{"id":45501,"date":"2023-08-07T07:17:21","date_gmt":"2023-08-07T07:17:21","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=45501"},"modified":"2026-03-17T12:25:43","modified_gmt":"2026-03-17T12:25:43","slug":"lognormal-distribution-and-continuous-compounding","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/lognormal-distribution-and-continuous-compounding\/","title":{"rendered":"Lognormal Distribution and Continuous Compounding"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Simulation Methods (2024\/2025 CFA\u00ae Level I Exam \u2013 Quantitative Methods \u2013 Learning Module 6)\",\n  \"description\": \"This video lesson covers simulation methods in quantitative finance, focusing on portfolio return calculations, including expected value, variance, standard deviation, covariance, and correlation. It explains joint probability functions for risk assessment and introduces shortfall risk, the safety-first ratio, and Roy\u2019s safety-first criterion for optimal portfolio selection.\",\n  \"uploadDate\": \"2023-08-19T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/qLhvORG9ITo\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/qLhvORG9ITo\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/qLhvORG9ITo\",\n  \"duration\": \"PT37M18S\"\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/images\/lognormal-distribution\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM06_IMG1.jpg\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM06_IMG1.jpg\",\n  \"caption\": \"Lognormal Distribution\",\n  \"width\": 450,\n  \"height\": 400,\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\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Properties of lognormal distributions compared to normal distributions\",\n    \"text\": \"Which of the following is true about lognormal distributions compared to normal distributions?\\n\\nA. They are skewed to the right.\\n\\nB. They can take on negative values.\\n\\nC. They are less suitable for describing asset prices than asset returns.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"A. They are skewed to the right.\\n\\nLognormal distributions only take positive values and are typically skewed to the right, making them useful for modeling variables such as asset prices.\\n\\nB is incorrect because lognormal distributions cannot take negative values.\\n\\nC is incorrect because lognormal distributions are commonly used to model asset prices, while asset returns are often modeled using normal distributions.\"\n    }\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/qLhvORG9ITo\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>A random variable \\(Y\\) is lognormally distributed if its natural logarithm, In \\(Y\\), is normally distributed. The opposite is true. If ln \\(Y\\) is normally distributed, then \\(Y\\) is lognormally distributed.<\/p>\n<p>The lognormal distribution is positively skewed, meaning it&#8217;s skewed to the right and has a long right tail. In this distribution, values are bounded by 0. Typically, the mean is greater than the mode.<\/p>\n<p>Consider the following graph of two probability density functions (pdfs) of two lognormal distributions.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"450\" height=\"400\" class=\"size-full wp-image-45502 aligncenter\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM06_IMG1.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM06_IMG1.jpg 450w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM06_IMG1-300x267.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM06_IMG1-400x356.jpg 400w\" sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><\/p>\n<p>Like the normal distribution, two parameters \u2013 the mean and variance of the associated normal distribution \u2013 fully describe the lognormal distribution.<\/p>\n<div style=\"text-align: center; margin: 25px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 10px 18px; border: 2px solid #1a73e8; border-radius: 999px; color: #1a73e8; text-decoration: none; font-weight: 500; background-color: #f5f9ff; white-space: nowrap;\" href=\"https:\/\/analystprep.com\" target=\"_blank\" rel=\"noopener\"> Build clarity on lognormal distributions with focused CFA Level I practice <\/a><\/div>\n<h2>Expressions for Mean and Variance of Lognormal Distribution<\/h2>\n<p>Assume that \\(X\\) is normally distributed with the mean \\(\\mu\\) and variance \\(\\sigma^2\\). Also, define the variable \\(Y=e^X\\).<\/p>\n<p>Then \\(\\ln{Y}=\\ln{\\left(e^X\\right)}=X\\) is lognormally distributed with the following mean and variance expressions:<\/p>\n<p>$$<br \/>\\text{Mean}=\\mu_L=e^{\\left(\\mu+\\frac{1}{2}\\sigma^2\\right)} \\\\<br \/>\\text{Variance}=\\sigma_L^2=e^{2u+\\sigma^2}\\left(e^{\\sigma^2}-1\\right)<br \/>$$<\/p>\n<h2>Why the Lognormal Distribution is Used to Model Stock Prices<\/h2>\n<p>The lognormal distribution works well for modeling asset prices that cannot be negative because it has a lower bound at zero.<\/p>\n<p>When the continuously compounded returns on a stock 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<h2>Continuously Compounded Rate of Return<\/h2>\n<p>Remember that given the investment horizon from time \\(t=0\\) to time \\(t=T\\), the continuously compounded return of a stock is given by:<\/p>\n<p>$$ r_{0,T}=\\ln{\\left(\\frac{P_T}{P_0}\\right)} $$<\/p>\n<p>If we apply the exponential function on both sides of the equation, we have the following:<\/p>\n<p>$$ P_T=P_0e^{r_{0,T}} $$<\/p>\n<p>Note that \\(\\frac{P_T}{P_0}\\) can be written as:<\/p>\n<p>$$ \\frac{P_T}{P_0}=\\left(\\frac{P_T}{P_{T-1}}\\right)\\left(\\frac{P_{T-1}}{P_{T-2}}\\right)\\ldots\\left(\\frac{P_1}{P_0}\\right) $$<\/p>\n<p>If we take natural logarithm on both sides of the above equation:<\/p>\n<p>$$ \\begin{align*}<br \/>ln \\left(\\frac{P_T}{P_0}\\right) &amp; =\\ln{\\left(\\left(\\frac{P_T}{P_{T-1}}\\right)\\left(\\frac{P_{T-1}}{P_{T-2}}\\right)\\ldots\\left(\\frac{P_1}{P_0}\\right)\\right)} \\\\ \\Rightarrow r_{0,T} &amp; =r_{T-1,T}+r_{T-2,T-1}+\\ldots+r_{0,1}<br \/>\\end{align*} $$<\/p>\n<p>Therefore, the continuously compounded return to time \\(T\\) equals the sum of one-period continuously compounded returns.<\/p>\n<p>Remember that a linear combination of normal random variables is also normal. Therefore, if the shorter period returns \\(r_{T-1,T}, r_{T-2,T-1},\\ldots,r_{0,1}\\) are normally distributed or approximately normal, then \\(r_{0,T}\\) will also be approximately normal.<\/p>\n<p>Furthermore, if we assume that the one-period continuously compounded returns \\(r_{T-1, T}, r_{T-2, T-1},\\ldots,r_{0,1}\\) are independently and identically distributed (i.i.d) random variables with a mean of \\(\\mu\\) and variance of \\(\\sigma^2\\), then:<\/p>\n<ul>\n<li>The expected value of the continuously compounded return over a holding period of \\(T\\) periods is given by: $$ E\\left(r_{0, T}\\right)=E\\left(r_{T-1, T}\\right)+E\\left(r_{T-2, T-1}\\right)+\\ldots+E\\left(r_{0,1}\\right)=\\mu T $$<\/li>\n<li>The variance of the continuously compounded return over a holding period is given by: $$ \\sigma^2\\left(r_{0, T}\\right)=\\sigma^2T $$<\/li>\n<\/ul>\n<p>The standard deviation of the continuously compounded returns, also known as volatility, is given by:<\/p>\n<p>$$ \\sigma(r_{0,T})=\\sigma \\sqrt{T} $$<\/p>\n<p>In other words, if \\(r_{T-1,T}, r_{T-2,T-1},\\ldots,r_{0,1}\\) are normally distributed with the mean of \\(\\mu\\) and variance of \\(\\sigma^2\\) then \\(r_{0,T}\\) is normally distributed with the mean of \\(\\mu T\\) and variance of \\(\\sigma^2T\\).<\/p>\n<p>Let us go back to the formula:<\/p>\n<p>$$ P_T=P_0e^{r_{0,T}} $$<\/p>\n<p>If \\(X\\) is normally distributed with the mean \\(\\mu\\) and variance \\(\\sigma^2\\) and that \\(Y=e^X\\) then, \\(\\ln{Y}=\\ln{\\left(e^X\\right)}=X\\) is lognormally distributed. Assuming we apply this intuition in the above formula, it would be easy to see that we can model \\(P_T\\) as a lognormally distributed random variable since \\(r_{0, T}\\) is approximately normally distributed.<\/p>\n<h3>Volatility and Continuously Compounded Returns<\/h3>\n<p>Volatility measures the standard deviation of the continuously compounded returns on the underlying asset. Conventionally, it is usually annualized.<\/p>\n<p>We calculate volatility using the historical series of continuously compounded returns. Another method is converting daily holding returns into continuously compounded daily returns and then calculating annualized volatility.<\/p>\n<p>We base annualizing volatility on 250 trading days in a year, which is an estimate of the business days the financial markets operate. The formula we use for annualizing volatility is:<\/p>\n<p>$$ \\sigma(r_{0,T})=\\sigma \\sqrt{T} $$<\/p>\n<p>For example, if the daily volatility is 0.05, then the annual volatility is:<\/p>\n<p>$$ \\sigma(r_{0,T})=0.05\\times \\sqrt{250}=0.79 $$<\/p>\n<p><strong>Example: Lognormal Distribution and Continuous Compounding<\/strong><\/p>\n<p>Jess Kasuku is analyzing the stock of ABC Company, which is listed on the London Stock Exchange under the ABC ticker symbol. Kasuku wants to understand how the stock&#8217;s price changed during a particular week when significant developments in the global economy impacted the UK stock market. To do this, she calculates the stock&#8217;s volatility for that week using the closing prices shown in Table 1.<\/p>\n<p>$$ \\textbf{Table 1: ABC Company Daily Closing Prices} \\\\<br \/>\\begin{array}{c|c}<br \/>\\textbf{Day} &amp; \\textbf{Closing Price (GBP)} \\\\ \\hline<br \/>\\text{Monday} &amp; 75 \\\\ \\hline<br \/>\\text{Tuesday} &amp; 78 \\\\ \\hline<br \/>\\text{Wednesday} &amp; 72 \\\\ \\hline<br \/>\\text{Thursday} &amp; 70 \\\\ \\hline<br \/>\\text{Friday} &amp; 68<br \/>\\end{array} $$<\/p>\n<p>Using the information in Table 1, calculate the annualized volatility of ABC Company&#8217;s stock for that week, assuming 250 trading days in a year.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>Step 1:<\/strong> Calculate the continuously compounded daily returns for each day using the formula \\(ln{\\left(\\frac{\\text{Ending Price}}{\\text{Beginning Price}}\\right)}\\):<\/p>\n<p>$$ \\begin{align*}<br \/>r_1 &amp; =ln\\left(\\frac{78}{75}\\right)=0.03922 \\\\<br \/>r_2 &amp;=ln \\left(\\frac{72}{78}\\right)=-0.08004 \\\\<br \/>r_3 &amp; =ln \\left(\\frac{70}{72}\\right)=-0.02817 \\\\<br \/>r_4 &amp; =ln \\left(\\frac{68}{70}\\right)=-0.02899<br \/>\\end{align*} $$<\/p>\n<p><strong>Step 2:<\/strong> Calculate the mean of the continuously compounded daily returns:<\/p>\n<p>$$ \\begin{align*} \\mu &amp; =\\frac{r_1+r_2+r_3+r_4}{4} \\\\<br \/>&amp; =\\frac{0.03922+\\left(-0.08004\\right)+\\left(-0.02817\\right)+\\left(-0.02899\\right)}{4} \\\\<br \/>&amp; =-0.024495 \\end{align*} $$<\/p>\n<p><strong>Step 3:<\/strong> Calculate the variance of the continuously compounded daily returns:<\/p>\n<p>$$ \\begin{align*}<br \/>\\sigma^2 &amp; =\\frac{\\left(r_1-\\mu\\right)^2+\\left(r_2-\\mu\\right)^2+\\left(r_3-\\mu\\right)^2+\\left(r_4-\\mu\\right)^2}{4} \\\\<br \/>&amp; = \\frac {<br \/>{<br \/>\\left[<br \/>\\left(0.03922-\\left(-0.24495\\right)\\right)^2 +<br \/>\\left(-0.08004-\\left(-0.024495\\right)\\right)^2 \\\\ +<br \/>\\left(-0.02817-\\left(-0.024495\\right)\\right)^2 +<br \/>\\left(-0.02899-\\left(-0.024495\\right)\\right)^2<br \/>\\right]<br \/>}<br \/>}{4} \\\\<br \/>&amp; =\\frac{0.007179}{4}=0.001795 \\end{align*} $$<\/p>\n<p><strong>Step 4:<\/strong> Calculate the standard deviation of the continuously compounded daily returns:<\/p>\n<p>$$ \\begin{align*}<br \/>\\sigma &amp; =\\sqrt{\\sigma^2} \\\\<br \/>&amp; =\\sqrt{0.001795}=0.042363 \\end{align*} $$<\/p>\n<p><strong>Step 5:<\/strong> Annualize the volatility by multiplying the daily volatility by the square root of the number of trading days in a year.<\/p>\n<p>We know that:<\/p>\n<p>$$ \\begin{align*}<br \/>\\sigma(r_{0,T}) &amp;=\\sigma \\sqrt T \\\\<br \/>\\therefore\\sigma_{\\text{annualized}} &amp; =\\sigma_{\\text{daily}}\\times\\sqrt{250} \\\\<br \/>&amp; =0.042363\\times\\sqrt{250} \\\\<br \/>&amp; =0.6698\\approx67\\% \\end{align*} $$<\/p>\n<p>So, the annualized volatility of ABC Company&#8217;s stock for that week was 67.23 percent.<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Which of the following is true about lognormal distributions compared to normal distributions?<\/p>\n<ol type=\"A\">\n<li>They are skewed to the right.<\/li>\n<li>They can take on negative values.<\/li>\n<li>They are less suitable for describing asset prices than asset returns.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>The correct answer is A<\/strong>.<\/p>\n<p>Lognormal distributions are continuous probability distributions that only take positive values and are often skewed to the right.<\/p>\n<p><strong>B is incorrect<\/strong> because lognormal distributions only take on positive values.<\/p>\n<p><strong>C is incorrect<\/strong> because there is no evidence to suggest that lognormal distributions are less suitable for describing asset prices than asset returns.<\/p>\n<\/blockquote>\n<div style=\"text-align: center; margin: 40px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 20px; border: 2px solid #1a73e8; border-radius: 999px; color: #1a73e8; text-decoration: none; font-weight: 600; background-color: #f5f9ff;\" href=\"https:\/\/analystprep.com\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a>\n<p style=\"font-size: 15px; margin-top: 12px; color: #555;\">Master continuous compounding and distribution concepts with step-by-step explanations and exam-style questions.<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>A random variable \\(Y\\) is lognormally distributed if its natural logarithm, In \\(Y\\), is normally distributed. The opposite is true. If ln \\(Y\\) is normally distributed, then \\(Y\\) is lognormally distributed. The lognormal distribution is positively skewed, meaning it&#8217;s skewed&#8230;<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-45501","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Lognormal Distribution &amp; Continuous Compounding | CFA<\/title>\n<meta name=\"description\" content=\"Understand lognormal distributions, their mean and variance expressions, and their application in modeling stock prices.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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