{"id":17444,"date":"2021-07-09T22:33:10","date_gmt":"2021-07-09T22:33:10","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17444"},"modified":"2024-04-06T11:26:57","modified_gmt":"2024-04-06T11:26:57","slug":"describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/","title":{"rendered":"Backward Induction"},"content":{"rendered":"\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/dqDjJ-eiqNI\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p><em><strong>Backward induction<\/strong><\/em> involves working backward from maturity to time 0 to determine a bond\u2019s value at each node. It makes the following assumptions:<\/p>\r\n<ol type=\"i\">\r\n\t<li>The value of a bond is known at maturity (Final coupon payment + Face value).<\/li>\r\n\t<li>The value at each node is the expected present value of two possible values from the next period.<\/li>\r\n\t<li>The discount rate is the forward rate for that node.<\/li>\r\n<\/ol>\r\n<h4>Formula: Bond Value at a Node<\/h4>\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:<\/p>\r\n<ul>\r\n\t<li>\\(V_u\\) and \\(V_d\\) are the bond values if the higher and lower forward rates, respectively, are realized in one year.<\/li>\r\n\t<li>\\(i\\) is the forward interest rate at a particular node.<\/li>\r\n\t<li>\\(C\\) is the coupon rate, that is independent of interest rates.<\/li>\r\n<\/ul>\r\n<h4>Example: Backward Induction<\/h4>\r\n<p>A two-year bond with no embedded options pays 8% annual coupons. Given the following interest rate tree, find the values of the bond at each node, i.e., \\(V_{1,u},\\) \\(V_{1,d}\\) and \\(V_0\\).<\/p>\r\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26498\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg\" alt=\"interest rate tree\" width=\"1590\" height=\"1521\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-300x287.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-1024x980.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-768x735.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-1536x1469.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-400x383.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Solution<\/strong><\/p>\r\n<p>$$ \\begin{align*} V_{1,u} &amp;=0.5\\times\\left[\\frac{100+8}{1.09}+\\frac{100+8}{1.09}\\right]=$99.08 \\\\ V_{1,d}&amp; =0.5\\times\\left[\\frac{100+8}{1.06}+\\frac{100+8}{1.06}\\right]=$101.89 \\text{ and}\\\\ V_0 &amp;=0.5\\times\\left[\\frac{99.08+8}{1.04}+\\frac{101.89+8}{1.04}\\right]=$104.31 \\end{align*} $$<\/p>\r\n<p>The resulting interest rate tree is as shown below:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26499\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Resulting-Interest-Rate-Tree.jpg\" alt=\"Resulting Interest Rate Tree\" width=\"1590\" height=\"1474\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Resulting-Interest-Rate-Tree.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Resulting-Interest-Rate-Tree-300x278.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Resulting-Interest-Rate-Tree-1024x949.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Resulting-Interest-Rate-Tree-768x712.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Resulting-Interest-Rate-Tree-1536x1424.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Resulting-Interest-Rate-Tree-400x371.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Exhibit 1 is a binomial interest rate tree related to a bond with annual coupons of 4% with a face value of $100 that matures in three years.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26500\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-Binomial-Interest-Rate-Tree.jpg\" alt=\"Exhibit 1 - Binomial Interest Rate Tree\" width=\"1590\" height=\"1611\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-Binomial-Interest-Rate-Tree.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-Binomial-Interest-Rate-Tree-296x300.jpg 296w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-Binomial-Interest-Rate-Tree-1011x1024.jpg 1011w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-Binomial-Interest-Rate-Tree-768x778.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-Binomial-Interest-Rate-Tree-1516x1536.jpg 1516w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Exhibit-1-Binomial-Interest-Rate-Tree-400x405.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Based on Exhibit 1, the price of the bond is <em>closest<\/em> to:<\/p>\r\n<ol type=\"A\">\r\n\t<li>$101.90.<\/li>\r\n\t<li>$103.87.<\/li>\r\n\t<li>$105.51.<\/li>\r\n<\/ol>\r\n<h3><strong>Solution<\/strong><\/h3>\r\n<p><strong>The correct answer is C.<\/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>The cash flow at Time 3 is 104, the redemption of par value ($100) plus the final coupon payment ($4)<\/p>\r\n<p>Node values for time two are determined as follows:<\/p>\r\n<p>$$ \\begin{align*} &amp; \\text{Upper node: }\\frac{104}{1.03}=100.97 \\\\ &amp; \\text{Middle node: }\\frac{104}{1.02}= 101.96 \\\\ &amp; \\text{Lower node: }\\frac{104}{1.01}= 102.97 \\end{align*} $$<\/p>\r\n<p><strong>Time 1<\/strong><\/p>\r\n<p>$$ \\begin{align*} &amp; \\text{Upper node: } 0.5\\times\\left(\\frac{100.97+4}{1.035}+\\frac{101.96+4}{1.035}\\right)= 101.90 \\\\ &amp; \\text{Lower node: } 0.5\\times\\left(\\frac{101.96+4}{1.025}+\\frac{102.97+4}{1.025}\\right)= 103.87 \\end{align*} $$<\/p>\r\n<p><strong>Time 0<\/strong><\/p>\r\n<p>$$ \\text{Node value: } 0.5\\times\\left(\\frac{101.90+4}{1.013}+\\frac{103.87+4}{1.013}\\right)= 105.51 $$<\/p>\r\n<p>Therefore, the price of the bond is $105.51.<\/p>\r\n<p>This can be shown in the interest rate tree framework.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26501\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-framework.jpg\" alt=\"interest rate tree framework\" width=\"1590\" height=\"1437\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-framework.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-framework-300x271.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-framework-1024x925.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-framework-768x694.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-framework-1536x1388.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree-framework-400x362.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\r\n<\/blockquote>\r\n<p>Reading 29: The Arbitrage-Free Valuation Framework<\/p>\r\n<p><em>LOS 29 (e) Describe the backward induction valuation methodology and calculate the value of a fixed-income instrument given its cash flow at each node.<\/em><\/p>\r\n\n            <div \n                class=\"elfsight-widget-pricing-table elfsight-widget\" \n                data-elfsight-pricing-table-options=\"%7B%22layout%22%3A%22grid%22%2C%22skin%22%3A%22skin5%22%2C%22mainColor%22%3A%22rgb%2851%2C%20129%2C%20234%29%22%2C%22styleColumnBorderRadius%22%3Atrue%2C%22styleColumnBorderWidth%22%3Atrue%2C%22useHorizontalScroll%22%3Afalse%2C%22headTitle%22%3A%22Plan%20name%22%2C%22headFeatures%22%3A%5B%7B%22text%22%3A%22Feature%201%22%7D%2C%7B%22text%22%3A%22Feature%202%22%7D%5D%2C%22headTextColor%22%3A%22rgb%2823%2C%2025%2C%2026%29%22%2C%22headBackgroundColor%22%3A%22rgb%28247%2C%20247%2C%20247%29%22%2C%22toggleVisible%22%3Afalse%2C%22toggleItems%22%3A%5B%5D%2C%22toggleDefaultItem%22%3A%22%22%2C%22toggleColor%22%3A%22%22%2C%22columns%22%3A%5B%7B%22title%22%3A%22Practice%20Package%22%2C%22titleCaption%22%3A%22for%20Level%20II%20of%20the%20CFA%5Cu00ae%20Exam%20offered%20by%20AnalystPrep%22%2C%22features%22%3A%5B%7B%22text%22%3A%22%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22%22%7D%2C%7B%22text%22%3A%22%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22%22%7D%2C%7B%22text%22%3A%22Question%20Bank%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Vignette-style%20practice%20questions%20organized%20by%20topic%22%7D%2C%7B%22text%22%3A%22Mock%20Exams%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%222%20a.m.%20and%202%20p.m.%20mock%20exams%22%7D%2C%7B%22text%22%3A%22Performance%20Tracking%20Tools%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22View%20your%20performance%20in%20attractive%20charts%22%7D%2C%7B%22text%22%3A%225%26nbsp%3BAsk-a-tutor%20Questions%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Ask%20five%20questions%20to%20our%20tutors%20via%20live%20chat%22%7D%5D%2C%22price%22%3A%22249%22%2C%22priceCurrency%22%3A%22USD%22%2C%22button%22%3A%22Buy%20Now%22%2C%22buttonLink%22%3A%22https%3A%5C%2F%5C%2Fanalystprep.com%5C%2Fshop%5C%2Fpractice-package-for-frm-part-ii%5C%2F%22%2C%22buttonCaption%22%3A%22%22%2C%22mainColor%22%3A%22rgb%2851%2C%20129%2C%20234%29%22%2C%22priceOptions%22%3A%5B%5D%2C%22buttonOptions%22%3A%5B%5D%2C%22isFeatured%22%3Afalse%2C%22contentDivider%22%3Anull%2C%22pricePrefix%22%3A%22%22%2C%22pricePostfix%22%3A%22%22%2C%22priceCaption%22%3A%22for%20a%2012-month%20access%22%2C%22picture%22%3A%22https%3A%5C%2F%5C%2Felfsight.com%5C%2Fassets%5C%2Fpricing-table%5C%2Fexample-grid-1.jpg%22%2C%22buttonTargetBlank%22%3Afalse%2C%22ribbonText%22%3A%22%22%2C%22ribbonBackgroundColor%22%3A%22%22%2C%22ribbonTextColor%22%3A%22%22%7D%2C%7B%22title%22%3A%22Learn%20%2B%20Practice%20Package%22%2C%22titleCaption%22%3A%22for%20Level%20II%20of%20the%20CFA%5Cu00ae%20Exam%20offered%20by%20AnalystPrep%22%2C%22features%22%3A%5B%7B%22text%22%3A%22%3Cb%3EVideo%20Lesson%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%2250%20hours%20of%20video%20lessons%20by%20Prof.%20James%20Forjan%2C%20CFA%2C%20PhD%22%7D%2C%7B%22text%22%3A%22%3Cb%3EStudy%20Notes%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Over%201%2C500%20pages%20worth%20of%20study%20notes%20covering%20each%20learning%20objective%20one-by-one%22%7D%2C%7B%22text%22%3A%22%3Cb%3EQuestion%20Bank%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Vignette-style%20practice%20questions%20organized%20by%20topic%22%7D%2C%7B%22text%22%3A%22%3Cb%3EPrintable%20Mock%20Exams%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%222%20a.m.%20and%202%20p.m.%20mock%20exams%22%7D%2C%7B%22text%22%3A%22%3Cb%3EPerformance%20Tracking%20Tools%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22View%20your%20performance%20in%20attractive%20charts%22%7D%2C%7B%22text%22%3A%22%3Cb%3E5%26nbsp%3BAsk-a-tutor%20Questions%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Ask%20five%20questions%20to%20our%20tutors%20via%20live%20chat%22%7D%5D%2C%22price%22%3A%22399%22%2C%22priceCurrency%22%3A%22USD%22%2C%22button%22%3A%22Buy%20Now%22%2C%22buttonLink%22%3A%22https%3A%5C%2F%5C%2Fanalystprep.com%5C%2Fshop%5C%2Fpractice-package-for-frm-part-ii-2%5C%2F%22%2C%22buttonCaption%22%3A%22%22%2C%22mainColor%22%3A%22rgb%2851%2C%20129%2C%20234%29%22%2C%22priceOptions%22%3A%5B%5D%2C%22buttonOptions%22%3A%5B%5D%2C%22isFeatured%22%3Atrue%2C%22contentDivider%22%3Anull%2C%22pricePrefix%22%3A%22%22%2C%22pricePostfix%22%3A%22%22%2C%22priceCaption%22%3A%22for%20a%2012-month%20access%22%2C%22buttonTargetBlank%22%3Afalse%2C%22picture%22%3A%22https%3A%5C%2F%5C%2Felfsight.com%5C%2Fassets%5C%2Fpricing-table%5C%2Fexample-grid-2.jpg%22%2C%22ribbonText%22%3A%22Most%20Popular%22%2C%22ribbonBackgroundColor%22%3A%22%22%2C%22ribbonTextColor%22%3A%22%22%7D%2C%7B%22title%22%3A%22Unlimited%20Package%22%2C%22titleCaption%22%3A%22for%20all%20three%20Levels%20of%20the%20CFA%5Cu00ae%20Exam%20offered%20by%20AnalystPrep%22%2C%22features%22%3A%5B%7B%22text%22%3A%22Video%20Lessons%20for%20All%20Three%20Levels%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Over%20100%20hours%20of%20video%20lessons%20by%20Prof.%20James%20Forjan%2C%20CFA%2C%20PhD%22%7D%2C%7B%22text%22%3A%22Study%20Notes%20for%20All%20Three%20Levels%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Study%20notes%20for%20each%20curriculum%20reading%2C%20organized%20by%20learning%20objective%22%7D%2C%7B%22text%22%3A%22Question%20Banks%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22All%20of%20our%20question%20banks%20are%20updated%20each%20year%20automatically%20on%20the%20platform%20to%20reflect%20curriculum%20updates.%20Those%20include%20vignette-type%20questions%20for%20level%20II%20and%20III%20and%20essay-type%20questions%20for%20the%20level%20III%20a.m.%20session.%22%7D%2C%7B%22text%22%3A%22Mock%20Exams%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Over%2010%20mock%20exams%20for%20all%20three%20levels.%20Also%20available%20in%20Printable%20PDFs.%22%7D%2C%7B%22text%22%3A%22Performance%20Tracking%20Tools%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22View%20your%20performance%20in%20attractive%20charts%22%7D%2C%7B%22text%22%3A%22Unlimited%26nbsp%3BAsk-a-tutor%20Questions%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Ask%20questions%20to%20our%20tutors%20via%20live%20chat%22%7D%5D%2C%22price%22%3A%22699%22%2C%22priceCurrency%22%3A%22USD%22%2C%22button%22%3A%22Buy%20Now%22%2C%22buttonLink%22%3A%22https%3A%5C%2F%5C%2Fanalystprep.com%5C%2Fshop%5C%2Funlimited-package-for-frm-part-i-part-ii%5C%2F%22%2C%22buttonCaption%22%3A%22%22%2C%22mainColor%22%3A%22rgb%2851%2C%20129%2C%20234%29%22%2C%22priceOptions%22%3A%5B%5D%2C%22buttonOptions%22%3A%5B%5D%2C%22isFeatured%22%3Afalse%2C%22contentDivider%22%3Anull%2C%22pricePrefix%22%3A%22%22%2C%22pricePostfix%22%3A%22%22%2C%22priceCaption%22%3A%22for%20lifetime%20access%22%2C%22buttonTargetBlank%22%3Afalse%2C%22picture%22%3A%22https%3A%5C%2F%5C%2Felfsight.com%5C%2Fassets%5C%2Fpricing-table%5C%2Fexample-grid-3.jpg%22%2C%22ribbonText%22%3A%22%22%2C%22ribbonBackgroundColor%22%3A%22%22%2C%22ribbonTextColor%22%3A%22%22%7D%5D%2C%22elements%22%3A%5B%7B%22complexGroup%22%3A%22picture%22%2C%22name%22%3A%22Picture%22%2C%22pictureEnable%22%3Atrue%7D%2C%7B%22complexGroup%22%3A%22title%22%2C%22name%22%3A%22Title%22%2C%22titleEnable%22%3Atrue%2C%22titleCaptionColor%22%3A%22%22%2C%22titleTextColor%22%3A%22%22%2C%22titleFontSize%22%3A24%2C%22titleFontWeight%22%3A400%7D%2C%7B%22complexGroup%22%3A%22features%22%2C%22name%22%3A%22Features%22%2C%22featuresEnable%22%3Atrue%2C%22featuresStyle%22%3A%22striped%22%2C%22featuresIconColor%22%3A%22%22%2C%22featuresTextColor%22%3A%22%22%2C%22featuresAlign%22%3A%22center%22%2C%22featuresFontSize%22%3A13%7D%2C%7B%22complexGroup%22%3A%22price%22%2C%22name%22%3A%22Price%22%2C%22priceEnable%22%3Atrue%2C%22priceCaptionColor%22%3A%22%22%2C%22priceTextColor%22%3A%22%22%2C%22priceFontSize%22%3A32%2C%22priceFontWeight%22%3A600%7D%2C%7B%22complexGroup%22%3A%22button%22%2C%22name%22%3A%22Button%22%2C%22buttonEnable%22%3Atrue%2C%22buttonType%22%3A%22outline%22%2C%22buttonColor%22%3A%22%22%2C%22buttonTextColor%22%3A%22%22%2C%22buttonSize%22%3A%22medium%22%2C%22buttonCaptionColor%22%3A%22%22%7D%5D%2C%22widgetId%22%3A%223%22%7D\" \n                data-elfsight-pricing-table-version=\"2.6.1\"\n                data-elfsight-widget-id=\"elfsight-pricing-table-3\">\n            <\/div>\n            \r\n\r\n","protected":false},"excerpt":{"rendered":"<p>Backward induction involves working backward from maturity to time 0 to determine a bond\u2019s value at each node. It makes the following assumptions: The value of a bond is known at maturity (Final coupon payment + Face value). The value&#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-17444","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 v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Backward Induction - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"Backward induction involves working backward from maturity to time 0 to determine a bond\u2019s value at each node.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Backward Induction - CFA, FRM, and Actuarial Exams Study Notes\" \/>\n<meta property=\"og:description\" content=\"Backward induction involves working backward from maturity to time 0 to determine a bond\u2019s value at each node.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/\" \/>\n<meta property=\"og:site_name\" content=\"CFA, FRM, and Actuarial Exams Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2021-07-09T22:33:10+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-04-06T11:26:57+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1590\" \/>\n\t<meta property=\"og:image:height\" content=\"1521\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"AnalystPrep Team\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"AnalystPrep Team\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/\"},\"author\":{\"name\":\"AnalystPrep Team\",\"@id\":\"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e\"},\"headline\":\"Backward Induction\",\"datePublished\":\"2021-07-09T22:33:10+00:00\",\"dateModified\":\"2024-04-06T11:26:57+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/\"},\"wordCount\":386,\"image\":{\"@id\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg\",\"articleSection\":[\"CFA Level II Study Notes\",\"Fixed Income\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/\",\"url\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/\",\"name\":\"Backward Induction - CFA, FRM, and Actuarial Exams Study Notes\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/study-notes\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg\",\"datePublished\":\"2021-07-09T22:33:10+00:00\",\"dateModified\":\"2024-04-06T11:26:57+00:00\",\"author\":{\"@id\":\"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e\"},\"description\":\"Backward induction involves working backward from maturity to time 0 to determine a bond\u2019s value at each node.\",\"breadcrumb\":{\"@id\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#primaryimage\",\"url\":\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg\",\"contentUrl\":\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg\",\"width\":1590,\"height\":1521,\"caption\":\"interest rate tree\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/analystprep.com\/study-notes\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Backward Induction\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/analystprep.com\/study-notes\/#website\",\"url\":\"https:\/\/analystprep.com\/study-notes\/\",\"name\":\"CFA, FRM, and Actuarial Exams Study Notes\",\"description\":\"Question Bank and Study Notes for the CFA, FRM, and Actuarial exams\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/analystprep.com\/study-notes\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e\",\"name\":\"AnalystPrep Team\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g\",\"caption\":\"AnalystPrep Team\"},\"url\":\"https:\/\/analystprep.com\/study-notes\/author\/analystprep-team\/\"}]}<\/script>\n<meta property=\"og:video\" content=\"https:\/\/www.youtube.com\/embed\/dqDjJ-eiqNI\" \/>\n<meta property=\"og:video:type\" content=\"text\/html\" \/>\n<meta property=\"og:video:duration\" content=\"3014\" \/>\n<meta property=\"og:video:width\" content=\"480\" \/>\n<meta property=\"og:video:height\" content=\"270\" \/>\n<meta property=\"ya:ovs:adult\" content=\"false\" \/>\n<meta property=\"ya:ovs:upload_date\" content=\"2021-07-09T22:33:10+00:00\" \/>\n<meta property=\"ya:ovs:allow_embed\" content=\"true\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Backward Induction - CFA, FRM, and Actuarial Exams Study Notes","description":"Backward induction involves working backward from maturity to time 0 to determine a bond\u2019s value at each node.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/","og_locale":"en_US","og_type":"article","og_title":"Backward Induction - CFA, FRM, and Actuarial Exams Study Notes","og_description":"Backward induction involves working backward from maturity to time 0 to determine a bond\u2019s value at each node.","og_url":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/","og_site_name":"CFA, FRM, and Actuarial Exams Study Notes","article_published_time":"2021-07-09T22:33:10+00:00","article_modified_time":"2024-04-06T11:26:57+00:00","og_image":[{"width":1590,"height":1521,"url":"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg","type":"image\/jpeg"}],"author":"AnalystPrep Team","twitter_card":"summary_large_image","twitter_misc":{"Written by":"AnalystPrep Team","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/"},"author":{"name":"AnalystPrep Team","@id":"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e"},"headline":"Backward Induction","datePublished":"2021-07-09T22:33:10+00:00","dateModified":"2024-04-06T11:26:57+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/"},"wordCount":386,"image":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#primaryimage"},"thumbnailUrl":"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg","articleSection":["CFA Level II Study Notes","Fixed Income"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/","url":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/","name":"Backward Induction - CFA, FRM, and Actuarial Exams Study Notes","isPartOf":{"@id":"https:\/\/analystprep.com\/study-notes\/#website"},"primaryImageOfPage":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#primaryimage"},"image":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#primaryimage"},"thumbnailUrl":"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg","datePublished":"2021-07-09T22:33:10+00:00","dateModified":"2024-04-06T11:26:57+00:00","author":{"@id":"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e"},"description":"Backward induction involves working backward from maturity to time 0 to determine a bond\u2019s value at each node.","breadcrumb":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#primaryimage","url":"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg","contentUrl":"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/interest-rate-tree.jpg","width":1590,"height":1521,"caption":"interest rate tree"},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-the-backward-induction-valuation-methodology-and-calculate-the-value-of-a-fixed-income-instrument-given-its-cash-flow-at-each-node\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/study-notes\/"},{"@type":"ListItem","position":2,"name":"Backward Induction"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/study-notes\/#website","url":"https:\/\/analystprep.com\/study-notes\/","name":"CFA, FRM, and Actuarial Exams Study Notes","description":"Question Bank and Study Notes for the CFA, FRM, and Actuarial exams","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/study-notes\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e","name":"AnalystPrep Team","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g","caption":"AnalystPrep Team"},"url":"https:\/\/analystprep.com\/study-notes\/author\/analystprep-team\/"}]},"og_video":"https:\/\/www.youtube.com\/embed\/dqDjJ-eiqNI","og_video_type":"text\/html","og_video_duration":"3014","og_video_width":"480","og_video_height":"270","ya_ovs_adult":"false","ya_ovs_upload_date":"2021-07-09T22:33:10+00:00","ya_ovs_allow_embed":"true"},"_links":{"self":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts\/17444","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/comments?post=17444"}],"version-history":[{"count":20,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts\/17444\/revisions"}],"predecessor-version":[{"id":37708,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts\/17444\/revisions\/37708"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/media?parent=17444"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/categories?post=17444"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/tags?post=17444"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}