Multifactor Risk Models and Heterotic CAPM

Abstract

ZK: To my children Mirabelle and Maximilien Kakushadze WY: To my parents Albert and Ribena Yu AbstractWe give a complete algorithm and source code for constructing general multifactor risk models (for equities) via any combination of style factors, principal components (betas) and/or industry factors. For short horizons we employ the Russian-doll risk model construction to obtain a nonsingular factor covariance matrix. This generalizes the heterotic risk model construction to include arbitrary non-industry risk factors as well as industry risk factors with generic "weights". The aim of sharing our proprietary know-how with the investment community is to encourage organic risk model building. The presentation is intended to be essentially self-contained and pedagogical. So, stop wasting money and complaining, start building risk models and enjoy! 4

ETF Risk Models

“…Heterotic risk models [Kakushadze, 2015b], [Kakushadze and Yu, 2016a] based on such industry classifications sizably outperform statistical risk models. One can also include non-industry style factors, which are based on stocks' estimated/measured properties, e.g., size, value, growth, momentum, volatility, liquidity, etc.…”

“…Given a time-series of stock returns 𝑅 𝑖𝑠 (where 𝑠 = 1, … , 𝑇 labels the times (e.g., trading days) in the time-series) and the 𝑁 × 𝐾 factor loadings matrix 𝛺 𝑖𝐴 (were 𝑖 = 1, … , 𝑁 labels stocks, while 𝐴 = 1, … , 𝐾 labels risk factors), one can construct a model covariance matrix 𝛤 𝑖𝑗 (which replaces the sample covariance matrix 𝐶 𝑖𝑗 ) using the explicit algorithm and source code given in [Kakushadze and Yu, 2016a]. The factor loadings 𝛺 𝑖𝐴 can be thought of (up to normalizations and/or factor rotations) as weights with which individual stocks contribute into the factors.…”

“…However, their number is limited, it is basically of order 10 or so. 11 Furthermore, contrary to an apparent common misconception, style factors generally are poor proxies for modeling correlations and add little to no value [Kakushadze and Yu, 2016a].…”

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We discuss how to build ETF risk models. Our approach anchors on i) first building a multilevel (non-) binary classification/taxonomy for ETFs, which is utilized in order to define the risk factors, and ii) then building the risk models based on these risk factors by utilizing the heterotic risk model construction of [Kakushadze, 2015b] (for binary classifications) or general risk model construction of [Kakushadze and Yu, 2016a] (for non-binary classifications). We discuss how to build an ETF taxonomy using ETF constituent data.

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*K-means and cluster models for cancer signatures

“… 30 In the financial context, these are known as statistical risk models [9] . For a discussion and literature on multifactor risk models, see, e.g., [30] , [31] and references therein. For prior works on fixing the number of statistical risk factors, see, e.g., [32] , [33] .…”