A Comparison of Universal andMean-Variance Efficient PortfoliosShane M. HaasSeptember 6, 20011 SummaryA portfolio is an allocation of wealth among investment opportunities, or assets.In this presentation, we will consider two portfolio strategies, as summarizedbelow:• Universal Portfolios [3]:– Objective: To “track” the return of the best possible constant per-centage allocation of wealth (a.k.a. constant rebalanced portfolio(CRP)) chosen with knowledge of future asset returns– Algorithm: Return-weighted average of all CRPs– Main Assumptions: No statistical assumptions are made on asset re-turns; asymptotic analysis, however, requires several sequence-basedassumptions on asset returns– Performance: Yields average return of all CRPs; return is greaterthan the geometric mean of stock returns (i.e. value line index);asymptotically tracks the return of the best CRP under many sequence-based assumptions• Mean-Variance Efficient Portfolios [8],[9],[5]:– Objective: To minimize the variance of portfolio returns, while main-taining a desired average return– Algorithm: A solution to a constrained optimization problem– Main Assumptions: Assumes asset returns from different periods areindependent and identically distributed (IID) random vectors; re-quires estimates of the distribution’s mean and covariance matrix.– Performance: Tracks objective portfolio if IID assumptions hold; oth-erwise, no guaranteeBoth strategies use asset returns from previous investment periods to build theportfolio for the current investment period.12 Investment DynamicsA portfolio describes a strategy for allocating and re-allocating wealth amongM assets over a particular time horizon, to attain certain objectives. The timehorizon and objectives vary from investor to investor depending on their pref-erences. In this summary, we assume that the investor falls into one of twocategories:• The investor wants a portfolio whose return tracks the return of the bestconstant percentage allocation of wealth given future stock returns.• The investor wants a desired average return, but wishes to minimize thereturn variance.Universal portfolios might be more suited to the first type of investor, whilemean-variance efficient portfolios to the latter.We simplify matters by dividing the time horizon into N periods. Let pn,mand an,mbe the price and amount owned, respectively, of asset m at the be-ginning of period n. The wealth of the investor at the beginning of period nishn= an,1pn,1+ · · · + an,Mpn,M(1)We define the factor by which the wealth increased from the beginning to theend of period n as Rn= hn+1/hn, assuming that the price of each asset atthe end of a period is equal to its price at the beginning of the next period.Furthermore, if we assume that no new wealth is added to the portfolio, thewealth at the end of a period must equal the wealth at the beginning of thenext, i.e.hn+1= an+1,1pn+1,1+ · · · + an+1,Mpn+1,M(2)= an,1pn+1,1+ · · · + an,Mpn+1,M(3)Notice that from (3)Rn=an,1pn+1,1hn+ · · · +an,Mpn+1,Mhn(4)=an,1pn,1hnpn+1,1pn,1+ · · · +an,Mpn,Mhnpn+1,Mpn,M(5)= bn,1Xn,1+ · · · + bn,MXn,M(6)= btnXn(7)where we definebn,m=an,mpn,mhn(8)as the fraction of wealth in asset m at the beginning of period n,Xn,m=pn+1,mpn,m(9)2as the factor by which asset m increased in price from the beginning to the endof period n, btn= (bn,1. . . bn,M), and Xtn= (Xn,1. . . Xn,M).A constant-rebalanced portfolio (CRP) is an investment strategy where thepercentage allocation of wealth in each asset is constant over time, i.e. b1=b2= · · · = bN= b. A buy-and-hold (B&H) portfolio is a constant rebalancedportfolio that places and leaves all the initial wealth in a single asset, i.e. b =em= (0, 0, . . . , 0, 1, 0, . . . , 0)tis the m-th standard Euclidean basis vector.We define the factor by which the wealth has increased from the beginningof the first period to the end of the N-th period asSN=hN+1h1(10)=hN+1hNhNhN−1· · ·h2h1(11)=NYn=1Rn(12)=NYn=1btnXn(13)Assuming that btnXn> 0 for n = 1, . . . , N, we can further manipulate the totalreturn SNto giveSN= exp NXn=1log btnXn!(14)= exp (NWN) (15)where log() denotes the natural logarithm, andWN=1NNXn=1log btnXn(16)is called the “doubling” rate of the portfolio.3 Universal Portfolios3.1 Investment ObjectiveThe goal of the universal portfolio [3] is to “track” the return of the best constantrebalanced portfolio (CRP) chosen after future asset outcomes are revealed.That is, the universal portfolio tries to trackS∗N= maxb∈BSN(b) = maxb∈BWN(b) (17)where B = {b ∈ RM: bm≥ 0, m = 1, 2, . . . , M ;PMm=1bm= 1} is the setof allowable CRPs. Denote a CRP that returns S∗Nas b∗N. Notice that the3restriction bm≥ 0 prohibits the use of short sales, i.e. selling an asset beforebuying it.3.2 Universal Portfolio AlgorithmThe universal portfolio algorithm is the return-weighted average of all CRPs [3]:Initialize:ˆb1=1M, · · · ,1MtFor n > 1:ˆbn=RBbSn−1(b)dbRBSn−1(b)dbThe universal portfolio is the weighted average of all CPRs, with greateremphasis placed on those portfolios with larger returns.3.3 PerformanceThe universal portfolio’s return is the average of all CRPs’ returns [3]:ˆSN=NYn=1ˆbtnXn=RBSN(b)dbRBdb(18)As a consequence of Jensen’s inequality, it is also greater than the geometricmean of stock returns (i.e. value line index) [3]:ˆSN≥ MYm=1SN(em)!1/M(19)Under many assumptions, the returns of the universal and best CRP port-folio have the same asymptotic growth rate in the exponent [3]:ˆSNS∗N∼ r2πN!M−1(M − 1)!|J∗|1/2(20)in the sense that the ratio of the left and right hand sides equals one for large N .Here, J∗Nis a “sensitivity matrix” defined as the curvature (Hessian) of WN(b)at its maximum, and J∗is this curvature for large N. The asymptotic propertyin (20) holds if the stock sequence X1, X2, . . . , satisfies:• a ≤ Xn,m≤ q, m = 1, . . . , M,n = 1, 2, . . ., for some 0 < a ≤ q < ∞• The sensitivity matrix J∗N→ J∗for some positive definite matrix J∗• The best CRP b∗N→ b∗for some b∗in the interior of the portfolio simplexBand there exists a function W (b) such that4• The doubling rate WN(b) % W (b)∗(i.e. converges uniformly and mono-tonically) for any portfolio b ∈ B• W (b) is strictly concave• W (b) has bounded third partial derivatives• W (b) achieves its maximum at b∗in the interior of the portfolio simplexBFor proofs of these