Dynamic Equicorrelation

Rob Engle and Bryan Kelly

QFE Seminar, April 27, 2009

Correlations Vary Over Time: Option Implieds

Source: Driessen, Maenhout & Vilkov, (J. of Finance, Forthcoming)

Correlations Vary Over Time: DCC (Engle 2002)

Source: VLAB, http://vlab.stern.nyu.edu/

The Difficulty With Covariances...

• Since early ’80s, attempts have been made to estimatemultivariate GARCH models

• Specifications so complex that traditional models are difficultto estimate for more than a few assets

• Finance applications often require large cross sections• Portfolio selection• Derivatives (basket options, CDOs, etc.)• Risk management

DCC: Problem Solved?

• Engle (2002) introduces Dynamic Conditional Correlation(DCC)

• Massive parameter reduction: an entire matrix evolution canbe described by a few parameters

• Computational burden as N increases: must calculate inverseand determinant of N × N matrices many thousands of timesin likelihood maximization• A pain for a moderate systems• Infeasible for very large systems?

• Other concerns• Storing correlation matrices• Digesting massive output: N(N − 1)/2 series

Dynamic Equicorrelation (DECO)

• Difficult problem? Change the problem:• All assets share the same correlation each period, but this

“equicorrelation” varies through time

• What does it buy?• Analytic inverse and determinant - likelihood simple to

compute for system of any dimension• Entire correlation evolution summarized by single time series

Outline

• Equicorrelation in action

• Model and theoretical properties

• DECO amid extant covariance models

• Monte Carlo evaluation

• Correlations among the S&P 500

Examples of DECO in Finance

DECO at a Glance

• Dynamic Equicorrelation takes the form

Rt = (1− ρt)In + ρtJn =

1 ρt · · ·

ρt. . . ρt

... ρt 1

• Equicorrelation arises naturally in many financial applications

DECO in Action: Homogeneous 1-Factor Systems

• Life in a one-factor world

rj = βj rm + εj , σ2j = β2

j σ2m + vj

• If cross sectional dispersion of βj is small and εj ’s have similarvariance in cross section (time-variation welcome), systemwell-described by Dynamic Equicorrelation

• To price CDOs, an LHP assumption often made: Each loanhas same var, the same covar with market and the sameidiosyncratic var.

• In fact, LHP implies equicorrelation

ρ =β2σ2

m

β2σ2m + v

• One correlation if firms in same industry, another in differentindustries - accommodated by Block DECO generalization

DECO in Action: Homogeneous 1-Factor Systems

• Life in a one-factor world

rj = βj rm + εj , σ2j = β2

j σ2m + vj

• If cross sectional dispersion of βj is small and εj ’s have similarvariance in cross section (time-variation welcome), systemwell-described by Dynamic Equicorrelation

• To price CDOs, an LHP assumption often made: Each loanhas same var, the same covar with market and the sameidiosyncratic var.

• In fact, LHP implies equicorrelation

ρ =β2σ2

m

β2σ2m + v

• One correlation if firms in same industry, another in differentindustries - accommodated by Block DECO generalization

DECO in Action: Homogeneous 1-Factor Systems

• Life in a one-factor world

rj = βj rm + εj , σ2j = β2

j σ2m + vj

• If cross sectional dispersion of βj is small and εj ’s have similarvariance in cross section (time-variation welcome), systemwell-described by Dynamic Equicorrelation

• To price CDOs, an LHP assumption often made: Each loanhas same var, the same covar with market and the sameidiosyncratic var.

• In fact, LHP implies equicorrelation

ρ =β2σ2

m

β2σ2m + v

• One correlation if firms in same industry, another in differentindustries - accommodated by Block DECO generalization

DECO in Action: Basket Options• Dispersion trades: long option on a basket, short options on

components• With delta hedging, value of strategy depends solely on

correlations• Let basket weights given by w , covariance matrix of

components S , variance of basket is

σ2 = w ′Sw .

• We only know about the implied variances, not covariances.Common to assume all correlations equal

σ2 =n∑

j=1

w 2j s2

j + ρ∑i 6=j

wiwjsi sj

• Which can be solved for the implied correlation

ρ =σ2 −

∑nj=1 w 2

j s2j∑

i 6=j wiwjsi sj.

DECO in Action: Basket Options• Dispersion trades: long option on a basket, short options on

components• With delta hedging, value of strategy depends solely on

correlations• Let basket weights given by w , covariance matrix of

components S , variance of basket is

σ2 = w ′Sw .

• We only know about the implied variances, not covariances.Common to assume all correlations equal

σ2 =n∑

j=1

w 2j s2

j + ρ∑i 6=j

wiwjsi sj

• Which can be solved for the implied correlation

ρ =σ2 −

∑nj=1 w 2

j s2j∑

i 6=j wiwjsi sj.

DECO in Action: Portfolio Choice

• Elton and Gruber (1973): Averaging pairwise correlations canreduce estimation noise and deliver superior portfolios

• Ledoit and Wolf (2003, 2004): Bayesian shrinkage toequicorrelated target improves portfolios

DECO in Action: Institutions, Information andComovement

Source: Morck, Yeung & Yu, (J. of Financial Economics, 2000)

DECO Model and Theoretical Properties

Simple Likelihood Inputs

Rt = (1− ρt)In + ρtJn =

1 ρt · · ·

ρt. . . ρt

... ρt 1

LemmaThe inverse and determinant are

R−1t =

1

1− ρtIn +

−ρt

(1− ρt)(1 + [n − 1]ρt)Jn

det(Rt) = (1− ρt)n−1(1 + [n − 1]ρt).

R−1t exists iff ρt 6= 1 and ρt 6= −1

n−1 , and Rt is positive definite iff

ρt ∈ ( −1n−1 , 1).

The Model: DECO From DCC

• Decompose return covariance Vart−1(r̃t) = DtRtDt

• Work with de-volatilized returns rt = D−1t r̃t , so that

Vart−1(rt) = Rt

• DCC (Engle 2002; Aielli 2006)

Qt = Q̄(1− α− β) + αQ̃12t−1rt−1r ′t−1Q̃

12t−1 + βQt−1

RDCCt = Q̃

− 12

t QtQ̃− 1

2t

(Q̃t replaces the off-diagonal elements of Qt with zeros)

• DECO sets ρt equal to the average pairwise DCC correlation.

RDECOt = (1− ρt)In + ρtJn×n

ρt =1

n(n − 1)

(ι′RDCC

t ι− n)

=2

n(n − 1)

∑i>j

qi ,j ,t√qi ,i ,tqj ,j ,t

The Model: DECO From DCC

• Decompose return covariance Vart−1(r̃t) = DtRtDt

• Work with de-volatilized returns rt = D−1t r̃t , so that

Vart−1(rt) = Rt

• DCC (Engle 2002; Aielli 2006)

Qt = Q̄(1− α− β) + αQ̃12t−1rt−1r ′t−1Q̃

12t−1 + βQt−1

RDCCt = Q̃

− 12

t QtQ̃− 1

2t

(Q̃t replaces the off-diagonal elements of Qt with zeros)

• DECO sets ρt equal to the average pairwise DCC correlation.

RDECOt = (1− ρt)In + ρtJn×n

ρt =1

n(n − 1)

(ι′RDCC

t ι− n)

=2

n(n − 1)

∑i>j

qi ,j ,t√qi ,i ,tqj ,j ,t

The Model: DECO From DCC

• Decompose return covariance Vart−1(r̃t) = DtRtDt

• Work with de-volatilized returns rt = D−1t r̃t , so that

Vart−1(rt) = Rt

• DCC (Engle 2002; Aielli 2006)

Qt = Q̄(1− α− β) + αQ̃12t−1rt−1r ′t−1Q̃

12t−1 + βQt−1

RDCCt = Q̃

− 12

t QtQ̃− 1

2t

(Q̃t replaces the off-diagonal elements of Qt with zeros)

• DECO sets ρt equal to the average pairwise DCC correlation.

RDECOt = (1− ρt)In + ρtJn×n

ρt =1

n(n − 1)

(ι′RDCC

t ι− n)

=2

n(n − 1)

∑i>j

qi ,j ,t√qi ,i ,tqj ,j ,t

The Model (Cont’d)

Assumption

The matrix Q̄ is positive definite, α + β < 1, α > 0 and β > 0.

LemmaCorrelation matrices from every realization of a DECO process arepositive definite and the process is mean reverting.

Proof: From last lemma, sufficient for p.d. (hence invertibility) to have

ρt ∈ ( −1n−1

, 1) ∀t. To this end, note that Qt is a weighted average of positive

definite matrices and is therefore positive definite, and quadratic form RDCCt is

also. It follows that ι′RDCCt ι > 0, which implies that ρt >

−1n−1

. ρt < 1 since

these are correlations.

Two-Stage DECO Estimation

1. Stock-by-stock GARCH models to de-volatize returns

2. Estimate DECO on standardized returns

Caveat: Correlation targeting

Estimating DECO

• Use Gaussian Quasi-ML

r̃t ∼ N(0,Ht), rt = D−1t r̃t , Vart−1(rt) = Rt

• Decompose log likelihood

L = − 1

T

∑t

(log |Ht |+ r̃ ′tH−1t r̃t)

= − 1

T

∑t

(log |Dt |2 + r̃ ′tD−2

t r̃t − r ′trt)

− 1

T

∑t

(log |Rt |+ r ′tR−1

t rt)

• Important theorem: two-stage estimator will be consistent!(White 1994, Engle 2002, Engle and Sheppard 2001)

Analytic Correlation Likelihood - Key to Large CrossSections

• Payoff from making the equicorrelation assumption:Computation vastly simplified, now may use many assets inyour covariance system

LDECOCorr (θ̂, φ) = − 1

T

∑t

(log |RDECOt |+ r̂ ′tRDECO

t−1

r̂t)

= − 1

T

∑t

[log

([1− ρt ]n−1[1 + (n − 1)ρt ]

)

+1

1− ρt

(∑i

(r̂ 2i ,t)− ρt

1 + (n − 1)ρt(∑

i

r̂i ,t)2

)]

Non-Equicorrelated Returns?

TheoremIf DCC is a QMLE, then DECO is also consistent andasymptotically normal.

Proof Sketch: Show DECO score has expected value zero as long as DCC scorehas expected value zero. The expectation of the DECO score is

E [∂ log f DECO

2,t (r̃ , θ∗, φ)

∂φk] = E

[Et−1[

∂ log f DECO2,t (r̃ , θ∗, φ)

∂ρt]∂ρt

∂φk

]. (1)

∂ log f DECO2,t (r̃ , θ∗, φ)

∂ρt= (1− ρt)

−2(1 + [n − 1]ρt)−2

[(n − 1)(1− ρt)

2(1 + [n − 1]ρt)

−(n − 1)(1− ρt)(1 + [n − 1]ρt)2 + (1 + [n − 1]ρt)

2∑

i

r 2i,t

−(1 + [n − 1]ρ2t )(∑

i

ri,t)2

].

When DCC is consistent,∑

i r2i,t and

(∑i ri,t

)2have (t − 1)-conditional

expectations of n and∑

i,j ρi,j,t = n(n − 1)ρt + n, respectively.

Et−1[∂ log f DECO2,t (r̃ , θ̂, φ)/∂ρt ] reduces to zero, and as a result (1) is also zero.

Non-Equicorrelated Returns?

• Implication: arbitrary dimension DCC model can be estimatedvia DECO, this could be infeasible with DCC alone

• How? Estimate DECO to find α and β, then generate DCCfits

Qt = Q̄(1− α− β) + αQ̃12t−1rt−1r ′t−1Q̃

12t−1 + βQt−1

RDCCt = Q̃

− 12

t QtQ̃− 1

2t

Block DECO

• More flexible structure with the tractability and robustness ofDECO

• Example: industry model - each industry has a single DECOparameter and each industry pair has a singlecross-equicorrelation parameter

Rt =

(1− ρ1,1,t )In1

0 · · ·

0. . . 0

.

.

. 0 (1− ρK,K,t )InK

+

ρ1,1,tJn1

ρ1,2,tJn1×n2· · ·

ρ2,1,tJn2×n1

. . .

.

.

. ρK,K,tJnK

TheoremLike DECO, Block DECO remains consistent and asymptoticallynormal when block structure violated

2-Block DECO

R =

[(1− ρ1,1)In1 0

0 (1− ρ2,2)In2

]+

[ρ1,1Jn1×n1 ρ1,2Jn1×n2

ρ2,1Jn2×n1 ρ2,2Jn2×n2

]

Lemma

R−1 =

[b1In1 0

0 b2In2

]+

[c1Jn1×n1 c3Jn1×n2

c3Jn2×n1 c2Jn2×n2

]

det(R) = (1−ρ1,1)n1−1(1−ρ2,2)n2−1[(1+[n1−1]ρ1,1)(1+[n2−1]ρ2,2)−ρ21,2n1n2

]

Also: Conditions for existence, positive definiteness, stationarity, etc.

2-Block DECO

• For more blocks - difficult analytics, but cozily falls intocomposite likelihood framework

• More information in block composite likelihood than DCCversion - potentially more efficient

• What is composite likelihood???

2-Block DECO

• For more blocks - difficult analytics, but cozily falls intocomposite likelihood framework

• More information in block composite likelihood than DCCversion - potentially more efficient

• What is composite likelihood???

Digression: Using Composite Likelihood

• Composite likelihood splices together likelihood of subsets ofassets

• In DCC, a subset is a pair of stocks, i and j

• In Block DECO, a subset is all the stocks in pair of blocks iand j

DECO Amid Literature

Related Literature

• Two types of approaches to estimating time-varyingcovariances in large systems

1. Factor GARCH (Engle, Ng, Rothschild 1992; Engle 2008)

2. Composite likelihood (Engle, Shephard, Sheppard, 2008)

Factor (Double) ARCH

• Impose factor structure on system

rt = BFt + εt

Var(rt) = BVar(Ft)B ′ + Var(εt)

• Univariate GARCH dynamics in factors and residuals cangenerate time-varying correlations while keeping the residualcorrelation matrix constant through time

• Benefits

1. Feasibility for large numbers of assets - only estimate n+KGARCH (regression) models

2. Full likelihood, potential for efficiency

• Limitations

1. Dont have factors?2. Mis-specification - dynamics in residual correlations?

Factor (Double) ARCH

• Impose factor structure on system

rt = BFt + εt

Var(rt) = BVar(Ft)B ′ + Var(εt)

• Univariate GARCH dynamics in factors and residuals cangenerate time-varying correlations while keeping the residualcorrelation matrix constant through time

• Benefits

1. Feasibility for large numbers of assets - only estimate n+KGARCH (regression) models

2. Full likelihood, potential for efficiency

• Limitations

1. Dont have factors?2. Mis-specification - dynamics in residual correlations?

Composite Likelihood

Qt = Q̄(1− α− β) + αQ̃12t−1rt−1r

′t−1Q̃

12t−1 + βQt−1

RDCCt = Q̃

− 12

t QtQ̃− 1

2t

• Model DCC for pairs of assets

• Modeling any pair will give consistent estimates of α and β(though noisy)

• Randomly select subset of all pairs to improve accuracy - apartial likelihood technique

• Benefits

1. Very flexible - no structural assumption required

• Limitations

1. Partial likelihood - never efficient

Composite Likelihood

Qt = Q̄(1− α− β) + αQ̃12t−1rt−1r

′t−1Q̃

12t−1 + βQt−1

RDCCt = Q̃

− 12

t QtQ̃− 1

2t

• Model DCC for pairs of assets

• Modeling any pair will give consistent estimates of α and β(though noisy)

• Randomly select subset of all pairs to improve accuracy - apartial likelihood technique

• Benefits

1. Very flexible - no structural assumption required

• Limitations

1. Partial likelihood - never efficient

Where Does DECO Fit?

• Fundamental Trade off• Factor ARCH - strict structural assumptions• Composite Likelihood - abandons useful information

• DECO flexibly balances this trade off• Structural models (like factor structures) can be estimated as

part of the first stage, and DECO can clean up correlationdynamics in residuals

• With blocks or first-stage structure, can be as well-specified ascomposite likelihood, yet more efficient

Monte Carlos

Performance: DECO as DGP

• As a first check, we ask How does DECO do when correctlyspecified?

• Simulate DECO processes using various

1. Time series dimensions2. Cross section sizes3. Parameter (α , β ) values

Table 1: DECO as Generating Process

Performance: DCC as DGP

• How does DECO do when incorrectly specified?

• Simulate DCC processes

• Standard deviation of pairwise correlations large, ∼ 0.33

Table 2: DCC as Generating Process

Correlation Among the S&P 500

Daily S&P 500, 1995-2008

• Stocks included if traded over entire sample and a member ofS&P 500 at some point in that time

• Final count: 466 stocks

• First-stage volatility models: GJR Asymmetric GARCH

• Second-Stage (Correlation) Models

1. DECO2. 10-Block DECO (block assignments based on industry)3. DCC

Correlation Estimates: Daily S&P 500, 1995-2008

Qt = Q̄(1− α− β) + αQ̃12t−1rt−1r

′t−1Q̃

12t−1 + βQt−1

DECO Estimates: Daily S&P 500, 1995-2008

Variations: Factor GARCH and Changing the BlockStructure ex post

• In first stage GARCH standardization, can include factorregressions to extract factor-based component of correlation(this is “factor double ARCH”)

• Once correlation parameters estimated with any DECO model,can vary block structure ex post

DECO Estimates: CAPM Residuals

Out-of-Sample Hedging

• Pre-estimation window, 1995-1999

• Forecast one-day ahead, form minimum variance portfolios

• Calculate sample variance of portfolios

• Which model delivers lower variance?

• Re-estimate model parameters every 22 days

Minimum Variance Portfolios

• Two hedge portfolios of interest: global minimum varianceportfolio (GMV ) and minimum variance portfolio subject toexpected return of at least q (MVq)

• GMV portfolio weights solve

minωω′Σω s.t. ω′ι = 1.

• MVq portfolio solves s.t. additional constraint ω′µ ≥ q

ωGMV =1

AΣ−1ι

ωMV =C − qB

AC − B2Σ−1ι+

qA− B

AC − B2Σ−1µ,

A = ι′Σ−1ι, B = ι′Σ−1µ C = µ′Σ−1µ

• µ is historical mean, q = 10% annual

Table 4: S&P 500 O.S. Hedging

Conclusion

• DECO: estimating covariance models of arbitrary dimension

• Consistent even when equicorrelation is violated

• Block DECO loosens structure yet retains simplicity androbustness

• Good descriptor of correlation in the S&P 500

Figure 1. DECO and DCC After Removing Factors