{"id":12838,"date":"2026-06-13T08:11:43","date_gmt":"2026-06-13T04:11:43","guid":{"rendered":"https:\/\/eoninvestment.com\/?p=12838"},"modified":"2026-06-13T19:33:17","modified_gmt":"2026-06-13T15:33:17","slug":"vasicek-credit-var","status":"publish","type":"post","link":"https:\/\/eoninvestment.com\/ka\/2026\/06\/13\/vasicek-credit-var\/","title":{"rendered":"<span class=\"eon-recovered-georgian-title\" data-no-translation=\"\">Vasicek-\u10d8\u10e1 Credit VaR \u10db\u10dd\u10d3\u10d4\u10da\u10d8<\/span>"},"content":{"rendered":"<p class=\"wp-block-paragraph\">The Vasicek Value at Risk model is a special case of the Gaussian Copula model.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To recall, in the previous posts we started with a simple idea: every company has a hidden credit score. If this hidden score deteriorates beyond a certain threshold, the company defaults.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then we saw that the hidden credit score consists of two factors: one common factor shared by the companies in the portfolio, and one company-specific factor. When the common factor deteriorates, many companies can default together. This is the logic behind modeling default correlation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">After that, in the Excel model, we used Monte Carlo simulation to show how correlation between defaults works in practice through the Gaussian Copula simulation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Vasicek Credit VaR shows how default correlation works for a large homogeneous portfolio \u2014 for example, a portfolio consisting of 1,000+ similar credit instruments.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The interesting point is that in the Excel model, simulation is no longer required. Vasicek transforms the model into an algebraic formula, because a large number of similar exposures allows the specific-risk component to be diversified away.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">What Credit VaR Measures<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Credit VaR asks a simple question:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">How large can credit losses become in a bad scenario?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For example, a 1-year 99.9% Credit VaR of $5.13 million means that, over one year, with 99.9% confidence, the maximum credit loss is $5.13 million.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There is a difference between Market VaR and Credit VaR.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The difference is the source of the loss.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In Market VaR, the question is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">How much can we lose because the market prices of securities change?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In Credit VaR, the question is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">How much can we lose because borrowers default?<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Vasicek\u2019s Default Formula<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The formula is:<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"92\" data-attachment-id=\"12949\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2026\/06\/13\/vasicek-credit-var\/image-789\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-12.png?fit=526%2C97&amp;ssl=1\" data-orig-size=\"526,97\" data-comments-opened=\"0\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;,&quot;alt&quot;:&quot;&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-12.png?fit=526%2C97&amp;ssl=1\" class=\"wp-image-12949\" style=\"width: 500px;\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-12.png?resize=500%2C92&#038;ssl=1\" alt=\"\" srcset=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-12.png?w=526&amp;ssl=1 526w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-12.png?resize=300%2C55&amp;ssl=1 300w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>V(X,T) \u2013 is the default rate in the bad scenario;<\/li>\n\n\n\n<li>Q(T) \u2013 is the cumulative probability of default over time (T);<\/li>\n\n\n\n<li>p \u2013 is the correlation between hidden credit scores;<\/li>\n\n\n\n<li>X \u2013 is the confidence level.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">After we calculate V(X,T) , Credit VaR is:<\/p>\n\n\n\n<div class=\"wp-block-math has-large-font-size\">\n<math display=\"block\"><semantics><mrow><mtext>Credit&nbsp;VaR<\/mtext><mo>=<\/mo><mi>L<\/mi><mo>\u2217<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>R<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2217<\/mo><mi>V<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>X<\/mi><mo separator=\"true\">,<\/mo><mi>T<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">text{Credit VaR} = L * (1-R) * V(X,T)<\/annotation><\/semantics><\/math>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>L \u2013 is the size of the portfolio;<\/li>\n\n\n\n<li>R \u2013 is the recovery rate.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Hull\u2019s Example in Excel<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">To summarize, the model looks like this:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"700\" height=\"244\" data-attachment-id=\"12884\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2026\/06\/13\/vasicek-credit-var\/image-782\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-5.png?fit=1525%2C532&amp;ssl=1\" data-orig-size=\"1525,532\" data-comments-opened=\"0\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;,&quot;alt&quot;:&quot;&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-5.png?fit=1024%2C357&amp;ssl=1\" class=\"wp-image-12884\" style=\"width: 700px;\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-5.png?resize=700%2C244&#038;ssl=1\" alt=\"\" srcset=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-5.png?w=1525&amp;ssl=1 1525w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-5.png?resize=300%2C105&amp;ssl=1 300w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-5.png?resize=1024%2C357&amp;ssl=1 1024w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-5.png?resize=768%2C268&amp;ssl=1 768w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-5.png?resize=1200%2C419&amp;ssl=1 1200w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The interesting point is that if events develop according to the worst 0.1% scenario, the loss is 6.4 times larger than the average expected loss, as shown in the final line.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Sensitivity Analysis<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Now let us look at how three different factors affect the expected loss in a bad scenario.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. Impact of changing the correlation rate<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"199\" data-attachment-id=\"12887\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2026\/06\/13\/vasicek-credit-var\/image-783\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-6.png?fit=853%2C339&amp;ssl=1\" data-orig-size=\"853,339\" data-comments-opened=\"0\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;,&quot;alt&quot;:&quot;&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-6.png?fit=853%2C339&amp;ssl=1\" class=\"wp-image-12887\" style=\"width: 500px;\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-6.png?resize=500%2C199&#038;ssl=1\" alt=\"\" srcset=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-6.png?w=853&amp;ssl=1 853w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-6.png?resize=300%2C119&amp;ssl=1 300w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-6.png?resize=768%2C305&amp;ssl=1 768w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. Impact of cumulative default probability \/ credit rating<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"115\" data-attachment-id=\"12890\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2026\/06\/13\/vasicek-credit-var\/image-784\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-7.png?fit=864%2C198&amp;ssl=1\" data-orig-size=\"864,198\" data-comments-opened=\"0\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;,&quot;alt&quot;:&quot;&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-7.png?fit=864%2C198&amp;ssl=1\" class=\"wp-image-12890\" style=\"width: 500px;\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-7.png?resize=500%2C115&#038;ssl=1\" alt=\"\" srcset=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-7.png?w=864&amp;ssl=1 864w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-7.png?resize=300%2C69&amp;ssl=1 300w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-7.png?resize=768%2C176&amp;ssl=1 768w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3. Impact of the confidence level<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"498\" height=\"99\" data-attachment-id=\"12892\" data-permalink=\"https:\/\/eoninvestment.com\/ka\/2026\/06\/13\/vasicek-credit-var\/image-785\/\" data-orig-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-8.png?fit=876%2C175&amp;ssl=1\" data-orig-size=\"876,175\" data-comments-opened=\"0\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;,&quot;alt&quot;:&quot;&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-8.png?fit=876%2C175&amp;ssl=1\" class=\"wp-image-12892\" style=\"width: 498px;\" src=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-8.png?resize=498%2C99&#038;ssl=1\" alt=\"\" srcset=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-8.png?w=876&amp;ssl=1 876w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-8.png?resize=300%2C60&amp;ssl=1 300w, https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-8.png?resize=768%2C153&amp;ssl=1 768w\" sizes=\"auto, (max-width: 498px) 100vw, 498px\" \/><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\"><a href=\"https:\/\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/23.7_vasicek_credit_var_learning_model.xlsx\">Excel \u2014 Vasicek VaR Model<\/a><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u10d0\u10d3\u10d0\u10de\u10e2\u10d8\u10e0\u10d4\u10d1\u10e3\u10da\u10d8\u10d0 \u10ec\u10e7\u10d0\u10e0\u10dd\u10d3\u10d0\u10dc:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Options, Futures, and Other Derivatives<\/em> \u2014 John C. Hull<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><a href=\"https:\/\/i0.wp.com\/eoninvestment.com\/wp-content\/uploads\/2026\/06\/image-5.png?ssl=1\"><\/a><\/figure>","protected":false},"excerpt":{"rendered":"<p>Vasicek-\u10d8\u10e1 Value at Risk \u10db\u10dd\u10d3\u10d4\u10da\u10d8 \u10ec\u10d0\u10e0\u10db\u10dd\u10d0\u10d3\u10d2\u10d4\u10dc\u10e1 Gaussian Copula-\u10e1 \u10d9\u10d4\u10e0\u10eb\u10dd \u10d5\u10d4\u10e0\u10e1\u10d8\u10d0\u10e1. \u10d2\u10d0\u10e1\u10d0\u10ee\u10e1\u10d4\u10dc\u10d4\u10d1\u10da\u10d0\u10d3, \u10ec\u10d8\u10dc\u10d0 \u10e9\u10d0\u10dc\u10d0\u10ec\u10d4\u10e0\u10d4\u10d1\u10e8\u10d8 \u10d3\u10d0\u10d5\u10d8\u10ec\u10e7\u10d4\u10d7 \u10db\u10d0\u10e0\u10e2\u10d8\u10d5\u10d8 \u10d8\u10d3\u10d4\u10d8\u10d7, \u2013 \u10dc\u10d4\u10d1\u10d8\u10e1\u10db\u10d8\u10d4\u10e0 \u10d9\u10dd\u10db\u10de\u10d0\u10dc\u10d8\u10d0\u10e1 \u10d0\u10e5\u10d5\u10e1 \u10e4\u10d0\u10e0\u10e3\u10da\u10d8 \u10e1\u10d0\u10d9\u10e0\u10d4\u10d3\u10d8\u10e2\u10dd \u10e5\u10e3\u10da\u10d0, \u10e0\u10dd\u10db\u10da\u10d8\u10e1 \u10d9\u10dd\u10dc\u10d9\u10e0\u10d4\u10e2\u10e3\u10da \u10d6\u10e6\u10d5\u10d0\u10e0\u10d6\u10d4 \u10db\u10d4\u10e2\u10d0\u10d3 \u10d2\u10d0\u10e3\u10d0\u10e0\u10d4\u10e1\u10d4\u10d1\u10d0 \u10d8\u10ec\u10d5\u10d4\u10d5\u10e1 \u10db\u10d8\u10e1 \u10d2\u10d0\u10d3\u10d4\u10e4\u10dd\u10da\u10e2\u10d4\u10d1\u10d0\u10e1. \u10e8\u10d4\u10db\u10d3\u10d4\u10d2, [\u2026]<\/p>","protected":false},"author":12998525,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"custom-post-template","format":"standard","meta":{"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_wpcom_ai_launchpad_first_post":false,"_eon_insights_quality":"","_eon_insights_decision":"","_eon_insights_destination":"","_eon_insights_merge_group":"","_eon_insights_required_changes":"","_eon_insights_start_order":0,"_eon_featured_insight":0,"_eon_insights_map_version":"","_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":true},"categories":[784237235,1,37372],"tags":[],"eon_topic":[],"eon_learning_path":[],"eon_collection":[],"eon_level":[],"eon_content_type":[],"eon_source":[],"eon_idea":[784445690],"eon_framework":[],"class_list":["post-12838","post","type-post","status-publish","format-standard","hentry","category-options-futures-other-derivatives-john-c-hull-ka","category-uncategorized","category-risk","eon_idea-risk"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - 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