![The Principal Eigenportfolio is connected to the concept of the Market Portfolio (Mod-
ern Portfolio Theory). If the entries of the principal eigenvector V
(1)
i are positive for all i,
the principal eigenportfolio has positive weights. This follows the Perron-Frobenius The-
orem.
Perron-Frobenius Theorem
Any square matrix Ai,j 0 with positive entries has a unique eigenvector with positive
entries (up to a multiplication by a positive scalar), and the corresponding eigenvalue has
multiplicity one and is strictly greater than the absolute value of any other eigenvalue.
That is
∃ λ∗
0, V ∗
0, ||V ∗
||2 = 1 s.t. Aλ∗
= λ∗
V ∗
. V ∗
→ right eigenvector
∀ λ, |λ| ≤ λ∗
V ∗
is unique up to scaling
The Perron-Frobenius vector applying Collotz-Wielandt formula
λ∗
= max
x≥0,x6=0
min
xi 6=0
[Ax]i
xi
= min
x0
max
[Ax]i
xi
This result can be extended to non-negative matrices
9th Annual Big Data Finance Conference Dec 2021](https://image.slidesharecdn.com/statisticalclusteringandportfoliomanagement-211209031157/85/Statistical-Clustering-and-Portfolio-Management-6-320.jpg)
![Mean-variance optimization:
I Monthly rebalancing using a 6-month estimation window
I HPCA statistical-based factor model (blue) outperforms the HPCA GICS-based factor
model (red) and the shrinkage estimator (salmon)
See [Fabozzi, Focardi and Kolm, 2010], [Fabozzi, Kolm and Focardi, 2006], [?].
9th Annual Big Data Finance Conference Dec 2021](https://image.slidesharecdn.com/statisticalclusteringandportfoliomanagement-211209031157/85/Statistical-Clustering-and-Portfolio-Management-26-320.jpg)
Statistical Clustering and Portfolio Management
- 1. Statistical Clustering, Hierarchical PCA and Portfolio Management Marco Avellaneda & Juan A. Serur Courant Institute of Mathematical Sciences, New York University 9th Annual Big Data Finance Conference Dec 2021 9th Annual Big Data Finance Conference Dec 2021
- 2. Introduction Quantitative factor analysis has become increasingly important in portfolio management. Some prominent works Sharpe → CAPM ˜ ri − rF = βi,M (˜ rM − rF ) (1) Ross → APT ˜ ri − rF = N X j=1 βi,j (˜ ri − rF ) (2) Fama-French (explicit factors: value, size, profitability, etc.) 9th Annual Big Data Finance Conference Dec 2021
- 3. Types of factor models Models based on explicit factors such as momentum, value, size, quality, etc. CAPM Fama-French Three-Factor Model Fama-French Five-Factor Model Models based on mathematical factors like statistical features extracted from assets’ returns using Principal Component Analysis Maximum Likelihood Among others... Some advantages of mathematical factor models: Don’t make assumptions on the drivers of price movements Rely on market data without additional information Some challenges, like choosing the number K factor. Much of the work in this area is related to Random Matrix Theory. 9th Annual Big Data Finance Conference Dec 2021
- 4. Principal Component Analysis (PCA) In a universe consisting of N stocks and T observations, we consider the N × N empirical correlation matrix, C = 1 T RRt (3) where R is the T × N matrix of standardized returns.1 PCA calculates the eigenvalues and eigenvectors of the correlation matrix ranked in de- creasing order by eigenvalues. Accordingly, the first eigenvector solves the variational problem V (1) = argmax {V t CV : ||V ||2 = 1} (4) where ||.||2 represents the Euclidean space. 9th Annual Big Data Finance Conference Dec 2021
- 5. First Eigenportfolio ≈ Market Portfolio The first eigenvector is key to describe the statistics of the system Describes it statistically (as the direction of maximum variance) and financially Define the principal eigenportfolio as the portfolio with weights θi = c V (1) i σi , where c = 1 N P j=1 V (1) j σj (5) If F is the return of the principal eigenportfolio, then the first-order optimality condition for a portfolio that maximizes the Sharpe Ratio over all competing portfolios investing in the same N stocks can be represented as r − E(r) = βr (F − E(F)) + r (6) The Principal Eigenportfolio is connected to the concept of the Market Portfolio (Modern Portfolio Theory). 9th Annual Big Data Finance Conference Dec 2021
- 6. The Principal Eigenportfolio is connected to the concept of the Market Portfolio (Mod- ern Portfolio Theory). If the entries of the principal eigenvector V (1) i are positive for all i, the principal eigenportfolio has positive weights. This follows the Perron-Frobenius The- orem. Perron-Frobenius Theorem Any square matrix Ai,j 0 with positive entries has a unique eigenvector with positive entries (up to a multiplication by a positive scalar), and the corresponding eigenvalue has multiplicity one and is strictly greater than the absolute value of any other eigenvalue. That is ∃ λ∗ 0, V ∗ 0, ||V ∗ ||2 = 1 s.t. Aλ∗ = λ∗ V ∗ . V ∗ → right eigenvector ∀ λ, |λ| ≤ λ∗ V ∗ is unique up to scaling The Perron-Frobenius vector applying Collotz-Wielandt formula λ∗ = max x≥0,x6=0 min xi 6=0 [Ax]i xi = min x0 max [Ax]i xi This result can be extended to non-negative matrices 9th Annual Big Data Finance Conference Dec 2021
- 7. PCA in Stock Markets Difficult to find a financial explanation in higher-order eigenportfolios The first eigenvector has all the entries positive (good proxy of the market portfolio) 9th Annual Big Data Finance Conference Dec 2021
- 8. Hierarchical PCA Partition the stock universe into clusters Strong beliefs on the “intra-cluster data” Weak beliefs on the “inter-cluster data” Define b “benchmark portfolios” associated with clusters Set I(i) = K if stock i is in cluster K A correlation matrix Ĉ which incorporates the modeler’s beliefs in a parsimonious fashion is given by: Ĉi,j = ( Ci,j if I(i) = I(j) βi βj ρ̂I(i),I(j) otherwise. (7) From the orthogonality of the eigenportfolios in the same sector, we can derive a simple formula for the regression coefficients: βi = Corr(Xi , FI(i) ) = √ λ1,I(i) V I(i) i (8) 9th Annual Big Data Finance Conference Dec 2021
- 9. If the benchmark portfolio for a cluster is its 1st eigenportfolio, then Fk = 1 √ λ1,k X i:I(i)=k V k i Xi (9) where Xi are the standardized returns, V k i is the 1st eigenvector of the PCA factorized matrix of cluster k, and λ1,k is the 1st eigenvalue. 𝑪𝑳𝑼𝑺𝑻𝑬𝑹𝑺 𝑴𝑶𝑫𝑰𝑭𝑰𝑬𝑫 𝑴𝑨𝑻𝑹𝑰𝑿 𝑪𝒊,𝒋 𝜷𝒊𝜷𝒋𝝆𝑰(𝒊),𝑰(𝒋) 𝜷𝒊𝜷𝒋𝝆𝑰(𝒊), 𝜷𝒊𝜷𝒋𝝆𝑰(𝒊),𝑰(𝒋) 𝑪𝒊,𝒋 𝜷𝒊𝜷𝒋𝝆𝑰(𝒊),𝑰(𝒋) 𝑪𝒊,𝒋 9th Annual Big Data Finance Conference Dec 2021
- 10. HPCA matrix presents a more clear, distinct block structure Darker areas → low correlation (inter-clusters) Lighter areas → higher correlation (intra-clusters) 0 Figure: Original (left) and modified (right) correlation matrices estimated with the SP 500 returns’ constituents and GICS clusters, from 2010 to 2019. 9th Annual Big Data Finance Conference Dec 2021
- 11. Examples: Global Stocks Analysis of four major global equity markets: United States, Europe, Emerging markets and China. Sector (GICS) USA Europe Emerging Mkts. China Communication 24 42 59 10 Consumer Discretionary 64 66 114 84 Consumer Staples 32 43 92 23 Energy 28 22 56 5 Financials 64 109 277 19 Health Care 60 54 54 50 Industrials 69 115 89 97 Information Technology 69 33 121 88 Materials 28 49 121 81 Real Estate 31 33 44 22 Utilities 28 30 46 19 Total 497 596 1049 498 Table: Numbers of companies considered in the study by GICS sectors and regions. 9th Annual Big Data Finance Conference Dec 2021
- 12. US Stocks The curve of cumulative explained variance of PCA rises faster HPCA has lower concentration given a number of components HPCA is a less greedy algorithm 9th Annual Big Data Finance Conference Dec 2021
- 13. US Stocks Clear higher-order portfolios Single-cluster portfolios Multi-cluster portfolios (i.e., portfolios of portfolios) 9th Annual Big Data Finance Conference Dec 2021
- 14. China Stocks Compared to the other markets, the spread between the two lines is greater Less diversity level 9th Annual Big Data Finance Conference Dec 2021
- 15. China Stocks Clear higher-order eigenportfolios Almost all are multi-cluster portfolios (i.e., portfolios of portfolios) 9th Annual Big Data Finance Conference Dec 2021
- 16. European Stocks Similar behavior to the case of the US The analysis with European countries (instead of GICS) delivered similar behavior 9th Annual Big Data Finance Conference Dec 2021
- 17. European Stocks Mix between single- and multi-cluster portfolios 9th Annual Big Data Finance Conference Dec 2021
- 18. Emerging Markets Behavior is more similar to the case of China than the case of the US The analysis with Emerging countries (instead of GICS) delivered similar behavior 9th Annual Big Data Finance Conference Dec 2021
- 19. Emerging Markets Most of the portfolios are multi-cluster 9th Annual Big Data Finance Conference Dec 2021
- 20. Statistically Generated Clusters Stocks belonging to the same GICS or country share common factors that capture –to some extend– their joint dynamics They are very easy to interpret However, they have some shortcomings Stock markets and their components change almost continuously For risk and portfolio management, practitioners seek a trade off between stable and adaptive clusters I It goes against the diversification of a seemingly diversified strategy I Trading strategies such as sector/country rotation may be affected 9th Annual Big Data Finance Conference Dec 2021
- 21. Description of the Algorithm Given M eigenvectors of PCA, we construct 2M clusters Set a {+1, −1} M-vector. For each stock i, each entry represents the sign of the eigenvector (representing a quadrant) The new space is divided into M quadrants 9th Annual Big Data Finance Conference Dec 2021
- 22. Results Like in the US case, the curve of cumulative explained variance of PCA rises faster The difference here is even higher (here it is for more than 2600 stocks!) 9th Annual Big Data Finance Conference Dec 2021
- 24. Application to Portfolio Management First eigenportfolio and the market portfolio (SPY) First eigenportfolio does a good job of tracking SPY! 9th Annual Big Data Finance Conference Dec 2021
- 25. Statistical Clustering Factor Model The model correlation matrix using K components of HPCA reads C = ÔΛ̂ÔT + ζ2 (10) ζ2 j = N X i=K+1 λ(i) (O (i) j )2 (11) where ζ2 is the (uncorrelated) idiosyncratic risk. The factor expected returns are E(r) = K X i=i β (K) j F(K) + i (12) The number K of factors are chosen with the eRank. Let the SVD of the T × N matrix of standardized log-return R = UDV (13) The associated probability distribution is Pj = σj kσk1 for j = 1, ..., Q for Q = M ∧ T (14) where k.k1 is the L − 1 norm. Then, the effective rank is defined as eRank(R) = exp{H(P1, P2, ..., PQ )} (15) where H is the Shannon entropy 9th Annual Big Data Finance Conference Dec 2021
- 26. Mean-variance optimization: I Monthly rebalancing using a 6-month estimation window I HPCA statistical-based factor model (blue) outperforms the HPCA GICS-based factor model (red) and the shrinkage estimator (salmon) See [Fabozzi, Focardi and Kolm, 2010], [Fabozzi, Kolm and Focardi, 2006], [?]. 9th Annual Big Data Finance Conference Dec 2021
- 27. References Avellaneda, M. (2019) Hierarchical PCA and Applications to Portfolio Management. NYU Courant Working Paper. Fabozzi, F.J., Focardi, S.M. and Kolm, P.N. (2010) Quantitative Equity Investing: Techniques and Strategies. Hoboken, NJ: John Wiley Sons, Inc. Fabozzi, F.J., Kolm, P.N. and Focardi, S.M. (2002) Robust Financial Modeling of the Equity Market: From CAPM to Cointegration. Hoboken, NJ: John Wiley Sons, Inc. Kakushadze, Z. (2015) Heterotic Risk Models. Wilmott Magazine 2015(80): 40-55. Kakushadze, Z. and Yu, W. (2017) Statistical Risk Models. Journal of Investment Strategies 6(2): 1-40. Lopez de Prado, M. (2016) Building Diversified Portfolios that Outperform Out-of-Sample. Journal of Portfolio Management 42(4): 59-69. Lopez de Prado, M. (2019) Ten Applications of Financial Machine Learning. Available at SSRN. 9th Annual Big Data Finance Conference Dec 2021