A Comparison of Universal and Mean-Variance Efficient Portfolios p. 1/28

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1 A Comparison of Universal and Mean-Variance Efficient Portfolios Shane M. Haas Research Laboratory of Electronics, and Laboratory for Information and Decision Systems Massachusetts Institute of Technology A Comparison of Universal and Mean-Variance Efficient Portfolios p. 1/28

2 Universal and MVE Portfolios Investment Dynamics Universal Portfolios Mean-Variance Efficient (MVE) Portfolios Examples Conclusions A Comparison of Universal and Mean-Variance Efficient Portfolios p. 2/28

3 Investment Problem Fixed parameters: M = number of assets N = number of periods to invest Market determines: p n,m = price of asset m at beginning of period n You choose: a n,m = number of asset m owned at beginning of per. n To control: h n = wealth at beginning of per. n A Comparison of Universal and Mean-Variance Efficient Portfolios p. 3/28

4 Wealth Assume: No new wealth added to portfolio Prices at the end of a period are equal those at beginning of next Wealth at beginning of period n: h n = a n,1 p n,1 + + a n,m p n,m Wealth at beginning of period n + 1: h n+1 = a n,1 p n+1,1 + + a n,m p n+1,m A Comparison of Universal and Mean-Variance Efficient Portfolios p. 4/28

5 Period Returns Return for period n: R n = h n+1 /h n = a n,1p n,1 h n p n+1,1 p n,1 + + a n,mp n,m h n p n+1,m p n,m = b n,1 X n,1 + + b n,m X n,m = b t nx n where b n,m = a n,mp n,m h n, b n = (b n,1... b n,m ) t X n,m = p n+1,m p n,m, X n = (X n,1... X n,m ) t A Comparison of Universal and Mean-Variance Efficient Portfolios p. 5/28

6 Cumulative Returns Return from period 1 to N: S N = h N+1 = = = h ( 1 hn+1 N n=1 N n=1 ) ( hn h N h N 1 R n b t nx n ) ( h2 h 1 ) A Comparison of Universal and Mean-Variance Efficient Portfolios p. 6/28

7 Doubling Rate Assuming that b t nx n > 0 for n = 1,..., N: ( N ) S N = exp log b t nx n where W N = 1 N n=1 = exp (NW N ) N log b t nx n n=1 is called the doubling rate of the portfolio. A Comparison of Universal and Mean-Variance Efficient Portfolios p. 7/28

8 Constant Rebalanced Portfolios Constant Rebalanced Portfolio (CRP): b 1 = b 2 = = b N = b Buy-and-Hold (B&H) Portfolio: b = e m = (0, 0,..., 0, 1, 0,..., 0) t Best CRP: b N = argmax b B S N (b) = argmax b B W N (b) where B = {b R M : b m 0, m = 1, 2,..., M; M m=1 b m = 1} A Comparison of Universal and Mean-Variance Efficient Portfolios p. 8/28

9 CRP Example [Blum97] Two Stocks: Stock #1: X n,1 = 1, n = 1,..., N { 12 n = even Stock #2: X n,2 = 2 n = odd Portfolios: S N 2 for any B&H portfolio Return of b = ( 1 2, 1 2 )t increases exponentially by 9/8 every two days: R n = { ( 12 ) (1) + ( 12 ) ( 12 ) = 3 4 n = even ( 12 ) (1) + ( 12 ) (2) = 3 2 n = odd A Comparison of Universal and Mean-Variance Efficient Portfolios p. 9/28

10 Equally Weighted CRP [Wang00] Equally weighted CRP: b = ( 1 M,, 1 M )t Assume X n are IID random vectors with E[X n ] = X and V = E[(X n E(X n ))(X n E(X n )) t ]: Expected return: E(R n ) = 1 M M m=1 X m Variance of return: ( var(r n ) = 1 1 M M = M m=1 ( M 2 M + ( 1 M M 2 v m,m ) ) 1 M 2 M M 2 M M m=1 k m v m,k ) ( M 2 ) M (avg. var.) + (avg. cov.) A Comparison of Universal and Mean-Variance Efficient Portfolios p. 10/28

11 Universal Portfolios Origin: T. Cover, "Universal Portfolios", Mathematical Finance, 1991 Objective: Track return of best CRP chosen after asset returns revealed (b N ) Algorithm: Return-weighted average of all CRPs Performance: Yields average return of all CRPs Return exceeds geometric mean of asset returns Asymptotically tracks best CRP return A Comparison of Universal and Mean-Variance Efficient Portfolios p. 11/28

12 Universal Portfolio Algorithm Return-weighted average of all CRPs: Initialize: ˆb1 = ( 1 M,, 1 M ) t where For n > 1: ˆbn = B bs n(b)db B S n(b)db B = {b R M : b m 0, m = 1, 2,..., M; M m=1 b m = 1} A Comparison of Universal and Mean-Variance Efficient Portfolios p. 12/28

13 Performance Regardless of the stock return sequence: Portfolio return is the average of all CRPs returns Ŝ N = N n=1 ˆbt n X n = B S n(b)db B db Return is greater than geometric mean Ŝ N ( M m=1 S N (e m ) ) 1/M A Comparison of Universal and Mean-Variance Efficient Portfolios p. 13/28

14 Asymptotic Performance Under many assumptions: Ŝ N S N ( 2π N ) M 1 (M 1)! J 1/2 where J = Hessian of W N (b) at its maximum for large N S N = return of best CRP b N and means the ratio of left- and right-hand sides equals one for large N A Comparison of Universal and Mean-Variance Efficient Portfolios p. 14/28

15 Assumptions for Asymptotic Perf. The stock sequence X 1, X 2,..., satisfies: a X n,m q, m = 1,..., M,n = 1, 2,..., for some 0 < a q < J N J for some positive definite matrix J b N b for some b in the interior of the B and there exists a function W (b) such that W N (b) W (b) 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 B A Comparison of Universal and Mean-Variance Efficient Portfolios p. 15/28

16 Outline of Proof Show LHS RHS: Expand W N (b) in Taylor series about the best CRP b N to three terms: First term is zero because gradient is zero at b N Second term is related to J N Third term is bounded Manipulate expression to look like a Gaussian CDF Bound the Gaussian CDF Show LHS RHS: Follows from Laplace s method of integration A Comparison of Universal and Mean-Variance Efficient Portfolios p. 16/28

17 Mean-Variance Efficient Portfolios Origins: Markowitz (1952) and Sharpe (1963) (1990 Nobel Laureates) Objective: To minimize the variance of portfolio returns, while maintaining a desired average return Algorithm: A solution to a constrained optimization problem Requires estimates of asset return mean and covariance matrix Performance: Tracks objective portfolio if asset returns are IID Otherwise, no guarantee A Comparison of Universal and Mean-Variance Efficient Portfolios p. 17/28

18 Objective of MVE Portfolio Minimize: Subject to: b t nv n b n 1) M m=1 b n,m = 1 where 2) b t n X n = R n X n = estimate of asset return mean V n = estimate of asset return covariance matrix R n = desired average return A Comparison of Universal and Mean-Variance Efficient Portfolios p. 18/28

19 The MVE Portfolio Mean and covariance estimators: X n = 1 Λ V n = 1 Λ n 1 k=1 n 1 k=1 λ n k 1 X k λ n k 1 (X k X n )(X k X n ) t where 0 < λ 1 and Λ = n 1 k=1 λn k 1 The MVE Portfolio: b n = w 0 (1 R n ) + w 1 Rn where w 0 and w 1 are portfolios that yield on average 0% and 100% returns, respectively A Comparison of Universal and Mean-Variance Efficient Portfolios p. 19/28

20 The 0% and 100% Avg. Ret. Ports. w 0 = 1 D ( c1 Vn 1 1 c 2 Vn 1 X n ) w 1 = 1 D ( c1 Vn 1 1 c 2 Vn 1 X n ) + 1 D ( c3 V 1 n X n c 2 V 1 n 1 ) where 1 t = (1,..., 1) c 1 = X nv t n 1 X n c 2 = X nv t n 1 1 c 3 = 1 t Vn 1 1 D = c 1 c 3 c 2 2 A Comparison of Universal and Mean-Variance Efficient Portfolios p. 20/28

21 Performance If asset returns are IID then: Estimators converge to their true values by LLN MVE portfolio asymptotically achieves objective Otherwise, no guarantee A Comparison of Universal and Mean-Variance Efficient Portfolios p. 21/28

22 Examples Data: Daily adjusted closing prices (yahoo.finance.com) January 2,1991 to August 8, 2001 (2,679 days) Missing data interpolated Examples: Universal and constant rebalanced portfolios Mean-Variance efficient portfolios Comparison: MVE portfolio allows short sales, but univ. portfolio does not No transaction costs Investor preferences A Comparison of Universal and Mean-Variance Efficient Portfolios p. 22/28

23 Univ. and CRPs of C & MSFT 38 Cumulative Returns for CRPs of C and MSFT Cum. Ret. from to CRPs Univ. Port. B&H C B&H MSFT Best CRP Portfolio Weight in C A Comparison of Universal and Mean-Variance Efficient Portfolios p. 23/28

24 Univ. and CRPs of MSFT & ORCL 90 Cumulative Returns for CRPs of MSFT and ORCL Cum. Ret. from to CRPs Univ. Port. B&H MSFT B&H ORCL Best CRP Portfolio Weight in MSFT A Comparison of Universal and Mean-Variance Efficient Portfolios p. 24/28

25 Univ. and CRPs of C & GE Cum. Ret. from to Cumulative Returns for CRPs of C and GE CRPs Univ. Port. B&H C B&H GE Best CRP Portfolio Weight in C A Comparison of Universal and Mean-Variance Efficient Portfolios p. 25/28

26 MVE Portfolios MV Eff. Port. B&H Ports. Mean Variance Efficient Portfolios MSFT ORCL Return Mean VFINX XOM PFE GE C WMT IBM Return Std. Dev. A Comparison of Universal and Mean-Variance Efficient Portfolios p. 26/28

27 Comparison Cumulative Return Parameters: Num. Rand. Port. = 5000 Des. MVE Port. Ret. = 0.12% Forgetting Fact. = 1.00 Daily Return Statistics: mean (std) [hi lo] Univ. Port. = 0.12 (1.38) [ ]% Best Port. = 0.22 (3.34) [ ]% MVE Port. = 0.10 (1.23) [ ]% VFINX B&H = 0.06 (0.97) [ ]% Eq. Wt. Port. = 0.12 (1.32) [ ]% Comparison of Portfolio Returns Univ. Port. Best CRP MVE Port. VFINX Eq. Wt. CRP Time [Day] A Comparison of Universal and Mean-Variance Efficient Portfolios p. 27/28

28 Conclusions Universal portfolio: Attempts to track return of best CRP given knowledge of future asset returns Yields average return of all CRPs Return greater than geometric mean Asymptotically tracks return of best CRP MVE portfolio: Attempts to minimize return variance while maintaining a desired average return Obtains objective in IID markets A Comparison of Universal and Mean-Variance Efficient Portfolios p. 28/28