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

Transcription

1 1 IDENTIFYING BROAD AND NARROW FINANCIAL RISK FACTORS VIA CONVEX OPTIMIZATION: PART II Alexander D. Shkolnik MMDS Workshop. June 22, joint with Jeffrey Bohn and Lisa Goldberg.

2 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 2 Overview Simulations (synthetic data). Performance metrics (finance). benchmark: PCA. Theoretical considerations for N > T.

3 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 3 Simulations (synthetic data) N = 125 and N = 500 securities. T = 250 observations (one year of daily data). K = 2 broad factors (16% ann. vol., 4% ann. vol.) κ N log N and κ N countries. 400 simulations (realizations of a sample covariance matrix). Convex program parameters: θ = (γ, λ), γ = 1 N (see Candès, Li, Ma & Wright (2011)), λ set to to control the number of recovered broad factors.

4 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 4 Input to algorithm (N = 125, T = 250)

5 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 5 Ordered by country

6 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 6 True vs recovered broad factor matrix L K

7 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 7 True vs recovered narrow factor matrix L κ

8 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 8 True vs recovered specific risk matrix

9 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 9 Measuring performance on simulated data Typically, for estimator Σ (θ), consider Σ (θ) Σ for some norm. Instead, consider the variance of returns to portfolio w. Var Σ (w) = w Σw (1) The portfolio risk forecasting ratio is computed as R Σ (w θ) = Var Σ(θ)(w) Var Σ (w). (2) Ratios R LK, R Lκ and R Δ are defined analogously. Other estimators: K (θ) and κ (θ).

10 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 10 Test portfolios Let e = (1,, 1). Equally weighted portfolio is given by w = e N (3) Minimum variance (long only) portfolio is the solution w (θ) of min w w Σ (θ) w (4) subject to w e = 1, (5) w 0. (6) Minimum variance (long/short): solves (4) subject to w e = 1, w (θ) = Σ 1 (θ) e e Σ 1 (θ) e. (7)

11 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 11 Number of factors: equally weighted portfolio N Metric DSLR: (λ 10 3 ) Ground Truth PCA K, κ 500 K (θ) (stdev) (0.83) (0.44) (0.00) κ (θ) n/a (stdev) (0.21) (0.86) (0.00) 2 22 T = 250 (observations).

12 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 12 Performance: equally weighted portfolio N Metric DSLR: (λ 10 3 ) PCA Ground Truth Ann. Vol. 500 R Σ (stdev) (0.08) (0.08) (0.08) R LK (stdev) (0.09) (0.09) (0.10) R Lκ n/a (stdev) (0.09) (0.05) (0.00) R Δ (stdev) (0.01) (0.01) (0.01)

13 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 13 Performance: optimized portfolio N Metric DSLR: (λ 10 3 ) PCA Ground Truth Ann. Vol. 500 R Σ (stdev) (0.06) (0.06) (0.07) R LK (stdev) (0.08) (0.10) (0.13) R Lκ n/a (stdev) (0.10) (0.07) (0.00) R Δ (stdev) (0.02) (0.03) (0.05)

14 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 14 Summary of findings SLRD ourperforms classical PCA on risk forecasts. Accurate sparse component recovery implies: accurate R Δ forecast, accurate κ (θ) estimates. Inaccurate K (θ) estimates does not imply poor risk forecast. N > T?

15 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 15 κ (θ) estimates (N > T ) With high probability CPW a recovers correctly the rank of and sparsity support of. Rate of convergence N T arises from bounds on (θ) 2 and (θ) l. (8) The error on (θ) l may be improved to order log N T. (9) When sparsity pattern is correct, κ (θ) is correct. a (Chandrasekaran, Parrilo & Willsky 2012)

16 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 16 K (θ) estimates (N > T ) With high probability CPW a recovers correctly the rank of and sparsity support of. Rate of convergence N T arises from bounds on (θ) 2 (10) Cannot estimate number of broad factors accurately for N > T. Numerical experiments suggest (θ) l may be order log N T. (11) a (Chandrasekaran et al. 2012)

17 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 17 Risk forecasts Equally weighted: accuracy of portfolio risk forecast depends on R Σ (w θ) 1 (θ) l (θ) l (12) ( log N ) = O. (13) T Optimized: same bound holds with different constants. Similar for factor and specific risk forecasts. Support for accuracy of risk forecasts in log (N ) T 0 regime.

18 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 18 Conclusions The approach shows promise in extracting narrow factors that highlight country/industry relationships in data. Financial applications require a reformulation of available low rank and sparse decomposition methods. low-rank + sparse + diagonal decomposition, sparse eigenvectors (narrow factors). Finance oriented performance metrics may lead to an alternative analysis of estimator consistency.

19 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 19 Ongoing & future work Theoretical performance guarantees for risk forecasting ratios. Data-driven methods of selecting optimal paramters θ = (λ, γ ). Alternative convex programs guided by performance metrics. Scale algorithms to tens of thousands of securities.

20 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 20 Questions.

21 Identifying Broad and Narrow Financial Risk Factors via Convex Optimization 21 R Δ (specific risk) forecast MTFA a decomposes a given matrix S into a low rank L κ = XGX component and a diagonal component Δ exactly if max x X {0 N } x x 2 < 1 2. (14) where X is the space spanned by the narrow factor exposures in X. A sufficient condition for (14) is x 2 i < i j x 2 j for all i and x X. (15) Robust recovery for factors that are not too narrow. a (Saunderson, Chandrasekaran, Parrilo & Willsky 2012)

22 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),