{"id":29833,"date":"2021-09-09T13:22:30","date_gmt":"2021-09-09T13:22:30","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29833"},"modified":"2026-03-08T11:02:40","modified_gmt":"2026-03-08T11:02:40","slug":"relationship-between-normal-distribution-and-lognormal-distribution","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/uncategorized\/relationship-between-normal-distribution-and-lognormal-distribution\/","title":{"rendered":"Relationship between Normal Distribution and Lognormal Distribution"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-134.jpg\",\n  \"caption\": \"The Lognormal Distribution\",\n  \"width\": 1463,\n  \"height\": 1100,\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  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2025 Level I CFA\u00ae Exam \u2013 Quantitative Methods \u2013 Module 4)\",\n  \"description\": \"This CFA\u00ae Level I Quantitative Methods lesson covers Common Probability Distributions in depth. Topics include discrete and continuous random variables, cumulative distribution functions, uniform, Bernoulli, binomial, normal, lognormal, Student\u2019s t, chi-square, and F-distributions. The video also explains standardization, interval probabilities, shortfall risk, Roy\u2019s safety-first criterion, continuously compounded returns, and Monte Carlo simulation concepts essential for exam success.\",\n  \"uploadDate\": \"2021-11-09\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/TXO9ODcLWiU\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=TXO9ODcLWiU\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/TXO9ODcLWiU\",\n  \"duration\": \"PT1H0M47S\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Which statement about lognormal distributions is least accurate?\",\n    \"text\": \"Which of the following statements regarding the properties of lognormal distributions is the least accurate?\\n\\nA. Lognormal distributions are skewed to the right.\\nB. Lognormal distributions are more often used to model asset prices than standard distributions.\\nC. Lognormal distributions can take negative values.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"C. Lognormal distributions can take negative values. Lognormal distributions are bounded below by zero and therefore cannot take negative values, making them suitable for modeling asset prices.\"\n    }\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/TXO9ODcLWiU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>A variable \\(X\\) is said to have a lognormal distribution if \\(Y = ln(X)\\) is normally distributed, where \u201cln\u201d denotes the natural logarithm. In other words, when the logarithms of values form a normal distribution, we say that the original values have a lognormal distribution.<\/p>\n<p><!--more--><\/p>\n<p>Let\u2019s consider this:<\/p>\n<p>$$Y=e^X$$<\/p>\n<p>Where \\(e\\) is the exponential constant.<\/p>\n<p>If we take natural logs on both sides, \\(lnY = ln\\ e^X\\), which leads us to \\(lnY = X\\). Therefore, if \\(X\\) has a normal distribution, then \\(Y\\) has a lognormal distribution.<\/p>\n<p>The lognormal distribution is positively skewed, with many small values and just a few large values. Consequently, the mean is greater than the mode in most cases.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-17024\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-134.jpg\" alt=\"the-lognormal-distribution\" width=\"1463\" height=\"1100\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-134.jpg 1463w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-134-300x226.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-134-768x577.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-134-1024x770.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-134-400x301.jpg 400w\" sizes=\"auto, (max-width: 1463px) 100vw, 1463px\" \/><\/p>\n<h2><strong>Why the Lognormal Distribution is Used to Model Stock Prices<\/strong><\/h2>\n<p>Since the lognormal distribution is bound by zero on the lower side, it is perfect for modeling asset prices that cannot take negative values. On the other hand, the normal distribution cannot be used for the same purpose because it has a negative side.<\/p>\n<p>When the returns on a stock (continuously compounded) follow a normal distribution, the stock prices follow a lognormal distribution. Note that even if returns do not follow a normal distribution, the lognormal distribution is still the most appropriate for stock prices.<\/p>\n<p>The probability density function of the distribution is:<\/p>\n<p>$$ f\\left( x \\right) =\\frac { 1 }{ x\\sqrt { 2\\pi { \\sigma }^{ 2 } } } { e }^{ -\\frac { { \\left( lnx-\\mu \\right) }^{ 2 } }{ \\sqrt { 2{ \\sigma }^{ 2 } } } } $$<\/p>\n<p>The Black-Scholes-Merton model used to price options, which we will see in-depth in level II, uses the lognormal distribution as its foundation.<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Which of the following statements regarding the properties of lognormal distributions is the <em>least accurate<\/em>?<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>Lognormal distributions are skewed to the right.<\/li>\n<li>Lognormal distributions are more often used to model asset prices than standard distributions.<\/li>\n<li>Lognormal distributions can take negative values.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>C<\/strong>.<\/p>\n<p>Lognormal distributions are bounded by 0 because they cannot take negative values. Since normal distributions can take negative values, and asset prices cannot be negative, lognormal distributions are more suitable for describing distributions of asset prices.<\/p>\n<p><strong>Exam tip<\/strong>: If you have negative values, your data isn&#8217;t lognormal.<\/p>\n<p><strong>A is incorrect<\/strong>. It is a true statement. Lognormal distributions are bound by zero. This implies that they do not have negative values and are thus skewed to the right.<br \/>\n<strong>B is incorrect<\/strong>. Lognormal distributions are bound by zero and thus cannot take negative values.<\/p>\n<\/blockquote>","protected":false},"excerpt":{"rendered":"<p>A variable \\(X\\) is said to have a lognormal distribution if \\(Y = ln(X)\\) is normally distributed, where \u201cln\u201d denotes the natural logarithm. In other words, when the logarithms of values form a normal distribution, we say that the original&#8230;<\/p>\n","protected":false},"author":10,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-29833","post","type-post","status-publish","format-standard","hentry","category-uncategorized","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>Normal vs. Lognormal Distribution | CFA Level 1<\/title>\n<meta name=\"description\" content=\"A normal distribution is symmetrical, while a lognormal distribution skews positively. 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