{"id":85222,"date":"2021-04-26T14:06:49","date_gmt":"2021-04-26T18:06:49","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=85222"},"modified":"2023-03-14T09:16:05","modified_gmt":"2023-03-14T13:16:05","slug":"cardinality-constrained-portfolios-optimization-approach-algorithm","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/cardinality-constrained-portfolios-optimization-approach-algorithm\/","title":{"rendered":"Cardinality-Constrained Portfolios: Optimization Approach &#038; Algorithm"},"content":{"rendered":"\n<p>Every portfolio can be partitioned into multiple asset groups defined by asset classes, sectors, styles,&nbsp;or&nbsp;other features. A\u202f<strong>cardinality-constrained<\/strong>\u202fportfolio caps the number of stocks to be traded within each of these groups. These limitations arise from real-world scenarios faced by fund managers who seek to satisfy certain investment mandates or achieve their asset allocation objectives.&nbsp;<\/p>\n\n\n\n<p>As an example, suppose you want to construct\u202fa portfolio by investing in stocks across\u202fm\u202fsectors you favor. And,&nbsp;in each sector,&nbsp;you select up to\u202f<em>k<\/em>\u202fstocks and each sector should not constitute more than\u202f<em>q<\/em>%\u202fof your portfolio. Moreover, you don\u2019t know\u202fwhich\u202fand\u202fhow\u202fmany\u202fstocks should be included yet. You\u2019ll also need to determine the&nbsp;portfolio\u202fweights\u202fbased on your risk-return tradeoff. Now, imagine you can do all that automatically by running an algorithm.&nbsp;<\/p>\n\n\n\n<p>In this\u202f<a href=\"https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3272588\" target=\"_blank\" rel=\"noreferrer noopener\">paper<\/a>, we develop a new approach to solve cardinality-constrained portfolio optimization problems with different constraints and objectives. In particular, our approach extends both Markowitz and conditional value at risk (CVaR) optimization models with cardinality constraints. A continuous relaxation method is proposed for the NP-hard objective, which allows for very efficient algorithms with standard convergence guarantees for nonconvex problems.&nbsp;<\/p>\n\n\n\n<p>For smaller cases, where brute force search is feasible to compute the globally optimal cardinality-constrained portfolio, the new approach finds the best portfolio for the&nbsp;cardinality-constrained Markowitz model and a very good local minimum for the cardinality-constrained&nbsp;CVaR&nbsp;model.&nbsp;<\/p>\n\n\n\n<p>As the total number of assets grows, brute-force exhaustive search quickly becomes prohibitively expensive. For instance, choosing 10 assets out of 30 requires solving more than 30 million optimization problems over the subsets, so even a seemingly simple case can be completely unmanageable.&nbsp;Our algorithm can solve problems of this scale on an average of 20 runs. We find feasible portfolios that are nearly as efficient as their non-cardinality constrained counterparts.&nbsp;<\/p>\n\n\n\n<p>We are given a total of\u202fn\u202fcandidate assets and a certain selection criterion\u202ff(w). The portfolio weights satisfy the simplex constraint:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/image-113-1100x84.png\" alt=\"Cardinality-Constrained Portfolios: Optimization Approach &amp; Algorithm\" class=\"wp-image-85243 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/84;\" \/><\/figure>\n\n\n\n<p>Combinatorial constraints restrict the number of stocks to purchase, within specified subgroups and\/or across the entire portfolio. The portfolio weights in each group are represented by the vector\u202fw\u1d62.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/image-114.png\" alt=\"Cardinality-Constrained Portfolios: Optimization Approach &amp; Algorithm\" class=\"wp-image-85244 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>This means that no more than\u202fk\u1d62\u202fstocks can be included in each group\u202fi\u202f. And the sum of portfolio weights within each group\u202fi\u202fis bounded between\u202fp\u1d62\u202fand\u202fq\u1d62.&nbsp;<\/p>\n\n\n\n<p>The constrained optimization problem is of the form:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/image-115.png\" alt=\"Cardinality-Constrained Portfolios: Optimization Approach &amp; Algorithm\" class=\"wp-image-85248 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>For example, for mean-variance portfolio, we have&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/image-116.png\" alt=\"Cardinality-Constrained Portfolios: Optimization Approach &amp; Algorithm\" class=\"wp-image-85249 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>And the conditional value-at-risk (CVaR) model minimizes the&nbsp;CVaR&nbsp;superquantile&nbsp;over portfolio selections:&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/image-117.png\" alt=\"Cardinality-Constrained Portfolios: Optimization Approach &amp; Algorithm\" class=\"wp-image-85251 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>where the quantile \u03b2 is related to the&nbsp;CVaR&nbsp;value \u03b1 by&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/image-118.png\" alt=\"Cardinality-Constrained Portfolios: Optimization Approach &amp; Algorithm\" class=\"wp-image-85254 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>To solve the problem, we first relax the problem by introducing an auxiliary variable\u202fv\u202fwhich we force to be close to\u202fw\u202fusing a quadratic penalty term. The optimization problem becomes&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/image-119.png\" alt=\"Cardinality-Constrained Portfolios: Optimization Approach &amp; Algorithm\" class=\"wp-image-85255 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>In turn, we apply a sophisticated projection method and use proximal alternating linearized minimization<strong>\u202f(PALM)\u202f<\/strong>with alternating updates on\u202fw\u202fand\u202fv. PALM converges to stationary points even in our nonconvex setting.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/image-120.png\" alt=\"Cardinality-Constrained Portfolios: Optimization Approach &amp; Algorithm\" class=\"wp-image-85256 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>In addition, a fast&nbsp;and&nbsp;iterative shrinkage-thresholding algorithm (<strong>FISTA<\/strong>) is applied to replace the proximal gradient step, which is empirically shown to help speed up the convergence.&nbsp;<\/p>\n\n\n\n<p>What sets our approach apart is that&nbsp;<\/p>\n\n\n\n<p>(i) we formulate the problem as a\u202fcontinuous\u202foptimization problem over a highly\u202fnonconvex\u202fset induced by the intersection of\u202f<strong>cardinality<\/strong>\u202fand\u202f<strong>simplex<\/strong>\u202fconstraints.&nbsp;<\/p>\n\n\n\n<p>(ii) we develop a relaxation method using auxiliary variables and create an efficient projection map onto the nonconvex set.&nbsp;<\/p>\n\n\n\n<p>These innovations allow recently developed techniques for structured&nbsp;nonsmooth&nbsp;nonconvex optimization to bear on the problem.&nbsp;<\/p>\n\n\n\n<p>The incorporation of cardinality constraints allows us to quantify and visualize their impact on risk and return. Among other effects, as fewer sectors and stocks are allowed to be included, the portfolio becomes less diversified, shifting the efficient frontier downward.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/image-121.png\" alt=\"Cardinality-Constrained Portfolios: Optimization Approach &amp; Algorithm\" class=\"wp-image-85257 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"has-text-align-center\"><em>The mean-variance efficient frontier for unconstrained portfolio (solid line), cardinality constrained portfolios (dashed). The green (resp. red) frontier includes 6 (resp. 5) sectors and allows at most 2 stocks from each sector.&nbsp;Source:&nbsp;Paper by&nbsp;Tim&nbsp;Leung, available at&nbsp;<a href=\"https:\/\/ieeexplore.ieee.org\/document\/8796164\/\">https:\/\/ieeexplore.ieee.org\/document\/8796164\/<\/a>.&nbsp;See Reference section for details.&nbsp;<\/em><\/p>\n\n\n\n<p>Reference&nbsp;<\/p>\n\n\n\n<p>J. Zhang, T. Leung, and A.&nbsp;Aravkin, \u202f<strong>A Relaxed Optimization Approach for Cardinality-Constrained Portfolios<\/strong>, Proceedings of the 18th IEEE European Control Conference (ECC), pp.2885\u20132892, 2019 [<a href=\"https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3272588\" target=\"_blank\" rel=\"noreferrer noopener\">pdf<\/a>] [<a href=\"https:\/\/ieeexplore.ieee.org\/document\/8796164\/\" target=\"_blank\" rel=\"noreferrer noopener\">DOI<\/a>]&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Join Tim Leung, UW CFRM Director, for a review of a new approach to solve cardinality-constrained portfolio optimization problems with different constraints and objectives. Find insight on both Markowitz and conditional value at risk (CVaR) optimization models with cardinality constraints. <\/p>\n","protected":false},"author":189,"featured_media":85223,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,338,341,352,344],"tags":[14917,9612,9619,9618,14916,1688,14915,494,9617,14918,9613],"contributors-categories":[13668],"class_list":{"0":"post-85222","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-ibkr-quant-news","9":"category-quant-development","10":"category-quant-north-america","11":"category-quant-regions","12":"tag-cardinality-constraints","13":"tag-cardinality-constrained-portfolios","14":"tag-iterative-shrinkage-thresholding-algorithm-fista","15":"tag-mean-variance-efficient-frontier","16":"tag-palm","17":"tag-portfolio-optimization","18":"tag-proximal-alternating-linearized-minimization","19":"tag-quant","20":"tag-relaxed-optimization-approach-for-cardinality-constrained-portfolios","21":"tag-simplex-constraints","22":"tag-trading-algorithm","23":"contributors-categories-computational-finance-risk-management-university-of-washington"},"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 v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Cardinality-Constrained Portfolios: Optimization Approach &#038; Algorithm<\/title>\n<meta name=\"description\" content=\"Join Tim Leung, UW CFRM Director, for a review of a new approach to solve cardinality-constrained portfolio optimization problems with different...\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.interactivebrokers.com\/campus\/wp-json\/wp\/v2\/posts\/85222\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cardinality-Constrained Portfolios: Optimization Approach &amp; Algorithm | IBKR Quant Blog\" \/>\n<meta property=\"og:description\" content=\"Join Tim Leung, UW CFRM Director, for a review of a new approach to solve cardinality-constrained portfolio optimization problems with different constraints and objectives. 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