{"id":17242,"date":"2021-07-08T23:48:41","date_gmt":"2021-07-08T23:48:41","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17242"},"modified":"2026-02-25T08:47:52","modified_gmt":"2026-02-25T08:47:52","slug":"explain-what-is-meant-by-arbitrage-free-valuation-of-a-fixed-income-instrument","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/explain-what-is-meant-by-arbitrage-free-valuation-of-a-fixed-income-instrument\/","title":{"rendered":"Arbitrage-Free Valuation of a Fixed Income Security"},"content":{"rendered":"<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"VideoObject\",\r\n  \"name\": \"The Arbitrage-Free Valuation Framework (2025 Level II CFA\u00ae Exam \u2013 Fixed Income \u2013 Module 2)\",\r\n  \"description\": \"CFA Level II Fixed Income lesson on the Arbitrage-Free Valuation Framework covering arbitrage-free valuation, law of one price, binomial interest rate trees, backward induction, tree calibration to a term structure, zero-coupon curve pricing versus lattice pricing, pathwise valuation, Monte Carlo simulation, and term structure models including Vasicek, CIR, and Ho-Lee.\",\r\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/dqDjJ-eiqNI\/maxresdefault.jpg\",\r\n  \"uploadDate\": \"2024-03-24\",\r\n  \"duration\": \"PT50M13S\",\r\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=dqDjJ-eiqNI\",\r\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/dqDjJ-eiqNI\"\r\n}\r\n<\/script>\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/dqDjJ-eiqNI\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<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\": \"Is there an arbitrage opportunity for the 8% annual coupon bond sold in Montreal and Ottawa?\",\r\n    \"text\": \"The yield of an 8% annual coupon bond that matures in five years is 5.2% in Montreal. The same bond sells for CAD 112.06 per CAD 100 face value in Ottawa.\\n\\nWhich of the following statements is most likely correct?\\n\\nThere is no arbitrage opportunity.\\n\\nAn arbitrage opportunity can be utilized by selling the bond in Montreal and buying in Ottawa.\\n\\nAn arbitrage opportunity can be exploited by buying the bond in Montreal and selling it in Ottawa.\",\r\n    \"answerCount\": 3,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"There is no arbitrage opportunity.\",\r\n      \"commentCount\": 0,\r\n      \"upvoteCount\": 0,\r\n      \"url\": \"#acceptedAnswer\"\r\n    },\r\n    \"suggestedAnswer\": [\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"An arbitrage opportunity can be utilized by selling the bond in Montreal and buying in Ottawa.\",\r\n        \"commentCount\": 0,\r\n        \"upvoteCount\": 0,\r\n        \"url\": \"#suggestedAnswerA\"\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"An arbitrage opportunity can be exploited by buying the bond in Montreal and selling it in Ottawa.\",\r\n        \"commentCount\": 0,\r\n        \"upvoteCount\": 0,\r\n        \"url\": \"#suggestedAnswerB\"\r\n      }\r\n    ]\r\n  }\r\n}\r\n<\/script>\r\n\r\n\r\n<p><em><strong>Arbitrage free valuation<\/strong><\/em> is an approach that determines bond values based on the assumption that arbitrage opportunities do not exist. The arbitrage-free valuation model is based on the <em><strong>law of one price<\/strong><\/em>, which states that two goods, which are perfect substitutes, must have equivalent market prices in the absence of transaction costs; otherwise, an <em><strong>arbitrage opportunity<\/strong><\/em> would exist.<\/p>\r\n<p>An arbitrage opportunity is a deal that involves no cash outlay but generates risk-free profits.<\/p>\r\n<h2>Types of Arbitrage Opportunities<\/h2>\r\n<p>There are two types of arbitrage opportunities: <em><strong>value additivity<\/strong><\/em> and <em><strong>dominance<\/strong><\/em>.<\/p>\r\n<h3>Value Additivity<\/h3>\r\n<p>The concept of<em><strong> value additivity<\/strong><\/em> suggests that the value of the whole can differ from the value obtained by adding the values of parts. For example, the value of a bond could be less than the value of the sum of its individual cash flows. In these circumstances, an arbitrage opportunity exists because a trader could buy the bond and at the same time sell<span id=\"t1f_12\" class=\"t s4_12\"> claims to the individual cash flows.\u00a0<\/span><\/p>\r\n<div style=\"text-align:center; margin:22px 0;\">\r\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\"\r\n     style=\"display:inline-flex; align-items:center; justify-content:center; padding:12px 24px; border:2px solid #1a73e8; border-radius:999px; color:#1a73e8; font-weight:600; text-decoration:none;\">\r\n    Analyze arbitrage free valuation using cash flow replication\r\n  <\/a>\r\n<\/div>\r\n<h4>Example: Value Additivity<\/h4>\r\n<p>The following information relates to Asset A and Portfolio B.<\/p>\r\n<p>$$ \\begin{array}{c|c|c} &amp; \\textbf{Price Today} &amp; \\textbf{Payoff in One Year} \\\\ \\hline \\text{Asset A} &amp; $0.9346 &amp; $1 \\\\ \\hline \\text{Portfolio B} &amp; $93 &amp; $107 \\end{array} $$<\/p>\r\n<p>Asset A is a risk-free discount bond that pays off $1 in one year and is priced at $0.9346 today (1\/1.07).<\/p>\r\n<p>Portfolio B consists of 107 units of Asset A with a payoff of $107 in a year and the price of the asset today is $93.<\/p>\r\n<p>In this case, a dealer can make a riskless profit by:<\/p>\r\n<ul>\r\n\t<li>Selling $107 units of Asset A for \\(107\\times$0.9346=$100\\).<\/li>\r\n\t<li>Buying Portfolio B at $93.<\/li>\r\n\t<li><strong>Arbitrage profit \\(= $100-$93=$7\\).<\/strong><\/li>\r\n<\/ul>\r\n<h3>Dominance<\/h3>\r\n<p><em><strong>Dominance<\/strong> <\/em>is a situation where one security consistently offers a better yield than another security, despite both having the <em>same characteristics<\/em>, including the buying price. <span id=\"t1r_12\" class=\"t s4_12\">A security X is said to be dominant over <\/span><span id=\"t1s_12\" class=\"t s4_12\">another security Y when both can be purchased at the same <\/span><span id=\"t1t_12\" class=\"t s4_12\">price at t = 0, but X will yield a <\/span><span id=\"t1u_12\" class=\"t s4_12\">higher return in every state of the world.\u00a0<\/span><\/p>\r\n<p>To make a riskless profit when dominance is present, <span id=\"t1v_12\" class=\"t s4_12\">an investor could buy the<\/span><span id=\"t1w_12\" class=\"t s4_12\"> dominant security and sell the <\/span><span id=\"t1x_12\" class=\"t s4_12\">dominated security.\u00a0<\/span><\/p>\r\n<h4>Example: Dominance<\/h4>\r\n<p>Consider two risk-free zero-coupon bonds A and B that mature one year from today.<\/p>\r\n<p>$$ \\begin{array}{c|c|c} \\textbf{Asset} &amp; \\textbf{Price Today} &amp; \\textbf{Payoff in One Year} \\\\ \\hline A &amp; 100 &amp; 105 \\\\ \\hline B &amp; 300 &amp; 330 \\end{array} $$<\/p>\r\n<p>Since both assets are risk-free, they have the same discount factor. Therefore, a dealer can generate arbitrage profit by:<\/p>\r\n<ul>\r\n\t<li>Selling three units of A (Proceeds = $300).<\/li>\r\n\t<li>Purchasing one unit of B with the sell proceeds for $300.\r\n\r\n<ul>\r\n\t<li>This investment requires <strong>zero cash outlay today<\/strong>.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p>At maturity:<\/p>\r\n<ul>\r\n\t<li>Asset B will generate a cash inflow of $330.<\/li>\r\n\t<li>The two units of Asset A sold will produce a cash outflow of $315.<\/li>\r\n\t<li><strong>Arbitrage profit generated = \\($15 ($330-$315)\\).<\/strong><\/li>\r\n<\/ul>\r\n<h2>Implications of Arbitrage-Free Valuation<\/h2>\r\n<h3>Stripping and Reconstitution<\/h3>\r\n<p>The arbitrage-free model values a bond, assuming that it can be converted into a series of zero-coupon bonds. This gives rise to two possibilities: stripping and reconstitution.<\/p>\r\n<p><em><strong>Stripping<\/strong> <\/em>is a process where periodic coupon payments of an existing security are converted into tradeable zero-coupon securities. The strips can be traded in the market at a discount and are redeemed at face value. For example, a \u00a0three-year 2% coupon Treasury issue could be viewed as a package of seven zero-coupon instruments (6 semiannual coupon payments, one of which is made at maturity, and one principal value payment at maturity).<\/p>\r\n<p><em><strong>Reconstitution<\/strong> <\/em>is the reverse of stripping, where the coupon strips and principal strips are reassembled into the original security.<\/p>\r\n<h4>Example: Stripping and Reconstitution<\/h4>\r\n<p>A three-year bond with a 10% annual coupon and $1,000 face value is priced at $1,000. The Treasury spot rates are given in the following table:<\/p>\r\n<p>$$ \\begin{array}{c|c} \\textbf{Year} &amp; \\textbf{Spot Rates} \\\\ \\hline 1 &amp; 7\\% \\\\ \\hline 2 &amp; 8\\% \\\\ \\hline 3 &amp; 7\\% \\end{array} $$<\/p>\r\n<p>The discounted price of this bond is:<\/p>\r\n<p>$$ \\frac{100}{1.07}+\\frac{100}{{1.08}^2}+\\frac{1,100}{\\left(1.07\\right)^3}= 1,077.12 $$<\/p>\r\n<p>Notice from above that a dealer can:<\/p>\r\n<ul>\r\n\t<li>Buy the security at the current market price of $1,000.<\/li>\r\n\t<li>Strip the security and earn a value of $1,077.12 using Treasury spot rates.<\/li>\r\n\t<li>There exists an arbitrage opportunity to make a riskless profit of $77.12 ($1,077.12-$1,000).<\/li>\r\n<\/ul>\r\n<p>If the price of the bond were $1,100 rather than $1,000, the dealer would:<\/p>\r\n<ul>\r\n\t<li>Buy up the Treasury strips for a value of $1,077.12.<\/li>\r\n\t<li>Repackage as a synthetic Treasury security for $1,100.<\/li>\r\n\t<li>This process, known as reconstitution, provides the dealer with a profit of $22.88 (profit from reconstitution = \\($1,100 &#8211; $1,077.12 = $22.88\\)).<\/li>\r\n\t<li>Arbitragers would either bid prices up or down until such arbitrage profits could no longer be earned.<\/li>\r\n<\/ul>\r\n<h3>Key Takeaways<\/h3>\r\n<p>Arbitrage free valuation:<\/p>\r\n<ul>\r\n\t<li>Maintains the <strong>value additivity<\/strong> principle.\r\n\r\n<ul>\r\n\t<li>Stripping and reselling bonds.<\/li>\r\n\t<li>Reconstitution of bonds.<\/li>\r\n\t<li>The value of strips and reconstituted bonds should be the same; otherwise, arbitrage opportunities would exist.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Eliminates <strong>dominance<\/strong>.\r\n\r\n<ul>\r\n\t<li>One bond cannot be priced more attractively as compared to an otherwise identical bond.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>The yield of an 8% annual coupon bond that matures in five years is 5.2% in Montreal. The same bond sells for CAD 112.06 per CAD 100 face value in Ottawa.<\/p>\r\n<p>Which of the following statements is <em>most likely<\/em> correct?<\/p>\r\n<ol type=\"A\">\r\n\t<li>There is no arbitrage opportunity.<\/li>\r\n\t<li>An arbitrage opportunity can be utilized by selling the bond in Montreal and buying in Ottawa.<\/li>\r\n\t<li>An arbitrage opportunity can be exploited by buying the bond in Montreal and selling it in Ottawa.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is A.<\/strong><\/p>\r\n<p>The price for the bond in Montreal is determined as follows:<\/p>\r\n<p>$$ P=\\frac{8}{1.052}+\\frac{8}{{1.052}^2}+\\frac{8}{{1.052}^3}+\\frac{8}{{1.052}^4}+\\frac{108}{{1.052}^5}=$112.06 $$<\/p>\r\n<p>The price in Montreal matches the price in Ottawa. Therefore, no arbitrage opportunity exists in the market.\u2003<\/p>\r\n<\/blockquote>\r\n<p>Reading 29: The Arbitrage-Free Valuation Framework<\/p>\r\n<p><em>LOS 29 (a) Explain what is meant by arbitrage-free valuation of a fixed-income instrument.<\/em><\/p>\r\n\n            <div \n                class=\"elfsight-widget-pricing-table elfsight-widget\" \n                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\n                data-elfsight-pricing-table-version=\"2.6.1\"\n                data-elfsight-widget-id=\"elfsight-pricing-table-3\">\n            <\/div>\n            \r\n<p>\r\n<\/p>\r\n<div style=\"text-align:center; margin:40px 0 10px 0;\">\r\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\"\r\n     style=\"display:inline-block; padding:14px 34px; background:#1a73e8; color:#ffffff; border-radius:999px; font-weight:700; text-decoration:none;\">\r\n    Start Free Trial\r\n  <\/a>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Arbitrage free valuation is an approach that determines bond values based on the assumption that arbitrage opportunities do not exist. The arbitrage-free valuation model is based on the law of one price, which states that two goods, which are perfect&#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-17242","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>Arbitrage-Free Valuation of Fixed Income | CFA Level II<\/title>\n<meta name=\"description\" content=\"Explains arbitrage-free valuation of fixed income securities, including value additivity, dominance principles, and bond stripping and reconstitution.\" \/>\n<meta name=\"robots\" content=\"index, follow, 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