Risk Factors and DiversificationDynamics

Peter ChristoffersenRotman School of Management, University of Toronto,

Copenhagen Business School, andCREATES, University of Aarhus

13rd Lectureon Thursday

Motivation and Overview• For portfolio risk management we need a

dynamic model of asset or risk factor returns.• The model needs to be able to handle fairly large

dimensions (say 50 assets or factors)• The model needs to capture dynamics in

volatility, correlation and also conditional non-normality.

• Two applications:– 1. Fama/French/Carhart U.S. equity factor dynamics– 2. International equity market diversification dynamics

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First Application:U.S. Equity Market Factors

• The Fama and French (1993) and Carhart (1997)four factors are pervasive in empirical assetpricing and in applied portfolio allocation.

• Traditional analysis focuses on constant lineardependence (correlation) between the factorswhich is typically low. Normality implicitlyassumed.

• Let us consider dynamic nonlinear dependenceand non-normal distributions.

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Preview of Findings

• An asymmetric Student t copula worksreasonably well in capturing the non-normaldistribution in weekly factor returns

• The economic value of accounting for non-normality in portfolio allocation is significant

• Portfolio risk measured by Expected Shortfall(ES) is much larger when using the non-normal distribution

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Data

• Weekly equity factor returns from July 5, 1963through December 31, 2010 from Ken French’sdata library.– Market excess return factor (Market)– Book-to-market factor (Value)– Size factor (Size)– Momentum factor (Momentum)

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Threshold Correlations

• Let us illustrate nonlinear (nonnormal)dependence between the factors via thresholdcorrelations

• Where u is the threshold and F is theempirical CDF.

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Fact

or D

ynam

ics:

ACF

of R

etur

ns a

nd A

bsol

utes

Volatility Dynamics

• The strong evidence of volatility dynamics inthe factors is modeled using asymmetricGARCH models allowing for a leverage effect:

• Let us also allow for simple expected returndynamics via:

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ACF of Residuals and Absolute Residuals

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Univariate Factor Distributions• We shall use the Hansen (1994) skewed t

distribution for the residual of each factor

• Where

• When combined with the AR-GARCH returndynamics we get the return distributions

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QQ Plots of Residuals versus Skewed t

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AR-GARCH Estimates via MLE.Normal and Skewed t Distributions

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Now Link the Univariate ModelsUsing Copulas

• Patton (2006) provides the conditional version ofSklar’s (1959) theorem:

• Where we have defined the univariate probability

• Note that we have already modeled all themarginal distributions

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The Skewed t Copula• The copula PDF is constructed from the skewed t

distribution in Demarta and McNeil (2005) given by

• Where• KX is the modified Bessel function of the third kind.

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Copu

la C

onto

ur P

lots

Dynamic Copula Correlations• Use Engle’s (2002) DCC model for z to capture

dynamics in the dependence across factors

• The normalization gives the copula correlation

• When z equals the return residual, ε, then weget the simple linear correlation DCC modelwith a multivariate Skewed t distribution.

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MLE on ct

Dynamic Copula Correlations

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Threshold Correlations for Factor Residuals

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Economic Implications

• Portfolio selection exercise where myopicCRRA investor takes positions in factors via

• Subject to the margin constraint

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Measuring Economic Value

• Use certainty equivalent of the averagerealized utility out of sample

• Where• This follows Pastor and Stambaugh (JFE, 2000)

and Patton (JFEC, 2004).

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Out of Sample Results:MR=20%. RA=7 or 10.

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Out of Sample Results:MR=50%. RA=7 or 10.

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Reverse Threshold Correlations for weekly returns

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Summary of First Application

• An asymmetric Student t copula worksreasonably well in capturing the non-normaldistribution in weekly factor returns

• The economic value of accounting for non-normality in portfolio allocation is significant

• Portfolio risk measured by Expected Shortfall(ES) is much larger when using the non-normal distribution

• Be careful with static factor models

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Second Application:International Equity Indexes

• Weekly data• Dataset 1: 16 Developed Markets (DM) Jan 12, 1973

to June 12, 2009 (1901 obs)• Dataset 2: IFCG 13 Emerging Markets (EM) Jan 6,

1989 to July 25, 2008 (1021 obs)• Dataset 3: IFCI: 17 Emerging Markets July 7, 1995 to

June 12, 2009 (728 obs)• We often combine Developed and Emerging Markets• Univariate AR(2)-NGARCH(1,1) models

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Descriptive Statistics. DMs

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Descriptive Statistics. EMs

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Country Risk. 1989-2008

• NGARCH models capture volatility dynamics ineach country well.

• Emerging market (EM) countries have muchhigher volatility than do developed market(DM) countries.

• Kurtosis is also higher much higher in EMsthan in DMs.

• But what about portfolio risk? First take a lookat simple rolling linear correlations.

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Rolling Correlations

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Non-Stationary Copula Correlations• Skewed t copula with DOF ν and asymmetry λ.• Copula shocks:• New copula correlation dynamic:

• Copula correlationtrend model:

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Composite Likelihood Estimation• Problem: High dimensions. Even when using

correlation targeting, the numerical complexity is highdue to the frequent inversion of the correlation matrix.Biased estimates.

• Solution: Composite Likelihood Estimation (Engle,Shephard, Sheppard 2008)

where ct(*) denotes the bivariate copula distributionfor country i and j.

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Dynamic Asymmetric t Copula (DAC)

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Black: Average DAC correlationwith DMsDark Grey: Average DACcorrelation with EMsLight Grey: Average DACcorrelation with All

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Portfolio Risk:Average Intra- and Inter-Regional Copula Correlations.

Copula correlation increasesare pervasive.

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CopulacorrelationRange: 90th

and 10th

percentile

The lowestcopulacorrelationsare alsoincreasing.

Conditional Diversification Benefit (CDB)for Nonnormal Returns

• ES Definition:• Upper bound: Lower bound:

• Define:

• Choose weights wt to maximize CDB• Compare with Gauss CDB with q=.50:

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ConditionalDiversificationBenefit (CDB)

You cannotavoidincreasingcorrelations byportfoliooptimization

Tail Dependence

• The non-normality in the copula implies thatthe second-moment based measures ofdiversification no longer suffice.

• We use the tail dependence measure:

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Dynamic TailDependence DACModel

Averages acrosscountry pairs.

Tail dependence israpidly rising inDMs

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Thresholdcorrelations fromempirical shocksand from modelgenerated datausing actualparameterestimates.

Also DAC modelwith calibratedλ=-1.

Summary• Correlation between markets has trended upward.

Developed market correlations are very high now• The correlation between emerging markets and have

also been trending upwards over time but they are ata lower level

• Non-normality important for joint distribution• Tail dependence has increased sharply for developed

markets• Diversification potential of emerging markets lies in

the tails.• Are volatility and correlation related?• Do market development, integration, and volatility

impact EM correlations?48