PM ] 3 D ec 2 01 5 Shrinkage = Factor Model

Shrunk sample covariance matrix is a factor model of a special form combining some (typically, style) risk factor(s) and principal components with a (block-)diagonal factor covariance matrix. As such, shrinkage, which essentially inherits out-of-sample instabilities of the sample covariance matrix, is not an alternative to multifactor risk models but one out of myriad possible regularization schemes. We give an example of a scheme designed to be less prone to said instabilities. We contextualize this within multifactor models. 1 Zura Kakushadze, Ph.D., is the President of Quantigic Solutions LLC, and a Full Professor at Free University of Tbilisi. Email: zura@quantigic.com 2 DISCLAIMER: This address is used by the corresponding author for no purpose other than to indicate his professional affiliation as is customary in publications. In particular, the contents of this paper are not intended as an investment, legal, tax or any other such advice, and in no way represent views of Quantigic Solutions LLC, the website www.quantigic.com or any of their other affiliates. In his seminal work on mutual fund performance, Sharpe [1966] eloquently posits: “The key element in the portfolio analyst’s view of the world is his emphasis on both expected return and risk. ” Construction of a trading portfolio for equities schematically can be thought of as consisting of two steps. First, one comes up with some expected returns for stocks in the trading universe. This drives the “reward” part of the portfolio. Second, one constructs the portfolio holdings based on these expected returns. It is mainly this stage that deals with the “risk” part. Many incarnations of this second step, including mean-variance optimization [Markowitz, 1952] and its numerous variations, Sharpe ratio [Sharpe, 1994] maximization, etc., require inverting a covariance matrix of returns. When the number of stocks in a portfolio is large and the number of available (relevant) observations in the historical time series of returns is limited, the sample covariance matrix (SCM) based on these historical returns is (badly) singular. Thus, if M+1 is the number of observations in the time series and N is the number of stocks in the portfolio, SCM is singular if M < N . Furthermore, unless M ≫ N , which is rarely – if ever – the case in practice, the off-diagonal elements of SCM are not out-of-sample stable. One method often used for mitigating these issues is the so-called shrinkage [Ledoit and Wolf, 2004]. It is often regarded as an “alternative” to multifactor risk models. However, as we discuss below, shrunk SCM is in fact a factor model of a special form based on a combination of some risk factors and principal components. The idea behind shrinkage is simple. Instead of using SCM Cij, one uses its weighted linear combination with another matrix (“shrinkage target”), call it ∆ij : C̃ij = q ∆ij + (1− q)Cij (1) Here the weight (“shrinkage constant”) 0 ≤ q ≤ 1. The matrix ∆ij is assumed to be positive-definite and (relatively) stable out-of-sample. We must have C̃ii = Cii, so ∆ii = Cii. A priori ∆ij can be otherwise arbitrary, in which case the “shrunk” matrix C̃ij can be thought of as a regularization of SCM Cij . Thus, when q ≪ 1, C̃ approaches C. Furthermore, even when C is singular, C̃ is invertible for q > 0. 3 Deliberately omitting many important details, that is. 4 Albeit, some elements of “risk management” can be (and at times are) incorporated into the expected returns, e.g., sector or industry neutrality. 5 Let Ris be the time series of our returns, i = 1, . , N , s = 0, 1, . ,M . Let Xis = Ris−Ri be the serially demeaned returns, where Ri is the time-series mean of Ris. In matrix notation SCM is given by C = 1 M XX . (We are assuming M ≫ 1, so the difference between the unbiased estimate with M in the denominator vs. the maximum likelihood estimate with M+1 in the denominator is immaterial for our purposes.) Since at most M columns of the matrix X are linearly independent (as its column sums ∑M s=0 Xis = 0), the rank of the matrix Cij is at most M if M < N . 6 This statement is often regarded as stemming from empirical evidence. However, it is wellunderstood theoretically. We can always rotate our serially demeaned returns Xis to an orthogonal basis and rescale them to have unit serial variances. Then the true covariance matrix is the N ×N identity matrix. Pursuant to the Bai-Yin theorem [Bai and Yin, 1993], the smallest and largest eigenvalues of SCM have the limits λmin = (1 − √ y) and λmax = (1 + √ y), where y = N/M is fixed and N,M → ∞. So for M,N ≫ 1 we must have M ≫ N for all eigenvalues to be close to 1. 7 These are typically, but not necessarily, style factors.