Dependence and VaR Estimation:An Empirical Study of Chinese Stock Markets using Copula. Baoliang Li WISE, XMU Sep. 2009

Transcription

1 Dependence and VaR Estimation:An Empirical Study of Chinese Stock Markets using Copula Baoliang Li WISE, XMU Sep. 2009

2 Outline Question: Dependence between Assets Correlation and Dependence Copula:Basics and Extensions Dependence between SHSI and SZCI Portfolio VaR: Copula-based MC Method Future Study

3 Questions: Dependence between Assets Dependence structure beween assets is important in financial decision-making. Dependence Structure of SH and SZ Stocks Tail Dependence? VaR Estimation using Copula Better Performance?

4 Correlation and Dependence Correlation is an important concept in finance. Optimal Porfolio Construction (Markowitz, 952) Pricing: CAPM and APT (Sharpe, 963; Roll, 97) Portfolio VaR Estimation (Tsay,2006) Pros: Simple to Calculate Easy to understand Cons: Finite Variance ( May not exist under fat tail distribution) Linear (Correlation will change under nonlinear transformation) Tail Dependence (important for VaR, see below for different tail behavior)

5 Bivariate Normal (Corr=0.8)

6 Same Corr, Same Marginal Distribution, Normal(L) and Gumbel(R), (Embrechts,2002)

7 Correlation and Dependence Correlation works well under multivariate normal distribution (or more generally, elliptical distribution) Finance theory explicitly or implicitly assumes asset returns follow multivariate normal distribution (Embrechts,2002) Yet, Empirical evidences show this is not true (Mandelbrot,963;Ang and Chen 2002)

8 How to Describe association between asset returns? Copula is a better alternative. (Sklar, 959) Advantages of Copula (Embrechts,2002; Patton, 2006; Nelson,2006) : More reasonable multivariate distribution Separating dependence from marginals Can use to study tail behavior

9 Copula: Basics and Extensions Definition and Properties Measures of Dependence Extension:Conditional Copula Inference: Estimation and Selection of Copulas Application of Copula in Finance

10 Definition: Copula Function (,, d) = (,, d d) = P F( X) F( x),, Fd ( Xd) Fd ( xd) = C( F( x),, Fd( xd) ) = Cu (,, u ) H x x P X x X x ( ) d (,, ) = ( ),, ( ) ( ) C u u H F u F u n n n

11 Properties of Copula Copula is essentially an multivariate distribution function. Same properties:

12 Sklar s Theorm (959) Multivariate distribution can be separated into two parts: marginals and Copula Copula links Marginals to form new multivariate distribution Remarks: Most important Theorem in Copula Continuous vs Discrete Variable

13 Copula Density (,, x ) h x d = ( ( ),, ( )) ( ) ( ) ( ( ),, d ( d) ) i( i) ( ) ( ) 2 C F x Fd xd F x Fd xd F x F x x x d d d c F x F x f x = d i= ( ),, ( ) ( ) c F x F x = d d d (,, x ) h x i= f i d ( x ) i

14 Examples:Normal Copula and t Copula (different behavior at (0,0) and (,)) Gaussian t u2 0 0 u u2 0 0 u

15 Symmetrised Joe-Clayton Copula (different behavior at (0,0) and (,)) 5 5 symmetrised Joe-Clayton symmetrised Joe-Clayton u2 0 0 u u2 0 0 u

16 Measures of Dependence Rank Correlation Coeff Spearman s rho Kendall s tau Tail Dependence: ( ( ) ( ) ) λu = lim P F X > u G Y > u u λ + 0 ( ( ) ( ) ) L = lim P F X < u G Y < u u

17 Tail Dependence U = lim = lim = lim u u ( ( ) ( ) ) ( ( ) >, ( ) > ) P G( Y) > u λ = lim P F X > u G Y > u ( ) ( > ( ), > ( )) PY ( > G ( u) ) u u P F X u G Y u P X F u Y G u ( ) 2 u+ C u, u u λ + L u 0 u 0 ( ( ) ( ) ) ( ( ) <, ( ) < ) P G( Y) < u = lim P F X < u G Y < u = lim = lim = lim ( ) ( < ( ), < ( )) PY ( < G ( u) ) u 0 u P F X u G Y u P X F u Y G u (, ) C u u u

18 Tail Coeff of Normal, t and SJC Normal Copula: T Copula: λ = λ = 0 U U L v+ L ( v ) λ = λ = 2 t + ρ / + ρ SJC Copula: τ = 2 2 U / κ τ L = 2 / γ

19 Extensions: Conditional Copula Extension to accommodate time series Conditional Copula (Patton, 200) ( ) = ( ) ( ) ( ) H x, y I C F x I, G y I I t t t t t t t t Remarks: Conditional Variables should be the same for marginals and Copula Sklar s theorem for conditional Copula

20 Dynamic Conditional Copula Patton (200, 2004) : Exchange rate Jondeau and Rockinger(2006): International stock markets Chiou and Tsay(2008): U.S. and Taiwan

21 Parameters Estimation ( ) ( ) n n d ( ) ( ) ( ) l x; θ,, θ, θ = log c F x ; θ,, F x ; θ ; θ + f x ; θ d j d dj d i ij i j= j= i= EML CML (Genest, 995;Shih andlouis,995) IFM (Joe and Xu,996;Joe,997) Remarks: Pros and Cons

22 IFM:Inference Function Method (, ; θ ) = (, ; θ ) ( ; θ ) ( ; θ ) h x y I c u v I f x I g y I t t h t t c t t f t t g T ( ) ( ˆ ) ˆ θ = arg max L θ = arg max log f x ; θ f f t t f t= T ( ) ( ˆ ) ˆ θ = arg max L θ = arg max log g y ; θ g g t t f t= T ( ) ( ) ( ) ( ) ˆ θ = arg max L θ = arg max log c F x ; ˆ θ, G y ; ˆ θ ; θ c c t t t f t t g c t=

23 Selection of Copulas How to select among different copulas? Goodness of Fit (Genest, 2006) Information Criteria (Manner,2007 ) AIC = 2log(max. likelihood) + 2k BIC = 2log(max. likelihood) + k log( T ) Joe (997): Computable and Explainable

24 Applications in Finance Risk management: (Li, 999;Embrechts et al,999,2002; McNeil, 2005; Alexander,2008) Portfolio Construction: (Patton, 2004) Pricing: Structured products (Cherubini and Luciano,2000; Chiou and Tsay,2008) Contagion: RS-Copula (Rodriguez,2007)

25 Dependence between SHSI and SZCI Copulas in Domestic Academic Study Model Specification Marginal Specification Copula Specification Empirical Results Compare with DCC Model

26 Copulas in Domestic Academic Study Time lag (First introduced by Zhang, 2002) Focus on introduction of Copula (Wei, 2004) Empirical Application to Chinese stock markets (Gong and Li, 2005;others ): Marginals: Empirical Dist, t Dist, GARCH Copula: Static Copula Strange: they always get what they want!! These can be improved to accommodate stylized facts of Chinese stock markets

27 Data Preliminary Analysis SHSI and SZCI: Jan. 2,997-Mar. 30,2009 Fat tail Autocorr Volatility clustering Leverage effect

28 Data: Leverage effect 7 6 深圳 A 股指数上海 A 股指数 Relative Daily Index Closings 5 Index Value 深圳 A 股指数日对数收益率 0.05 Return 上海 A 股指数日对数收益率 0.05 Return

29 Fat tail 0. 正态 QQ 图 : 深圳 A 股 0. 正态 QQ 图 : 上海 A 股 Quantiles of Input Sample Quantiles of Input Sample Standard Normal Quantiles Standard Normal Quantiles

30 Autocorr Sample ACF of Returns: 深圳 A 股 Sample ACF of Returns: 上海 A 股 Sample Autocorrelation Sample Autocorrelation Lag Lag

31 Volatility clustering Sample ACF of Squared Returns: 深圳 A 股 Sample ACF of Squared Returns: 上海 A 股 Sample Autocorrelation Sample Autocorrelation Lag Lag

32 Marginal Specification ARMA-GJR-GARCH-t (Bollerslev,Christoffersen and Diebold,2006) Mean: ARMA to alleviate autocorrelation Variance: GARCH to include Volatility Clustering Innovation term: t distribution to consider fat tail Leverage: GJR-GARCH or EGARCH

33 Copula Specification Normal Copula ( u ) ( u ) Cnorm ( u u ) dss 2 ( ρ ) Φ Φ s 2ρss 2 + s2, 2 = exp 2 2 2π ρ 2 Parameter: Correlation (Same as traditional) Zero tail dependence Use as Benchmark

34 Copula Specification t Copula: (, ) = ( ), ( ) ( ) C u u t t u t u t 2 v, v v 2 Natural extension of Normal Copula Symmetric tail dependence

35 Copula Specification Symmetrised Joe-Clayton Copula { } κ γ κ γ CJC ( u, u2) = ( u) + ( u2) C u, u = 0.5 C u, u + 0.5C u, u + u + u ( ) ( ) ( ) SJC 2 JC 2 JC 2 2 κ = /log2 ( 2 τu ) γ /log ( τ ) Allows asymmetric tail dependence / γ τ, τ 0, = ( ) 2 L U L / κ

36 Dynamic Copula Empirics show that correlation coeff is timevarying (Engel, 2002) Dynamic conditional copula (Patton 2006) Normal Cop and transplant to t Cop: ρ ω βρ α 0 t =Λ + t + Φ t j Φ 0 j= 2t j ( ) u ( u )

37 Dynamic Copula Symmetrised Joe-Clayton Copula 0 τ =Λ ω + β τ + α u u Ut U U Ut U t j 2t j 0 j= 0 τ =Λ ω + β τ + α u u Lt L L Lt L t j 2t j 0 j= here Λ ( x) = + e x

38 Empirical results Marginal estimation Copula estimation Comparison and Evaluation

39 Marginal: SZCI

40 Marginal: SHSI

41 Diagnostic test

42 QQ-plot of Standard Residuals

43 Copula Estimation

44 Copula Estimation 2

45 Correlation of Normal Copula Normal copula time-varying constant

46 Corr and Tail Dependence of t Cop rho of t copula time-varying constant tail dependence of t copula time-varying constant

47 Upper and Lower Tail of SJC Cop 0.9 SJC copula - Average tail dependence SJC copula - Difference between lower and upper tail

48 Selection of Copulas

49 Compare with DCC-MGARCH

50 Conclusions High time-varying Corr Coeff Approximate symmetric tail dependence Economic Explanation: Same regulation, rapid capital and information flow; Homogenuous investors, Policy markets, Closed investment environment, limited investment intruments,lack of shortselling mechanism Economic Implication: Little benefit from Diversification in SH and SZ; VaR (Tail dependence)

51 Portfolio VaR: Copula-based MC Method Traditional Methods: HS, Analytic and MC; DCC Method using Copulas Modification based on dependence measures MC using Copula Backtesting: Compare with traditional Methods Empirical Results

52 Traditional Methods Analytic or DCC: m VaR = VaR + 2 ρ VaR VaR t it ijt it jt i= i< j VaR =.65 w h + ( w) h + 2 w( w) ρ h h 2 2 t t 2t t t 2t

53 Pitfalls Individual asset: fat tail vs normal dist Dependence: asymmetric dependence (Ang and Chen, 2002)

54 Modification based on dependence measures Kendall s tau m VaR = VaR + 2 τ VaR VaR t it ijt it jt i= i< j Tail dependence m L t = it + 2 λijt it jt i= i< j VaR VaR VaR VaR Remarks: No theoretical foundation

55 MC using Copula = ( ) ( ) ( ) ( ) Pr Z z = Pr wx+ wy z t wx+ wy z t ( ) ( ) (, ) C F x I G y I I dxdy t t t t t t t w zt yt + w w = cfxi ( ( ) ( ) ) t t t, GyI t t It ifxi t( t ) dxgyi t( t ) dy ( * ) t Pr Z VaR = p

56 MC using Copula Hard to solve * VaR t Using MC method

57 Backtesting Procedure Kupiec s test (995) Christoffersen s test (998) Loss Function (998) 2 ( ) ( ˆ ) LRKupiec = 2ln L p / L π χ ( ) ( ˆ ) 2 LRcc = 2ln L p / L Π χ 2 L t+ (, ) (, ) f r VaR r < VaR = g r VaR r > VaR pt, + t+ pt, + t+ pt, + t+ pt, + t+

58 Empirical Results:95% VaR t Copula analytic historical simulation Monte Carlo DCC

59 Backtest: 95% VaR

60 Empirical Results:95% VaR 0 5 t Copula analytic historical simulation Monte Carlo DCC

61 Backtest: 99% VaR

62 Future Study Theory: Extension to High-dimensional Copula Selection of Copulas Application: There are potential applications where Correlation is used

63 High-dimensional Copula:4 assets, 2 industries, portfolio (Hierarchical Copula,Berg and Aas,2008)

64 Application obstacles Diagnostic test Computer sofeware