Dynamic Equicorrelation - NYUpages.stern.nyu.edu/~bkelly/deco_QFE_slides.pdf · Equicorrelation...
Transcript of Dynamic Equicorrelation - NYUpages.stern.nyu.edu/~bkelly/deco_QFE_slides.pdf · Equicorrelation...
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