{"id":3446,"date":"2019-09-06T12:00:00","date_gmt":"2019-09-06T12:00:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=3446"},"modified":"2026-03-02T17:42:02","modified_gmt":"2026-03-02T17:42:02","slug":"calculate-interpret-convexity","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/calculate-interpret-convexity\/","title":{"rendered":"Calculate and Interpret Convexity"},"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 on Understanding Fixed-Income Risk and Return explores the key aspects of bond investment analysis. It covers sources of bond returns, various duration measures (Macaulay, modified, effective, and key rate), portfolio duration, money duration, and price value of a basis point (PVBP). It also explains convexity, interest rate risk, yield curve sensitivity, credit spread and liquidity impacts, and compares empirical and analytical duration. Practical calculations and interpretations are emphasized throughout.\",\n  \"uploadDate\": \"2022-06-01T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/ys7hMfL_EIs\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/ys7hMfL_EIs\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/ys7hMfL_EIs\",\n  \"duration\": \"PT1H1M\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/54d-h.png\",\n  \"caption\": \"Image showing approximate convexity\",\n  \"width\": 974,\n  \"height\": 591,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\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\": \"A bond has a convexity of 120. What is the convexity effect if there is a sudden interest rate change of 50 basis points?\",\n    \"text\": \"A bond has a convexity of 120. What is the convexity effect if there is a sudden interest rate change of 50 basis points?\\n\\nA. 0.003\\nB. 0.0015\\nC. 0.3\",\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 0.0015. The convexity effect is calculated as \u00bd \u00d7 convexity \u00d7 (\u0394yield)\u00b2. Using a convexity of 120 and a yield change of 0.005, the convexity effect is approximately \u00bd \u00d7 120 \u00d7 (0.005)\u00b2 = 0.0015.\",\n      \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/calculate-interpret-convexity\/\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\n}\n<\/script>\n\n\n\n<p>\n  <iframe loading=\"lazy\"\n    src=\"\/\/www.youtube.com\/embed\/ys7hMfL_EIs\"\n    width=\"611\"\n    height=\"343\"\n    allowfullscreen=\"allowfullscreen\">\n  <\/iframe>\n<\/p>\n\n\n\n<p>The Modified Duration provides an estimate of the percentage price change for a bond given a change in its yield-to-maturity. A secondary effect is measured by the convexity statistic.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Approximate Convexity<\/strong><\/h2>\n\n\n\n<p>The true relationship between the bond price and the yield-to-maturity (YTM) is a curved (convex) line. The duration (in particular, money duration) estimates the change in bond price along with the straight line that is tangent to the curved line. For small yield-to-maturity changes, there is little difference between the lines. However, as the change in YTM grows larger, the difference becomes significant.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"974\" height=\"591\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/54d-h.png\" alt=\"Image showing approximate convexity\" class=\"wp-image-10110\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/54d-h.png 974w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/54d-h-300x182.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/54d-h-768x466.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/54d-h-400x243.png 400w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><\/figure>\n<\/div>\n\n\n<p>A convexity measure is used to improve the estimate of the percentage price change.<\/p>\n\n\n\n<p>$$\\%\u0394PV^{FULL}\u2248(\\text{-AnnModDur}\u00d7\u0394Yield)+(\\frac{1}{2}\u00d7\\text{AnnConvexity}\u00d7(\u0394Yield)^2)$$<\/p>\n\n\n\n<p>The second bracketed expression is the convexity adjustment:<\/p>\n\n\n\n<p>$$ \\text{Convexity effect} \u2248 \\frac{1}{2}\u00d7\\text{AnnConvexity}\u00d7(\u0394Yield)^2 $$<\/p>\n\n\n\n<p>The convexity adjustment is the annual convexity statistic, AnnConvexity, times one-half, multiplied by the change in the yield-to-maturity squared. This amount adds to the linear estimate provided by the duration alone, which brings the adjusted estimate very close to the actual price on the curved line. The sign \u201c\u2248\u201d is used because the resulting adjusted price is just an estimate.<\/p>\n\n\n\n<div style=\"background:#f3f4f6; padding:16px 14px; border-radius:12px; margin:20px 0; text-align:center;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-flex; align-items:center; justify-content:center; padding:12px 18px; border:2px solid #1d4ed8; border-radius:999px; color:#1d4ed8; text-decoration:none; font-weight:600; font-size:16px; line-height:1; background:#ffffff; white-space:nowrap;\">\n    Refine your convexity calculations with a free CFA trial\n  <\/a>\n<\/div>\n\n\n<p>\n\n\n\n<\/p>\n<h2><strong>Effective Convexity<\/strong><\/h2>\n<p>The effective convexity of a bond is a curve convexity statistic that measures the secondary effect of a change in a benchmark yield curve. Note that for bonds with somewhat unpredictable cash flows, we use effective duration to measure interest rate risk. Similarly, we use the effective convexity to measure the change in price resulting from a change in the benchmark yield curve for securities with uncertain cash flows. In fact, effective convexity is a secondary curve convexity statistic that measures the secondary effect of a change in the benchmark yield curve.<\/p>\n<p>$$\\text{Effective convexity}=\\frac{PV_{-}+PV_{+}-2PV_{0}}{(\u0394Curve)^2PV_{0}}$$<\/p>\n<p>Where:<\/p>\n<p>PV<sub>\u2013<\/sub>\u00a0= Price if yield curve declines by Yield (parallel shift)<\/p>\n<p>PV<sub>+<\/sub>\u00a0= Price if yield curve\u00a0 increases by Yield (parallel shift)<\/p>\n<p>PV<sub>0<\/sub> = Initial bond price<\/p>\n<p>\u0394Curve = Change in yield on the benchmark curve<\/p>\n<p>The main difference between approximate and effective convexity is that approximate convexity is based on a yield to maturity change, whereas effective convexity is based on a benchmark yield change.<\/p>\n<p>Here are a few quick facts about bonds with embedded options:<\/p>\n<ul>\n<li>when the benchmark yield is high, prices of callable and non-callable bonds change almost in tandem with interest rate changes;<\/li>\n<li>when the benchmark yield is low, the value of the call option (to the issuer) increases;<\/li>\n<li>putable bonds always have positive convexity;<\/li>\n<li>callable bonds exhibit negative convexity.<\/li>\n<\/ul>\n<h2><strong>Zero-coupon Bonds<\/strong><\/h2>\n<p>For a zero-coupon bond, the exact convexity statistic in terms of periods is given by:<\/p>\n<p>$$\\text{Convexity}_{\\text{zero-coupon bond}}=\\frac{[N-\\frac{t}{T}]\u00d7[N+1-\\frac{t}{T}]}{(1+r)^2}$$<\/p>\n<p>Where:<\/p>\n<p><em>N <\/em>= number of periods to maturity as of the beginning of the current period;<\/p>\n<p><em>t\/T <\/em>= the fraction of the period that has gone by; and<\/p>\n<p><em>r <\/em>= the yield-to-maturity per period.<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>A bond has a convexity of 120. What is the convexity effect if there is a sudden interest rate change of 50 basis points?<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li data-tadv-p=\"keep\">0.003<\/li>\n<li data-tadv-p=\"keep\">0.0015<\/li>\n<li data-tadv-p=\"keep\">0.3<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>\\(\\text{Convexity effect} \u2248 \\frac{1}{2}\u00d7\\text{AnnConvexity}\u00d7(\u0394Yield)^2\\)<\/p>\n<p>\\(0.0015\u2248 \\frac{1}{2}\u00d7120\u00d7(0.005)^2\\)<\/p>\n<\/blockquote>\n\n\n<div style=\"background:#f3f4f6; padding:22px 18px; border-radius:14px; margin:30px 0; text-align:center;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-flex; align-items:center; justify-content:center; padding:14px 24px; border-radius:999px; background:#1d4ed8; color:#ffffff; text-decoration:none; font-weight:700; font-size:17px; line-height:1;\">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"margin-top: 12px; font-size: 15px; color: #333;\">\n    Practice convexity, duration &#038; fixed\u2011income problems with full solutions.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The Modified Duration provides an estimate of the percentage price change for a bond given a change in its yield-to-maturity. A secondary effect is measured by the convexity statistic. Approximate Convexity The true relationship between the bond price and the&#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-3446","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>Calculate and Interpret Bond Convexity | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Convexity measures the non-linear effect of yield changes on bond prices, capturing the curvature of the price-yield relationship. 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