日本統計学会誌, 第45巻, 第2号, 329頁-352頁

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1 45, 2, Market Risk Aggregation Using Copula and Its Application to Financial Practice Toshinao Yoshiba 2009 We investigate how a copula between risk factors takes portfolio diversification effect on market risk in both stressed and normal situations when the portfolio is comprised of stock and bond securities. In Japanese market over the past ten years, the movement of stock price and interest rate had shown positive correlation; in this situation a financial industry-standard practice of risk aggregation (such as variance-covariance method) provides large diversification effect on portfolio market risk, compared with the simple sum of stock risk and bond risk. In the European debt crisis since 2009, however, stock price plunge and interest rate surge occur at the same time; in such a stressed situation the diversification effect is considered to be limited. Hence, we consider copulas representing the dependency between risk factors in a stressed situation within the industry-standard framework of time series models. Devising copulas to capture the stressed situations even in normal periods as well as examining data in stressed both periods and areas, we measure the impact of the copulas in a stressed situation on the diversification effect and carry several implications. : ( toshinao.yoshiba@boj.or.jp).

2 value-at-risk VaR expected shortfall ES VaR ES VaR ES diversification effect copula Basel Committee on Banking Supervision, Joint Forum (2010)

3 VaR ES 1 5 1) 5 1 (2014) Engle (2002) DCC dynamic conditional correlation GARCH generalized autoregressive conditional heteroscedasticity Patton (2006) GAS Generalized Autoregressive Score Creal et al. (2013) conditional approach 2) 1 1 1) 5 Bloomberg 2) Christoffersen et al. (2012) Smith et al. (2012) McNeil et al. (2005) Alexander and Sheedy (2008) Choi and Min (2011) (2014)

4 VCV HS Azzalini and Capitanio (2003) 1 t 3) t µ σ λ ν 4 (2.1) 4) f(x) = 2 ( ) ( ) x µ σ t ν T ν+1 λ x µ ν + 1 σ σ (x µ) 2 /σ 2 + ν (2.1) t ν ( ) ν t T ν+1 ( ) ν + 1 t t t ν ) (2009) Miura and Oue (2000) (2014) VaR ES 4) λ 0 t ν t t Azzalini and Capitanio (2003) Hansen (1994) Fernández and Steel (1998) 1 t Barndorff-Nielsen (1977) generalized hyperbolic t (2011) t Aas and Haff (2006)

5 % (µ) (σ) (λ) (ν) 99% ( ) ( ) (0.153) (0.382) ( ) ( ) (0.129) (0.283) 99% 1% 1% 1 t t 2 10% t C (u 1,..., u d ) Pr(X 1 x 1,..., X d x d ) Pr(X 1 x 1,..., X d x d ) = C(Pr(X 1 x 1 ),, Pr(X d x d )) (2.2) C(u 1,..., u d ) [0, 1] d d c(u 1,..., u d ) 5) c(u 1,..., u d ) = d C(u 1,..., u d ) u 1 u d (2.3) 5) C(u 1,..., u d )

6 Kolmogorov Smirnov Anderson Darling Kolmogorov Smirnov Anderson Darling D = AD = D = AD = p = p = p = p = Q Q t H 0 t p 2 Q Q 2 x 11,..., x 1N x 21,..., x 2N t u 11,..., u 1N u 21,..., u 2N pseudo sample 6) [0, 1] [0, 1] (u 11, u 21 ),, (u 1N, u 2N ) 1/N [0, 1] [0, 1] (u 1, u 2 ) C NP (u 1, u 2 ) c NP (u 1, u 2 ) θ c (u 1,..., u d ; θ) 6) empirical copula (2012) t

7 λ U λ L γ exp{ (( ln u 1 ) γ + ( ln u 2 ) γ ) 1/γ } 2 2 1/γ 0 α (u α 1 + u α 2 1) 1/α 0 2 «1/α δ 1 δ ln 1 + (e δu 1 1)(e δu 2 1) 0 0 (e δ 1) N i=1 ln c (u i1,..., u id ; θ) (u 11, u 21 ),, (u 1N, u 2N ) tail dependency 1 U 1 u 2 U 2 u lower tail dependency λ L (2.4) C(u 1, u 2 ) λ L = lim u 0 Pr(U 2 u U 1 u) = lim u 0 C(u, u) u (2.4) U 1, U 2 1 λ U = lim u 1 Pr(U 2 > u U 1 > u) = lim u 1 1 2u + C(u, u) 1 u (2.5) λ U Gumbel Clayton Frank 2 1 7) 2 7) (2005)

8 u 2 1 u 2 2 8) 1 rotated copula, reflected copula 9) t t t t t ν ν 3 10) ν 2 t 3 t ν 11) t ν 1 ρ 1 ρ 2 2 θ : (1 θ) 8) c C (u 1, u 2 ) c(u 1, u 2 ) = c C (u 1, 1 u 2 ) 270 copula rotated by 270 degrees 9) 2 survival copula ĉ (u 1, u 2 ) c (u 1, u 2 ) = ĉ (1 u 1, 1 u 2 ) 10) t ν 2 ν ν 3 11) ρ ν T ν+1 ( p (1 ρ) (ν + 1) /(1 + ρ)) ν

9 t λ U = λ L ρ Φ(Φ 1 (u 1 ), Φ 1 (u 2 ) ρ) 0 q t ρ, ν T ν(tν 1 (u 1 ), Tν 1 (1 ρ)(ν+1) (u 2 ) ρ) 2T ν+1 (1+ρ) θ, ρ 1, ρ 2 θφ(φ 1 (u 1 ), Φ 1 (u 2 ) ρ 1 ) +(1 θ)φ(φ 1 (u 1 ), Φ 1 (u 2 ) ρ 2 ) Φ(, ρ) 0 1 ρ 2 T ν(, ρ) ν ρ 2 t Φ 1 ( ) 1 T ν ( ) 1 ν t Tν 1 ( ) 0 ρ 1 ρ 2 θ 12) % VaR 97.5% ES AS AB ln S t ln S t + S 5 r t r t + r P V T = 5 r t P V = AS S te S S t + AB e (rt+ r)t e r tt = AS S AB( r)t (AS + AB) (2.6) (2.6) 1% 99%VaR (2.6) 2.5% 97.5%ES ( S, r) 99%VaR S 1 r 99 99%VaR 1 99% S 1% r 99% (2.6) 12) R optim L-BFGS-B ρ 1,ρ 2 θ 3 EM 3 2

10 VaR ES VaR (99%) ES (97.5%) VaR VaR = AS( S 1% ) + AB( r 99% )T (2.7) 4 VaR 50.8 VaR 13) VaR 97.5%ES 2.5% S 2.5% 2.5% r 97.5% VaR (2.6) ES ES = AS( S 2.5% ) + AB( r 97.5% )T (2.8) ES ES VaR VaR ES ES diversification effect = (2.9) ) 13) VaR Daníelsson et al. (2013) jointly regular varying 14) Kojadinovic et al. (2011)

11 339 5 VaR ES VaR (99%) ES (97.5%) VCV % % HS % % % % VaR ES BIC Schwarz Bayesian information criterion BIC 15) 3.1 VaR VCV HS 2 99%VaR 97.5%ES 5 VCV VaR 55% ES 59% HS VaR 41% ES 42% VCV HS VCV Ω VCV VCV Ω t 16) Ω VaR (2.7) VaR = (AS( S 1% ), AB( r 99% )T )Ω(AS( S 1% ), AB( r 99% )T ) (3.1) ES (2.8) ES = (AS( S 2.5% ), AB( r 97.5% )T )Ω(AS( S 2.5% ), AB( r 97.5% )T ) (3.2) 15) Tsafack (2009) AIC Akaike information criterion BIC BIC l(ξ) p N 2 ln l(ξ) + p ln N 2 (2010) AIC BIC 16)

12 VaR ES 44% HS 41 42% VCV t [0,1] BIC ) 6 17) t BIC t BIC

13 341 6 BIC γ γ α α δ ρ t ρ ν ρ ρ θ BIC 1 6 BIC BIC BIC VaR ES 7 18) 18) t Yoshiba (2015) (10 6, ) 500 [10 7, 10 6 ], [1 10 6, ] 100 Christoffersen et al. (2012) t 10 (2013) 50

14 VaR ES 100 VaR ES VaR ES 7 5 VCV HS 2 9

15 343 7 VaR ES VaR ES (99%) (97.5%) % % % % % % % % % % % % % % t % % % % ( ) % % ( ) 41 49% VCV % 3.1 VCV 41 42% HS % 19) 6 BIC VaR ES 46% 48% t 49% t t ν t ν t 19) HS 18) (2013)

16 % (µ) (σ) (λ) (ν) 99% ( ) ( ) (0.234) (0.869) ( ) ( ) (0.141) (0.247) 6 θ = 14.5% ρ 1 = VaR 16% ES 18% ) /10/1 2012/10/1 [0,1] IBEX35 99% 8 8 [0,1] 5 20) (2012) (2012) (a) (b) (c) (a) (b) (2013)

17 ) 9 BIC t ) BIC BIC 1 VaR ES % 10 ES 29% 2 N 2 21) 1 2 c(u 1, u 2 ) c(u 1, u 2 ) = c(u 1, 1 u 2 ) c(u 1, u 2 ) = c(1 u 1, u 2 ) 22) BIC McNeil et al. (2005)

18 BIC ( ) γ ( ) γ ( ) α ( ) α δ ρ t ρ ν ρ ρ θ VaR ES VaR ES (99%) (97.5%) % % % % % % % % % % % % % % t % % % % ( ) % % 1/N VaR ES N 10 7 VaR 39.0 VaR 44.4 VaR 44.7

19 t 7 t t t t 3% /4/1 1991/3/ % t BIC t % (µ) (σ) (λ) (ν) 99% ( ) ( ) (0.359) (1.103) ( ) ( ) (0.205) (0.252)

20 BIC ( ) γ ( ) γ ( ) α ( ) α δ ρ t ρ ν ρ ρ θ ) BIC t 48% 7 17% BIC t 23) 11

21 VaR ES VaR ES (99%) (97.5%) % % % % % % % % % % % % % % t % % % % ( ) % % 10 17% VaR VaR 6.6 ES ) 1 VaR ES 10 ES VaR VCV VCV VCV 2 VaR ES 24) (2013) VaR VaR

22 VaR ES 4 t BIC 5 t ν t t ρ Ω vine hierarchal Archimedean copula nested Archimedean copula t t t Joe (2014) Brockmann and Kalkbrener (2010)

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24 Daníelsson, J., Jorgensen, B. N., Samorodnitsky, G., Sarma, M. and de Vries, C. G. (2013). Fat tails, VaR and subadditivity, Journal of Econometrics, 172(2), Engle, R. (2002). Dynamic conditional correlation, Journal of Business & Economic Statistics, 20(3), Fernández, C. and Steel, M. F. J. (1998). On Bayesian modeling of fat tails and skewness, Journal of the American Statistical Association, 93(441), Hansen, B. (1994). Autoregressive conditional density estimation, International Economic Review, 35(3), (2014). VaR ES Joe, H. (2014). Dependence Modeling with Copulas: CRC Press/Chapman & Hall. (2012)., No.2012-J-6. Kojadinovic, I., Yan, J. and Holmes, M. (2011). Fast large-sample goodness-of-fit tests for copulas, Statistica Sinica, 21(2), (2009). VaR, IMES Discussion Paper Series No.2009-J-27. (2015)., IMES Discussion Paper Series No.2015-J-12. McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques, and Tools, Princeton University Press. Miura, R. and Oue, S. (2000). Statistical methodologies for the market risk measurement, Asia-Pacific Financial Markets, 7(4), (2011). t 40(2), Okimoto, T. (2008). New evidence of asymmetric dependence structures in international equity markets, Journal of Financial and Quantitative Analysis, 43(3), (2014)., 44(1), Patton, A. (2006). Modelling asymmetric exchange rate dependence, International Economic Review, 47(2), (2010). 29(3), Smith, M. S., Gan, Q. and Kohn, R. J. (2012). Modelling dependence using skew t copulas: Bayesian inference and applications, Journal of Applied Econometrics, 27(3), (2005)., 24( 2), Tsafack, G. (2009). Asymmetric dependence implications for extreme risk management, Journal of Derivatives, 17(1), (2012). 21 Vol.III, HP, 5, , (2013)., No.13-J-4. Yoshiba, T. (2015). Maximum likelihood estimation of skew-t copulas with its applications to stock returns, Research Memorandum No.1195, resm1195.pdf

03.Œk’ì

03.Œk’ì HRS KG NG-HRS NG-KG AIC Fama 1965 Mandelbrot Blattberg Gonedes t t Kariya, et. al. Nagahara ARCH EngleGARCH Bollerslev EGARCH Nelson GARCH Heynen, et. al. r n r n =σ n w n logσ n =α +βlogσ n 1 + v n w