This article presents a new methodology for combining market-cap index (or “index”), factors, and alpha-seeking funds. The procedure complements a traditional optimization process rather than replacing it.

A Bayesian framework is used to model the combination of investors’ beliefs on manager skill and the observed track records of funds. The investor sets priors on Sharpe ratios or information ratios (IRs) in excess of the index and factor strategies, which has the advantage that these are important statistics used in evaluating fund managers and in asset allocation.

The intuition behind this framework is that the higher the conviction an investor has on alpha-seeking funds, the higher the prior mean and tighter the prior standard deviation on the active funds’ IRs and the more the active risk budget will be allocated to alpha versus factor strategies.

Factor strategies, on the other hand, have economic rationales and long histories. They can be implemented in low-cost and transparent vehicles, such as exchange-traded funds (ETFs). Therefore, investors’ degree of confidence in factors is likely to be different from their conviction in alpha-seeking strategies. Alpha should be delivering returns in excess of factor exposures in order for funds to be allocated to alpha-seeking strategies.

The authors derive the posterior distribution of Sharpe ratios subject to an imposed prior mean and variance, and they explicitly model management costs.

The authors illustrate the methodology with equities, but the methodology is generalizable to other index/factors/alpha in other asset classes and across a multi-asset portfolio.

In their illustration of the method, they use the S&P 500 Index as the market-cap index.

They use standard long-only factor investment strategies that track the minimum-volatility, momentum, value, small size, and quality factors—all of which are easily accessible through ETFs. They regress the factor index returns onto the index, and the residuals are then used to represent the factors.

The authors use the Morningstar database for US listed equity mutual funds as potential alpha-seeking funds. They compute monthly frequency active returns as the gross fund return minus the stated primary specified benchmark index. The excess returns of the alpha-seeking funds relative to commonly used systematic factors in the academic literature form their priors for the alpha-seeking funds. The methodology accommodates any well-defined priors. In this case, the authors take advantage of the long histories of published academic factors to inform the priors on the factors but they use investable factor indexes for investment.

The authors use a set of different prior beliefs on factors and alpha IRs and compare them with the resulting posterior and predictive means and variances. The prior mean IRs are set as the full data averages. They bootstrap using no-U-turn sampling (NUTS) to generate posterior and predictive information ratios, alphas, and the covariance matrix. They use automatic differentiation variational inference (ADVI) to initialize the sampling, setting the target acceptance ratio to 80%. For their posterior predictive analysis, they generate 50,000 simulated returns.

Finally, they construct mean–variance portfolios using the predictive moments: They maximize portfolio active net-of-fee return subject to a long-only constraint and limit the number of holdings to 10 or fewer.

Their illustrative portfolio predicts an excess return of 3.8% and an active risk of 2.0%, representing an excess return-to-active risk ratio of 1.9 above the large-cap benchmark. In this illustration, the portfolio allocates to only one factor, momentum, with no direct allocation to the market factor. Minimum volatility acts as a stabilizer: As active risk increases, the optimizer uses the higher risk budget to hold more aggressive positions in the alpha-seeking funds and the high-risk positions are funded by larger positions in lower-risk minimum volatility.

The authors demonstrate how investors can incorporate prior information on the three return sources (index, factors, and alpha) and implement the combination within existing portfolio construction techniques.