Universal and MVE PortfoliosInvestment ProblemWealthPeriod ReturnsCumulative ReturnsDoubling RateConstant Rebalanced PortfoliosCRP Example [Blum97]Equally Weighted CRP [Wang00]Universal PortfoliosUniversal Portfolio AlgorithmPerformanceAsymptotic PerformanceAssumptions for Asymptotic Perf.Outline of ProofMean-Variance Efficient PortfoliosObjective of MVE PortfolioThe MVE PortfolioThe 0% and 100% Avg. Ret. Ports.PerformanceExamplesUniv. and CRPs of C & MSFTUniv. and CRPs of MSFT & ORCLUniv. and CRPs of C & GEMVE PortfoliosComparisonConclusionsA Comparison of Universal andMean-Variance Efficient PortfoliosShane M. HaasResearch Laboratory of Electronics, andLaboratory for Information and Decision SystemsMassachusetts Institute of TechnologyA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 1/28Universal and MVE PortfoliosInvestment DynamicsUniversal PortfoliosMean-Variance Efficient (MVE) PortfoliosExamplesConclusionsA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 2/28Investment ProblemFixed parameters:M = number of assetsN = number of periods to investMarket determines:pn,m= price of asset m at beginning of period nYou choose:an,m= number of asset m owned at beginning of per. nTo control:hn= wealth at beginning of per. nA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 3/28WealthAssume:No new wealth added to portfolioPrices at the end of a period are equal those atbeginning of nextWealth at beginning of periodn:hn= an,1pn,1+ · · · + an,Mpn,MWealth at beginning of period n + 1:hn+1= an,1pn+1,1+ · · · + an,Mpn+1,MA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 4/28Period ReturnsReturn for period n:Rn= hn+1/hn=an,1pn,1hnpn+1,1pn,1+ · · · +an,Mpn,Mhnpn+1,Mpn,M= bn,1Xn,1+ · · · + bn,MXn,M= btnXnwherebn,m=an,mpn,mhn, bn= (bn,1. . . bn,M)tXn,m=pn+1,mpn,m, Xn= (Xn,1. . . Xn,M)tA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 5/28Cumulative ReturnsReturn from period 1 to N:SN=hN+1h1=hN+1hNhNhN−1· · ·h2h1=NYn=1Rn=NYn=1btnXnA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 6/28Doubling RateAssuming that btnXn> 0 for n = 1, . . . , N:SN= exp NXn=1log btnXn!= exp (NWN)whereWN=1NNXn=1log btnXnis called the “doubling” rate of the portfolio.A Comparison of Universal and Mean-Variance Efficient Portfolios – p. 7/28Constant Rebalanced PortfoliosConstant Rebalanced Portfolio (CRP):b1= b2= · · · = bN= bBuy-and-Hold (B&H) Portfolio:b = em= (0, 0, . . . , 0, 1, 0, . . . , 0)tBest CRP:b∗N= argmaxb∈BSN(b) = argmaxb∈BWN(b)whereB = {b ∈ RM: bm≥ 0, m = 1, 2, . . . , M;PMm=1bm= 1}A Comparison of Universal and Mean-Variance Efficient Portfolios – p. 8/28CRP Example [Blum97]Two Stocks:Stock #1: Xn,1= 1 , n = 1, . . . , NStock #2: Xn,2=(12n = even2 n = oddPortfolios:SN≤ 2 for any B&H portfolioReturn of b = (12,12)tincreases exponentially by 9/8every two days:Rn=(12(1) +1212=34n = even12(1) +12(2) =32n = oddA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 9/28Equally Weighted CRP [Wang00]Equally weighted CRP: b = (1M, · · · ,1M)tAssume Xnare IID random vectors with E[Xn] =¯X andV = E[(Xn− E(Xn))(Xn− E(Xn))t]:Expected return: E(Rn) =1MPMm=1¯XmVariance of return:var(Rn) =1M 1MMXm=1vm,m!+M2− MM21M2− MMXm=1MXk6=mvm,k=1M(avg. var.) +M2− MM2(avg. cov.)A Comparison of Universal and Mean-Variance Efficient Portfolios – p. 10/28Universal PortfoliosOrigin: T. Cover, "Universal Portfolios", MathematicalFinance, 1991Objective: Track return of best CRP chosen after assetreturns revealed (b∗N)Algorithm: Return-weighted average of all CRPsPerformance:Yields average return of all CRPsReturn exceeds geometric mean of asset returnsAsymptotically “tracks” best CRP returnA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 11/28Universal Portfolio AlgorithmReturn-weighted average of all CRPs:Initialize:ˆb1=1M, · · · ,1MtFor n > 1:ˆbn=RBbSn(b)dbRBSn(b)dbwhereB = {b ∈ RM: bm≥ 0, m = 1, 2, . . . , M;MXm=1bm= 1}A Comparison of Universal and Mean-Variance Efficient Portfolios – p. 12/28PerformanceRegardless of the stock return sequence:Portfolio return is the average of all CRPs’ returnsˆSN=NYn=1ˆbtnXn=RBSn(b)dbRBdbReturn is greater than geometric meanˆSN≥ MYm=1SN(em)!1/MA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 13/28Asymptotic PerformanceUnder many assumptions:ˆSNS∗N∼ r2πN!M−1(M − 1)!|J∗|1/2whereJ∗= Hessian of WN(b) at its maximum for large NS∗N= return of best CRP b∗Nand ∼ means the ratio of left- and right-hand sides equalsone for large NA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 14/28Assumptions for Asymptotic Perf.The stock sequence X1, X2, . . . , satisfies:a ≤ Xn,m≤ q, m = 1, . . . , M,n = 1, 2, . . ., for some0 < a ≤ q < ∞J∗N→ J∗for some positive definite matrix J∗b∗N→ b∗for some b∗in the interior of the Band there exists a function W (b) such thatWN(b) % W (b)∗for any portfolio b ∈ BW (b) is strictly concaveW (b) has bounded third partial derivativesW (b) achieves its maximum at b∗in the interior of the BA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 15/28Outline of ProofShow LHS ≥ RHS:Expand WN(b) in Taylor series about the best CRPb∗Nto three terms:First term is zero because gradient is zero at b∗NSecond term is related to J∗NThird term is boundedManipulate expression to look like a Gaussian CDFBound the Gaussian CDFShow LHS ≤ RHS:Follows from Laplace’s method of integrationA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 16/28Mean-Variance Efficient PortfoliosOrigins: Markowitz (1952) and Sharpe (1963) (1990Nobel Laureates)Objective: To minimize the variance of portfolio returns,while maintaining a desired average returnAlgorithm:A solution to a constrained optimization problemRequires estimates of asset return mean andcovariance matrixPerformance:Tracks objective portfolio if asset returns are IIDOtherwise, no guaranteeA Comparison of Universal and Mean-Variance Efficient Portfolios – p. 17/28Objective of MVE PortfolioMinimize: btnVnbnSubject to: 1)PMm=1bn,m= 12) btn¯Xn=¯Rnwhere¯Xn= estimate of asset return meanVn= estimate of asset return covariance matrix¯Rn= desired average returnA Comparison of Universal and
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