{"id":59787,"date":"2020-09-16T11:59:03","date_gmt":"2020-09-16T15:59:03","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=59787"},"modified":"2022-11-21T09:46:19","modified_gmt":"2022-11-21T14:46:19","slug":"kurtosis-its-application-in-risk-evaluation","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/kurtosis-its-application-in-risk-evaluation\/","title":{"rendered":"Kurtosis &#038; its Application in Risk Evaluation"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">When analysing the performance of financial securities, we give a lot of importance to the mean and the standard deviation as measures of the average return and risk, respectively.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">However, the risk-adjusted performance of financial securities also depends upon the risks which arise due to the shape of the distribution of their returns. These are the higher-moment risks such as the skewness and kurtosis risks, which need to be taken into consideration for proper evaluation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In this blog, we discuss the concept of kurtosis and its application in understanding the risk profiles of financial securities. In addition, we also glance over some common misconceptions regarding the calculation and interpretation of kurtosis.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"what-is-kurtosis\"><strong>What is kurtosis?<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Kurtosis is a&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/statistics-probability-distribution\/\">statistical<\/a>&nbsp;measure which quantifies the degree to which a distribution of a random variable is likely to produce extreme values or outliers relative to a normal distribution.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From extreme values and outliers, we mean observations that cluster at the tails of the probability distribution of a random variable. In other words, kurtosis measures the &#8216;tailedness&#8217; of distribution relative to a normal distribution.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Along with variance and skewness, which measure the dispersion and symmetry, respectively, kurtosis helps us to describe the &#8216;shape&#8217; of the distribution.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/blog.quantinsti.com\/algorithmic-trading-maths\/#Math-Concepts\">Mathematically<\/a>, the kurtosis of a distribution of a random variable X, with a mean \u03bc and standard deviation \u03c3 is defined as the ratio of the fourth moment to the square of the variance<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2020\/09\/quantInsti-kurtosis-2.png\" alt=\"\" class=\"wp-image-59804 lazyload\" width=\"75\" height=\"56\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 75px; aspect-ratio: 75\/56;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">ie.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2020\/09\/quantInsti-kurtosis-1.png\" alt=\"\" class=\"wp-image-59805 lazyload\" width=\"480\" height=\"106\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/09\/quantInsti-kurtosis-1.png 480w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/09\/quantInsti-kurtosis-1-300x66.png 300w\" data-sizes=\"(max-width: 480px) 100vw, 480px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 480px; aspect-ratio: 480\/106;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">From the above expression, we can already see that outliers will significantly impact this value.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is because for any outlier<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2020\/09\/quantInsti-kurtosis-3.png\" alt=\"\" class=\"wp-image-59806 lazyload\" width=\"106\" height=\"56\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 106px; aspect-ratio: 106\/56;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">the value<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2020\/09\/quantInsti-kurtosis-4.png\" alt=\"\" class=\"wp-image-59807 lazyload\" width=\"109\" height=\"50\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 109px; aspect-ratio: 109\/50;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">will tend to be large anyway, but then the outliers contribution to the sum is raised to the power of four<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">i.e.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2020\/09\/quantInsti-kurtosis-5.png\" alt=\"\" class=\"wp-image-59808 lazyload\" width=\"123\" height=\"76\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 123px; aspect-ratio: 123\/76;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">In other words, the key factor that would determine the value of kurtosis is the number and size of the outliers\/extreme values, which are reflected in heavier\/fatter tails of the distribution.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Contrary to popular perception, kurtosis does not measure the peakedness of the distribution, and the only unambiguous interpretation of kurtosis is with regard to the heaviness or lightness of tails of the distribution, relative to a normal distribution (we will see an example of this in a subsequent section).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"excess-kurtosis\"><strong>Excess kurtosis<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">There exists one more method of calculating the kurtosis called &#8216;<strong>excess kurtosis<\/strong>&#8216;. As kurtosis is calculated relative to the normal distribution, which has a kurtosis value of 3, it is often easier to analyse in terms of excess kurtosis.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">As the name suggests, it is the kurtosis value in excess of the kurtosis value of the normal distribution. This means that for a normal distribution with any mean and variance, the excess kurtosis is always&nbsp;<strong>0<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2020\/09\/quantInsti-kurtosis-6.png\" alt=\"\" class=\"wp-image-59819 lazyload\" width=\"574\" height=\"114\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/09\/quantInsti-kurtosis-6.png 574w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2020\/09\/quantInsti-kurtosis-6-300x60.png 300w\" data-sizes=\"(max-width: 574px) 100vw, 574px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 574px; aspect-ratio: 574\/114;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Many resources refer to &#8216;<strong>excess kurtosis<\/strong>&#8216; as &#8216;<strong>kurtosis<\/strong>&#8216; and hence to avoid any confusion one must clarify this point beforehand.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In the next section, we will learn about the three categories of distributions based on the kurtosis.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Visit QuantInsti to download additional code: <a href=\"https:\/\/blog.quantinsti.com\/kurtosis\/\">https:\/\/blog.quantinsti.com\/kurtosis\/<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><small>Disclaimer: All investments and trading in the stock market involve risk. Any decisions to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.<\/small><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>QuantInsti discusses the concept of kurtosis and its application in understanding the risk profiles of financial securities. <\/p>\n","protected":false},"author":295,"featured_media":59790,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[339,338,350,341,344],"tags":[851,4923,4922,1226,1006,8397,494],"contributors-categories":[13654],"class_list":["post-59787","post","type-post","status-publish","format-standard","has-post-thumbnail","category-data-science","category-ibkr-quant-news","category-quant-asia-pacific","category-quant-development","category-quant-regions","tag-algo-trading","tag-computational-finance","tag-econometrics","tag-financial-engineering","tag-fintech","tag-kurtosis","tag-quant","contributors-categories-quantinsti"],"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v28.3) - 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\/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\n\t    \"@context\": \"https:\\\/\\\/schema.org\",\n\t    \"@graph\": [\n\t        {\n\t            \"@type\": \"NewsArticle\",\n\t            \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/kurtosis-its-application-in-risk-evaluation\\\/#article\",\n\t            \"isPartOf\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/kurtosis-its-application-in-risk-evaluation\\\/\"\n\t            },\n\t            \"author\": {\n\t                \"name\": \"Ashutosh Dave\",\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/person\\\/e462b9a6b7cd1a6e27f6e29fc69942c8\"\n\t            },\n\t            \"headline\": \"Kurtosis &#038; its Application in Risk Evaluation\",\n\t            \"datePublished\": \"2020-09-16T15:59:03+00:00\",\n\t 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