Mean–variance efficient portfolios with many assets: 50% short

Abstract

Any given set of asset parameters yields a specific mean–variance optimal tangency portfolio. Yet, when the number of assets is large, there are some general characteristics of optimal portfolios that hold ‘almost surely’. This paper investigates these characteristics. We analytically show that the proportion of assets held short converges to 50% as the number of assets grows. This is a fundamental and robust property of mean–variance optimal portfolios, and it does not depend on the parameter estimation method, the investment horizon, or on a special covariance structure. While it is known that optimal portfolios may all have positive weights in some special situations (e.g. uncorrelated assets), the analysis shows that these cases occupy a zero measure in the parameter space, and therefore should not be expected to be observed empirically. Thus, our analysis offers a general explanation for the empirical finding of many short positions in optimal portfolios.

Keywords:

• Portfolio analysis
• Portfolio optimization
• Mean-variance analysis
• Stochastic matrix analysis

Acknowledgements

We are grateful to the editors and two anonymous referees for their helpful comments and suggestions. Financial support from the Zagagi Fund is thankfully acknowledged.

Notes

†See, for example, Wigner (1951), Carmeli (1983) and Mehta (1991). Longstaff et al. (2001) employ a similar approach to investigate the covariance structure among forwards. Recently, several researchers have employed random matrix theory in the context of portfolio optimization (see, for example, Plerou et al. 1999, 2000, 2001, Laloux et al. 1999, 2000 and Bouchaud and Potters 2000). The main idea in this novel line of research is to compare the eigenvalue spectrum of the empirical covariance matrix with the eigenvalue spectrum of a random matrix, and to filter out the random or ‘noise’ elements. Sharifi et al. (2004) and Malevergne and Sornette (2004) discuss the stability properties of this filtering process.

‡The mean–variance framework is usually justified based on the assumption of normal return distributions or, alternatively, on the assumption of quadratic preference (see Chamberlain 1983, Owen and Rabinovitch 1983 and Berk 1997 for generalizations). Levy and Markowitz (1979) show that mean–variance analysis provides an excellent approximation for expected utility maximization even if preferences are not quadratic and the return distributions are not normal. Note that the analysis in this paper does not depend on the normality assumption.

§Note that the inverse must also have all the diagonal elements equal (denoted by a) and all the off-diagonal elements equal (denoted by b). From the condition (where C is the covariance matrix) we have and . These two equations for a and b lead to

†For example, if one takes all the stocks in the CRSP database with complete monthly return records for the 60-month period 2002–2006, one finds an average expected monthly return of 1.7%, and a slightly lower median expected return of 1.4%. The proportion of stocks with an expected return lower than the average expected return () is 57%. Panel A of figure A1 in the appendix shows the empirical distribution of expected returns. In the context of the equal correlation case with equal standard deviations, this implies that 57% of the stocks will be held short in the optimal mean–variance portfolio. 

‡

Thus, as , we have .

†Taking all the stocks in the CRSP database with complete monthly return records for the 60-month period 2002–2006, one finds an average μ/σ value of 0.175, and a median value of 0.172. The proportion of stocks with is 50.9%. Panel B of figure A1 in the appendix shows the empirical distribution of μ/σ.

‡The portfolio mean return and standard deviation are given by and . In calculating the Sharpe ratio one can use either the scaled (x) or the unscaled (ω) portfolio proportions. The scaled proportions are obtained by dividing ω _i by the sum , and this term cancels out in the numerator and the denominator of the Sharpe ratio.

§It is easiest to see that the Sharpe ratio is bounded when shortselling is excluded by analysing the simple case of equal correlations and equal standard deviations. In this case, if all portfolio weights are positive, the portfolio expected excess return is bounded from above by . The portfolio variance is given by . The minimal portfolio variance is obtained in the symmetric case of Thus, the minimal portfolio variance is . For large N we have , and the Sharpe ratio is therefore bounded from above by . A similar argument holds when the standard deviations are heterogeneous.

†Note that this is not in violation of the arbitrage pricing theory no-arbitrage condition—the expected return of each stock may be given by the APT framework. However, unless all excess returns are equal to zero, the market price of risk grows indefinitely with the number of assets (see also Chamberlain and Rothschild 1983).

†This is consistent with the empirical results showing that when the number of assets is large but shortselling is not allowed, typically only a very few assets are held in the optimal portfolio, and most assets have a portfolio weight of zero.

‡Of course, this statement is not mathematically rigorous. For an exact mathematical formulation, one has to make more specific assumptions about the covariance matrix, such as those in section 2 or 3.

§Note that even if the economy is structured, e.g. the CAPM holds and μ _j is linearly related to β _j , is almost independent of μ _j when the number of assets is large.

¶We choose a relatively narrow range for the correlations, because otherwise, as the number of assets becomes large, it becomes increasingly difficult to generate positive definite matrices by the procedure described above.

†Note that the derived optimal portfolio weights and the resulting optimal ‘market portfolio’, by definition, make the return parameters drawn consistent with the CAPM risk–return relationship.

‡

§The condition is or . The probability for this condition holding for a stock with randomly drawn parameters is 

¶We thank an anonymous referee for suggesting this analysis.

⊥This is similar to the procedure employed by Green and Hollifield (1992). However, as Green and Hollifield construct portfolios of up to 50 stocks, they require only a five-year period of complete monthly records (in order for a covariance matrix estimated from historical data to be non-singular, the number of time periods over which returns are observed must exceed the number of assets). We take a 20-year period because we construct portfolios of up to 200 stocks. While this introduces a survivorship bias, we do not believe that this bias plays any significant role in our empirical analysis.

†The empirical distribution is obtained by employing a non-parametric density estimate with a Gaussian kernel and the ‘normal reference rule’ (Scott 1992, p. 131).

†For example, Kandel (1984) shows that, for any set of N–1 assets, one can mathematically construct an Nth asset such that the mean–variance optimal portfolio is positively weighted (see theorem 1, p. 67 of Kandel (1984) and p. 1066 of Green and Hollifield (1992)). While such an Nth asset always exists mathematically, in large markets this asset may be very a typical and unrealistic. To see this, consider the 200 assets randomly selected from the CRSP file as described in section 2. What are the characteristics of the 201st asset which makes the optimal portfolio positively weighted (say, with an investment proportion of 1/201 in each asset)? Following the procedure of Kandel for characterizing this asset (Kandel 1984, p. 67), we find that the added asset should have a monthly standard deviation of at least 642%. The expected monthly return of this asset is 64,020% (!). Thus, while it is always possible to mathematically construct an Nth asset which makes the optimal portfolio positively weighted, this does not imply that it is reasonable to expect the existence of a positively weighted optimal portfolio for a general (or empirical) set of parameters.