Dynamic Equicorrelation

Bryan Kelly(Joint work with Rob Engle)

The Problem with Covariances...

Since early ‘80s, attempts have been made to estimate multivariate GARCH models

Specifications so complex that traditional models are difficult to estimate for more than a few assets

In finance, we want to work with large cross sections

Portfolio selection

Derivatives (basket options, CDOs, etc.)

Risk Management

DCC: Problem Solved?

Engle (2002) introduces Dynamic Conditional Correlation

Massive parameter reduction: an entire matrix evolution can be described by two parameters (sort of...)

Computational burdeneven for a few parameters: must calculate inverse and determinant of N x N matrices many thousands of times in 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)

Where to begin? Simplify 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 a single time series

Outline

Model and theoretical properties

DECO amid extant covariance models

Monte Carlo evaluation

Correlations among the S&P 500

Equicorrelation in action

Introducing DECO

Defining EquicorrelationAn equicorrelation matrix takes the form

Lemma 1:

Invertible and positive definite if and only if

Rt = (1! !t)In + !tJn =

!

""#

1! !t 0 · · ·

0. . . 0

... 0 1! !t

$

%%& +

!

""#

!t !t · · ·

!t. . .

... !t

$

%%&

R!1t =

11! !t

In +!!t

(1! !t)(1 + [n! 1]!t)Jn

det(Rt) = (1! !t)n!1(1 + [n! 1]!t).

!t ! ("1

n" 1, 1)

The Model

rt, n x 1 vector, unit variance, correlations Rt

DECO is born from the DCC process

Average pairwise DCC correlations

RDCCt = Q̃t

! 12 QtQ̃t

! 12

RDECOt = (1! !t)In + !tJn!n

!t =1

n(n! 1)

!"!RDCC

t "! n"

=2

n(n! 1)

#

i "=j,i>j

qi,j,t"qi,i,tqj,j,t

Qt = Q̄(1! !! ") + ! ˜Qt!1

12 rt!1r

"t!1

˜Qt!1

12 + "Qt!1.

The Model

Assumption 1:

Theorem 1: Correlation matrices generated by every realization of a DECO process are p.d. and mean reverting

Q̄ is p.d., ! + " < 1, ! > 0, " > 0.

Estimation

Gaussian (Quasi-) Maximum Likelihood

Assume returns are conditionally normal

Log likelihood can be decomposed into

Important theorem: two-stage estimator will be consistent!

L = ! 1T

!

t

"log |Dt|2 + r̃!tD

"2t r̃t ! r!trt

#! 1

T

!

t

"log |Rt| + r!tR

"1t rt

#

r̃t|t!1 ! N(0, Ht), Ht = DtRtDt, rt " D!1t r̃t

Estimation

Proceed in two easy steps

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

2. Estimate DECO on standardized returns

Data Is Non-Equicorrelated?

Have no fear, DECO will provide consistent estimates anyway

Theorem 2: As long as DCC (a very general, non-equicorrelated covariance model) is a consistent estimator of correlations, DECO will be too

How useful: arbitrary dimension DCC model can be estimated via DECO, this could be infeasible with DCC alone

Block DECO

More flexible structure with the tractability and robustness of DECO

Example: industry model - each industry has a single DECO parameter and each industry pair has a single cross-equicorrelation parameter

Rt =

!

""#

(1! !1,1,t)In1 0 · · ·

0. . . 0

... 0 (1! !K,K,t)InK

$

%%& +

!

""#

!1,1,tJn1 !1,2,tJn1!n2 · · ·

!2,1,tJn2!n1

. . .... !K,K,tJnK

$

%%&

Block DECO

Theorem 3: Two-block DECO has easy analytic inverses and determinants - thus as computationally feasible as DECO

R!1 =!

b1In1 00 b2In2

"+

!c1Jn1"n1 c3Jn1"n2

c3Jn2"n1 c2Jn2"n2

"det(R) = (1! !1,1)n1!1(1! !2,2)n2!1

!(1 + [n1 ! 1]!1,1)(1 + [n2 ! 1]!2,2)! !2

1,2n1n2

"

c1 =!1,1

!!2,2(n2 ! 1) + 1

"! !2

1,2n2

(!1,1 ! 1)![!1,1(n1 ! 1) + 1][!2,2(n2 ! 1) + 1]! n1n2!2

1,2

"

c2 =!2,2

!!1,1(n1 ! 1) + 1

"! !2

1,2n1

(!2,2 ! 1)![!1,1(n1 ! 1) + 1][!2,2(n2 ! 1) + 1]! n1n2!2

1,2

"

c3 =!1,2

n1n2!21,2 !

!!1,1(n1 ! 1) + 1

"!!2,2(n2 ! 1) + 1

"

Block DECO

For more blocks - difficult analytics, but cozily falls into composite likelihood framework

More information in block composite likelihood than DCC version - potentially more efficient

Theorem 4: like DECO, block DECO is a QML estimator of non-block-equicorrelated systems

Digression: Using Composite Likelihood

Composite likelihood splices together likelihood of subsets of assets

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 i and j

Rt =

!

"

#

$

Pairs of stocks

Pairs of Blocks

DECO Amid Current Literature

Related Models

Two types of approaches to estimating time-varying covariances 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 systemrt = BFt + !t

V ar(rt) = BV ar(Ft)B! + V ar(!t)

Factor (Double) ARCH

Impose factor structure on system

Univariate GARCH dynamics in factors can generate time-varying correlations while keeping the residual covariance matrix constant through time.

rt = BFt + !t

V ar(rt) = BV ar(Ft)B! + V ar(!t)

V art(rt) = BV art(Ft)B! + V ar(!t)

Ft ! GARCH

Factor (Double) ARCH

Impose factor structure on system

Univariate GARCH dynamics in factors and residuals can generate time-varying correlations while keeping the residual correlation matrix constant through time.

rt = BFt + !t

V ar(rt) = BV ar(Ft)B! + V ar(!t)

Ft, !t ! GARCH

V art(rt) = BV art(Ft)B! + V art(!t)

Factor (Double) ARCH

Benefits

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

2. Full likelihood, potential for efficiency

Limitations

1. Don’t know factors? Don’t have data?

2. Misspecification - dynamics in residual correlations?

Composite Likelihood DCC

Estimate DCC for arbitrary cross sections

Modeling any pair will give consistent estimates of

Randomly select subset of all pairs - a partial likelihood technique

RDCCt = Q̃t

! 12 QtQ̃t

! 12

Qt = Q̄(1! !! ") + !Q̃t

12 rt!1r

"t!1Q̃t

12 + "Qt!1

!,"

Composite Likelihood

Benefits

1. Very flexible - no structural assumption required

Limitations

1. Partial likelihood - never efficient

Fundamental Tradeoff

Factor ARCH - strict structural assumptions

Composite Likelihood - abandons useful information

Where Does DECO Fit?

Flexibly balances this tradeoff

Structural models (like factor structures) can be estimated as part of the first stage, and DECO can clean up correlation dynamics in residuals

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

Monte Carlos

Performance: DECO as DGP

As a first check, we ask “How does DECO do when correctly specified?”

Simulate DECO processes using various

1. Time series dimensions

2. Cross section sizes

3. Parameter ( ) 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

S&P 500, 1995-2008

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

Final count: 466 stocks

EstimationModel menu: Choose one of each...

First-Stage Model

1. Constant Factor

2. CAPM

3. Fama-French Three-Factor

4. 10 Industry Factors

Second-Stage (Correlation) Model

1. DECO

2. 10-Block DECO

3. DCC

Using Composite Likelihood

Composite likelihood splices together likelihood of subsets of assets

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 i and j

!

Table 3: Full Sample Results

Interpretation

Intuitively, DECO will outperform DCC when there is a dominating component of pairwise correlations inducing all pairwise correlations to move together

In this case, smoothing reduces noise without compromising structure

Out-of-Sample Forecasts

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

(G)MV Portfolios

Solution to Markowitz problem:

!GMV =1A

!!1"

!MV =C ! qB

AC !B2!!1" +

qA!B

AC !B2!!1µ,

A = !!!"1!

B = !!!"1µ

C = µ!!"1µ

A Twist: Varying Block Structure

No reason that best block structure for estimation should be best for your application

Once estimated (with any model) can vary block structure ex post

After we estimate each model, we will also look at how ex post changes in blocks affect hedges

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

Equicorrelation in Action

Equicorrelation Appeal

Life in a one-factor world

If cross sectional dispersion of βj is small and idiosyncrasies have similar variance each period, system well-described by Dynamic Equicorrelation

S&P data, perhaps surprisingly, well described by this case

rj = !jrm + ej , "2j = !2

j "2m + vj

Equicorrelation and Options (1)

Natural one-factor model: credit derivatives (esp. CDO’s)

Key feature in loan portfolios: correlation in default risk

Wall Street model: one correlation if firms in same industry, another in different industries.

More broadly, to price CDO’s, an LHP assumption often made: Each loan has same var, the same covar with market and the same idiosyncratic var.

In fact, LHP implies equicorrelation

! ="2#2

m

"2#2m + v

.

Equicorrelation and Options (2)

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

We only know about implied variance, not covariances - so assume all correlations are equal

!2 = w!Sw

! ="2 !

!nj=1 w2

j s2j!

i !=j wiwjsisj.!2 =

n!

j=1

w2j s2

j + "!

i !=j

wiwjsisj

Equicorrelation and Portfolio Choice

Elton and Gruber (1973): Averaging pairwise correlations can reduce estimation noise and deliver superior portfolios

Ledoit and Wolf (2003, 2004): Bayesian shrinkage to equicorrelated target improves portfolios

Conclusion

DECO: estimating covariance models of arbitrary dimension

Consistent even when equicorrelation is violated

Block DECO loosens structure yet retains simplicity and robustness

Good descriptor of correlation in the S&P 500