{"id":17672,"date":"2021-07-13T15:27:14","date_gmt":"2021-07-13T15:27:14","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17672"},"modified":"2026-04-02T09:19:40","modified_gmt":"2026-04-02T09:19:40","slug":"explain-expected-exposure-the-loss-given-default-the-probability-of-default-and-the-credit-valuation-adjustment","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/explain-expected-exposure-the-loss-given-default-the-probability-of-default-and-the-credit-valuation-adjustment\/","title":{"rendered":"Measures of Credit Risk"},"content":{"rendered":"<p><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\": \"Probability of survival using an annual hazard rate\",\r\n    \"text\": \"A zero-coupon $100 par corporate bond issued today matures in three years. If the bond\u2019s annual hazard rate is 1.5%, what is the probability that the bond survives over the next three years?\",\r\n    \"answerCount\": 3,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is 95.57%. The probability of survival is calculated by sequentially reducing the survival probability each year by the annual hazard rate. Year 1 survival is 98.5%, year 2 survival is approximately 97.02%, and year 3 survival is approximately 95.57%.\",\r\n      \"confidence\": 0.63\r\n    },\r\n    \"suggestedAnswer\": [\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"98.50%\",\r\n        \"confidence\": 0.12\r\n      },\r\n      {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"97.02%\",\r\n        \"confidence\": 0.25\r\n      }\r\n    ]\r\n  }\r\n}\r\n<\/script><\/p>\r\n\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/pHEcwZjtzl8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<p><em><strong>Credit risk<\/strong><\/em> is the risk of default or delay in making interest or principal payments on a loan. On the other hand, <em><strong>credit spread<\/strong> <\/em>is the difference between the yield to maturity of credit risky, zero-coupon bond, and a risk-free zero-coupon bond. Credit spread rewards exposure to credit risk.<\/p>\r\n<p>There are four credit risk measures for fixed income securities. These include:<\/p>\r\n<ul>\r\n\t<li>Loss given default.<\/li>\r\n\t<li>Probability of default.<\/li>\r\n\t<li>Expected loss.<\/li>\r\n\t<li>The present value of the expected loss.<\/li>\r\n<\/ul>\r\n<div style=\"margin: 20px 0;\"><a style=\"display: block; width: 100%; text-align: center; padding: 10px; border: 2px solid #2f5bea; border-radius: 40px; font-size: 16px; color: #2f5bea; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Practice credit risk questions with our free trial. <\/a><\/div>\r\n<h2>Expected Exposure and Loss Given Default<\/h2>\r\n<p><em><strong>Expected exposure (EE)<\/strong><\/em> is the amount that an investor or bondholder stands to lose at any given point in time in case of default. It does not factor in possible recovery. The <em><strong>recovery\u00a0rate<\/strong><\/em> is the proportion that can be recovered in a default event.<\/p>\r\n<p>$$ \\text{Loss severity} = 1 \u2013 \\text{Recovery rate.} $$<\/p>\r\n<p><em><strong>Loss given default (LGD)<\/strong><\/em> is the amount of loss to the investor if a default occurs. The loss given default is a positive function of the expected exposure and a negative recovery rate function.<\/p>\r\n<p>$$ \\begin{align*} LGD &amp; =\\text{ Loss severity}\\times \\text{Expected exposure} \\\\ LGD &amp; =\\left(1-\\text{Recovery rate}\\right)\\times \\text{Expected exposure} \\end{align*} $$<\/p>\r\n<h2><strong>Probability of Default<\/strong><\/h2>\r\n<p><em><strong>Probability of default (PD)<\/strong><\/em> is the likelihood of the bond issuer not paying the interest\/principal amounts when due. In other words, it is the probability of default in any given year.<\/p>\r\n<p><em><strong>The hazard rate<\/strong><\/em> is the initial probability of default. The probability of default in every subsequent year is the conditional probability of default given that default had not previously occurred.<\/p>\r\n<p>$$ PD_t=PS_{{t-1}}\\times \\text{Hazard rate} $$<\/p>\r\n<p>Where:<\/p>\r\n<p>\\(PD_t\\) is the probability of default at any given year \\(t\\).<\/p>\r\n<p>\\(PS_{t-1}\\) is the survival probability for the previous year.<\/p>\r\n<h2>Expected Loss<\/h2>\r\n<p>The <em><strong>expected loss<\/strong><\/em> varies depending on the state of the economy and other microeconomic factors. For example, in a boom phase, the value of assets is high, and the probability of default is low\u2014further, the value of the collateral increases. Therefore, LGD decreases, and so does the expected loss.<\/p>\r\n<p>$$ \\begin{align*} \\text{Expected loss} \\left(\\%\\right) &amp;= LGD\\times PD \\\\ \\text{Expected loss} \\left($\\right) &amp;=\\text{Exposure}\\times PD\\times LGD \\end{align*} $$<\/p>\r\n<h2>The Present Value of Expected Loss<\/h2>\r\n<p><em><strong>The present value of the expected loss<\/strong><\/em> is obtained by discounting the expected loss by risk-neutral probabilities. This is the preferred measure of credit risk as it includes the probability of default, loss given default, time value of money, and the risk premium in its calculation. <em><strong>Credit valuation adjustment<\/strong><\/em> (CVA) is the aggregate of the present value of expected loss over the term of the bond.<\/p>\r\n<p>It is worth noting that the expected losses are computed using risk-neutral probabilities, and discounting is done at the risk-free rates for the relevant maturities.<\/p>\r\n<p>CVA can also be determined by taking the difference between the risk-free value and the value of the risky bond. It captures investors&#8217; compensation for bearing default risk.<\/p>\r\n<p>$$ \\text{CVA} = \\text{Price of a riskless bond}\\ &#8211; \\text{Price of the risky bond.} $$<\/p>\r\n<h4>Example: Credit Risk Measures<\/h4>\r\n<p>A zero-coupon corporate bond with a par value of $100 matures in four years. The risk-neutral probability of default (hazard rate) for the bond is 1%, and the recovery rate is 40%. The benchmark spot rate curve is constant at 4%. Calculate:<\/p>\r\n<ol style=\"list-style-type: lower-roman;\" type=\"a\">\r\n\t<li>Expected exposure (EE).<\/li>\r\n\t<li>Loss given default (LGD).<\/li>\r\n\t<li>Probability of survival (PS).<\/li>\r\n\t<li>Probability of default (PD).<\/li>\r\n\t<li>Credit valuation adjustment (CVA).<\/li>\r\n\t<li>Value of the risky bond.<\/li>\r\n<\/ol>\r\n<p>$$ \\begin{array}{c|c|c|c|c|c|c} \\text{Year} &amp; \\text{EE} &amp; \\text{LGD} &amp; \\text{PD} &amp; \\text{PS} &amp; \\text{EL}&amp; \\text{PV of EL} \\\\ \\hline 1 &amp; 88.900 &amp; 53.34 &amp; 1.0\\% &amp; 99.000\\% &amp; 0.5334 &amp; 0.5129 \\\\ \\hline 2 &amp; 92.456 &amp; 55.47 &amp; 0.9900\\% &amp; 98.010\\% &amp; 0.5492 &amp; 0.5078 \\\\ \\hline 3 &amp; 96.154 &amp; 57.69 &amp; 0.9801\\% &amp; 97.030\\% &amp; 0.5654 &amp; 0.5027 \\\\ \\hline 4 &amp; 100.00 &amp; 60.00 &amp; 0.9703\\% &amp; 96.060\\% &amp; 0.5822 &amp; 0.4976 \\\\ \\hline &amp; &amp; &amp; &amp; &amp; \\textbf{CVA} &amp; \\bf{2.0210} \\end{array} $$<\/p>\r\n<ol style=\"list-style-type: lower-roman;\" type=\"a\">\r\n\t<li><strong>Expected exposure (EE):<\/strong>\r\n<p>$$ \\begin{align*} EE_1 &amp; =\\text{Par value discounted for 3 years at }4\\%=\\frac{100}{\\left(1.04\\right)^3}=88.90 \\\\ EE_2 &amp;=\\text{Par value discounted for 2 years at } 4\\%=\\frac{100}{\\left(1.04\\right)^2}=92.456 \\\\ EE_3 &amp;=\\text{Par value discounted for 1 year at } 4\\%=\\frac{100}{\\left(1.04\\right)^1}=96.154 \\\\ EE_4 &amp;=\\text{Par value of the bond}=100 \\end{align*} $$<\/p>\r\n<\/li>\r\n\t<li><strong>Loss given default:<\/strong>\r\n<p>$$ \\begin{align*} \\text{Loss given default}&amp;=\\text{Expected exposure} \\times(1-\\text{Recovery rate}) \\\\ LGD_1 &amp;=88.90\\times\\left(1-40\\%\\right)=53.34 \\\\ LGD_2 &amp;=92.456\\times\\left(1-40\\%\\right)=55.47 \\\\ LGD_3 &amp;=96.154\\times\\left(1-40\\%\\right)=57.69 \\\\ LGD_4 &amp;=100\\times\\left(1-40\\%\\right)=60.00 \\end{align*} $$<\/p>\r\n<\/li>\r\n\t<li><strong>Probability of default and probability of survival:<\/strong>\r\n<p>$$ \\begin{align*} PD_t &amp;=PS_{t-1}\\times \\text{Hazard rate} \\\\ \\\\ \\text{Probability of survival}&amp; = 1-{\\text{Cumulative conditional} \\\\ \\text{probability of default}} \\end{align*} $$ $$ \\begin{align*} PD_1 &amp;=PS_0\\times \\text{Hazard rate}=100\\%\\times 1\\% \\\\ &amp; =1\\% \\\\ PS_1 &amp;=1-1\\% \\\\ &amp; =99\\% \\\\ PD_2 &amp;=PS_1\\times \\text{Hazard rate}=99\\%\\times1\\% \\\\ &amp; =0.99\\% \\\\ PS_2 &amp;=1-\\left(1\\%+0.99\\%\\right) \\\\ &amp; =98.01\\% \\\\ PD_3 &amp;=PS_2\\times \\text{Hazard rate}=98.01\\% \\times 1\\% \\\\ &amp; =0.9801\\% \\\\ PS_3 &amp;=1-\\left(1\\%+0.99\\%+0.9801\\%\\right) \\\\ &amp; =97.03\\% \\\\ PD_4 &amp;=PS_3\\times \\text{Hazard rate}\\\\ &amp; =97.03\\%\\times1\\%=0.9703\\% \\\\ PS_4 &amp;=1-\\left(1\\%+0.99\\%+0.9801\\%+0.9703\\%\\right)\\\\ &amp; =96.06\\% \\end{align*} $$<\/p>\r\n<\/li>\r\n\t<li><strong>The present value of the expected loss:<\/strong>\r\n<p>$$ \\begin{align*} \\text{Present value of expected loss}&amp;=\\text{LGD} \\times\\frac{\\text{PD}}{\\left(1+i\\right)^t} \\\\ \\text{PVEL}_1 &amp;=53.34\\times\\frac{1\\%}{1.04} =0.5129 \\\\ \\text{PVEL}_2 &amp;=55.47\\times\\frac{0.99\\%}{{1.04}^2} =0.5077 \\\\ \\text{PVEL}_3 &amp;=57.69\\times\\frac{0.9801\\%}{{1.04}^3} =0.5027 \\\\ \\text{PVEL}_4 &amp;=60\\times\\frac{0.9703\\%}{{1.04}^4} =0.4976 \\end{align*} $$<\/p>\r\n<\/li>\r\n\t<li><strong>Credit valuation adjustment (CVA):<\/strong>\r\n<p>CVA is the total present value of the expected loss<\/p>\r\n<p>Thus,<\/p>\r\n<p>$$ \\text{CVA} = 0.5129+0.5077+0.5027+0.4976=${2}.{021} $$<\/p>\r\n<\/li>\r\n\t<li><strong>Price of the risky bond:<\/strong>\r\n<p>$$ \\begin{align*} \\text{CVA} &amp; = \\text{Price of the riskless bond} \\\\ &amp; &#8211; \\text{Price of the risky bond} \\\\ \\text{Price of the risk-free bond} &amp; =\\frac{100}{{1.04}^4}=85.48 \\\\ \\text{Price of the risky bond}&amp;=\\text{Price of the risk-free bond} -\\text{CVA} \\\\ &amp; =85.48-2.021=$83.46 \\end{align*} $$<\/p>\r\n<\/li>\r\n<\/ol>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>A zero-coupon, $100 par corporate bond issued today matures in three years. If the bond&#8217;s annual hazard rate is 1.5%, the probability that the bond survives over the next three years is <em>closest to<\/em>:<\/p>\r\n<ol type=\"A\">\r\n\t<li>98.50%.<\/li>\r\n\t<li>97.02%.<\/li>\r\n\t<li>95.57%.<\/li>\r\n<\/ol>\r\n<h4><strong>Solution<\/strong><\/h4>\r\n<p><strong>The correct answer is C.<\/strong><\/p>\r\n<p>$$ \\begin{align*} \\text{Probability of} &amp; \\text{ survival } \\left(PS_t\\right) \\\\ &amp; = 1 -{\\text{Cumulative conditional probability of default}} \\end{align*} $$ $$ \\begin{align*} PD_1 &amp;=PS_0\\times \\text{Hazard rate}=100\\%\\times 1.5\\% \\\\ &amp; =1.5\\% \\\\ PS_1 &amp;=1-1.5\\% \\\\ &amp; =98.5\\% \\\\ PD_2 &amp;=PS_1\\times \\text{Hazard rate}=98.5\\%\\times1.5\\% \\\\ &amp; =1.4775\\% \\\\ PS_2 &amp;=1-\\left(1.5\\%+1.4775\\%\\right) \\\\ &amp;=97.0225\\% \\\\ PD_3 &amp;=PS_2\\times \\text{Hazard rate} \\\\ &amp; =97.0225\\%\\times1.5\\% \\\\ &amp; =1.4553375\\% \\\\ PS_3 &amp;=1-\\left(1.5\\%+1.4775\\%+1.4553375\\%\\right) \\\\ &amp; =95.57\\% \\end{align*} $$<\/p>\r\n<\/blockquote>\r\n<p>Reading 31: Credit Analysis Models<\/p>\r\n<p><em>LOS 31 (a) Explain expected exposure, the loss given default, the probability of default, and the credit valuation adjustment.<\/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\r\n<div style=\"text-align: center; margin: 40px 0;\"><a style=\"display: inline-block; padding: 10px 26px; background: #3f78d7; color: #fff; border-radius: 40px; font-size: 16px; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a>\r\n<p style=\"margin-top: 10px; max-width: 600px; margin-left: auto; margin-right: auto; font-size: 14px;\">Solve CFA Level II questions on expected exposure, LGD, PD, and expected loss.<\/p>\r\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Credit risk is the risk of default or delay in making interest or principal payments on a loan. On the other hand, credit spread is the difference between the yield to maturity of credit risky, zero-coupon bond, and a risk-free&#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-17672","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>Credit Risk Measures: PD, LGD, EE, and CVA | CFA L2<\/title>\n<meta name=\"description\" content=\"Learn how expected exposure, probability of default, loss given default, and credit valuation adjustment are used to measure credit risk in practice.\" \/>\n<meta name=\"robots\" content=\"index, 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