Principal Components Analysis (PCA) and random matrix theory (RMT) have become widespread tools for data analysis. PCA (Joliffe (2002) [6]) provides a mathematical and objective approach to extract economic information from the correlation matrices of asset returns. In this approach, the analyst extracts common risk factors from the eigenvectors and eigenvalues of the correlation matrix.

The first eigenvector of the correlation of stock returns corresponds to the solution of the variational problem

V ( 1 ) = a r g m a x     { V t R V ; | | V | | = 1 } . (1)

Here, R is the correlation matrix of daily returns and | | . | | is the Euclidean norm in R n , n being the total number of assets. Equation 1 shows that the principal eigenvector is represents the direction (line) which “captures the most variance” as described by the correlation matrix. The first eigenvector satisfies

R V ( 1 ) = λ ( 1 ) V ( 1 ) (2)

PCA also finds recursively additional (orthogonal) directions beyond V 1 which capture the most variance. The other eigenvectors and eigenvalues are computed in the same way as Eq. (1)) with the maximization to the sub-space orthogonal to the space spanned by the ones computed previously, i.e,

V ( k ) = a r g m a x     { V t R V ; | | V | | = 1 , V ( k ) t V ( l ) = 0 ,     1 ≤ l < k } . (3)

The eigenvalues satisfy λ ( 1 ) > λ ( 2 ) ≥ . . . ≥ λ ( n ) . Assume that the data corresponds to the daily returns of a group of stocks. The Karhunen-Loeve representation of the standardized returns is

X j = ∑ k = 1 n ‍ λ ( k ) V j ( k )   F ( k ) (4)

where

F ( k ) = 1 λ ( k ) ∑ i = 1 n ‍ V i ( k )   X i . (5)

By construction, F k are uncorrelated and have variance 1. Since these random variables are linear combinations of the daily standardized returns of the assets, we call them (standardized) “eigenportfolio (EP) returns”, with the caveat that the actual portfolio “weights” are obtained by dividing each entry of the eigenvector by the volatility of the asset (Avellaneda and Lee 2008, 2010) [1].²

PCA is a framework for learning about the common factors which affect the returns of a given group of assets. The first eigenportfolio, associated with the r.v. F 1 , is a common risk factor which explains the maximum variability. We can write a one-factor model for each asset, namely

X j = β j F ( 1 ) + ϵ j (6)

where β j is the regression coefficient of the standardized return on the first EP. The “residuals” ϵ j in equation 6 are uncorrelated with F 1 , which is nice. However, they are generally correlated for different stocks.

The regression coefficients satisfy

β j = λ ( 1 )   V j ( 1 ) ,         j = 1 , . . . , n . (7)

In the case of economic data, which is noisy, the consensus is to disregard EPs which correspond to low eigenvalues. In a celebrated paper, Laloux et al (2000) [7] proposed to use random matrix theory (RMT) to establish a cutoff in the number of EPs use to model the standardized returns, namely

X j = ∑ k = 1 m ‍ β j ( k )   F ( k )     + ϵ j (8)

where β j k are “factor loadings”and (with a slight abuse of notation) ϵ j are residuals obtained after “defactoring” relatively to the m eigenportfolios. The number m is a cutoff which is to be determined from the context.

According to [7], the eigenvalues of a pure noise matrix follow the Marcenko-Pastur distribution and have a spectrum which, for large matrices,is asymptotically bounded from above by λ + , M P = ( 1 + n / T ) 2 , where T is the number of observations. Asymptotics should hold in the limit n / T → γ (a constant) as n and T both tend to infinity. The way to use RMT to calculate the cutoff is to construct the correlation matrix R i , j ( m ) = C o r r ( ϵ i , ϵ j ) for m large enough and verify that its top eigenvalue is of the order of λ + , M P . One can also compare the empirical distribution of eigenvalues with the Marcenko-Pastur probability distribution.

PCA aided by RMT is an elegant approach to analyzing correlation matrices of financial data and can also be applied to may areas of science. The main strength of the method is that it can detect common risk factors based on a matrix of asset returns, without any additional information. In other works, PCA “lets the data speak for itself”. Generally speaking, PCA explains the most variability with the smallest number of factors. Most studies tend to justify the PCA approach by recognizing that it produces some factors which have ex-post economic interpretations, such as equating E P 1 with the Sharpe Market Portfolio (Boyle 2017) [2], or attempt to interpret higher-order EPs in terms of industry sectors [1]. In the case of fixed-income, the EPs are often identified with “parallel shifts”, or with long-term vs short-term oscillations of the yield curve (Litterman and Scheinkman, 1991) [8].

One of the frequent criticisms of PCA in Finance is that the common risk factors generated by higher-order eigenportfolios - aside from the first eigenportfolio - are difficult to interpret and appear to be unstable across time. We call this the identification problem. Because of it, many portfolio managers favor traditional factor models such as Barra; see Shkolnik et. al. (2016) [9] for alternative approaches to model financial correlations.

The identification problem in PCA reflects the uncertainty, or unreliability, of cross-asset correlations. From a practical point of view, as the size of trading universe increases, the correlations of assets which are not economically related (a tech stock with an energy stock, or with a foreign stock) are difficult to quantify and may be noisy. This could be due to several reasons: the lack of “explanation” for the relation between the stocks, or perhaps that their prices are not sampled simultaneously (e.g. if they are end-of-day prices in different time-zones) or that the number of observations is not large compared to the number of assets considered. For example, empirical correlations of price changes of out-of-the money options with different underlying assets may not be as reliable or significant as the data would suggest.

To mitigate the identification problem, we should seek a factor model which can recognize the economic nature or function of the asset as well as the statistical properties of returns. This lead us to the model described hereafter.

The hierarchical PCA (HPCA) applies to markets which can be partitioned into several sectors or asset-classes. Consider first an abstract market, in which the empirical data matrix of asset returns, with dimensions T × n , can be partitioned into “blocks of columns” labeled k = 1,2,..., b . These blocks have dimensions T × n k with k = 1,2,..., b . Each block represents data sampled from a sector. For simplicity, we assume that the indices of the securities are organized so that blocks which are adjacent to one another in the matrix and do not overlap. We have a few concrete situations in mind:

The blocks represent data of industry sectors for equities in the same economy (e.g. sectors associated with the 500 or so stocks in the S&P 500 index). In this case, the columns of a block correspond to the historical standardized returns of the stocks in the sector observed over T consecutive dates.

Define the function I j = k if asset j is in block k . According to Eq. (4) we can write, for each asset in the “big universe”,

X j = β j   F ( 1 , I ( j ) ) + ϵ j , (9)

where β j is the regression coefficient of the returns of asset j on the first factor of block I j and ϵ j is the residual.

We shall make the following assumption (“HPCA assumption”):

  I f       I ( i ) ≠ I ( j ) ,       t h e n       C o r r ( ϵ i , ϵ j ) = 0 . (10)

The assumption states that residuals are uncorrelated if their assets belong to different sectors. Equation (9) defines the asset statistics within each block exactly, and the model is completed by specifying the joint statistics of the factors F 1, k ,     k = 1,2,..., b . The HPCA assumption says nothing new regarding intra-block correlations, which are set equal to the empirical correlations between asset returns within the same sector or block. Of course, the intra-block correlations could be further denoised using RMT if necessary ([4]).

Using the HPCA assumption Eq. (10), the proposed model has the modified correlation matrix for asset returns:

R ~ i j = R i j       i f           I ( i ) = I ( j )

= β i   β j   ρ ¯ I ( i ) I ( j )       i f           I ( i ) ≠ I ( j ) (11)

where ρ ¯ k , k ' = C o r r ( F ( 1 , k ) , F ( 1 , k ' ) ) .

Proposition 1 Eq. (11) corresponds to a symmetric non-negative matrix with R ˜ i i = 1 for all i . In particular, it corresponds to the correlation matrix of a system of standardized random variables.

Proof. To check non-negative definiteness, note that for all θ   ∈   R n we have

θ t     R ~ θ = ∑ k = 1 b ‍ ∑ I ( i ) = I ( j ) = k ‍ θ i θ j ( R i j - β i β j ) + ∑ k , k ' = 1 b ‍ ( ∑ I ( i ) = k ‍ θ i β i )   ( ∑ I ( j ) = k ' ‍ θ j β j )     ρ ¯ k , k ' . (12)

For any k , the matrix R i j − β i β j restricted to sector k is identical to the sector correlation, except for the fact that the eigenvalue corresponding to V 1, k is set to zero. In particular, it is non-negative definite. Moreover, the matrix ρ ¯ k , k ' is also a correlation matrix, so it is non-negative definite. Since both summands are non-negative it follows that θ t   R ˜ θ ≥ 0 for all θ   ∈   R n .

A concrete implementation of the data model is achieved as follows: let ψ 1 ,..., ψ b be Gaussian random variables with mean zero and covariance matrix ρ ¯ , and let ζ i k     , i : I ( i ) = b ,     k = 1 , . . . , b be i.i.d. standardized Gaussian random variables which are independent of the ψ ’s. The data model is

X i = β i     ψ I ( i )     + ∑ { j : j ≥ 2 ,     I ( j ) = I ( i ) } ‍ γ i j     ζ j   I ( i ) (13)

The random variables need not be necessarily Gaussian: they can be multivariate Student-t, or they can be transforms of arbitrary distributions connected by a Gaussian or t-Copula; see for instance [5].

The multivariate distribution associated with HPCA presents an alternative model to the classical PCA (Eq. (8)). It has a tree structure: in the equity example discussed below, the top vertex corresponds to the “market”; there are 11 branches corresponding to industry sectors, and each of the 11 vertices has branches corresponding to the stocks in each sector.

Hierarchical models with more than two layers arise naturally. For instance, HPCA can be used to model “world portfolios”, in which the first layer consists of countries or regions, the second to industry sector indices in each country, and the third layer could describe the individual securities in each region/sector.

For another useful example, consider a stock market in which stocks belong to different industry sectors, and then, include columns associated with equity options returns. In this case, the tree has three layers because we can associate to each stock an additional sub-group: the block consisting of the returns of implied volatilities (on a constant delta/time-to-maturity grid) and the stock returns. Now the root corresponds to the full market, the first layer corresponds to industry sectors, the second layer corresponds to stocks and the third layer represents an individual name with all the associated option-implied volatilites.

A similar approach works for credit derivatives. In this case, the returns of the CDS with different tenors referencing each obligor constitute a block associated with an obligor. These blocks can be grouped by industry sectors or, alternatively, blocks could be generated according to membership in a credit index (CCX.IG, CDX.HY, CDX.HV), or both; [5].

In summary, if financial data can be grouped into blocks or sectors with clear economic interpretation, with multiple instruments associated with each block, we can generate a data model with tree-like structure from the HPCA assumption in Eq. (10). This approach combines information available for each asset (sector, sub-sector, reference obligor, option underlying asset) with the explanatory power of PCA. For simplicity, we will consider the analysis of a two-layer HPCA. Adding more layers is mathematically straightforward.

The HPCA assumption Eq. (10) gives rise to explicitly computable eigenvalues and eigenvectors for the matrix R ˜ defined in Eq. (11).

Proposition 2.

1. For each sector k = 1,..., b , let λ ( 1 , k )     >     λ ( 2 , k ) ≥     . . .     ≥ λ ( n k , k ) denote the n k eigenvectors of the sector correlation matrix, ordered from largest to smallest, and let V i , k be the corresponding eigenvectors. Define the n-dimensional vectors

W j ( i , k ) = V j ( i , k )           i f           I ( j ) = k

= 0                   i f           I ( j ) ≠ k , (14)

which correspond to the embedding of the sector-level eigenvectors, V ( i , k )     ∈     R n k , into the large space R n . The vectors W ( i , k ) ,     i = 1 , . . . , n k ,         k = 1 , . . . , b form an orthogonal basis of R n .

2. The subspace Ω of R n generated by the vectors W 1, k ,     k = 1,..., b , viz.

Ω = { ∑ k = 1 b ‍ α k W ( 1 , k ) : ( α 1 , . . . , α b ) ∈ R b } , (15)

is invariant under the action of R ˜ viewed as an operator from R n to R n .

M k , k ' : = λ ( 1 , k ) λ ( 1 , k ' ) ρ ¯ k , k ' . (16)

Let μ 1 ,..., μ b denote the eigenvalues of M , ranked in decreasing order, and let ( α k = α 1 k ,...., α b k )     k = 1,..., b represent the corresponding normalized eigenvectors (defined up to sign). The vectors

W ~ ( 1 , k ) = ∑ p = 1 b ‍ α p ( k )   W ( l , p ) (17)

are eigenvectors of R ˜ , with corresponding eigenvalues μ k , for k = 1,..., b .

This proposition completely characterizes the eigenvalues and eigenvectors of the HPCA correlation matrix relating them to the eigenvalues and eigenvectors of sector PCAs.³ Thus, the HPCA assumption eliminates the identification problem for common factors: “eigenportfolios” have concrete meanings attached to the information about the correlations of sectors. In the examples to follow, we shall compare HPCA with PCA and show that the former is an excellent substitute for the full empirical correlation matrices when we model multivariate financial data.

We consider data for n = 434 equities which are constituents of the S&P500 index. The data ranges from February 22, 2012 to February 16, 2018. We consider the correlation matrix of standardized stock returns, and define the sectors as General Industry Classification groups (GICs), so b = 11 ; see Table 1.

Cuadro 1

GIC ( k )	Description	Number of companies ( n k )
1	Consumer Discretionary	73
2	Consumer Staples	56
3	Energy	27
4	Financials	59
5	Health Care	51
6	Industrials	57
7	Information Technology	58
8	Materials	23
9	Real Estate	27
10	Telecommunication Services	3
11	Utilities	28

To further evaluate the HPCA, we considered both models (HPCA,PCA) with a cutoff m = 30 , and compared the multivariate statistics. We expect that after removing m ≈ 30 eigenvectors, the correlations of the residuals (both intra- and inter- sector) should be small.

Empirically, the top eigenvectors of the correlation matrices of residuals are approximately 6.8 (HPCA) and 7.7 (PCA), which correspond to an approximate average correlation of 7.3 / 434 = 1.7 % . We compared the histograms of the eigenvalues for the corresponding correlation matrices and found that they are very near each other. We also compared the histograms with a discretization of the Marcenko-Pastur distribution (mimicking the comparable histogram for the large-matrix limit), suggesting that the residuals behave like a random matrix in both models; see Fig. (6). The majority of the lines, in both cases, are below the Marcenko-Pastur cutoff λ + = 2.36 , as postulated by RMT, and have comparable sizes to the MP distribution. There are, nevertheless, some lines above the MP threshold in both models (which are essentially equal), but they decreasing in magnitude as λ increases,and could perhaps be interpreted as finite-size fluctuations.

This calculation suggests that using the full empirical correlation matrix is not more informative than using the HPCA model, which uses only the sector correlation matrices, and in which intra-sector correlations are derived from the correlations of the EV1 for different sectors. Clearly, the HPCA provides a simpler description of common risk factors than PCA. The HPCA is therefore a viable alternative to PCA in the analysis of multivariate data in Finance, which should be of interest for asset-allocation and portfolio risk-management.

Figure 14