Cleaning correlation matrices,

Random Matrix Theory &

HCIZ integrals

J.P Bouchaud

with: M. Potters, L. Laloux, R. Allez, J. Bun, S. Majumdar

http://www.cfm.fr

Portfolio theory: Basics

• Portfolio weights wi, Asset returns Xti

• If expected/predicted gains are gi then the expected gain ofthe portfolio is

G =∑

i

wigi

• Let risk be defined as: variance of the portfolio returns(maybe not a good definition !)

R2 =∑

ij

wiσiCijσjwj

where σ2i is the variance of asset i, and

Cij is the correlation matrix.

J.-P. Bouchaud

Markowitz Optimization

• Find the portfolio with maximum expected return for a givenrisk or equivalently, minimum risk for a given return (G)

• In matrix notation:

wC = GC−1g

gTC−1gwhere all gains are measured with respect to the risk-free

rate and σi = 1 (absorbed in gi).

• Note: in the presence of non-linear contraints, e.g.∑

i

|wi| ≤ A

a “spin-glass” problem! (see [JPB,Galluccio,Potters])

J.-P. Bouchaud

Markowitz Optimization

• More explicitly:

w ∝∑

αλ−1α (Ψα · g)Ψα = g +

∑

α(λ−1α − 1) (Ψα · g)Ψα

• Compared to the naive allocation w ∝ g:

– Eigenvectors with λ≫ 1 are projected out

– Eigenvectors with λ≪ 1 are overallocated

• Very important for “stat. arb.” strategies (for example)

J.-P. Bouchaud

Empirical Correlation Matrix

• Before inverting them, how should one estimate/clean cor-relation matrices?

• Empirical Equal-Time Correlation Matrix E

Eij =1

T

∑

t

XtiXtj

σiσj

Order N2 quantities estimated with NT datapoints.

When T < N , E is not even invertible.

Typically: N = 500−2000; T = 500−2500 days (10 years –Beware of high frequencies) −→ q := N/T = O(1)

J.-P. Bouchaud

Risk of Optimized Portfolios

• “In-sample” risk (for G = 1):

R2in = wTEEwE =

1

gTE−1g

• True minimal risk

R2true = wTCCwC =

1

gTC−1g

• “Out-of-sample” risk

R2out = wTECwE =

gTE−1CE−1g(gTE−1g)2

J.-P. Bouchaud

Risk of Optimized Portfolios

• Let E be a noisy, unbiased estimator of C. Using convexityarguments, and for large matrices:

R2in ≤ R2true ≤ R2out

• In fact, using RMT: R2out = R2true(1 − q)−1 = R2in(1 − q)−2,indep. of C! (For large N)

• If C has some time dependence (beyond observation noise)one expects an even worse underestimation

J.-P. Bouchaud

In Sample vs. Out of Sample

0 10 20 30Risk

0

50

100

150R

etur

n

Raw in-sample Cleaned in-sampleCleaned out-of-sampleRaw out-of-sample

J.-P. Bouchaud

Rotational invariance hypothesis (RIH)

• In the absence of any cogent prior on the eigenvectors, onecan assume that C is a member of a Rotationally Invariant

Ensemble – “RIH”

• Surely not true for the “market mode” ~v1 ≈ (1,1, . . . ,1)/√N ,

with λ1 ≈ Nρ but OK in the bulk (see below)

A more plausible assumption: factor model → hierarchical,block diagonal C’s (“Parisi matrices”)

• “Cleaning” E within RIH: keep the eigenvectors, play witheigenvalues

→ The simplest, classical scheme, shrinkage:

C = (1 − α)E + αI → λ̂C = (1 − α)λE + α, α ∈ [0,1]

J.-P. Bouchaud

RMT: from ρC(λ) to ρE(λ)

• Solution using different techniques (replicas, diagrams, freematrices) gives the resolvent GE(z) = N

−1Tr(E − zI) as:

GE(z) =∫dλ ρC(λ)

1

z − λ(1 − q+ qzGE(z)),

Note: One should work from ρC −→ GE

• Example 1: C = I (null hypothesis) → Marcenko-Pastur [67]

ρE(λ) =

√(λ+ − λ)(λ− λ−)

2πqλ, λ ∈ [(1 −√q)2, (1 + √q)2]

• Suggests a second cleaning scheme (Eigenvalue clipping, [Lalouxet al. 1997]): any eigenvalue beyond the Marcenko-Pastur

edge can be trusted, the rest is noise.

J.-P. Bouchaud

Eigenvalue clipping

λ < λ+ are replaced by a unique one, so as to preserve TrC =

N .

J.-P. Bouchaud

RMT: from ρC(λ) to ρE(λ)

• Solution using different techniques (replicas, diagrams, freematrices) gives the resolvent GE(z) as:

GE(z) =∫dλ ρC(λ)

1

z − λ(1 − q+ qzGE(z)),

Note: One should work from ρC −→ GE

• Example 2: Power-law spectrum (motivated by data)

ρC(λ) =µA

(λ− λ0)1+µΘ(λ− λmin)

• Suggests a third cleaning scheme (Eigenvalue substitution,Potters et al. 2009, El Karoui 2010): λE is replaced by the

theoretical λC with the same rank k

J.-P. Bouchaud

Empirical Correlation Matrix

0 1 2 3 4 5λ

0

0.5

1

1.5

ρ(λ)

DataDressed power law (µ=2)Raw power law (µ=2)Marcenko-Pastur

0 250 500rank

-2

0

2

4

6

8

κMP and generalized MP fits of the spectrum

J.-P. Bouchaud

Eigenvalue cleaning

0 0.2 0.4 0.6 0.8 1α

0

0.5

1

1.5

2

2.5

3

3.5

R2

Classical ShrinkageLedoit-Wolf ShrinkagePower Law SubstitutionEigenvalue Clipping

Out-of sample risk for different 1-parameter cleaning schemes

J.-P. Bouchaud

A RIH Bayesian approach

• All the above schemes lack a rigorous framework and are atbest ad-hoc recipes

• A Bayesian framework: suppose C belongs to a RIE, withP(C) and assume Gaussian returns. Then one needs:

〈C〉|Xti =∫

DCCP(C|{Xti})

with

P(C|{Xti}) = Z−1 exp[−NTrV (C, {Xti})

];

where (Bayes):

V (C, {Xti}) =1

2q

[logC + EC−1

]+ V0(C)

J.-P. Bouchaud

A Bayesian approach: a fully soluble case

• V0(C) = (1 + b) lnC + bC−1, b > 0: “Inverse Wishart”

• ρC(λ) ∝√

(λ+−λ)(λ−λ−)λ2

; λ± = (1 + b±√

(1 + b)2 − b2/4)/b

• In this case, the matrix integral can be done, leading exactlyto the “Shrinkage” recipe, with α = f(b, q)

• Note that b can be determined from the empirical spectrumof E, using the generalized MP formula

J.-P. Bouchaud

The general case: HCIZ integrals

• A Coulomb gas approach: integrate over the orthogonalgroup C = OΛO†, where Λ is diagonal.

∫DO exp

[

−N2q

Tr[logΛ + EO†Λ−1O + 2qV0(Λ)

]]

• Can one obtain a large N estimate of the HCIZ integral

F(ρA, ρB) = limN→∞

N−2 ln∫

DO exp[N

2qTrAO†BO

]

in terms of the spectrum of A and B?

J.-P. Bouchaud

The general case: HCIZ integrals

• Can one obtain a large N estimate of the HCIZ integral

F(ρA, ρB) = limN→∞

N−2 ln∫

DO exp[N

2qTrAO†BO

]

in terms of the spectrum of A and B?

• When A (or B) is of finite rank, such a formula exists in termsof the R-transform of B [Marinari, Parisi & Ritort, 1995].

• When the rank of A,B are of order N , there is a formula dueto Matytsin [94](in the unitary case), later shown rigorously

by Zeitouni & Guionnet, but its derivation is quite obscure...

J.-P. Bouchaud

An instanton approach to large N HCIZ

• Consider Dyson’s Brownian motion matrices. The eigenval-ues obey:

dxi =

√2

βNdW +

1

Ndt∑

j 6=i

1

xi − xj,

• Constrain xi(t = 0) = λAi and xi(t = 1) = λBi. The proba-bility of such a path is given by a large deviation/instanton

formula, with:

d2xidt2

= − 2N2

∑

ℓ 6=i

1

(xi − xℓ)3.

J.-P. Bouchaud

An instanton approach to large N HCIZ

• Constrain xi(t = 0) = λAi and xi(t = 1) = λBi. The proba-bility of such a path is given by a large deviation/instanton

formula, with:

d2xidt2

= − 2N2

∑

ℓ 6=i

1

(xi − xℓ)3.

• This can be interpreted as the motion of particles interactingthrough an attractive two-body potential φ(r) = −(Nr)−2.Using the virial formula, one finally gets Matytsin’s equations:

∂tρ+ ∂x[ρv] = 0, ∂tv+ v∂xv = π2ρ∂xρ.

J.-P. Bouchaud

An instanton approach to large N HCIZ

• Finally, the “action” associated to these trajectories is:

S ≈ 12

∫dxρ

[

v2 +π2

3ρ2]

− 12

[∫dxdyρZ(x)ρZ(y) ln |x− y|

]Z=B

Z=A

• Now, the link with HCIZ comes from noticing that the prop-agator of the Brownian motion in matrix space is:

P(B|A) ∝ exp−[N2

Tr(A−B)2] = exp−N2

[TrA2+TrB2−2TrAOBO†]

Disregarding the eigenvectors of B (i.e. integrating over O)

leads to another expression for P(λBi|λAj) in terms of HCIZthat can be compared to the one using instantons

• The final result for F(ρA, ρB) is exactly Matytsin’s expression,up to details (!)

J.-P. Bouchaud

Back to eigenvalue cleaning...

• Estimating HCIZ at large N is only the first step, but...

• ...one still needs to apply it to B = C−1, A = E = X†CXand to compute also correlation functions such as

〈O2ij〉E→C−1with the HCIZ weight

• As we were working on this we discovered the work of Ledoit-Péché that solves the problem exactly using tools from RMT...

J.-P. Bouchaud

The Ledoit-Péché “magic formula”

• The Ledoit-Péché [2011] formula is a non-linear shrinkage,given by:

λ̂C =λE

|1 − q+ qλE limǫ→0GE(λE − iǫ)|2.

• Note 1: Independent of C: only GE is needed (and is observ-able)!

• Note 2: When applied to the case where C is inverse Wishart,this gives again the linear shrinkage

• Note 3: Still to be done: reobtain these results using theHCIZ route (many interesting intermediate results to hope

for!)

J.-P. Bouchaud

Eigenvalue cleaning: Ledoit-Péché

Fit of the empirical distribution with V ′0(z) = a/z+b/z2+c/z3.

J.-P. Bouchaud

What about eigenvectors?

• Up to now, most results using RMT focus on eigenvalues

• What about eigenvectors? What natural null-hypothesis be-yond RIH?

• Are eigen-values/eigen-directions stable in time?

• Important source of risk for market/sector neutral portfolios:a sudden/gradual rotation of the top eigenvectors!

• ..a little movie...

J.-P. Bouchaud

What about eigenvectors?

• Correlation matrices need a certain time T to be measured

• Even if the “true” C is fixed, its empirical determinationfluctuates:

Et = C + noise

• What is the dynamics of the empirical eigenvectors inducedby measurement noise?

• Can one detect a genuine evolution of these eigenvectorsbeyond noise effects?

J.-P. Bouchaud

What about eigenvectors?

• More generally, can one say something about the eigenvec-tors of randomly perturbed matrices:

H = H0 + ǫH1

where H0 is deterministic or random (e.g. GOE) and H1random.

J.-P. Bouchaud

Eigenvectors exchange

• An issue: upon pseudo-collisions of eigenvectors, eigenvaluesexchange

• Example: 2 × 2 matrices

H11 = a, H22 = a+ ǫ, H21 = H12 = c,−→

λ± ≈ǫ→0 a+ǫ

2±√

c2 +ǫ2

4

• Let c vary: quasi-crossing for c→ 0, with an exchange of thetop eigenvector: (1,−1) → (1,1)

• For large matrices, these exchanges are extremely numerous→ labelling problem

J.-P. Bouchaud

Subspace stability

• An idea: follow the subspace spanned by P -eigenvectors:

|ψk+1〉, |ψk+2〉, . . . |ψk+P 〉 −→ |ψ′k+1〉, |ψ′k+2〉, . . . |ψ′k+P 〉

• Form the P × P matrix of scalar products:

Gij = 〈ψk+i|ψ′k+j〉

• The determinant of this matrix is insensitive to label per-mutations and is a measure of the overlap between the two

P -dimensional subspaces

– D = −1P ln |detG| is a measure of how well the first sub-space can be approximated by the second

J.-P. Bouchaud

Intermezzo

• Non equal time correlation matrices

Eτij =1

T

∑

t

XtiXt+τj

σiσj

N ×N but not symmetrical: ‘leader-lagger’ relations

• General rectangular correlation matrices

Gαi =1

T

T∑

t=1

Y tαXti

N ‘input’ factors X; M ‘output’ factors Y

– Example: Y tα = Xt+τj , N = M

J.-P. Bouchaud

Intermezzo: Singular values

• Singular values: Square root of the non zero eigenvaluesof GGT or GTG, with associated eigenvectors ukα and v

ki →

1 ≥ s1 > s2 > ...s(M,N)− ≥ 0

• Interpretation: k = 1: best linear combination of input vari-ables with weights v1i , to optimally predict the linear com-

bination of output variables with weights u1α, with a cross-

correlation = s1.

• s1: measure of the predictive power of the set of Xs withrespect to Y s

• Other singular values: orthogonal, less predictive, linear com-binations

J.-P. Bouchaud

Intermezzo: Benchmark

• Null hypothesis: No correlations between Xs and Y s:

Gtrue ≡ 0

• But arbitrary correlations among Xs, CX, and Y s, CY , arepossible

• Consider exact normalized principal components for the sam-ple variables Xs and Y s:

X̂ti =1√λi

∑

j

UijXtj; Ŷ

tα = ...

and define Ĝ = Ŷ X̂T .

J.-P. Bouchaud

Intermezzo: Random SVD

• Final result:([Wachter] (1980); [Laloux,Miceli,Potters,JPB])

ρ(s) = (m+ n− 1)+δ(s− 1) +

√(s2 − γ−)(γ+ − s2)

πs(1 − s2)with

γ± = n+m− 2mn± 2√mn(1 − n)(1 −m), 0 ≤ γ± ≤ 1

• Analogue of the Marcenko-Pastur result for rectangular cor-relation matrices

• Many applications; finance, econometrics (‘large’ models),genomics, etc. and subspace stability!

J.-P. Bouchaud

Back to eigenvectors

• Extend the target subspace to avoid edge effects:

|ψk+1〉, |ψk+2〉, . . . |ψk+P 〉 −→ |ψ′k−Q+1〉, |ψ′k+2〉, . . . |ψ′k+Q〉

• Form the P ×Q matrix of scalar products:

Gij = 〈ψk+i|ψ′k+j〉

• The singular values of G indicates how well the Q perturbedvectors approximate the initial ones

D = −1P

∑

i

ln si

J.-P. Bouchaud

Null hypothesis

• Note: if P and Q are large, D can be “accidentally” small

• One can compute D exactly in the limit P,Q → ∞, N → ∞,with fixed p = P/N , q = Q/N :

• Final result: (same problem as above!)

D = −∫ 1

0ds ln s ρ(s)

with:

ρ(s) =

√(s2 − γ−)(γ+ − s2)

πs(1 − s2)and

γ± = p+ q − 2pq ± 2√pq(1 − p)(1 − q), 0 ≤ γ± ≤ 1

J.-P. Bouchaud

Back to eigenvectors: perturbation theory

• Consider a randomly perturbed matrix:

H = H0 + ǫH1

• Perturbation theory to second order in ǫ yields:

D ≈ ǫ2

2P

∑

i∈{k+1,...,k+P}

∑

j 6∈{k−Q+1,...,k+Q}

(〈ψi|H1|ψj〉λi − λj

)2.

• The full distribution of s can again be computed exactly (insome limits) using free random matrix tools.

J.-P. Bouchaud

GOE: the full SV spectrum

• Initial eigenspace: spanned by [a, b] ⊂ [−2,2], b− a = ∆

Target eigenspace: spanned by [a− δ, b+ δ] ⊂ [−2,2]

• Two cases (set s = ǫ2ŝ):

– Weak fluctuations ∆ ≪ δ ≪ 1 → ρ(ŝ) is a semi circlecentered around ∼ δ−1, of width ∼

√∆

– Strong fluctuations δ ≪ ∆ ≪ 1 → ρ(ŝ) ∼ ŝmin/ŝ2, withŝmin ∼ ∆−1 and ŝmax ∼ δ−1.

J.-P. Bouchaud

The case of correlation matrices

• Consider the empirical correlation matrix:

E = C + η η =1

T

T∑

t=1

(XtXt − C)

• The noise η is correlated as:〈ηijηkl

〉=

1

T(CikCjl + CilCjk)

• from which one derives:

D ≈ 12TP

P∑

i=1

N∑

j=Q+1

λiλj

(λi − λj)2

.

(and a similar equation for eigenvalues)

J.-P. Bouchaud

Stability of eigenvalues: Correlations

Eigenvalues clearly change: well known correlation crises

J.-P. Bouchaud

Stability of eigenspaces: Correlations

D(τ) for a given T , P = 5, Q = 10

J.-P. Bouchaud

Stability of eigenspaces: Correlations

D(τ = T) for P = 5, Q = 10

J.-P. Bouchaud

Conclusion

• Many RMT tools available to understand the eigenvalue spec-trum and suggest cleaning schemes

• The understanding of eigenvectors is comparatively poorer

• The dynamics of the top eigenvector (aka market mode) isrelatively well understood

• A plausible, realistic model for the “true” evolution of C isstill lacking (many crazy attempts – Multivariate GARCH,

BEKK, etc., but “second generation models” are on their

way)

J.-P. Bouchaud

Bibliography

• J.P. Bouchaud, M. Potters, Financial Applications of Ran-dom Matrix Theory: a short review, in “The Oxford Hand-

book of Random Matrix Theory” (2011)

• R. Allez and J.-P. Bouchaud, Eigenvectors dynamics: generaltheory & some applications, arXiv 1108.4258

• P.-A. Reigneron, R. Allez and J.-P. Bouchaud, Principal re-gression analysis and the index leverage effect, Physica A,

Volume 390 (2011) 3026-3035.

J.-P. Bouchaud