[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-202013-105":59,"doc-detail-202013-en":123},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":116,"head_meta":118,"extra_data":120,"updated_unix":122},105,"en","portfolio-risk-in-multiple-frequencies","PORTFOLIO RISK IN MULTIPLE FREQUENCIES","","Portfolio risk, defined as the standard deviation of portfolio return introduced in Markowitz’s Modern Portfolio Theory, is a central metric for balancing return against uncertainty across multiple assets. Empirical correlation matrices estimated in practice contain intrinsic noise that must be filtered before risk evaluation. The paper develops risk engineering concepts for finance, applies PCA-based eigen-filtering for hedged portfolios, and extends risk measurement to trading at multiple frequencies. Back-testing on NASDAQ 100 stocks demonstrates that smarter portfolio risk management can improve investment performance.",{"@graph":69,"@context":115},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/portfolio-risk-in-multiple-frequencies/202013/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/portfolio-risk-in-multiple-frequencies/202013.png","ImageObject",300,407,{"name":92,"@type":93},"Ethan Miller","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-09-04",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109],{"name":110,"@type":111,"acceptedAnswer":112},"How does the paper extend traditional risk measurement for hedged portfolios?","Question",{"text":113,"@type":114},"After PCA-based noise filtering of the correlation matrix, the approach includes risk measurement for hedged portfolios and introduces an extension for trading at multiple frequencies, followed by three overlay risk management methods.","Answer","https://schema.org",{"og:url":83,"og:type":117,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":119,"canonical":83},"index,follow",{"doc_id":121,"site_id":62},202013,1788541064,{"code":4,"msg":5,"data":124},{"doc_id":121,"user_id":125,"nickname":92,"user_avatar":126,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":127,"file_id":128,"file_url":129,"file_type":130,"file_size":131,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":132,"language":133,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":134,"faqs":135,"seo_title":136,"seo_description":67,"update_tm":122,"read_time":41,"preview_page_count":4},687207017582,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","PORTFOLIO RISK IN MULTIPLE FREQUENCIES  \nMustafa U. Torun, Ali N. Akansu, and Marco Avellaneda  \nPortfolio risk, introduced by Markowitz in 1952, and defined as the standard deviation of the portfolio return, is an important metric in the Modern Portfolio Theory (MPT) . A popular method for portfolio selection is to manage the risk and return of a portfolio according to the cross-correlations of returns for various financial assets. In a real world scenario, estimated empirical financial correlation matrix contains significant level of intrinsic noise that needs to be filtered prior to risk calculations. In this paper, we present basic concepts of risk engineering in finance applications. Then, we extend our discussion to the eigenfiltering of measurement noise for hedged portfolios. Moreover, we extend risk measurement methods for trading in multiple frequencies. Finally, three novel risk management methods are proposed as an independent overlay of the underlying investment decision mechanism, i.e. the trading strategy. We highlight performance and merit of the risk engineering techniques introduced by presenting the back-testing results of an investment strategy for the stocks listed in the NASDAQ 100 index. It is shown in the paper that managing portfolio risk more intelligently may offer advantages for improved return on investment.  \nINTRODUCTION  \nPortfolio is a collection of investments in various financial assets (e.g. stocks, futures, options, commodities, bonds, etc.) with their allocated investment capitals. Two important aspects of a portfolio are its return and its risk. The main goal of a portfolio manager is to keep the ratio of portfolio return over portfolio risk as high as possible. The standard deviation of a portfolio return is a widely used metric for risk measurement. In Markowitz's celebrated MPT [1], a portfolio is formed, [i.e. capital](i.e. capital) allocation in each asset is decided, by minimizing the risk under the constraint that the expected portfolio return is constant. The portfolio with the minimum risk for a given expected return is called efficient portfolio, and the set of efficient portfolios form the Efficient Frontier introduced by Markowitz. The main objective in MPT is to keep the portfolio balanced according to cross-correlations of asset pairs. As an example, being simultaneously long (buying the assets with the assumption that the prices will increase in time) in two assets that are positively and highly correlated is a highly risky bet and it must be avoided. However, portfolio size, i.e.  \nIEEE SIGNAL PROCESSING MAGAZINE, SEPTEMBER 2011  \nthe number of assets in the portfolio, is much larger than just two in reality. In order to measure the risk of an N-asset portfolio, N (N 􀀀 1)=2 unknown asset pair correlations need tobe estimated to create the N-by-N empirical financial correlation matrix, C. Naturally, C contains significant amount of inherent noise that needs to be removed. There are several methods in the quantitative finance and signal processing fields, including Principal Component Analysis (PCA), offering efficient noise filtering prior to risk calculations [2–4] .  \nWe present PCA based noise filtering of C, followed by risk measurement for the case of hedged portfolios. In a hedged portfolio, each asset is traded against another asset, usually against an index-tracking asset such as an Exchange Traded Fund (ETF), in order to protect the investment from the volatility (fluctuations) of the market. Next, we argue that, in contrast to the traditional view, an investor may re-balance the portfolio (change the investment amount in each asset), measure and manage the risk at different sampling time intervals or time resolutions. Therefore, a novel extension of the traditional risk metric for trading in multiple frequencies is introduced. Then, three simple-to-implement and yet powerful risk management methods are proposed. The proposed methods can be embedded in any investment decision mechani","cbCaiuKFcQqz7rLm","https://ap.wps.com/l/cbCaiuKFcQqz7rLm","pdf",414488,12,"English","# Introduction\n# Portfolio Return and Risk","[{\"question\":\"How does the paper extend traditional risk measurement for hedged portfolios?\",\"answer\":\"After PCA-based noise filtering of the correlation matrix, the approach includes risk measurement for hedged portfolios and introduces an extension for trading at multiple frequencies, followed by three overlay risk management methods.\"}]","PORTFOLIO RISK IN MULTIPLE FREQUENCIES | PDF"]