IDENTIFYING BROAD AND NARROW FINANCIAL RISK FACTORS VIA CONVEX OPTIMIZATION: PART I

Transcription

1 1 IDENTIFYING BROAD AND NARROW FINANCIAL RISK FACTORS VIA CONVEX OPTIMIZATION: PART I Lisa Goldberg lrg@berkeley.edu MMDS Workshop. June 22, joint with Alex Shkolnik and Jeff Bohn.

2 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 2 Recovery problem Assumption: security returns are driven by a relatively small number of risk factors plus security specific return Return due to factors is undiversifiable Specific return can be diversified away R = Zζ + ε, (1) Consequence: a security return covariance matrix can be expressed as a sum Σ = L + Δ (2) Problem: given a sample covariance matrix, recover the true systematic and specific components.

3 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 3 Frameworks for financial return covariance matrix decomposition Fundamental analysis (Barra factors, identified by human analysis) Statistical methods (PCA factors, eigenvectors with large eigenvalues) Convex optimization (low rank plus sparse plus diagonal decompositions disentangle broad factors, narrow factors and firm-specific effects)

4 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 4 Frameworks for financial return covariance matrix decomposition Fundamental analysis (Barra factors, identified by human analysis) Statistical methods (PCA factors, eigenvectors with large eigenvalues) Convex optimization (low rank plus sparse plus diagonal decompositions disentangle broad factors, narrow factors and firm-specific effects)

5 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 5 Frameworks for financial return covariance matrix decomposition Fundamental analysis (Barra factors, identified by human analysis) Statistical methods (PCA factors, eigenvectors with large eigenvalues) Convex optimization (low rank plus sparse plus diagonal decompositions disentangle broad factors, narrow factors and firm-specific effects)

6 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 6 Practical considerations Attributes of risk factors Broad, dominant: market Broad: interest rates, credit, equity styles (in an equity-only model) Narrow: country, industry, currency Emerging: carbon, cyberterrorism, longevity, new industries Non-stationarity Impact of estimation error on optimization (in portfolio construction) Data frequency Emerging structure on different scales

7 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 7 Refined factor model R is a N-vector of security returns driven by K broad factor returns, κ narrow factor returns and N specific returns. The return generating process follows R = Zζ + ε = Y ψ + Xφ + ε, (3) Y is a N K (non-random) matrix of broad factor exposures. X is a N κ (non-random) matrix of narrow factor exposures. (ψ, φ, ε) are random, uncorrelated factor and specific returns. Model assumptions: K N, κ < N and X is sparse.

8 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 8 Refined return covariance matrix Since the factor and specific returns are mutually uncorrelated Σ = L K + L κ + Δ (4) = L + Δ = L K + S is the security returns covariance matrix decomposition. L K = Y F Y is a low rank matrix (rank K N). F is diagonal matrix of broad factor risks. L κ = XGX is (often) a rank κ, sparse matrix (since X is sparse). G is diagonal matrix of narrow factor risks. S = L κ + Δ where Δ is a diagonal matrix of specific risks.

9 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 9 Refined recovery problem For T observations of the security return N-vector R, form the N N sample covariance matrix Σ. (5) Given an input Σ, output components ( LK, L κ, Δ ) (L = L K + L κ, S = L κ + Δ). (6) Performance metrics: measure the effectiveness of the recovery on simulated data and use the results to improve the algorithm.

10 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 10 Given a sample covariance

11 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 11 recover Σ = L K + L κ + Δ

12 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 12 Estimators of L K, L κ, Δ and Σ We develop (convex optimization) estimators ( LK (θ), L κ (θ), Δ (θ) ) of (L K, L κ, Δ). (7) θ is a parameter of the estimator (e.g. for classical PCA, θ would be the number of eigenvalues/eigenvectors in the SVD). a Yields an estimator of Σ as Σ (θ) = L K (θ) + L κ (θ) + Δ κ (θ). (8) a Classical PCA takes the T -sample covariance Σ = N i=1 λ iu k u k and computes L (θ) = θ λ k u k u k, Δ (θ) = diag( Σ L (θ) ). k=1

13 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 13 Two-step decomposition scheme Step 1: CPW (Chandrasekaran, Parrilo & Willsky 2012) convex program extracts (θ) and (θ). min λ( ) ( + γ l1 l ( ) Σ ), (9) (, ) subject to 0. (10) Convert solutions into L K (θ) and S (θ). Step 2: MTFA (Saunderson, Chandrasekaran, Parrilo & Willsky 2012) extracts diagonal from S (θ). Return estimators: L K (θ), L κ (θ) and Δ (θ). min (L κ,δ) L κ (11) subject to S (θ) = L κ + Δ, (12) L κ 0, Δ diagonal. (13)

14 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 14 Empirical study State street data set: 25, 000 global equities. N = 500 securities. T = 250 observations (one year of daily data). K =? broad factors. κ N countries. Convex program parameters: θ = (γ, λ), γ = 1 N (see Candès, Li, Ma & Wright (2011)), λ is set to to control the number of recovered broad factors.

15 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 15 Sample correlation, Oct. 2015

16 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 16 Low rank recovery (correlation; K(θ) = 25)

17 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 17 Sparse recovery (correlation; K(θ) = 25)

18 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 18 Eigenvalue recovery, K(θ) = 25

19 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 19 Summary Security return covariance matrices are central to financial risk management. Practical considerations complicate the process of recovering systematic and diversifiable components of a covariance matrix from sample returns. To date, this recovery has relied on human-powered fundamental analysis. Initial empirical experiments suggest sparse low rank diagonal (SLRD) recovery, powered by convex optimization, may have the potential to supersede fundamental analysis. SLRD is flexible enough to allow eigenvalues of low rank and sparse components to interleave.

20 Identifying Broad and Narrow Financial Risk Factors via Convex References Candès, Emmanuel J, Xiaodong Li, Yi Ma & John Wright (2011), Robust principal component analysis?, Journal of the ACM (JACM) 58(3), 11. Chandrasekaran, Venkat, Pablo A Parrilo & Alan S Willsky (2012), Latent variable graphical model selection via convex optimization, The Annals of Statistics (with discussion) 40(4), Saunderson, James, Venkat Chandrasekaran, Pablo A Parrilo & Alan S Willsky (2012), Diagonal and low-rank matrix decompositions, correlation matrices, and ellipsoid fitting, SIAM Journal on Matrix Analysis and Applications 33(4),