Dynamic Equicorrelation Bryan Kelly (Joint work with Rob Engle) - - PowerPoint PPT Presentation

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 burden even 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 Equicorrelation

An equicorrelation matrix takes the form Lemma 1: Invertible and positive definite if and only if

Rt = (1 − ρt)In + ρtJn =     1 − ρt · · · ... . . . 1 − ρt     +     ρt ρt · · · ρt ... . . . ρt    

R−1

t

= 1 1 − ρt In + −ρt (1 − ρt)(1 + [n − 1]ρt)Jn det(Rt) = (1 − ρt)n−1(1 + [n − 1]ρt). ρt ∈ ( −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

RDCC

t

= ˜ Qt

− 1

2 Qt ˜

Qt

− 1

2

RDECO

t

= (1 − ρt)In + ρtJn×n

ρt = 1 n(n − 1)

• ι′RDCC

t

ι − n

• =

2 n(n − 1)

• i=j,i>j

qi,j,t √qi,i,tqj,j,t

Qt = ¯ Q(1 − α − β) + α ˜ Qt−1

1 2 rt−1r′

t−1

˜ Qt−1

1 2 + βQt−1.

The Model

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

¯ Q is p.d., α + β < 1, α > 0, β > 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 = − 1 T

• t
• log |Dt|2 + ˜

r′

tD−2 t

˜ rt − r′

trt

• − 1

T

• t
• log |Rt| + r′

tR−1 t rt

• ˜

rt|t−1 ∼ N(0, Ht), Ht = DtRtDt, rt ≡ D−1

t

˜ rt

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 − ρ1,1,t)In1 · · · ... . . . (1 − ρK,K,t)InK     +     ρ1,1,tJn1 ρ1,2,tJn1×n2 · · · ρ2,1,tJn2×n1 ... . . . ρK,K,tJnK    

Block DECO

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

R−1 = b1In1 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) − ρ2

1,2n1n2

• c1

= ρ1,1

• ρ2,2(n2 − 1) + 1
• − ρ2

1,2n2

(ρ1,1 − 1)

• [ρ1,1(n1 − 1) + 1][ρ2,2(n2 − 1) + 1] − n1n2ρ2

1,2

• c2

= ρ2,2

• ρ1,1(n1 − 1) + 1
• − ρ2

1,2n1

(ρ2,2 − 1)

• [ρ1,1(n1 − 1) + 1][ρ2,2(n2 − 1) + 1] − n1n2ρ2

1,2

• c3

= ρ1,2 n1n2ρ2

1,2 −

• ρ1,1(n1 − 1) + 1
• ρ2,2(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

• f 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 system

rt = BFt + ǫt V ar(rt) = BV ar(Ft)B′ + V ar(ǫt)

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 + ǫt V ar(rt) = BV ar(Ft)B′ + V ar(ǫt) V art(rt) = BV art(Ft)B′ + V ar(ǫt) 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 + ǫt V ar(rt) = BV ar(Ft)B′ + V ar(ǫt) Ft, ǫt ∼ GARCH V art(rt) = BV art(Ft)B′ + V art(ǫt)

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

• f

Randomly select subset of all pairs - a partial likelihood technique

RDCC

t

= ˜ Qt

− 1

2 Qt ˜

Qt

− 1

2

Qt = ¯ Q(1 − α − β) + α ˜ Qt

1 2 rt−1r′

t−1 ˜

Qt

1 2 + βQt−1

α, β

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

α, β

!

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

Estimation

Model 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

• f 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:

ωGMV = 1 AΣ−1ι ωMV = C − qB AC − B2 Σ−1ι + qA − B AC − B2 Σ−1µ, A = ι′Σ−1ι B = ι′Σ−1µ C = µ′Σ−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 = βjrm + ej, σ2

j = β2 j σ2 m + 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

• ften made: Each loan has same var, the same covar

with market and the same idiosyncratic var. In fact, LHP implies equicorrelation

ρ = β2σ2

m

β2σ2

m + v .

Equicorrelation and Options (2)

Dispersion trades: long option on a basket, short

• ptions 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

σ2 = w′Sw ρ = σ2 − n

j=1 w2 js2 j

• i=j wiwjsisj

. σ2 =

n

• j=1

w2

js2 j + ρ

• 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