How to combine a billion alphas

Abstract

We give an explicit algorithm and source code for computing optimal weights for combining a large number N of alphas. This algorithm does not cost O(N 3 ) or even O(N 2 ) operations but is much cheaper, in fact, the number of required operations scales linearly with N . We discuss how in the absence of binary or quasi-binary "clustering" of alphas, which is not observed in practice, the optimization problem simplifies when N is large. Our algorithm does not require computing principal components or inverting large matrices, nor does it require iterations. The number of risk factors it employs, which typically is limited by the number of historical observations, can be sizably enlarged via using position data for the underlying tradables.

*K-Means and Cluster Models for Cancer Signatures

“…One thing to keep in mind is that in *K-means one sifts through a large number P of aggregations, which can get computationally costly when clustering 2000+ stocks into 100+ clusters. 47 Another potential application is in the context of combining alphas (trading signals) -see, e.g., [Kakushadze and Yu, 2017a]. Yet another application is when we have a term structure, such as a portfolio of bonds (e.g., U.S. Treasuries or some other bonds) with varying maturities, or futures (e.g., Eurodollar futures) with varying deliveries.…”

Decoding stock market with quant alphas

“…23 So, why are all q AA ≫ 1? This is the case when [Kakushadze and Yu, 2017a]: i) N is large, and ii) there is no "clustering" in the vectors β iA . That is, we do not have vanishing or small values of β 2 iA for most values of the index i with only a small subset thereof having β 2 iA ∼ > 1.…”

“…10 E.g., turnover, etc. See [Kakushadze, 2014] and [Kakushadze and Yu, 2017a] for details. 11 Up to corrections suppressed by powers of 1/N , where N is the number of alphas.…”

Dead alphas as risk factors

“…As and taking the residuals, up to overall normalization factors), we can take a different route. We can take good alphas labeled by i ∈ J and i) either combine them with some weights (see, e.g., [Kakushadze and Yu, 2017a]) and trade the resultant portfolio, or ii) calculate the expected returns for stocks using the expected returns for alphas as in [Kakushadze and Yu, 2017c] and directly trade a portfolio of stocks based on these expected returns (without combining alphas). In the case i) above we get a stock portfolio which can be further optimized, e.g., by maximizing the Sharpe ratio [Sharpe, 1994] or via mean-variance optimization [Markowitz, 1952].…”

“…What we end up extracting from this data is essentially a risk factor loadings matrix using which we can neutralize (via, e.g., a regression) stock positions of good alphas or build a multifactor risk model for optimizing our stock portfolio. This stock portfolio is obtained i) either via combining good alphas with some weights (see, e.g., [Kakushadze and Yu, 2017a]), or ii) by calculating the expected returns for stocks using the expected returns for good alphas as in [Kakushadze and Yu, 2017c] and directly trading a portfolio of stocks based on these expected returns (without combining alphas).…”