Modeling Co-movements and Tail Dependency in the International Stock Market via Copulae

Transcription

1 Modeling Co-movements and Tail Dependency in the International Stock Market via Copulae Katja Ignatieva, Eckhard Platen Bachelier Finance Society World Congress June 2010, Toronto K. Ignatieva, E. Platen Dependency in the International Stock Market 1/25

2 Linear Portfolio value of portfolio w = (w 1,..., w d ) of assets S t = (S 1,t,..., S d,t ) : V t = d w j S j,t j=1 profit and loss (P&L) function: L t+1 = (V t+1 V t ) = Value-at-Risk at level α: d w j S j,t (e X j,t+1 1) j=1 X t+1 = (log S t+1 log S t ) VaR(α) = F 1 L (α) K. Ignatieva, E. Platen Dependency in the International Stock Market 2/25

3 The VaR depends on the distribution F X of the risk factor increments X = (X 1,..., X d ). 1 How to model the dependency among X 1,..., X d? 2 How does F X and the dependency among X 1,..., X d vary over time? K. Ignatieva, E. Platen Dependency in the International Stock Market 3/25

4 Traditional approach: Riskmetrics the conditional distribution of log-returns is multivariate normal: X t N(0, Σ t ) the covariance matrix Σ t is estimated by: ˆΣ t = (e λ 1) s<t e λ(t s) X t s X T t s decay factor λ (0 < λ < 1) is determined by backtesting λ = 0.94 provides best results (Morgan/Reuters, 1996) Drawbacks: does not allow to generate tail dependence does not allow heavy tails K. Ignatieva, E. Platen Dependency in the International Stock Market 4/25

5 Copula based approach the conditional distribution of log-returns is modelled with Copula C: X t C{F X1 (x 1 ),..., F Xd (x d ), θ t } F X1,..., F Xd are marginal distributions θ t dependence parameter Specify marginal distributions Specify dependence structure K. Ignatieva, E. Platen Dependency in the International Stock Market 5/25

6 Outline 1 Motivation 2 Copulae and Value-at-Risk 3 Copula Estimation 4 Empirical Analysis Specify marginals Specify dependence structure 5 Value-at-Risk applications 6 Conclusion K. Ignatieva, E. Platen Dependency in the International Stock Market 6/25

7 Copulae Theorem (Sklar s Theorem) Let F be a d-dimensional distribution function with marginals F 1..., F d. Then there exists a copula C with F (x 1,..., x d ) = C{F 1 (x 1 ),..., F d (x d )} (1) for every x 1,..., x d R. If F 1,..., F d are continuous, then C is unique. On the other hand, if C is a copula and F 1,..., F d are distribution functions, then the function F defined in (1) is a joint distribution function with marginals F 1,..., F d. K. Ignatieva, E. Platen Dependency in the International Stock Market 7/25

8 Generating tail dependence 1 Elliptical Copulae Gaussian Copula (no tail dependence) CΨ Ga(u 1,..., u d ) = Φ Ψ {Φ 1 (u 1 ),..., Φ 1 (u d )} where Φ Ψ d-dimensional standard normal cdf Student-t Copula (symmetric tail dependence) Cν,Ψ t (u 1,..., u d ) = t ν,ψ {tν 1 (u 1 ),..., tν 1 (u d )} where t d (ν, 0, Ψ) is Student-t cdf, Ψ is the correlation matrix, ν df 2 Archimedean Copulae, Mixture Copula Models Clayton (lower tail dependence) θ (0, ) Gumbel (upper tail dependence) θ (1, ) 3 Survival Copulae C (u 1, u 2 ) = 1 u 1 u 2 + C(1 u 1, 1 u 2 ) survival Clayton (upper tail dependence) survival Gumbel (low tail dependence) K. Ignatieva, E. Platen Dependency in the International Stock Market 8/25

9 Value-at-Risk with Copulae The process {X t } T t=1 of log-returns can be modelled as X j,t = μ j,t + σ j,t ε j,t with E[ε j,t ] = 0, E[ε 2 j,t ] = 1, j = 1,..., d and E[X j,t F t 1 ] = μ j,t E[(X j,t μ j,t ) 2 F t 1 ] = σ 2 j,t where F t is the available information at time t. ε t = (ε 1,t,..., ε d,t ) are standardized i.i.d. innovations with a joint distribution function F ε ε j, j = 1,..., d have continuous marginal distributions F j K. Ignatieva, E. Platen Dependency in the International Stock Market 9/25

10 VaR with Copulae For the log-returns {x j,t } T t=1, j = 1,..., d Value-at-Risk at level α is estimated: 1 determination of the innovations ˆε t (e.g. by degarching) 2 specification and estimation of marginal distributions F j (ˆε j ) 3 specification of a copula C and estimation of dependence parameter θ 4 simulation of innovations ε and losses L 5 determination of ˆVaR(α), the empirical α-quantile of F L. K. Ignatieva, E. Platen Dependency in the International Stock Market 10/25

11 Copula estimation The distribution of X = (X 1,..., X d ) with marginals F Xj (x j, δ j ), j = 1,..., d is given by: F X (x 1,..., x d ) = C{F X1 (x 1 ; δ 1 ),..., F Xd (x d ; δ d ); θ} and its density is given by f (x 1,..., x d ; δ 1,..., δ d, θ) = c{f X1 (x 1 ; δ 1 ),..., F Xd (x d ; δ d ); θ} where c is a copula density. d f j (x j ; δ j ) j=1 K. Ignatieva, E. Platen Dependency in the International Stock Market 11/25

12 Copula estimation For a sample of observations {x t } T t=1 and θ = (δ 1,..., δ d, θ) R d+1 the likelihood function is L(θ; x 1,..., x T ) = T f (x 1,t,..., x d,t ; δ 1,..., δ d, θ) t=1 and the corresponding log-likelihood function l(θ; x 1,..., x T ) = T t=1 log c{f X 1 (x 1,t ; δ 1 ),..., F Xd (x d,t ; δ d ); θ} + d j=1 log f j(x j,t ; δ j ) T t=1 Estimation methods: Exact Maximum Likelihood Inference for Margins Canonical Maximum Likelihood K. Ignatieva, E. Platen Dependency in the International Stock Market 12/25

13 Data Set Data used for regional indices S&P 500 Dow Jones EURO STOXX 50 FTSE 100 TOPIX Sample period from 01 January 1987 to 10 March 2006 K. Ignatieva, E. Platen Dependency in the International Stock Market 13/25

14 Specify Marginal Distribution Symmetric generalized hyperbolic (SGH) family of distributions: f X (x) = ( ) 1 α 1 + x 2 1 δσk λ ( α) 2π (δσ) 2 2 (λ 1 2 ) K λ 1 2 ( α ) 1 + x 2 (δσ) 2 K λ ( ) Bessel function λ and α are the shape parameters: α = 0 if λ 0 and δ = 0 if λ 0 Variance Gamma (VG) distribution: α = 0 and λ > 0 Student-t distribution: α = 0 and λ < 0 (consider λ 1 for ν = 2λ 2, std.dev. σ X = σ ν ν 2 ) Hyperbolic (HYP) distribution: λ = 1 Normal Inverse Gaussian (NIG) distribution: λ = 0.5 K. Ignatieva, E. Platen Dependency in the International Stock Market 14/25

15 Specify Marginal Distribution Normal vs. Empirical density Student t vs. Empirical density Normal density Empirical density Student t density Empirical density Figure: Logarithm of the histogram for the pooled data vs. normal density (left panel) and Student-t density (right panel). Pooled data is taken for indices S&P 500, Dow Jones EURO STOXX 50, FTSE100, TOPIX from 01 January 1987 to 10 March Estimated number of degrees of freedom for the Student-t distribution is ν = K. Ignatieva, E. Platen Dependency in the International Stock Market 15/25

16 Specify Marginal Distribution Goodness-of-fit testing: Anderson-Darling (AD) distance and the Kolmogorov-Simirnov (KS) AD = sup x R F s (x) ˆF (x), ˆF (x)(1 ˆF (x)) KS = sup F s (x) ˆF (x), x R F s (x) denotes the empirical sample distribution ˆF (x) is the estimated distribution. K. Ignatieva, E. Platen Dependency in the International Stock Market 16/25

17 Specify Marginal Distribution Normal Student t S&P 500 NIG HYP VG Normal Student t DJ EURO STOXX NIG HYP VG Normal Student t FTSE 100 NIG HYP VG Normal Student t TOPIX NIG HYP VG Figure: Box-plots for Anderson-Darling distance for modelling marginal distributions of the S&P 500, Dow Jones EURO STOXX 50, FTSE100, TOPIX with alternative residual distributions. K. Ignatieva, E. Platen Dependency in the International Stock Market 17/25

18 Model Selection: static case Akaike information criterion (AIC): AIC = 2L(α; x 1,..., x T ) + 2q favors: Student-t copula vs. mixture Gumbel & survival Gumbel for two-constituents portfolios where TOPIX is not included Student-t copula vs. mixture Gumbel & survival Gumbel model for a 3-constituent portfolio (S&P 500, DJ EURO STOXX 50, FTSE 100) Mixture Clayton & Gumbel model vs. Student-t copula for a 4-constituent portfolio (S&P 500, DJ EURO STOXX 50, FTSE 100, TOPIX) K. Ignatieva, E. Platen Dependency in the International Stock Market 18/25

19 Time-varying estimation Static case: estimate the dependence parameter at once based on the whole series of observations. Time-varying case: Estimate the dependence parameter by using subsets of size n of log-returns, that is a moving window of size n, {ˆX t } s t=s n+1 scrolling in time for s = n,..., T It generates a time-series for the dependence parameter {ˆθ t } T t=n and time-series of VaR: { ˆVaR t } T t=n. K. Ignatieva, E. Platen Dependency in the International Stock Market 19/25

20 Student-t dependence parameter time-varying Dependence parameter for (S&P 500, DJ ES 50, FTSE 100) using Student t copula, Student t marginals theta Time Dependence parameter for (S&P 500, DJ ES 50, FTSE 100, TOPIX) using Student t copula, Student t marginals theta Time Figure: Copula dependence parameter ˆθ estimated for a 3-constituent portfolio (S&P 500, Dow Jones EURO STOXX 50, FTSE 100) (upper panel) and 4-constituent portfolio constructed of (S&P 500, Dow Jones EURO STOXX 50, FTSE 100, TOPIX) (lower panel) using Student-t copula with Student-t marginals. K. Ignatieva, E. Platen Dependency in the International Stock Market 20/25

21 VaR for portfolio The one-day VaR at time t and significance level α is given by the α-quantile of the distribution of the P&L: VaR t (α) = F 1 L t+1 (α), The expected shortfall (ES) at time t is: ES t (α) = 1 N t+1 L t+1,i 1 N {Lt+1,i VaR t(α)}, t+1 i=1 N t+1 is the number of simulated portfolio returns with value less or equal than VaR t (α) and L t+1,i is the i th outcome of the N t+1 samples. K. Ignatieva, E. Platen Dependency in the International Stock Market 21/25

22 Backtesting Compare the estimated values for the VaR with the true realizations {L t } of the P&L function the exceedances ratio is given by ˆα = 1 T w T 1{L t < ˆVaR t (α)} t=w Table: Exceedances ratios (S&P 500, Dow Jones EURO STOXX 50, FTSE 100) Copula α ((α ˆα)/α)2 Student-t Gumbel & surv. Gumbel Riskmetrics (S&P 500, Dow Jones EURO STOXX 50, FTSE 100, TOPIX) Copula α ((α ˆα)/α)2 Student-t Clayton & Gumbel Riskmetrics K. Ignatieva, E. Platen Dependency in the International Stock Market 22/25

23 Estimated VaR Time VaR P&L and VaR for portfolio (S&P 500, DJ ES 50, FTSE 100) using Student t copula, Student t marginals P&L VaR 0.1 VaR 0.05 VaR Time VaR P&L and VaR for portfolio (S&P 500, DJ ES 50, FTSE 100) using Mix. Gumbel & surv. Gumbel, Student t marginals P&L VaR 0.1 VaR 0.05 VaR 0.01 Figure: P&L, VaR estimated at different confidence levels using Student-t copula (upper panel) and mixture model Gumbel & survival Gumbel with (lower panel) for a 3-constituent portfolio of (S&P 500, Dow Jones EURO STOXX 50 FTSE 100); Student-t marginals; exceedances at level α = K. Ignatieva, E. Platen Dependency in the International Stock Market 23/25

24 Summarize Results Summarize Results: Student-t assumption allows to better capture the dependent extreme values which can be observed in index log-returns Log-returns of the indices follow the Student-t distribution with about four degrees of freedom Dependence structure: Student-t is preferred over mixture Gumbel & surv. Gumbel for (S&P 500, Dow Jones EURO STOXX 50, FTSE 100) Mixture Clayton & Gumbel is preferred over Studnet-t for (S&P 500, Dow Jones EURO STOXX 50, FTSE 100, TOPIX) providing the best backtesting results. K. Ignatieva, E. Platen Dependency in the International Stock Market 24/25

25 Thank you very much! Thank you very much for your attention! K. Ignatieva, E. Platen Dependency in the International Stock Market 25/25