{"id":46763,"date":"2023-09-22T17:13:48","date_gmt":"2023-09-22T17:13:48","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=46763"},"modified":"2026-03-11T07:16:38","modified_gmt":"2026-03-11T07:16:38","slug":"bond-risk-and-return-using-duration-and-convexity","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/bond-risk-and-return-using-duration-and-convexity\/","title":{"rendered":"Bond Risk and Return Using Duration and Convexity"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Yield-Based Bond Convexity and Portfolio Properties (2025 CFA\u00ae Level I Exam \u2013 Fixed Income \u2013 Learning Module 12)\",\n  \"description\": \"CFA\u00ae Level I Fixed Income video lesson by AnalystPrep covering Yield-Based Bond Convexity and Portfolio Properties. This module explains how to define, calculate, and interpret modified duration, money duration, and price value of a basis point (PVBP), and analyzes how bond maturity, coupon, and yield level affect interest rate risk, with clear, exam-focused explanations.\",\n  \"uploadDate\": \"2023-12-03T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/VEg8_rxtZuI\/default.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=VEg8_rxtZuI\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/VEg8_rxtZuI\",\n  \"duration\": \"PT29M44S\"\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\": \"An investor purchases a \u00a35 million semi-annual 2.5% coupon bond with a yield-to-maturity of 1.75%, settling 01 July 2023 and maturing 01 July 2025. The bond\u2019s approximate convexity using a 1 bp increase and decrease in yield-to-maturity is closest to:\",\n    \"text\": \"An investor purchases a \u00a35 million semi-annual 2.5% coupon bond with a yield-to-maturity of 1.75%, settling 01 July 2023 and maturing 01 July 2025. The bond\u2019s approximate convexity using a 1 bp increase and decrease in yield-to-maturity is closest to:\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"4.9277.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"1.9464.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"19.1756.\"\n      }\n    ]\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/VEg8_rxtZuI?si=cr-InumqxyflHNDG\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\n<p>The percentage price change of a bond, given a specified change in yield, can be more accurately estimated using both the bond&#8217;s duration and convexity compared to using duration alone. We will give an example to illustrate this.<\/p>\n\n\n\n<p>Consider a 2-year, 4% semiannual coupon bond, settling on 10 June 2024, maturing on 10 June 2026, and yielding 4%\u2014thus, priced at par. Suppose the investor has a position in the bond with a par value of USD50 million, and the yield-to-maturity increases by 100 bps.<\/p>\n\n\n\n<div style=\"text-align:center; margin:20px 0;\">\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"display:inline-block; padding:10px 18px; border:2px solid #1e5bd8; color:#1e5bd8; border-radius:9999px; text-decoration:none; font-weight:600;\">\nPractice duration and convexity questions for CFA Level I\n<\/a>\n<\/div>\n\n\n<p><strong>Calculating the Actual Prices<\/strong><\/p>\n<p><span class=\"math display\">\\[PV_{0} = \\frac{2}{1.02} + \\frac{2}{{1.02}^{2}} + \\frac{2}{{1.02}^{3}} + \\frac{102}{{1.02}^{4}} = 100\\]<\/span><\/p>\n<p>100bp decrease in yield (3%)<\/p>\n<p><span class=\"math display\">\\[PV_{-} = \\frac{2}{1.015} + \\frac{2}{(1.015)^{2}}\\ + \\frac{2}{(1.015)^{3}} + \\frac{102}{(1.015)^{4}} = 101.9271923\\]<\/span><\/p>\n<p><span class=\"math display\">\\[\\%\\Delta PV^{Full} = \\frac{101.9271923}{100} &#8211; 1 = 1.9272\\%\\]<\/span><\/p>\n<p>100bp increase in yield (5%)<\/p>\n<p><span class=\"math display\">\\[PV_{+} = \\frac{2}{1.025} + \\frac{2}{(1.025)^{2}}\\ + \\frac{2}{(1.025)^{3}}\\ + \\frac{102}{(1.025)^{4}}\\ = 98.1190129\\]<\/span><\/p>\n<p><span class=\"math display\">\\[\\%\\Delta PV^{Full} = \\frac{98.1190129}{100} &#8211; 1 = &#8211; 1.8810\\%\\]<\/span><\/p>\n<p><strong>Calculating the Modified Duration<\/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{PV} &amp; \\textbf{Weights} &amp; \\textbf{Time to Receipt*Weight} &amp; \\textbf{Convexity of cashflows} \\\\ \\hline 1 &amp; 1.0000 &amp; 2.0000 &amp; 1.9608 &amp; 0.0196 &amp; 0.0196 &amp; 0.0377 \\\\ \\hline 2 &amp; 2.0000 &amp; 2.0000 &amp; 1.9223 &amp; 0.0192 &amp; 0.0384 &amp; 0.1109 \\\\ \\hline 3 &amp; 3.0000 &amp; 2.0000 &amp; 1.8846 &amp; 0.0188 &amp; 0.0565 &amp; 0.2174 \\\\ \\hline 4 &amp; 4.0000 &amp; 102.0000 &amp; 94.2322 &amp; 0.9423 &amp; 3.7693 &amp; 18.1146 \\\\ \\hline \\ &amp; \\ &amp; \\ &amp; \\textbf{100.0000} &amp; \\textbf{1.0000} &amp; \\textbf{3.8839} &amp; \\textbf{18.4805} \\\\\u00a0 \\end{array} \\]<\/p>\n<p><span class=\"math display\">\\[Annualized\\ Macaulay\\ duration\\ \\ = \\frac{3.8839}{2} = 1.94195\\]<\/span><\/p>\n<p><span class=\"math display\">\\[Annualized\\ convexity = \\frac{18.4805}{2^{2}} = 4.620125\\]<\/span><\/p>\n<p><span class=\"math display\">\\[Modified\\ duration = \\frac{1.94195}{1 + \\frac{0.04}{2}} = 1.9039\\]<\/span><\/p>\n<p>So, a 100bp increase (decrease) in yield-to-maturity results in <span class=\"math inline\">\\(\\%\\mathrm{\\Delta}PVFull\\ \\)<\/span>?\u20131.9039% (1.9039%)<\/p>\n<p><strong>Adding Convexity Adjustment to the Duration Estimate:<\/strong><\/p>\n<p><span class=\"math display\">\\[\\%\\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\\]<\/span><\/p>\n<p><span class=\"math display\">\\[\\%\\Delta P_{VFull} \\approx ( &#8211; 1.9039 \\times &#8211; 0.01) + \\left\\lbrack \\frac{1}{2} \\times 4.6201 \\times ( &#8211; 0.01)^{2} \\right\\rbrack = 1.9270\\%\\]<\/span><\/p>\n<p><span class=\"math display\">\\[\\%\\Delta P_{VFull} \\approx ( &#8211; 1.9039 \\times 0.01) + \\left\\lbrack \\frac{1}{2} \\times 4.6201 \\times (0.01)^{2} \\right\\rbrack = &#8211; 1.8808\\%\\]<\/span><\/p>\n<p>The results can be summarized in the following table:<\/p>\n<p>\\[ \\begin{array}{c|c|c|c|c|c} \\textbf{Change in yield} &amp; \\textbf{Actual} &amp; \\textbf{Estimated using} &amp; \\textbf{Difference from} &amp; \\textbf{Estimated using ModDur} &amp; \\textbf{Difference from} \\\\ &amp; \\%\\Delta PV_{\\text{Full}} &amp; \\textbf{ModDur} &amp; \\textbf{actual change} &amp; \\textbf{and Convexity} &amp; \\textbf{actual change} \\\\ \\hline -100bps &amp; 1.9272\\% &amp; 1.9039\\% &amp; -0.0233\\% &amp; 1.9270\\% &amp; -0.0002\\% \\\\ \\hline +100bps &amp; -1.8810\\% &amp; -1.9039\\% &amp; -0.0229\\% &amp; -1.8808\\% &amp; 0.0002\\% \\\\ \\end{array} \\]<\/p>\n<p>Notice the enhanced precision after adding the convexity adjustment, shown by the decreased difference from the actual change.<\/p>\n<p>Money duration and money convexity capture the first-order and second-order effects on the full price of a bond in currency units, respectively. The money convexity is calculated using the formula:<\/p>\n<p><span class=\"math display\">\\[\\text{MoneyCon} = \\text{AnnConvexity} \\times PV_{\\text{Full}}\\]<\/span><\/p>\n<p>The change in the bond&#8217;s full price using Money Duration and Money Convexity is calculated using the formula:<\/p>\n<p><span class=\"math display\">\\[\\Delta PV_{\\text{Full}} \\approx &#8211; (\\text{MoneyDur} \\times \\Delta\\text{Yield}) + \\left\\lbrack \\frac{1}{2} \\times \\text{MoneyCon} \\times (\\Delta\\text{Yield})^{2} \\right\\rbrack\\]<\/span><\/p>\n<blockquote>\n<p><strong>Question<\/strong><\/p>\n<p>An investor purchases a \u00a35 million semi-annual 2.5% coupon bond with a yield-to-maturity of 1.75%, settling 01 July 2023 and maturing 01 July 2025. The bond\u2019s ApproxCon using a 1 bp increase and decrease in yield-to-maturity is <em>closest to:<\/em><\/p>\n<ol type=\"A\">\n<li>1.9464<\/li>\n<li>4.9277<\/li>\n<li>19.1756<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>The correct answer is B:<\/strong><\/p>\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{1.25}{1.00875} + \\frac{1.25}{{1.00875}^{2}} + \\frac{1.25}{{1.00875}^{3}} + \\frac{101.25}{{1.00875}^{4}} = 101.467753\\]<\/span><\/p>\n<p><span class=\"math display\">\\[PV_{-} = \\frac{1.25}{1.00870} + \\frac{1.25}{(1.00870)^{2}} + \\frac{1.25}{(1.00870)^{3}} + \\frac{101.25}{(1.00870)^{4}} = 101.4875066\\]<\/span><\/p>\n<p><span class=\"math display\">\\[PV_{+} = \\frac{1.25}{1.00880} + \\frac{1.25}{(1.00880)^{2}} + \\frac{1.25}{(1.00880)^{3}} + \\frac{101.25}{(1.00880)^{4}} = 101.4480044\\]<\/span><\/p>\n<p><span class=\"math display\">\\[ApproxCon = \\frac{\\left( (101.4875066 + 101.4480044) &#8211; (2 \\times 101.467753) \\right)}{(0.0001)^{2} \\times 101.467753} = 4.9277\\]<\/span><\/p>\n<\/blockquote>\n\n\n<div style=\"text-align:center; margin:30px 0;\">\n\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"display:inline-block; padding:12px 24px; border-radius:9999px; background:#1e5bd8; color:#ffffff; font-weight:700; text-decoration:none;\">\nStart Free Trial \u2192\n<\/a>\n\n<p style=\"margin-top:12px; font-size:16px; line-height:1.5;\">\nStrengthen your CFA Level I fixed-income skills with exam-style practice on duration, convexity, and bond price sensitivity to interest rates.\n<\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The percentage price change of a bond, given a specified change in yield, can be more accurately estimated using both the bond&#8217;s duration and convexity compared to using duration alone. We will give an example to illustrate this. Consider a&#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-46763","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 Risk &amp; Return: Duration &amp; Convexity | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Explore how duration and convexity impact bond prices, enhancing accuracy in risk and return assessments with practical examples.\" \/>\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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