{"id":17549,"date":"2021-07-10T23:32:50","date_gmt":"2021-07-10T23:32:50","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17549"},"modified":"2026-05-07T16:01:43","modified_gmt":"2026-05-07T16:01:43","slug":"calculate-the-value-of-a-callable-or-putable-bond-from-an-interest-rate-tree","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/calculate-the-value-of-a-callable-or-putable-bond-from-an-interest-rate-tree\/","title":{"rendered":"Valuing Bonds with Embedded Options"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/1-Embedded-Options-Assuming-Interest-Rate-Volatility.jpg\",\r\n  \"caption\": \"Embedded-Options-Assuming-Interest-Rate-Volatility\",\r\n  \"width\": 1590,\r\n  \"height\": 1611,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" }\r\n}\r\n<\/script> <script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/2-Example-Callable-Bond-scaled.jpg\",\r\n  \"caption\": \"Example-Callable-Bond\",\r\n  \"width\": 2048,\r\n  \"height\": 1567,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" }\r\n}\r\n<\/script> <script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/3-Example-Putable-Bond-1024x784.jpg\",\r\n  \"caption\": \"Example Putable Bond\",\r\n  \"width\": 1024,\r\n  \"height\": 784,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" }\r\n}\r\n<\/script> <script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/4-Question-1-Interest-Rate-Tree-768x682.jpg\",\r\n  \"caption\": \"Question 1 \u2013 Interest Rate Tree\",\r\n  \"width\": 768,\r\n  \"height\": 682,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" }\r\n}\r\n<\/script> <script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/5-Queston-1-Callable-Bond-768x691.jpg\",\r\n  \"caption\": \"Question 1 \u2013 Callable Bond\",\r\n  \"width\": 768,\r\n  \"height\": 691,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": { \"@type\": \"Organization\", \"name\": \"AnalystPrep\" }\r\n}\r\n<\/script> <script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"QAPage\",\r\n  \"mainEntity\": {\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Given the following interest rate tree, the value of a two-year, 6% annual coupon bond with a par value of $100 and callable at par at the end of year 1 is closest to:\",\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is A.\\n\\nUsing the interest rate tree, the value at each node is computed as:\\nV = 0.5[(Vu + C) \/ (1 + i) + (Vd + C) \/ (1 + i)].\\nAt the up node: 97.696. At the down node: 100.474.\\nBecause v1,d > 100, the bond is called at $100.\\nFinal value: v0 = $101.30.\"\r\n    },\r\n    \"suggestedAnswer\": [\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"$101.30\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"$101.53\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"$102.64\"\r\n      }\r\n    ],\r\n    \"answerCount\": 3\r\n  }\r\n}\r\n<\/script><\/p>\r\n\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/5zZuTYmjxW8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p>The steps for valuing a bond with an embedded option in the presence of interest rate volatility are as follows:<\/p>\r\n<p><strong>Step 1<\/strong>: Generate an interest rate tree using the yield curve and interest rate volatility assumptions.<\/p>\r\n<p><strong>Step 2<\/strong>: Determine whether the embedded option will be exercised at each node.<\/p>\r\n<p><strong>Step 3<\/strong>: Calculate the present value of the bond using the backward induction method.<\/p>\r\n<p>Recall that backward induction involves starting from maturity and working backward from right to left to determine the bond\u2019s value at each node.<\/p>\r\n<div style=\"text-align: center; margin: 28px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 10px 22px; border: 1.5px solid #1a73e8; border-radius: 999px; color: #1a73e8; font-size: 15px; font-weight: 600; text-decoration: none; line-height: 1.3;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Practice callable bond valuation with our Free Trial <\/a><\/div>\r\n<h4>Example: Valuation of Bonds with Embedded Options Assuming Interest Rate Volatility<\/h4>\r\n<p>The following interest rate tree has been calibrated, assuming a 15% interest rate volatility and the applicable benchmark yield curve.<\/p>\r\n<p>We can determine the value of a three-year default-free 5% annual coupon bond, both callable and putable at par one year and two years from today.<\/p>\r\n<h4>Callable Bond<\/h4>\r\n<p><strong>Formula:<\/strong><\/p>\r\n<p>$$ V=0.5\\left[\\frac{V_u+C}{1+i}+\\frac{V_d+C}{1+i}\\right] $$<\/p>\r\n<p>Where<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26597\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/1-Embedded-Options-Assuming-Interest-Rate-Volatility.jpg\" alt=\"\" width=\"1590\" height=\"1611\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/1-Embedded-Options-Assuming-Interest-Rate-Volatility.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/1-Embedded-Options-Assuming-Interest-Rate-Volatility-296x300.jpg 296w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/1-Embedded-Options-Assuming-Interest-Rate-Volatility-1011x1024.jpg 1011w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/1-Embedded-Options-Assuming-Interest-Rate-Volatility-768x778.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/1-Embedded-Options-Assuming-Interest-Rate-Volatility-1516x1536.jpg 1516w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/1-Embedded-Options-Assuming-Interest-Rate-Volatility-400x405.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\r\n<p>V = Value of the bond at each node.<\/p>\r\n<p>C = Coupon payment.<\/p>\r\n<p>The bond value at each node must meet the call rule (call the bond if the price exceeds $100) for the option to be exercised.<\/p>\r\n<p>$$ \\begin{align*} V_{2,uu} &amp;=\\frac{105}{1.065}=98.59 \\\\ V_{2,ud} &amp;=\\frac{105}{1.055}=99.53 \\\\ V_{2,dd} &amp;=\\frac{105}{1.035}=101.45 \\\\ V_{2,dd} &amp;= $101.45 &gt; $100. \\end{align*} $$<\/p>\r\n<p>The bond will thus be called at $100. To continue with the valuation, the bond\u2019s value at this node will be taken as $100.<\/p>\r\n<p>$$ \\begin{align*} V_{1,u} &amp;=0.5\\left[\\frac{98.59+5}{1.05}+\\frac{99.53+5}{1.05}\\right]=$99.10 \\\\ V_{1,d} &amp;=0.5\\left[\\frac{99.53+5}{1.025}+\\frac{100+5}{1.025}\\right]=$102.21 \\end{align*} $$<\/p>\r\n<p>\\(V_{1,d} &gt; $100,\\) implying that the bond is called at $100.<\/p>\r\n<p>$$ V_0 =0.5\\left[\\frac{99.10+5}{1.015}+\\frac{100+5}{1.015}\\right]=$103.00 $$<\/p>\r\n<p>This can be shown in the following tree:<\/p>\r\n<h4><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26598\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/2-Example-Callable-Bond-scaled.jpg\" alt=\"2 - Example - Callable Bond\" width=\"2048\" height=\"1567\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/2-Example-Callable-Bond-scaled.jpg 2048w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/2-Example-Callable-Bond-300x230.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/2-Example-Callable-Bond-1024x784.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/2-Example-Callable-Bond-768x588.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/2-Example-Callable-Bond-1536x1175.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/2-Example-Callable-Bond-400x306.jpg 400w\" sizes=\"auto, (max-width: 2048px) 100vw, 2048px\" \/>Putable Bond<\/h4>\r\n<p>The value of the bond at each node is calculated using the formula:<\/p>\r\n<p>$$ V=0.5\\left[\\frac{V_u+C}{1+i}+\\frac{V_d+C}{1+i}\\right] $$<\/p>\r\n<p>Further, the embedded put option will be exercised at each node of the tree if the value of the bond is less than $100. This is known as the<em> <strong>put rule<\/strong><\/em>.<\/p>\r\n<p>$$ \\begin{align*} V_{2,uu} &amp;=\\frac{105}{1.065}=98.59 \\\\ V_{2,ud} &amp;=\\frac{105}{1.055}=99.53 \\\\ V_{2,dd} &amp;=\\frac{105}{1.035}=101.45 \\end{align*} $$<\/p>\r\n<p>Notice that both \\(V_{2,uu}\\) and \\(V_{2,ud}\\) are less than $100. In this case, the bond will be put at $100. To continue with the valuation, the values of the bond at these nodes will be taken as $100.<\/p>\r\n<p>$$ \\begin{align*} v_{1,u} &amp; =0.5\\left[\\frac{100+5}{1.05}+\\frac{100+5}{1.05}\\right]=$100\\\\ v_{1,d} &amp; =0.5\\left[\\frac{100+5}{1.025}+\\frac{101.45+5}{1.025}\\right]=$103.12 \\\\ v_0 &amp;=0.5\\left[\\frac{100+5}{1.015}+\\frac{103.12+5}{1.015}\\right]=$105.00 \\end{align*} $$<\/p>\r\n<p>This can be shown in the following tree:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26599\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/3-Example-Putable-Bond-scaled.jpg\" alt=\"3 - Example - Putable Bond\" width=\"2048\" height=\"1567\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/3-Example-Putable-Bond-scaled.jpg 2048w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/3-Example-Putable-Bond-300x230.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/3-Example-Putable-Bond-1024x784.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/3-Example-Putable-Bond-768x588.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/3-Example-Putable-Bond-1536x1175.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/3-Example-Putable-Bond-400x306.jpg 400w\" sizes=\"auto, (max-width: 2048px) 100vw, 2048px\" \/><\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Given the following interest rate tree:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26600\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/4-Question-1-Interest-Rate-Tree.jpg\" alt=\"4 - Question 1 - Interest Rate Tree\" width=\"1590\" height=\"1412\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/4-Question-1-Interest-Rate-Tree.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/4-Question-1-Interest-Rate-Tree-300x266.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/4-Question-1-Interest-Rate-Tree-1024x909.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/4-Question-1-Interest-Rate-Tree-768x682.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/4-Question-1-Interest-Rate-Tree-1536x1364.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/4-Question-1-Interest-Rate-Tree-400x355.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>The value of a two-year,6% annual coupon bond, with a par value of $100 and callable at par at the end of year 1 is <em>closest to<\/em>:<\/p>\r\n<ol type=\"A\">\r\n\t<li>$101.30.<\/li>\r\n\t<li>$101.53.<\/li>\r\n\t<li>$102.64.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is A.<\/strong><\/p>\r\n<p>The value of the bond at each node is calculated using the formula:<\/p>\r\n<p>$$ V=0.5\\left[\\frac{V_u+C}{1+i}+\\frac{V_d+C}{1+i}\\right] $$<\/p>\r\n<p>The call rule will then be used to establish if the embedded call option will be exercised at each node.<\/p>\r\n<p>$$ \\begin{align*} v_{1,u} &amp;=0.5\\left[\\frac{100+6}{1.085}+\\frac{100+6}{1.085}\\right]=$97.696 \\\\ v_{1,d} &amp;=0.5\\left[\\frac{100+6}{1.055}+\\frac{100+6}{1.055}\\right]=$100.474 \\end{align*} $$<\/p>\r\n<p>\\(v_{1,d}&gt;$100\\). Therefore, the bond is called at $100.<\/p>\r\n<p>$$ v_0=0.5\\left[\\frac{97.696+6}{1.035}+\\frac{100+6}{1.035}\\right]=$101.30 $$<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26601\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/5-Queston-1-Callable-Bond.jpg\" alt=\"5 - Queston 1 - Callable Bond\" width=\"1590\" height=\"1431\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/5-Queston-1-Callable-Bond.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/5-Queston-1-Callable-Bond-300x270.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/5-Queston-1-Callable-Bond-1024x922.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/5-Queston-1-Callable-Bond-768x691.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/5-Queston-1-Callable-Bond-1536x1382.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/5-Queston-1-Callable-Bond-400x360.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\r\n<\/blockquote>\r\n<p>Reading 30: Valuation and Analysis of Bonds with Embedded Options<\/p>\r\n<p><em>LOS 30 (f) Calculate the value of a callable or putable bond from an interest rate tree.<\/em><\/p>\r\n<p><\/p>\r\n<div style=\"text-align: center; margin: 40px 0 24px;\"><a style=\"display: inline-block; background-color: #1a73e8; color: #ffffff; padding: 14px 32px; border-radius: 999px; font-size: 17px; font-weight: bold; text-decoration: none; line-height: 1.3;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a>\r\n<div style=\"margin-top: 12px; font-size: 14px; color: #555; max-width: 720px; margin-left: auto; margin-right: auto; line-height: 1.5;\">Practice callable and putable bond valuation, interest rate trees, and backward induction techniques with CFA Level II exam-style questions.<\/div>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>The steps for valuing a bond with an embedded option in the presence of interest rate volatility are as follows: Step 1: Generate an interest rate tree using the yield curve and interest rate volatility assumptions. Step 2: Determine whether&#8230;<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,472],"tags":[],"class_list":["post-17549","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-fixed-income","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Valuing Callable and Putable Bonds with Trees<\/title>\n<meta name=\"description\" content=\"Learn how callable and putable bonds are valued using interest rate trees, embedded options, and interest rate volatility assumptions.\" \/>\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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