In this paper, we study the construction of US mean-variance efficient portfolios during the period 1999–2017. We construct mean-variance portfolios by maximizing the 10-factor U.S. Expected Return stock selection model (USER) alphas and constraining Tracking Error with respect to the S&P 500 benchmark using 3-factor risk model of Blume et al. [1]¹. The main finding of this paper is that the mean-variance efficient portfolios produce statistically significant portfolio excess returns in the US market.

The organization of the paper is as follows. The first section describes the construction of efficient portfolios, estimation of covariance matrix with multi-factor models, and the data used in construction of size ranked portfolios in estimating common risk factors. The second section describes the expected excess return model used in the study, statistical estimation method, and the data. The third section describes construction of tracking portfolios and presents portfolio statistics. The final section contains concluding remarks.

The Markowitz portfolio construction approach is based on the premise that mean and variance of future outcomes are sufficient for rational decision making under uncertainty, to identify the best opportunity set, efficient frontier, where returns are maximized for a given level of risk, or minimize risk for a given level of return. The reader is referred to Markowitz [14, 15] for the seminal discussion of portfolio construction and management. The two parameters needed are the portfolio expected return, E(R_p) is calculated by taking the sum of the security weights, w multiplied by their respective expected returns, and the portfolio standard deviation is the sum of the weighted covariances.

where, E = {μ₁, μ₂,…,μ_N} is N × 1 vector of expected security returns, (N is the number of candidate securities), Ω is the N × N covariance matrix, W = {w₁, w₂,…, w_N} is the vector of weights, and 1 is the unit column vector. Sum of weights in (3) indicates that the portfolio is fully invested.

One can construct infinite number of Mean-Variance efficient portfolios. Optimal portfolio choice decision will be determined by an investors' risk tolerance². Following Markowitz's [14, 16–18] general portfolio optimization objective function is:

where, λ is the coefficient of relative risk aversion of the investor³.

Accurate characterization of portfolio risk requires an accurate estimate of the covariance matrix of security returns.

Estimation of the covariance structure is almost always based on a linear return generating multi-factor model (MFM) in the form of:

The non-factor, or asset-specific return on security j e~j,t is the residual return of the security after removing the estimated impacts of the finite number of K factors where 1 ≤ K ≤ N. The term f~k,t is the rate of return of factor “k,” which is independent of securities and affects the security's return through its exposure coefficient β_jk. Under the assumption that the residual return e_j,t is not correlated across securities the covariance matrix of the securities is reduced to form:

B in (7) is the matrix of exposure coefficients, also referred as “loadings” in the literature, θ in (8) is the covariance matrix of the factors, and ⋎ in (9) is the covariance of the residuals.

The very first model used in the literature is Treynor's market model that led to development of the Capital Asset Pricing Model (CAPM)⁴. There is a rich volume of research covering multi factor models starting with King [4].

In this paper, we use a statistical risk model developed by Blume et al. [1], (BGG). Statistical factor models deduce the appropriate factor structure by analyzing the sample asset returns covariance matrix. There is no need to pre-define factors and compute exposures, as required by fundamental factor models. The only inputs are a time-series of asset returns and the number of desired factors. BGG has shown that return generating model based on factors analysis estimation is superior to commonly used multi-factor models used in the literature.

There are many approaches to security valuation and the creation of expected returns. We believe that asset managers use security analysis and stock selection models consisting of reported earnings, forecasted earnings and financial data⁵. Graham and Dodd [27] recommended that stocks be purchased on the basis of the price-earnings (P/E) ratio. The “low” PE investment strategy was discussed in Williams [28], the monograph that influenced Harry Markowitz and his thinking on portfolio construction. Bloch et al. [29] and Haugen and Baker [30, 31] advocated models incorporating earnings-to-price (EP), book-value-to-price, BP, cash flow-to-price, CP, sales-to-price, SP, and other fundamental data. Guerard et al. [32, 33] added price momentum (PM), price at t-1 divided by the price 12 months ago, t-12, and consensus temporary earnings forecast (CTEF) to expected returns modeling. They denoted the stock selection model as United States Expected Returns (USER). They reported, among other results, that: (1) the EP variable had a larger average weight than the BP variable; (2) the relative PE, denoted RPE, the EP relative to its 60-month average had a higher average weight than the PE variable; and (3) the composite earnings forecast variable, CTEF, had a larger weight than the RPE variable. In fact, in the USER model, only the price momentum variable, PM, had a higher weight than the CTEF variable (and only by one percent, at that)⁶.

In this paper, we use the same USER Model.

Given concerns about both outlier distortion and multicollinearity, Bloch et al. [29] tested the relative explanatory and predictive merits of alternative regression estimation procedures: OLS, robust regression using the Beaton and Tukey [42] bi-square criterion to mitigate the impact of outliers, latent root to address the issue of multicollinearity [see [43]], and weighted latent root, denoted WLRR, a combination of robust and latent root. The Guerard et al. [33] USER model test substantiated the Bloch et al. [29] approach, techniques, and conclusions that WLLR works best among the alternative linear predictive models.

We construct monthly long only, i.e., w_i ≥ 0, portfolios to track S&P 500 index with minimum tracking error by solving the following equation:

where λ is the relative risk aversion, 1- is the unit, vector X_t = {x_1,t, x_{2, t}, …, x_N,t} is the vector of binary variables that indicate if the security i is included in the portfolio in month t, Z_t = {|x_{1, t} − x_{1, t−1}|, |x_{2, t} − x_{2, t−1}|, …, |x_{N, t} − x_{N, t−1}|} is the vector of binary variables to account for security turnover, c is the transaction cost, p is the portfolio turnover limit percentage, and M is the maximum number of securities allowed in the portfolio. Variance of S&P 500 for the period in Equation (12) is estimated with Equation (10) using the factor loadings of the 3-factor model. We specifically solve Equation (12) with a relatively small number of securities (M), set at 50 and 100 or less, transactions cost (c), set 150 basis points each way, and portfolio turnover (p) set at 8 percent or less⁷.

In Table 1, we present portfolio statistics for each year for relative risk aversion of 0.01, 0.05, and 0.10. Excess return is the annual portfolio return net of truncations costs in excess of the annual return of S&P 500. Relative Sharpe ratios is portfolio Sharpe ratio divided by Share ratio of S&P 500. The active average excess returns of the USER model are statistically significant. Tracking error is not statistically significant for 100-security portfolio.

Investing with fundamental, expectations, and momentum variables is a good investment strategy over the long run. The use of multi-factor risk-control significantly improves portfolios performance relative to the benchmark. We considered long only portfolio construction in this study. Construction of realistic Long–Short portfolios are not feasible under these settings unless one assumes that securities are always available to borrow to short sell. However, there are various actively traded derivative securities based on S&P 500 index, the benchmark used in this study. Portfolios constructed in the study tracks the S&P Index with reasonably low tracking error. With the use of these derivative securities, it is possible to expand the opportunity set for investors.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation, to any qualified researcher.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

JG is employed by company McKinley Capital Management, LLC. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.