Is the Potential for International Diversification Disappearing? A Dynamic Copula Approach Peter Christoffersen University of Toronto Vihang Errunza McGill University Kris Jacobs University of Houston Hugues Langlois McGill University 9 th International Institute of Forecasters Workshop September 28 th, 212 Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 1/2
Motivation Rolling Linear Threshold Key Contribution Key Motivation Understanding the evolution of co-movements in international markets is crucial for asset pricing and portfolio selection Research Questions 1. How and has cross-country dependence changed through time? Cross-country linear correlations have not increased (Bekaert, Hodrick, and Zhang (29)) 2. Is correlation a satisfactory dependence measure in international markets? s are higher in down markets (Longin and Solnik (21), Ang and Bekaert (22), Ang and Chen (22)) 3. How does the diversification benefit of emerging markets compare to developed countries? Differences in the evolution of correlations? Differences in tail dependence? Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 2/2
Motivation Rolling Linear Threshold Key Contribution Key Average Linear Rolling on Weekly Returns 1989 28 16 Developed Markets Rolling Rolling.8.6.2.8.6.2 199 1995 2 25 21 13 Emerging Markets, IFCG 6 months 2 years 199 1995 2 25 21 All 29 Markets Rolling.8.6.2 199 1995 2 25 21 Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 3/2
Motivation Rolling Linear Threshold Key Contribution Key Average Threshold s on Weekly Returns.6 16 Developed Markets 1973 29.5 Threshold on Returns.3.2.1.6 1.5.5 1 13 Emerging Markets 1989 28 Threshold on Returns.5.3.2.1 Empirical Gaussian Distribution 1.5.5 1 Standard Deviations Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 4/2
Motivation Rolling Linear Threshold Key Contribution Key Key Contributions Our key contributions are 1. We develop a model which can be estimated on a large set of countries can accommodate for dynamic dependence a trend in correlation positive tail dependence univariate and multivariate asymmetries 2. We develop a diversification benefit measure that takes into account higher order moments Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 5/2
Motivation Rolling Linear Threshold Key Contribution Key Key Our key results are 1. Cross-country dependence has significantly increased over time Dependence for emerging markets is still a lot lower than for developed countries 2. We find overwhelming evidence of non-normalities in dependence Tail dependence is both positive and asymmetric for developed and emerging markets 3. We confirm with different panel regressions that dependence is positively linked to volatility although dependence is related to market integration, financial, and macro variables, the time trend is still significant and remains unexplained Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 6/2
Multivariate Volatility for Each Country Dependence Dynamic Asymmetric Copula Estimation for Many Countries Multivariate We decompose the conditional multivariate log-likelihood function as T N L = log ( ( )) T f i,t Ri,t + log ( ( ( ))) c t F1,t (R 1,t ), F 2,t (R 2,t ),..., F N,t RN,t t=1 i=1 }{{} t=1 }{{} VOLATILITY MODEL FOR COUNTRY i DEPENDENCE MODEL FOR N COUNTRIES where T is the number of weeks in our sample developed markets 1973-29 emerging markets 1989-28 investable emerging market 1995-29 N is the number of countries used in the estimation 16 developed markets 13 emerging markets 17 investable emerging markets Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 7/2
Multivariate Volatility for Each Country Dependence Dynamic Asymmetric Copula Estimation for Many Countries Volatility for Each Country We decompose the conditional multivariate log-likelihood function as T N L = log ( ( )) T f i,t Ri,t + log ( ( ( ))) c t F1,t (R 1,t ), F 2,t (R 2,t ),..., F N,t RN,t t=1 i=1 }{{} t=1 }{{} VOLATILITY MODEL FOR COUNTRY i DEPENDENCE MODEL FOR N COUNTRIES where f i,t ( Ri,t ) is given by a AR-NGARCH model R i,t = µ i,t + σ i,t z i,t σ 2 i,t = ω i + α i ( εi,t 1 γ i σ i,t 1 ) 2 + βi σ 2 i,t 1 2 sources of univariate asymmetry 1. leverage effect γ i 2. residual asymmetry z i,t comes from an asymmetric t distribution Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 8/2
Multivariate Volatility for Each Country Dependence Dynamic Asymmetric Copula Estimation for Many Countries Dependence We decompose the conditional multivariate log-likelihood function as T N L = log ( ( )) T f i,t Ri,t + log ( ( ( ))) c t F1,t (R 1,t ), F 2,t (R 2,t ),..., F N,t RN,t t=1 i=1 }{{} t=1 }{{} VOLATILITY MODEL FOR COUNTRY i DEPENDENCE MODEL FOR N COUNTRIES where c t (F 1,t (R 1,t ),...) comes from a skewed t copula with Ψ t a time-varying correlation matrix ν a degree-of-freedom parameter λ an asymmetry parameter Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 9/2
Multivariate Volatility for Each Country Dependence Dynamic Asymmetric Copula Estimation for Many Countries The Dynamic Asymmetric Copula The copula correlation matrix is time-varying At time t, it is given by a weighted average of 3 components Γ t = (1 β Γ α Γ ) [(1 φ Γ )Ω + φ Γ Υ t] + β Γ Γ t 1 + α Γ z t 1 z t 1 where Υ t captures a deterministic trend Υ t = δ2 t 2 1+δ 2 t 2 Γ t 1 is the lagged correlation matrix z t 1 z t 1 is the cross-product of copula shocks Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 1/2
Multivariate Volatility for Each Country Dependence Dynamic Asymmetric Copula Estimation for Many Countries Estimation for Many Countries Estimation on many countries is made possible by two improvements 1. We use a moment estimator for Ω where 1 T T t=1 z t z t ˆΩ = 1 T T t=1 z t z 1 t φ T Γ T t=1 Υ t 1 φ Γ is the sample copula correlation 2. From Engle, Shephard and Sheppard (28), we maximize the composite log-likelihood CL(θ) = T N t=1 i=1 j>i ln c t (η i,t, η j,t ; θ) }{{} Bivariate log-likelihood for countries i and j Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 11/2
Evolution of Evolution of Evolution of Regional Copula Evolution of Average Tail Dependence Implied Threshold 1989 28 16 Developed Markets.8.6.2 199 1995 2 25 21 13 Emerging Markets, IFCG.8.6.2 199 1995 2 25 21 All 29 Markets.8.6.2 199 1995 2 25 21 16 DMs vs 13 EMs Cross.8.6.2 199 1995 2 25 21 Dynamic Long run 1995 29 16 Developed Markets.8.6.2 199 1995 2 25 21 17 Emerging Markets, IFCI.8.6.2 199 1995 2 25 21 All 33 Markets.8.6.2 199 1995 2 25 21 16 DMs vs 17 EMs Cross.8.6.2 199 1995 2 25 21 Constant 9% C.I. Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 12/2
Evolution of Evolution of Regional Copula Evolution of Average Tail Dependence Implied Threshold Evolution of Copula for Developed Markets.8.6.2.8.6.2.8.6.2 1973 29 1975 198 1985 199 1995 2 25 21 1989 28 1975 198 1985 199 1995 2 25 21 1995 29 1975 198 1985 199 1995 2 25 21 Dynamic Long run Constant 9% C.I. Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 13/2
Evolution of Evolution of Regional Copula Evolution of Average Tail Dependence Implied Threshold Evolution of Regional Copula European Union (EU).8.6.2 European Union (EU) 9 95 5 1.8.6.2 Developed Non EU 9 95 5 1.8.6.2 Latin America 9 95 5 1.8.6.2 Emerging Eurasia 9 95 5 1 Developed Non EU Latin America Emerging Eurasia Developed Non EU.8.6.2 9 95 5 1.8.6.2 9 95 5 1.8.6.2 9 95 5 1 Latin America Emerging Eurasia Latin America.8.6.2 9 95 5 1.8.6.2 9 95 5 1 Emerging Eurasia Emerging Eurasia.8.6.2 9 95 5 1 Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 14/2
Evolution of Average Tail Dependence Evolution of Evolution of Regional Copula Evolution of Average Tail Dependence Implied Threshold Tail Dependence.3.2.1 Lower Tail Upper Tail 16 Developed Markets 199 1995 2 25 21 13 Emerging Markets Tail Dependence Tail Dependence.3.2.1.3.2.1 199 1995 2 25 21 All 29 Markets 199 1995 2 25 21 Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 15/2
Implied Threshold Evolution of Evolution of Regional Copula Evolution of Average Tail Dependence Implied Threshold.6 16 Developed Markets 1973 29.5 Threshold.3.2.1 Threshold.6.5.3.2.1 1.5.5 1 13 Emerging Markets 1989 28 Empirical Gaussian Distribution NS DCD t Copula NS DCD Skewed t Copula NS DCD Skewed t Copula with calibrated λ 1.5.5 1 Standard Deviations Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 16/2
A Measure A Measure Evolution of Diversification Benefit To take into account higher order moments in the portfolio return distribution, we construct a diversification benefit measure based on expected shortfall [ ] ES q t (R i,t) = E R i,t R i,t F 1 i,t (q) Note that We define ES q t = VaRq t (w t R t ) }{{} Perfect diversification ES q t (w t R t ) }{{} ES q t = Portfolio expected shortfall N w i,t ES q t (R i,t) } i=1 {{ } No diversification CDB t (w t, q) = ESq t ES q t (w t R t ) ES q t ES q t Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 17/2
A Measure A Measure Evolution of Diversification Benefit The conditional diversification benefit measure 1. lies between and 1 2. does not depend of expected returns The Special Case of Normality If returns are multivariate normal and q = 5%, then CDB t reduces to portfolio s volatility { }}{ wt Σ tw t CDB t(w t, q) = 1 wt σ t }{{} upper bound for portfolio s volatility Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 18/2
Evolution of Diversification Benefit A Measure Evolution of Diversification Benefit CDB 1.8.6.2 16 Developed Markets, 1973 29 16 Developed Markets, 1973 29 1.8 VolCDB.6 VolCDB EW CDB CDB EW.2 75 8 85 9 95 5 1 75 8 85 9 95 5 1 VolCDB 1 13 Emerging Markets, 1989 28 1 13 Emerging Markets, 1989 28.8.8 CDB.6 VolCDB.6.2.2 75 8 85 9 95 5 1 75 8 85 9 95 5 1 1 All 29 Markets, 1989 28 1 All 29 Markets, 1989 28.8.8 CDB.6 VolCDB.6.2.2 75 8 85 9 95 5 1 75 8 85 9 95 5 1 Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 19/2
1. We propose a new model capturing dynamic trending copula correlation, tail dependence, and multivariate asymmetries 2. We propose a conditional diversification benefit measure which takes into account higher order moments We find that 1. Cross-country dependence has significantly increased over time 2. But dependence for emerging markets is still lower than for developed countries Christoffersen, Errunza, Jacobs and Langlois (212) Is the Potential for International Diversification Disappearing? 2/2