{"id":17331,"date":"2021-07-09T14:24:53","date_gmt":"2021-07-09T14:24:53","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17331"},"modified":"2024-04-06T11:19:56","modified_gmt":"2024-04-06T11:19:56","slug":"calculate-the-arbitrage-free-value-of-an-option-free-fixed-rate-coupon-bond","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/calculate-the-arbitrage-free-value-of-an-option-free-fixed-rate-coupon-bond\/","title":{"rendered":"Arbitrage Free Value"},"content":{"rendered":"<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/dqDjJ-eiqNI\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n\r\n<p>Bonds can be valued either using the traditional valuation approach or the arbitrage-free valuation approach.<\/p>\r\n<p>Under the <em><strong>traditional valuation approach<\/strong><\/em>, a <i>single interest rate<\/i> is used to<span style=\"font-size: 1rem;\">\u00a0discount all of a bond&#8217;s cash flows. In this approach, all cash flows of a bond are considered the same, regardless of their timing. In other words, each cash flow is viewed as just a token from the same package, and therefore the <strong>same discount rate<\/strong> is applied to all cash flows. This is a fundamental flaw because each individual cash flow of a bond is always unique. The use of a single discount rate in valuation may result in mispricing, thereby creating arbitrage opportunities.\u00a0<\/span><\/p>\r\n<p>Under the <em><strong>arbitrage-free valuation approach<\/strong><\/em>, each cash flow is discounted at its <em>own discount rate<\/em> that takes into account the shape of the yield curve and the timing of the cash flow.<\/p>\r\n<p>Perhaps an example will help illustrate. Consider a three-year U.S. Treasury note with a 10% coupon rate, paid semiannually. Considering the cash flows per $100 of par value, we would have six payments of $3 (one payment after every six months) and a final principal payment of $100.<\/p>\r\n<ul>\r\n\t<li>The traditional valuation approach would discount each of these cash flows using the same discount rate without considering their timing nor the shape of the yield curve.\u00a0<\/li>\r\n\t<li>The arbitrage-free valuation approach would view each cash flow as a zero-coupon instrument maturing on the date the cash flow is received. The three-year Treasury note would be viewed as a package of seven zero-coupon instruments maturing over a three-year period at six-month intervals. This would deny market participants an opportunity to realize an arbitrage profit by \u201cstripping\u201d the bond and selling the individual cash flows at a higher aggregate value than it would cost to purchase the Treasury in the market.<\/li>\r\n<\/ul>\r\n<h2>Arbitrage-Free Valuation of an Option-Free, Fixed-Rate Coupon Bond<\/h2>\r\n<p>An arbitrage-free value is the present value of expected future values using Treasury spot rates for option-free bonds. Arbitrage-free valuation usually involves three main steps:<\/p>\r\n<ul>\r\n\t<li><em><strong>Step 1<\/strong>:<\/em> Estimate the future cash flows.<\/li>\r\n\t<li><em><strong>Step 2<\/strong>:<\/em> Determine the appropriate discount rates that should<br \/>\r\nbe used to discount each cash flow.<\/li>\r\n\t<li><em><strong>Step 3<\/strong>:<\/em> Calculate the present value of the expected future cash flows by applying the appropriate discount rates determined in Step 2.<\/li>\r\n<\/ul>\r\n<p>Thus, the following formula is used:<\/p>\r\n<p>$$ PV=\\frac{PMT}{\\left(1+S_1\\right)^1}+\\frac{PMT}{\\left(1+S_2\\right)^2}+\\ldots+\\frac{PMT+FV}{\\left(1+S_n\\right)^N} $$<\/p>\r\n<p>Where:<\/p>\r\n<p>\\(PMT\\) is the periodic coupon.<\/p>\r\n<p>\\(FV\\) is the face value.<\/p>\r\n<p>\\(S_1\\), \\(S_2,\\) and \\(S_N\\) are the spot rates for periods 1 to \\(N\\).<\/p>\r\n<h4>Example: Arbitrage-Free Valuation<\/h4>\r\n<p>An 8% semi-annual coupon bond is priced at $800. It has a remaining term to maturity of 2 years. Given the following benchmark spot rates, the value of the bond, if its face value is $1,000, is <em>closest<\/em> to:<\/p>\r\n<p>$$ \\begin{array}{c|c} \\textbf{Year} &amp; \\textbf{Spot Rates} \\\\ \\hline 0.5 &amp; 16\\% \\\\ \\hline 1 &amp; 17\\% \\\\ \\hline 1.5 &amp; 16\\% \\\\ \\hline 2 &amp; 15\\% \\end{array} $$<\/p>\r\n<p><strong>Solution<\/strong><\/p>\r\n<p>$$ \\text{Present value} =\\frac{40}{1.08}+\\frac{40}{{1.085}^2}+\\frac{40}{{1.08}^3}+\\frac{1,040}{{1.075}^4}=$881.52 $$<\/p>\r\n<p>Note that $40 is the semi-annual coupon.<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>An option free, 3-year 6% annual coupon bond priced at $100 has similar liquidity and risk to a Treasury bond whose par curve is shown in the table below.<\/p>\r\n<p>$$ \\textbf{Treasury Par Curve} \\\\ \\begin{array}{c|c} \\textbf{Term to Maturity (Years)} &amp; \\textbf{Par Rate} \\\\ \\hline 1 &amp; 3.00\\% \\\\ \\hline 2 &amp; 4.00\\% \\\\ \\hline 3 &amp; 5.00\\% \\end{array} $$<\/p>\r\n<p>The arbitrage-free price for the bond is <em>closest to<\/em>:<\/p>\r\n<ol type=\"A\">\r\n\t<li>$102.69.<\/li>\r\n\t<li>$102.76.<\/li>\r\n\t<li>$102.94.<\/li>\r\n<\/ol>\r\n<h4><strong>Solution<\/strong><\/h4>\r\n<p><strong>The correct answer is B.<\/strong><\/p>\r\n<p>We first calculate the implied one-year spot rates given the above term structure by bootstrapping.<\/p>\r\n<p>The one-year implied spot rate is 3%, as it is simply the one-year par yield.<\/p>\r\n<p>We can bootstrap the two-year implied spot rate, \\(r(2)\\), as follows:<\/p>\r\n<p>$$ \\begin{align*} 1 &amp;=\\frac{0.04}{1.03}+\\frac{(1+0.04)}{\\left(1+r\\left(2\\right)\\right)^2} \\\\ r\\left(2\\right)&amp;=4.02\\% \\end{align*} $$<\/p>\r\n<p>Similarly, the three-year spot rate can be bootstrapped by solving the equation:<\/p>\r\n<p>$$ \\begin{align*} 1 &amp;=\\frac{0.05}{1.03}+\\frac{0.05}{{1.0402}^2}+\\frac{1+0.05}{\\left(1+r\\left(3\\right)\\right)^3} \\\\ r\\left(3\\right) &amp;=5.07\\% \u2003\\end{align*} $$<\/p>\r\n<p>The implied spot rate curve:<\/p>\r\n<p>$$ \\begin{array}{c|c|c} \\textbf{Term to Maturity} &amp; \\textbf{Par Rate} &amp; \\textbf{Spot Rate} \\\\ \\hline 1 &amp; 3.00\\% &amp; 3.00\\% \\\\ \\hline 2 &amp; 4.00\\% &amp; 4.02\\% \\\\ \\hline 3 &amp; 5.00\\% &amp; 5.07\\% \\end{array} $$<\/p>\r\n<p>To calculate the arbitrage-free price, each cash flow is discounted using the same maturity spot rate as the date of the cash flow.<\/p>\r\n<p>$$ \\text{Arbitrage-free price } (P)=\\frac{6}{1.03}+\\frac{6}{{1.0402}^2}+\\frac{106}{{1.057}^3}=$102.76\u2003$$<\/p>\r\n<\/blockquote>\r\n<p>Reading 29: The Arbitrage-Free Valuation Framework<\/p>\r\n<p><em>LOS 29(b) Calculate the arbitrage-free value of an option-free, fixed-rate coupon bond.<\/em><\/p>\r\n\n            <div \n                class=\"elfsight-widget-pricing-table elfsight-widget\" \n                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Under the traditional valuation approach, a single interest rate is used to\u00a0discount all of a bond&#8217;s cash flows. In this approach, all cash flows of&#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-17331","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 v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Arbitrage Free Value - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"Learn the traditional valuation approach and the arbitrage-free valuation approach for bonds, including their differences and implications.\" \/>\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\/calculate-the-arbitrage-free-value-of-an-option-free-fixed-rate-coupon-bond\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Arbitrage Free Value - 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