DeBacco, Cassandra

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Elizabethtown College: JayScholar@ETown

Elizabethtown College JayScholar Summer Scholarship, Creative Arts and Research Projects (SCARP) Programs and Events Summer 2020 Post-Modern Portfolio Theory Cassandra DeBacco Elizabethtown College, debaccoc@etown.edu Follow this and additional works at: https://jayscholar.etown.edu/scarp  Part of the Mathematics Commons Recommended Citation DeBacco, Cassandra, "Post-Modern Portfolio Theory" (2020). Summer Scholarship, Creative Arts and Research Projects (SCARP). 8. https://jayscholar.etown.edu/scarp/8 This Presentation is brought to you for free and open access by the Programs and Events at JayScholar. It has been accepted for inclusion in Summer Scholarship, Creative Arts and Research Projects (SCARP) by an authorized administrator of JayScholar. For more information, please contact kralls@etown.edu. MPTModified MPTPMPTPost-Modern Portfolio TheoryCassy DeBaccoJuly 27, 2020Post-Modern Portfolio Theory 1 / 38MPTModified MPTPMPTHow Should I Invest My Money?AAPLBADISGEKOMSFT0.1 0.2 0.3 0.4volatility0.10.20.30.40.50.60.7mean returnPost-Modern Portfolio Theory 2 / 38MPTModified MPTPMPTModern Portfolio TheoryThe mean and variance for portfolioP = w1X1 + . . .+ wnXnis µP = µTw and  2P = wT⌃w.Modern Portfolio Theory (MPT) solves the quadratic programMinimize  2Psuch that µP   ↵and 1Tw = 1.Post-Modern Portfolio Theory 3 / 38MPTModified MPTPMPTSolution with Shorting AllowedIf short sales are allowed, then the optimal weights for aminimum variance portfolio arew =11T⌃ 11⌃ 11.The corresponding mean and variance areµP =µT⌃ 111T⌃ 11and  2P =11T⌃ 11.Post-Modern Portfolio Theory 4 / 38MPTModified MPTPMPTMinimum Variance PortfolioPost-Modern Portfolio Theory 5 / 38MPTModified MPTPMPTE cient FrontierPost-Modern Portfolio Theory 6 / 38MPTModified MPTPMPTE cient FrontierPost-Modern Portfolio Theory 7 / 38MPTModified MPTPMPTE cient FrontierPost-Modern Portfolio Theory 8 / 38MPTModified MPTPMPTE cient FrontierPost-Modern Portfolio Theory 9 / 38MPTModified MPTPMPTE cient FrontierPost-Modern Portfolio Theory 10 / 38MPTModified MPTPMPTE cient Frontier with a Risk-Free AssetPost-Modern Portfolio Theory 11 / 38MPTModified MPTPMPTE cient Frontier with a Risk-Free AssetPost-Modern Portfolio Theory 12 / 38MPTModified MPTPMPTE cient Frontier with a Risk-Free AssetPost-Modern Portfolio Theory 13 / 38MPTModified MPTPMPTE cient Frontier with a Risk-Free AssetPost-Modern Portfolio Theory 14 / 38MPTModified MPTPMPTE cient Frontier with a Risk-Free AssetPost-Modern Portfolio Theory 15 / 38MPTModified MPTPMPTE cient Frontier with a Risk-Free AssetPost-Modern Portfolio Theory 16 / 38MPTModified MPTPMPTSome Shortcomings of MPTMPT can produce crazy portfoliosVariance as a measure of risk is debatableProceeds from short sales cannot be reinvested in practicePost-Modern Portfolio Theory 17 / 38MPTModified MPTPMPTMPT Can Produce Crazy PortfoliosDow 30 returns for 2017Post-Modern Portfolio Theory 18 / 38MPTModified MPTPMPTIs Smaller Variance Really What We Want?BA, CVX, and Portfolio Normal DistributionPost-Modern Portfolio Theory 19 / 38MPTModified MPTPMPTProceeds from the Short Sale are ReinvestedBased on Data for BA and CVX in 2018119% BA and -19% CVXPost-Modern Portfolio Theory 20 / 38MPTModified MPTPMPTModified MPTFor n assets, modify MPT to minimize  2P such thatµTw + 1.5rnXi=1R( wi)| {z }µP  ↵nXi=1[wi] = 1,where[x] =8<:x x   0 0.5x x < 0 and R(x) =8<:x x   00 x < 0 .Post-Modern Portfolio Theory 21 / 38MPTModified MPTPMPTExample Illustrating ConstraintsPrincipal: $1000Spot Prices: Asset 1: $100 Asset 2: $100Long $500 in asset 1 (w1 = 0.5) and short $1000 of asset 2(w2 =  1).w1 + 0.5( w2) = 0.5 + 0.5(+1) = 1.Margin: $0|{z}depositfor asset 1+ $500|{z}depositfor asset 2+ $1000| {z }proceeds fromshort sale= $15001.5 [R( w1) +R( w2)] = 1.5 [R( 0.5) +R(1)] = 1.5Post-Modern Portfolio Theory 22 / 38MPTModified MPTPMPTExampleWeights for 2019 Portfolio with and without the new constraintPost-Modern Portfolio Theory 23 / 38MPTModified MPTPMPTCompare with MPT (2 Assets)Based on Data for BA and CVX in 2018.Post-Modern Portfolio Theory 24 / 38MPTModified MPTPMPTCompare with MPT (2 Assets)Based on Data for BA and CVX in 2018.Post-Modern Portfolio Theory 25 / 38MPTModified MPTPMPTSolutionSolutions must be found numerically, but the existence of manysimilar local minima can make the global minimum di cult tofind.Post-Modern Portfolio Theory 26 / 38MPTModified MPTPMPTProbability Density Function for ReturnsBased on Data for BA and CVX in 2018.Post-Modern Portfolio Theory 27 / 38MPTModified MPTPMPTProbability Density Function for ReturnsBased on Data for BA and CVX in 2018.Post-Modern Portfolio Theory 28 / 38MPTModified MPTPMPTValue at RiskBased on Data for BA and CVX in 2018.Post-Modern Portfolio Theory 29 / 38MPTModified MPTPMPTValue at RiskIf the asset returns are normally distributed, thenValue at Risk = P (portfolio return < 0)= P✓return  µp p< µp p◆= P✓Z <  µP P◆=  ✓ µP P◆,where   is the CDF for the standard normal distribution.Note that minimizing the value at risk is equivalent tomaximizingµP Por minimizing PµP, the coe cient of variationfor the portfolio.Post-Modern Portfolio Theory 30 / 38MPTModified MPTPMPTPost-Modern Portfolio Theory (PMPT)Minimize PµP=pwT⌃wµTw + 1.5rPni=1R( wi)such thatµP   ↵ andnXi=1[wi] = 1, where[x] =8<:x x   0 0.5x x < 0 and R(x) =8<:x x   00 x < 0 .Post-Modern Portfolio Theory 31 / 38MPTModified MPTPMPTOptimization Problem for PMPTBased on Data for BA and CVX in 2018.Post-Modern Portfolio Theory 32 / 38MPTModified MPTPMPTAlmost Closed-Form Solutionw = ⌃ 1 [µ  1.5rH( w)] t, t > 0,whereH(x) =8<:1 x   00 x < 0is the Heaviside step function and t is chosen so thatnXi=1[wi] = 1.If r > 0, then we can use fixed point iteration to quickly findthe solution for any r usingwk = ⌃ 1 [µ  1.5rH( wk 1)] t, w0 = ⌃ 1µt0.Post-Modern Portfolio Theory 33 / 38MPTModified MPTPMPTMPT vs PMPTMPTPMPTPost-Modern Portfolio Theory 34 / 38MPTModified MPTPMPTQuestions?Post-Modern Portfolio Theory 35 / 38MPTModified MPTPMPTPMPT WeightsNote that the weights are piecewise linear in ↵ and, except forGE, they achieve exact values of zero for larger values of ↵.Post-Modern Portfolio Theory 36 / 38MPTModified MPTPMPTr = 0.02It looks like the contours are still lines even with r > 0.Perhaps we can get a closed form solution here too?Setting r = 0.02 doesn’t seem to do much to the portfolio inthis case.Post-Modern Portfolio Theory 37 / 38MPTModified MPTPMPTBack TestingThere may not be room for this in your presentation, but it’sthe kind of thing that a person from the audience may ask, soit might be good to prepare a slide on it.Pictures comparing the results for all three models would begood to have.Post-Modern Portfolio Theory 38 / 38

Post-Modern Portfolio Theory

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Post-Modern Portfolio Theory

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