is the percentage price gain from this fall in interest rate?<\/p>\n\n\n\n<p>$$\\%\u0394PV^{FULL}\u2248(-24.500\u00d7-0.0010)+(\\frac{1}{2}\u00d7775.0\u00d7(-0.0010)^2 )$$<\/p>\n\n\n\n<p>$$\u22480.0245+0.0004\u22450.0249$$<\/p>\n\n\n\n<p>The modified duration alone underestimates the gain to be 2.45%. The convexity adjustment adds 4 basis points.<\/p>\n\n\n\n<div style=\"text-align: center; margin: 22px 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: flex; align-items: center; justify-content: center;\n       width: 100%; padding: 10px 18px;\n       border: 2px solid #1e5bd8; color: #1e5bd8;\n       border-radius: 9999px; text-decoration: none; font-weight: 600;\">\n      Practice duration and convexity price changes with our free trial\n    <\/a>\n  <\/div>\n<\/div>\n\n\n<h4 class=\"Text001\"><strong><span class=\"Style2AltBaslikChar\"><span lang=\"EN-US\">Example<\/span><\/span><\/strong><\/h4>\n<p>: Change in Price of the Bond when Interest Rate Increase and Increase<\/span><\/span><\/strong><\/h4>\n<p>A pension scheme holds a large position in a 6.5% annual coupon payment government bond that matures on 10th March 2034. The bond\u2019s yield-to-maturity is 6.75% for settlement on 15<sup>th<\/sup> May 2019, stated as the effective annual rate. That settlement date is 65 days into the 360-day year using the 30\/360-day count convention.<\/p>\n<p>(a) Calculate the full price of the bond per 100 of par value.<\/p>\n<p>The full price of the bond is 98.845543 per 100 par value.<\/p>\n<p>$$PV_0=[\\frac{6.5}{1.0675^1} +\\frac{6.5}{1.0675^2} +\u22ef+\\frac{100+6.5}{1.0675^15} ]\u00d71.0675^{65\/360}=98.845543$$<\/p>\n<p>(b) Calculate the approximate modified duration and approximate convexity using a 1 bp increase and decrease in the yield-to-maturity.<\/p>\n<p>$$PV_+=[\\frac{6.5}{1.0676^1} +\\frac{6.5}{1.0676^2} +\u22ef+\\frac{100+6.5}{1.0676^15} ]\u00d71.0676^{65\/360}=98.755130$$<\/p>\n<p>$$PV_-=[\\frac{6.5}{1.0674^1} +\\frac{6.5}{1.0674^2} +\u22ef+\\frac{100+6.5}{1.0674^15} ]\u00d71.0674^{65\/360}=98.936070$$<\/p>\n<p>The approximate modified duration is 9.1527.<\/p>\n<p>$$\\text{Approx. ModDur}=\\frac{98.936070-98.755130}{2\u00d70.0001\u00d798.845543}=9.1527$$<\/p>\n<p>The approximate convexity is 115.3315.<\/p>\n<p>$$\\text{Approx. conv}=\\frac{98.936070+98.755130-(2\u00d798.845543)}{0.0001^2\u00d798.845543}=115.3315$$<\/p>\n<p>(c) Calculate the estimated convexity-adjusted percentage price change resulting from a 100 bp increase in the yield-to-maturity.<\/p>\n<p>The convexity-adjusted percentage price drop resulting from a 100 bp increase in the yield-to-maturity is estimated to be 8.576%. Notably, modified duration alone estimates the percentage drop to be 9.1527%. The convexity adjustment adds 57.67 bps.<\/p>\n<p>(100 bps = 1% = 0.0100)<\/p>\n<p>$$\\%\u0394PV^{FULL}\u2248(-9.1527\u00d70.0100)+(\\frac{1}{2}\u00d7115.3315\u00d7(-0.0100)^2 )$$<\/p>\n<p>$$\u2248-0.091527+0.005767=-0.08576$$<\/p>\n<p>(d) How does the estimated percentage price change compare with the actual change, assuming the yield-to-maturity jumps to 7.75% on that settlement date?<\/p>\n<p>The new full price if the yield-to-maturity shoots from 6.75% to 7.75% on 15<sup>th<\/sup> May 2019 is 90.344807.<\/p>\n<p>$$PV^{FULL}=[\\frac{6.5}{1.0775^1} +\\frac{6.5}{1.0775^2} +\u22ef+\\frac{100+6.5}{1.0775^15} ]\u00d71.0775^{65\/360}=90.344807$$<\/p>\n<p>$$\\%\u0394PV^{FULL}\u2248\\frac{90.344807-98.845543}{98.845543}=-0.086000$$<\/p>\n<p>As these calculations show, the actual percentage change in the bond price is \u20138.6%. The convexity-adjusted estimate is \u20138.576%, whereas the estimated change using modified duration alone is \u20139.1527%. As such, it is evident that convexity adjustment is paramount.<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>An investment bank holds a considerable position in a 7% annual coupon paying bond. The bond\u2019s yield-to-maturity is 8%. The settlement date is 83 days into the 360- day year. The approximate modified duration is 9 years and approximate convexity is 105. What is the estimated convexity-adjusted percentage price change resulting from a 100 bps decrease in the yield-to-maturity?<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li data-tadv-p=\"keep\">0.0875<\/li>\n<li data-tadv-p=\"keep\">0.0925<\/li>\n<li data-tadv-p=\"keep\">0.0953<\/li>\n<\/ol>\n<p class=\"Style2AltBaslik\" style=\"line-height: 130%; margin: 12.0pt 0cm 12.0pt 0cm;\"><strong><span lang=\"EN-US\">Solution<\/span><\/strong><\/p>\n<p class=\"Text001\"><span lang=\"EN-US\">The correct answer is <strong>C<\/strong>.<\/span><\/p>\n<p class=\"Text001\">\\(\\%\u0394PV^{FULL}\u2248(-9.00\u00d7-0.0100)+(\\frac{1}{2}\u00d7105.00\u00d7(-0.01)^2 )\\)<\/p>\n<p class=\"Text001\">\\(\u22480.09+0.0053\u22480.0953\\)<\/p>\n<p class=\"Text001\"><span lang=\"EN-US\">The convexity-adjusted percentage price drop resulting from a 100 bps decrease in the yield-to-maturity is estimated to be 9.53%. The modified duration alone underestimates the gain to be 9.00%, and the convexity adjustment adds 53.0 bps.<\/span><\/p>\n<\/blockquote>\n\n\n<div style=\"text-align: center; margin: 30px 0;\">\n  <a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">\n    Practice bond price sensitivity, duration, and convexity adjustments with CFA Level I exam-style questions.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The change in the price of a bond can be summarized as follow: $$\\text{Change in price} = \\text{Duration effect} + \\text{Convexity effect} $$ $$\u2248(\\text{-AnnModDur}\u00d7\u0394Yield)+(\\frac{1}{2}\u00d7\\text{AnnConvexity}\u00d7(\u0394Yield)^2)$$ Example: Change in Price of the Bond when Interest Rate Falls Suppose the yield-to-maturity is expected&#8230;<\/p>\n","protected":false},"author":1,"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-3464","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>Duration &amp; Convexity | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how duration and convexity formulas estimate bond price changes. 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