{"id":17739,"date":"2021-07-14T22:27:29","date_gmt":"2021-07-14T22:27:29","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17739"},"modified":"2026-01-23T19:42:05","modified_gmt":"2026-01-23T19:42:05","slug":"calculate-the-value-of-a-bond-and-its-credit-spread-given-assumptions-about-the-credit-risk-parameters","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/calculate-the-value-of-a-bond-and-its-credit-spread-given-assumptions-about-the-credit-risk-parameters\/","title":{"rendered":"Value of a Bond and its Credit Spread"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Credit Valuation Adjustment (CVA) of a Corporate Bond\",\n    \"text\": \"A $1,000 par, 6% annual coupon corporate bond matures five years from today. The bond is currently priced with a credit spread of 150 bps over the benchmark par rate of 3%. The bond\u2019s CVA is closest to:\\n\\nA. $43.63\\n\\nB. $60.00\\n\\nC. $1,084.86\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"A. $43.63\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"B. $60.00\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"C. $1,084.86\"\n      }\n    ]\n  }\n}\n<\/script><br \/>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/value-of-the-bond-backward-induction\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-through-Backward-Induction-1024x784.jpg\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-through-Backward-Induction-1024x784.jpg\",\n  \"caption\": \"Value of the bond through backward induction\",\n  \"width\": 1024,\n  \"height\": 784,\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    \"url\": \"https:\/\/analystprep.com\/\"\n  }\n}\n<\/script><\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/pHEcwZjtzl8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The <em><strong>arbitrage-free framework<\/strong><\/em> is applied for credit analysis of a risky bond, assuming that interest rates are volatile. A <em><strong>binomial interest rate tree<\/strong><\/em> is constructed assuming no arbitrage. The tree is then verified if it has been correctly calibrated and used to value corporate bonds.<\/p>\n<h2>Fixed Coupon Corporate Bonds<\/h2>\n<p>A fixed coupon corporate bond can be evaluated using the binomial interest rate tree in the following steps:<\/p>\n<ul>\n<li>Calculate the value of the bond assuming no default (VND).<\/li>\n<li>Determine the credit valuation adjustment (CVA).<\/li>\n<li>Work out the bond&#8217;s fair value, where fair value = VND \u2013 CVA.<\/li>\n<li>Determine the yield to maturity (YTM) of the bond using fair value.<\/li>\n<li>Calculate the credit spread of the bond using YTM.<\/li>\n<li>Work out the credit spread, where credit spread = YTM of the risky bond \u2013 Benchmark YTM.<\/li>\n<\/ul>\n<h3>Example 1: Zero-volatility Benchmark Rates<\/h3>\n<p>Consider a four-year zero-coupon corporate bond with a par value of $1,000 and a flat government bond yield curve at 5%. The risk-neutral probability of default (hazard rate) for each date of the bond is 2%, and the recovery rate is 40%.<\/p>\n<ol style=\"list-style-type: lower-roman;\" type=\"A\">\n<li>The fair value bond is <em>closest<\/em> to:<\/li>\n<li>The YTM of the bond is <em>closest<\/em> to:<\/li>\n<li>The credit spread of the bond is <em>closest<\/em> to:<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<ol style=\"list-style-type: lower-roman;\" type=\"a\">\n<li><strong>Fair value<\/strong>\n<p>$$ \\begin{align*} \\text{Value of the bond assuming no default (VND)} &amp;= \\frac{1000}{\\left(1+0.05\\right)^4} \\\\ &amp; = 822.70 \\end{align*} $$<\/p>\n<p>Recall the calculation of CVA from a previous LOS.<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c|c|c|c} \\textbf{Year} &amp; \\textbf{EE} &amp; \\textbf{LGD} &amp; \\textbf{Hazard } &amp; \\textbf{PD} &amp; \\textbf{PS} &amp; \\textbf{EL} &amp; \\textbf{PV} \\\\ &amp; &amp; &amp; \\textbf{rate} &amp; &amp; &amp; &amp; \\textbf{of} \\\\ &amp; &amp; &amp; {} &amp; &amp; &amp; &amp; \\textbf{EL} \\\\ \\hline 1 &amp; 863.84 &amp; 518.30 &amp; 2\\% &amp; 2.0000\\% &amp; 98.000\\% &amp; 10.37 &amp; 9.8724 \\\\ \\hline 2 &amp; 907.03 &amp; 544.22 &amp; 2\\% &amp; 1.9600\\% &amp; 96.040\\% &amp; 10.67 &amp; 9.6750 \\\\ \\hline 3 &amp; 952.38 &amp; 571.43 &amp; 2\\% &amp; 1.9208\\% &amp; 94.119\\% &amp; 10.98 &amp; 9.4815 \\\\ \\hline 4 &amp; 1,000.00 &amp; 600.00 &amp; 2\\% &amp; 1.8824\\% &amp; 92.237\\% &amp; 11.29 &amp; 9.2919 \\\\ \\hline &amp; &amp; &amp; &amp; &amp; &amp; \\textbf{CVA} &amp; \\bf{38.321} \\end{array} $$<\/p>\n<ul>\n<li>EE = Expected exposure<\/li>\n<li>LGD = Loss given default<\/li>\n<li>PD = Probability of default<\/li>\n<li>PS = Probability of survival<\/li>\n<li>EL = Expected loss<\/li>\n<\/ul>\n<p>$$ \\begin{align*} \\text{Fair value} &amp; =\\text{VND} -\\text{CVA} \\\\\u00a0 &amp; = 822.70 \u2013 38.32 =$784.38 \\end{align*} $$<\/p>\n<\/li>\n<li><strong>YTM<\/strong>\n<p>YTM of the bond can be determined as the rate that solves the equation:<\/p>\n<p>$$ \\begin{align*} 784.38 &amp; =\\frac{1000}{\\left(1+\\text{YTM}\\right)^4} \\ \\text{YTM} &amp; = 6.26\\% \\end{align*} $$<\/p>\n<\/li>\n<li><strong>Credit spread<\/strong>\n<p>$$ \\begin{align*} \\text{Credit spread} &amp; = \\text{YTM of the risky bond}\\ \u2013 \\text{Benchmark YTM} \\\\\u00a0 &amp; = 6.26\\% &#8211; 5\\% = 1.26\\% \\end{align*} $$<\/p>\n<p>Thus, the compensation for credit risk received by the investor can be expressed in terms of:<\/p>\n<ul>\n<li>The CVA, 38.32, which is the present value per 100 of par value on today.<\/li>\n<li>A credit spread of 126 basis points, which is an annual percentage rate for four years. \u2003<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3>Example 2: Volatility Assumption<\/h3>\n<p>In the previous example, we have determined the fair value and the credit spread of a risky bond, assuming a flat benchmark yield curve. This example will use the binomial interest rate tree to calculate the value of the bond assuming no default (VND) and the expected exposure in a volatile interest rate environment.<\/p>\n<p>Given the spot rate curve for the annual payment benchmark Treasury, we can derive the discount factors and forward rates as per the following table:<\/p>\n<p>$$ \\begin{array}{c|c|c|c} \\textbf{Maturity} &amp; \\textbf{Spot Rates} &amp; \\textbf{Discount Factors (DF)} &amp; \\textbf{Forward Rates} \\\\ \\hline 1 &amp; 4.00\\% &amp; 0.96154 &amp; 4.000\\% \\\\ \\hline 2 &amp; 5.00\\% &amp; 0.90703 &amp; 6.010\\% \\\\ \\hline 3 &amp; 6.00\\% &amp; 0.83962 &amp; 8.029\\% \\end{array} $$<\/p>\n<p>Using the above forward rates, an iterative process has been used to generate the following binomial tree, assuming an interest rate volatility of 15%.<\/p>\n<p>$$ \\begin{array}{ccc} \\textbf{Time 0} &amp; \\textbf{Time 1} &amp; \\textbf{Time 2} \\\\\u00a0 \u00a0&amp; &amp; 10.69\\% \\\\\u00a0 &amp; 6.89\\% &amp; \\\\\u00a0 \u00a04.000\\% &amp; &amp; 7.86\\% \\\\\u00a0 \u00a0&amp; 5.12\\% &amp; \\\\\u00a0 \u00a0&amp; &amp; 5.90\\% \\end{array} $$<\/p>\n<p>Consider a 5%, three-year corporate bond with a par value of $1,000. The risk-neutral probability of default for this bond has been estimated as 2%, and the recovery rate as 40%<\/p>\n<p>We can determine the fair value of the bond as follows:<\/p>\n<p>The first step is to determine the value of the bond, assuming no default (VND). This is done through backward induction.<\/p>\n<p><em><strong><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter size-full wp-image-26608\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-through-Backward-Induction.jpg\" alt=\"Value of the Bond through Backward Induction\" width=\"1590\" height=\"1217\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-through-Backward-Induction.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-through-Backward-Induction-300x230.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-through-Backward-Induction-1024x784.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-through-Backward-Induction-768x588.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-through-Backward-Induction-1536x1176.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Value-of-the-Bond-through-Backward-Induction-400x306.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>N\/B:<\/strong> <\/em>The values in the above tree are subject to rounding errors<\/p>\n<p>Year 3 cashflows are the principal (1,000) plus the coupon of 50.<\/p>\n<p>Year 2 values for the bond are obtained as follows:<\/p>\n<p>$$ \\begin{align*} \\frac{1050}{1.1069} &amp;=948.59 \\\\ \\\\ \\frac{1050}{1.0786} &amp;=973.48 \\\\ \\frac{1050}{1.0590} &amp; =991.50 \\end{align*} $$<\/p>\n<p>Year 1 values are obtained as follows:<\/p>\n<p>$$ \\begin{align*} \\frac{1}{2}\\times\\frac{\\left(948.59+50\\right)+\\left(973.48+50\\right)}{1.0689} &amp; =945.86 \\\\ \\\\ \\frac{1}{2}\\times\\frac{\\left(973.48+50\\right)+\\left(991.50+50\\right)}{1.0512} &amp;= 982.20 \\end{align*} $$<\/p>\n<p>Year 0 Value (VND) are obtained as follows:<\/p>\n<p>$$ \\frac{1}{2}\\times\\frac{\\left(945.86+50\\right)+\\left(982.20+50\\right)}{1.04}=975.03 $$<\/p>\n<p>VND can also be determined by discounting the bond&#8217;s yearly cashflows at the relevant spot rates:<\/p>\n<p>$$ VND =\\frac{50}{1.04}+\\frac{50}{\\left(1.05\\right)^2}+\\frac{1,050}{\\left(1.06\\right)^3}=$975.03 $$<\/p>\n<p>However, the binomial tree is key for calculating the expected exposure at each node when computing the CVA.<\/p>\n<p>$$ \\begin{align*} &#038; \\text{Expected exposure at each year} \\\\ &#038; =\\sum{\\text{Value in node i at time t}\\times \\text{Probability}}+\\text{Coupon for year t} \\end{align*} $$<\/p>\n<p>The next step is to calculate CVA.<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c|c|c|c} \\textbf{Year} &amp; \\textbf{EE} &amp; \\textbf{LGD} &amp; \\textbf{PD} &amp; \\textbf{PS} &amp; \\textbf{EL} &amp; \\textbf{Discount} &amp; \\textbf{PV of} \\\\ &amp; &amp; &amp; &amp; &amp; &amp; \\textbf{Factors} &amp; \\textbf{EL} \\\\ \\hline 1 &amp; 1,014.03 &amp; 608.42 &amp; 2.000\\% &amp; 98.00\\% &amp; 12.17 &amp; 0.9615 &amp; 11.70 \\\\ \\hline 2 &amp; 1,021.76 &amp; 613.05 &amp; 1.960\\% &amp; 96.04\\% &amp; 12.02 &amp; 0.9070 &amp; 10.90 \\\\ \\hline 3 &amp; 1,050.00 &amp; 630.00 &amp; 1.921\\% &amp; 94.12\\% &amp; 12.10 &amp; 0.8396 &amp; 10.16 \\\\ \\hline &amp; &amp; &amp; &amp; &amp; &amp; \\textbf{CVA} &amp; \\bf{32.76} \\end{array} $$<\/p>\n<p>$$ \\begin{align*} &amp; {\\text{Expected exposure for each year}} \\\\ &amp; = \\sum{\\text{(Value at each node }}\\times {\\text{Probability}}) +{\\text{Coupon}} \\end{align*} $$<\/p>\n<p>$$ \\begin{align*} \\text{Expected exposure for year one} &amp; =\\left(945.88\\times0.5\\right)+\\left( 982.18\\times0.50\\right) \\\\ &amp; +50 \\\\ &amp; =1,014.03 \\\\ \\text{Expected exposure for year two} &amp; = (948.57\\times0.25)+\\left( 973.48\\times0.5\\right)\\\\ &amp; +\\left(991.51\\times0.25\\right) +50 \\\\ &amp; =1,021.76 \\\\ \\text{Expected exposure for year three} &amp; = 1,050 \\end{align*} $$<\/p>\n<p>$$ \\begin{align*} \\text{Loss given default (LGD)} &amp; = \\left(1-\\text{Recovery rate}\\right)\\times \\text{Expected exposure} \\\\ LGD_1 &amp; = 1,014.03\\times\\left(1-0.40\\right)=608.42 \\\\ LGD_2 &amp;= 1,021.76\\times\\left(1-0.40\\right)=613.05 \\\\ LGD_3 &amp;=1,050\\times\\left(1-0.40\\right)=630.00 \\end{align*} $$<\/p>\n<p>The probability of default is calculated using the formula:<\/p>\n<p>$$ PD_t=PS_{t-1}\\times \\text{Hazard rate} $$<\/p>\n<p>Where:<\/p>\n<p>$$ \\begin{align*} \\text{Probability of survival } (PS_t) &amp;= 1- {\\text{Cumulative conditional} \\\\ \\text{probability of default}} \\\\ \\text{Expected loss} &amp; = LGD\\times PD \\\\ \\text{PV of expected loss} &amp;=LGD\\times\\frac{PD}{\\left(1+i\\right)^t} \\\\ \\end{align*} $$<\/p>\n<p>CVA = sum of the present value of the expected loss for each period.<\/p>\n<p>CVA = $32.76<\/p>\n<p>VND = $975.03<\/p>\n<p>$$ \\begin{align*} \\text{The fair value of the bond} &amp; = VND\\ &#8211; CVA \\\\ \\text{Fair value}&amp;= 975.03-32.76=$942.27 \\end{align*} $$<\/p>\n<p>Note that changes in the interest rate volatility have minimal effect on a corporate bond&#8217;s fair value. The volatility assumption has more weight on bonds with <strong>embedded options<\/strong>.<\/p>\n<p>Similar to a fixed-coupon corporate bond, the arbitrage-free framework can also be used to analyze a <strong>floater<\/strong> as follows:<\/p>\n<ol type=\"i\">\n<li>Determine the VND given the quoted margin<\/li>\n<li>Calculate the CVA<\/li>\n<li>Fair value = VND &#8211; CVA<\/li>\n<li>Determine the discount margin by trial and error.<\/li>\n<\/ol>\n<blockquote>\n<h2>Question<\/h2>\n<p>A $1,000 par, 6% annual coupon corporate bond matures five years from today. The bond is currently priced with a credit spread of 150 bps over the benchmark par rate of 3%. The bond&#8217;s CVA is <em>closest to<\/em>:<\/p>\n<ol type=\"A\">\n<li>$43.63.<\/li>\n<li>$60.00.<\/li>\n<li>$1,084.86.<\/li>\n<\/ol>\n<h4><strong>Solution<\/strong><\/h4>\n<p><strong>The correct answer is A.<\/strong><\/p>\n<p>$$ CVA = VND- \\text{Price of risky bond} $$<\/p>\n<p>VND is the value of the cash flows arising from the bond, discounted at the benchmark par rate<\/p>\n<p>In this case,<\/p>\n<p>$$ VND=\\frac{60}{1.03}+\\frac{60}{{1.03}^2}+\\frac{1060}{{1.03}^3}=$1,084.86 $$<\/p>\n<p>Price of the risky bond using credit spread = 3% benchmark rate + 1.5% = 4.5%. We will therefore discount the bond&#8217;s cash flows assuming a YTM of 4.5%<\/p>\n<p>$$ \\begin{align*} \\text{Price of risky bond} &amp;=\\frac{60}{1.045}+\\frac{60}{{1.045}^2}+\\frac{1060}{{1.045}^3}=$1,041.23 \\\\ CVA &amp;=$1,084.86-$1,041.23=$43.63 \\end{align*} $$<\/p>\n<\/blockquote>\n<p>Reading 31: Credit Analysis Models<\/p>\n<p><em>LOS 31 (e) Calculate the value of a bond and its credit spread, given assumptions about the credit risk parameters.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The arbitrage-free framework is applied for credit analysis of a risky bond, assuming that interest rates are volatile. A binomial interest rate tree is constructed assuming no arbitrage. The tree is then verified if it has been correctly calibrated and&#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-17739","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>Bond Valuation and Credit Spread Calculation | CFA Level II<\/title>\n<meta name=\"description\" content=\"Learn how to value risky bonds and calculate credit spreads using arbitrage-free pricing, credit risk assumptions, and binomial interest rate trees.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, 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