asymmetric dependence for large dimensions: the canonical ... · ppge/uff joint with andréasheinen...

Motivation-Contribution Copulas-Canonical Vine Model Data-Results VaR Conclusions

Asymmetric dependence for large dimensions:the Canonical Vine Autoregressive Model

Alfonso ValdesogoPPGE/UFF

joint with

Andréas HeinenTHEMA, Université de Cergy-Pontoise

São Paulo, 14 August 2014

Heinen & Valdesogo Canonical Vine Autoregressive Model


Outline

Motivation - Contribution

Copulas - Canonical Vine

Model

Data - Results

Value-at-Risk

Conclusions



Jan95 Nov97 Oct00 Aug03 Jul06 May09 Jun12−20

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15SP&500

Jan95 Nov97 Oct00 Aug03 Jul06 May09 Jun12−20

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15HEALTH

Low Volatility Period

Low Volatility Period

High Volaility Period






Multivariate volatility models: dimension

This paper is about high-dimensional multivariate modelsfor returns using time varying canonical vine copulas.One remaining challenge in multivariate GARCH models is tocome up with flexible and parsimonious models for largesets of assets.GARCH model with a fat-tailed (asymmetric) distributionand/or leverage effect provides a good statistical description ofreturns of one stock.Problem: in multivariate case, have to ensure that variancecovariance is sdp.Need complicated multivariate non-linear constraints.Even bigger problem: Large-dimensional case (e.g. 100assets), most approaches break down.One viable alternative: scalar DCC model, but: “one size fitsall” dynamics.



Stylized facts

Well-known that returns are not normally distributed (fat tails,asymmetry).Longin & Solnik (1995) find evidence that correlations are notconstant in a CCC model, but that they have risen over 30years and that they depend on economic variables (interestrates).Dependence is not constant over time.Longin & Solnik (2001) find that there is more asymptoticdependence for negative returns than for positive returns.More dependence in times of crisis than in good times.



Exceedance correlation:

Corr(y1, y2|y1 ≤ θ1, y2 ≤ θ2)︸︷︷︸Small or negative returns

Corr(y1, y2|y1 ≥ θ1, y2 ≥ θ2)︸︷︷︸Large or positive returns

.

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70.2

0.25

0.3

0.35

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0.45

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0.55

0.6

0.65

0.7Exceedance Correlation: SP&500 − HEALTH



What do we do in this paper?

We propose a new model, the CAVA:that separates the joint distribution into models for themarginals and for the dependence.for marginals, can take any GARCH model you like (or othermodels as well!).the dependence is captured by a very flexible multivariatecopula: canonical vine.

accounts for asymmetric dependence (more dependencewhen returns are below mean) and time variation.the structure for the canonical vine is consistent with a CAPMor market sector model.

that can be estimated with an effort proportional to thedimension of the problem.that provides VaR estimates consistent with asymmetricdependence.



What are copulas?

Copulas are functions that join multivariate probabilitydistribution functions to their one-dimensional marginaldistributions functions,

There exist whole catalogs of copulas...... less so for multivariate caseEssentially: Gaussian, Student tHere: use new very flexible copula: canonical vine copula.

F1(y1) F2(y2) F3(y3)

Copula functionC(F1(y1), F2(y2), F3(y3))

F (y1, y2, y3)

''OOOOOOOOOO

�� wwoooooooooo

��

1



Contour plots, the difference is only in the underlying copula

−2 −1 0 1 2−2

−1

0

1

2Normal copula, ρ = 0.5

−2 −1 0 1 2−2

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0

1

2Student’s t copula, ρ = 0.5, ν = 3

−2 −1 0 1 2−2

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0

1

2Rotated Gumbel copula, κ = 1.5

−2 −1 0 1 2−2

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0

1

2

Mixed normal copula, ρ1 = 0.95, ρ

2 = 0.05



Exceedance correlation

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

0.1

0.2

0.3

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0.5

0.6

0.7

Normal copula, ρ = 0.5

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Student’s t copula, ρ = 0.5, ν = 3

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Rotated Gumbel copula, θ = 1.5

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

0.1

0.2

0.3

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0.7

Mixed normal copula, ρ1 = 0.95, ρ

2 = 0.05



Idea of Vine copula

Due to Bedford and Cooke (AoStats, 2002), Aas, Czado,Frigessi and Bakken (IME, 2007).Example of normal: Correlation matrix needs to besemi-definite positive and have ones on the diagonal.Can be decomposed wlog into as many unrestricted partialcorrelations, according to a regular vine structure.example, correlation matrix R :

R =

1 ρ12 ρ13ρ12 1 ρ23ρ13 ρ23 1

equivalent to ρ12, ρ13, ρ23|1



Canonical Vine

1!"#$%&'(2!"#$%&'(

3!"#$%&'(4!"#$%&'( 5!"#$%&'(

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#####################################

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15

12!"#$%&'( 13!"#$%&'(15!"#$%&'(

14!"#$%&'($$$$$$$$$$$$$$

23|1

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24|1

25|1

23|124|1

25|1&&&&&&&&&&

34|12

35|12

34|12 35|1245|123

1



The dependence structure: a nonlinear CAPM

Small dimensional case, can use a full canonical vine modeland decompose the joint distribution to the fullest level ofgenerality.This leads to estimating n(n − 1)/2 bivariate copulas for across section of n stocks.This means we would have to estimate (95)(94)/2 = 4465bivariate copulas!Instead, rely on CAPM structure: Model copula (possiblydynamic) of each stock with the market.Alternative model: dependence on market and on sector.Gaussian copula for the remaining dependence.



We use a data set of returns of 95 stocks from the S&P500.The stocks are chosen from 10 different sectors, for whichStandard and Poor reports returns.The sector indexes are Energy, Industrial, Health, Financial,Utilities, Materials, Consumer Discretionary, ConsumerStaples, Information Technology and TelecomFor each sector we pick the 5 largest and the 5 smallest stocksin terms of market capitalization data of June 2012, except forthe Telecom sector, where there are only 5 stocks that arepresent during all of our sample period.We use weekly data from January 1, 1995 to June 6, 2012,which gives us 909 returns. In addition to the stocks we usethe S&P500 and the 10 sectorial stock indexes.



Sample with market cap

ENER 10,59 IND 10,09 HEALTH 11,61 FINAN 14,05 UTIL 3,71XOM 29,06 GE 16,27 JNJ 12,20 WFC 9,62 SO 9,05CVX 15,26 UTX 5,46 PFE 11,61 JPM 7,36 EXC 7,06SLB 6,68 MMM 4,78 MRK 8,15 BAC 4,81 DUK 6,73OXY 5,32 CAT 4,60 ABT 6,79 C 4,65 D 6,61COP 5,25 UNP 4,27 UNH 4,23 AXP 3,73 NEE 6,07SUN 0,38 SNA 0,29 XRAY 0,37 PBCT 0,24 PNW 1,23NBR 0,31 AVY 0,24 CVH 0,32 LM 0,21 POM 0,99NFX 0,31 PBI 0,23 PDCO 0,26 ZION 0,20 GAS 0,98RDC 0,30 R 0,18 PKI 0,21 AIV 0,19 TEG 0,95TSO 0,25 RRD 0,16 THC 0,14 FHN 0,12 TE 0,85

MAT 3,42 C DISCR 11,09 C STAP 12,13 IT 20,20 TEL 3,12DD 10,94 MCD 6,67 WMT 15,10 AAPL 21,72 T 53,37DOW 9,10 DIS 6,03 PG 11,53 MSFT 10,02 VZ 31,25PX 7,47 HD 5,74 KO 11,40 IBM 9,10 CTL 6,11NEM 6,09 CMCSA 4,65 PEP 7,15 ORCL 6,14 S 2,15MOS 4,90 SBUX 3,00 MO 4,49 INTC 5,33 FTR 0,88ATI 0,80 HAR 0,21 MKC 0,45 LSI 0,15BMS 0,76 BIG 0,19 TAP 0,41 FLIR 0,14OI 0,76 GT 0,18 TSN 0,38 TER 0,11SEE 0,73 WPO 0,17 SWY 0,31 JDSU 0,10X 0,69 DV 0,14 STZ 0,22 MOLX 0,09



Copula estimates

The Gaussian copula and the Student t copula are not alwaysselected.Moreover, in 17% of all cases the copula that we selectpresents some kind of asymmetry.Most of this asymmetry can be found in the dependence ofstocks and sectors with the S&P500 index return.Not all sectors present the same kind of asymmetry in thedependence.Most of the selected time varying copulas are found for thedependence of stocks and sectors with the S&P500 indexreturn.



Copula estimates

Model τ/ω Dof τ Prob α β

HEALTH RGumbel 0.64 0.10 0.85JNJ RGumbel 0.38 0.05 0.90PFE RGumbel 0.43 0.07 0.87MRK Gaussian 0.54 0.09 0.86ABT RGumbel 0.37 0.07 0.89UNH Clayton 0.17XRAY Gumbel 0.16CVH Gaussian 0.26PDCO Frank 0.09PKI Frank 0.32THC Clayton 0.15C STAP Student t 0.66 6.27 0.07 0.90WMT Gaussian 0.49PG Gaussian 0.33 0.03 0.95KO Gaussian 0.42 0.15 0.76PEP Student t 0.23 7.88MO Frank 0.21MKC RGumbel 0.11TAP Frank 0.17TSN Frank 0.16SWY Gaussian 0.31STZ Gaussian 0.18



We evaluate the performance of our model in terms of itsability to generate good estimates of Value at Risk.The criterions we use to judge performance are the likelihoodratio test of correct unconditional coverage of Kupiec (95) andthe likelihood ratio test of conditional coverage ofChristoffersen (1998).We consider 1000 random portfolios of 2 stocks, 10 stocks and50 stocks.We use the Historical DCC model as a benchmark for theevaluation of our model



Value at Risk

Panel A: 2-Assets PortfoliosUnconditional Coverage Conditional Coverage

10% 5% 2.5% 1% 10% 5% 2.5% 1%CAVA 71% 75% 80% 84% 71% 78% 87% 86%Hsim 57% 66% 81% 92% 61% 70% 86% 91%

Panel B: 10-Assets PortfoliosUnconditional Coverage Conditional Coverage

10% 5% 2.5% 1% 10% 5% 2.5% 1%CAVA 69% 73% 82% 86% 71% 76% 87% 87%Hsim 49% 59% 76% 91% 53% 63% 82% 90%

Panel C: 50-Assets PortfoliosUnconditional Coverage Conditional Coverage

10% 5% 2.5% 1% 10% 5% 2.5% 1%CAVA 65% 71% 78% 84% 69% 74% 84% 85%Hsim 47% 53% 71% 89% 53% 59% 80% 88%



Conclusions

We propose a model for the joint distribution of a large set ofreturns that

is flexiblecan accommodate multivariate departures from normality

We show that the dependence in cross-section of assets can bewell modeled by focusing on time-varying asymmetricdependence between returns and market and sectors.In terms of out-of-sample VaR we show that the CAVA modelperforms better than the Filtered Historical DCC.


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