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1 c i.i.d MSCI World YTD 2008 HFRI Weighted Composite Equity Hedge Relative Value Macro

2 /1 2009/ i.i.d. 2 3 VaR ES 4 3 VaR ES HFR Bloomberg HFR Hedge Fund Research Inc HFR Equity Hedge Macro Relative Value Event-driven 4 Equity Hedge Macro Relative Value Event-driven

3 91 2. Malkiel and Sara 2005 Getmanky et al Miura et al LTCM 4 LTCM 2.1 Shapiro-Wilk Jarque Bera 2007 (2.1) H 0 : X i Φ(µ,σ 2 ), Φ(µ,σ 2 ): µ σ 2 X t Ljung-Box Breusch Pagan Ljung-Box p 1 5 p X t = ρ ix t 1 + ε t, (2.2) i=1 ε t Φ(0,σs), 2 H 0 : ρ 1 = = ρ p =0, H 1 : ρ i 0, i=1,...,p. Breusch Pagan X t = X t 1 + ε t, (2.3) σt 2 = g(α 0 + α 1X t 1), σt 2 : ε t H 0 : α 1 =0, H 1 : α 1 0.

4 Rejected Ratio LTCM LTCM /4 2009/ /4 2004/ /4 2009/ /1 2006/12

5 Relative Value Relative Value LTCM Relative Value Event-driven Equity Hedge Macro Shapiro Wilk Jarque Bera Relative Value Event-driven Macro Ljung Box Relative Value Macro Relative Value Macro Getmanky et al Breusch Pagan 3. VaR ES 4 VaR ES VaR ES McNeil et al VaR VaR VaR VaR α α 0 1 α VaR L l 1 α l (3.1) VaR α inf{l R : P (L>l) 1 α} =inf{l R : F L(l) α}. L α α =0.95 VaR 3.2 ES ES VaR

6 VaR ES ES i T : R R T T (y) inf{x R : T (x) y} ii F F F α (0,1) F α (3.2) q α F (α)=inf{x R : F (x) α} 3.2 α ES L E( L ) < F L (3.3) ES α 1 1 α 1 α q u(f L)du = 1 1 α 1 α VaR u(l)du q u(f L)=F L (u) F L VaR α=0.95 ES 3.3 VaR ES 4 VaR ES VaR ES Cornish Fishser Hakamada et al Takahashi and Yamamoto 2009, 2010 VaR ES VaR ES (3.4) VaR i α = µ i + σ iφ 1 (α), ES i φ(φ 1 (α)) α = µ i + σ i, 1 α µ i : σ i : VaR ES VaR ES k T α k 1 < 1 α k T T, Histrical VaR i α = L i k 1 (L i k 1 L i k){t (1 α) (k 1)}, k 1 Histrical ES i t=1 α = L i { t (1 α)t + 1 k 1 } (3.5) L i k, (1 α)t L i t : t i {L i 1,...,L i T } : {L i 1,...,L i T }

7 95 Cornish Fisher VaR ES Zangari 1996 q =Φ 1 (1 α), (3.6) Z i cf = q + (q2 1)S i 6 + (q3 3q)K i 24 Cornish Fisher VaR i α = µ i σ izcf i, Cornish Fisher ES i φ(zcf) i α = µ i σ i φ(zcf i ), S i : i K i : i (2q3 5q)S 2 i 36, 1 Macro Cornish Fisher VaR ES (3.7) p X π jφ(µ j,σj 2 ), j=1 p π j =1. j=1 EM p Papageorgiou et al VaR ES 2 Relative Value 1. Cornish-Fisher Macro

96 59 1 2011 2. p =5 Relative Value Johnson VaR ES Johnson Kaplan and Knowles 2004 Perez 2004 (3.8) F (X)=Φ(Z), ( X ξ ) Z = γ + δ g, λ X : g ln(y + y 2 +1), S U unbounded distribution (3.

8 p =5 Relative Value Johnson VaR ES Johnson Kaplan and Knowles 2004 Perez 2004 (3.8) F (X)=Φ(Z), ( X ξ ) Z = γ + δ g, λ X : g ln(y + y 2 +1), S U unbounded distribution (3.9) g(y)= ( y ) ln, S B bounded distribution 1 y Z Papageorgiou et al VaR ES 3 Relative Value Johnson unbound 3.4 VaR VaR VaR VaR 6 a 95 VaR b 1 36 VaR 5 6 VaR 5 VaR VaR VaR VaR

9 97 3. Johnson unbound Relative Value 6. VaR 1 a 36 b 1 VaR LTCM ES ES (3.10) I t I {Lt>VaRα}, R t (L t ES α)i t. E[R t]=0 L i.i.d. R i.i.d. R

10 BCa BCa VaR VaR ES VaR VaR VaR ES VaR VaR VaR ES 7 VaR Equity Hedge Event-driven Relative Value LTCM VaR Relative Value LTCM VaR 5.64 VaR Macro LTCM VaR 2 VaR Macro 2 Macro ES VaR LTCM Relative Value Event-driven ES VaR ES Equity Hedge Macro ES ES 8 VaR Macro ES Macro Relative Value VaR 6 VaR ES Equity Hedge Macro VaR 5 9 Cornish Fisher VaR Cornish Fisher Relative Value 6 10 VaR Relative Value VaR VaR 6.47 ES ES 3 95 VaR ES 95 VaR 99 VaR

11 99 11 Johnson VaR LTCM Relative Value VaR ES VaR 5 VaR Macro i.i.d. Equity Hedge Event-driven LTCM Relative Value i.i.d. 7. VaR 8. VaR

12 Cornish Fisher VaR 10. VaR 11. Johnson unbound VaR

13 i.i.d. VaR ES Johnson Macro i.i.d. Relative Value VaR i.i.d. VaR ES ES Relative Value A. ES BCa r 1,...,r T i.i.d. R 1,...,R T 1 i r 1,...,r T R1,...,R T ii (A.1) ˆµ (b)= 1 T Rt (b). T B µ 1,...,µ T iii ẑ 0 ˆα ( ) n (A.2) z 0 =Φ 1 1 B {ˆµ i=1 b < ˆµ}, ˆα = (ˆµ ( ) ˆµ (i) ) 3 { n } 3. 6 i=1 (ˆµ ( ) ˆµ (i)) 2 2 i=1 ˆµ = 1 T T r i, i=1 x (b)={r1(b),...,r T (b)}, b=1,2,...,b, ˆµ (b)= 1 T Ri (b), ˆµ (i) = 1 n R j I {i j}, T n 1 i=1 iv 1 2α j=1 ˆµ ( ) = 1 n n ˆµ (i), j=1 (A.3) (Ĝ 1 {ˆα(α)},Ĝ 1 {ˆα(1 α)}). ( ) ẑ 0 + z α ˆα(α)=Φ ẑ 0 +, Φ(z α)=α, 1 ˆα(ẑ 0 + z α) Ĝ 1 (α) { B ˆµ (b) Bα },

14 GCI Getmansky,M.,Lo,A.andMakarov,I An econometric analysis of serial correlation and illiquidity in hedge-fund returns, The Journal of Financial Economics, 74, Hakamada, T., Takahashi, A. and Yamamoto, K Selection and performance analysis of Asia- Pacific hedge funds, Journal of Alternative Investments, 10 3, Kaplan,P.D.andKnowles,J.A Kappa: A generalized downside risk-adjusted performance measure, Journal of Performance Measurement, 8, EM MCMC, McNeil, A. J., Frey, R. and Embrechts, P Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, Princeton, New Jersey Miura, R., Aoki, Y. and Yokouchi, D A note on statistical models for individual hedge fund returns, Mathematical Methods of Operations Research, 69 3, I 1 64, Papageorgiou, N., Remillard, B. and Hocquard, A Replicating the properties of hedge fund returns, The Journal of Alternative Investments, 11 2, Perez, P. G An approach to the non-normal behavior of hedge fund indices using Johnson distributions, presented to Finance Department, ESSEC in Paris and submitted for publication. Takahashi, A. and Yamamoto, K Hedge fund replication. The Recent Trend of Hedge Fund Strategies, Nova Science Publishers (forthcoming). Takahashi, A. and Yamamoto, K Generating a target payoff distribution with the cheapest dynamic portfolio: An application to hedge fund replication, 22nd Australasian Finance and Banking Conference 2009, Available at SSRN: VaR, Zangari, P A VaR methodology for portfolios that include options, RiskMetrics Monitor, 1st quarter, 4 12.

15 Proceedings of the Institute of Statistical Mathematics Vol. 59, No. 1, (2011) 103 Backtesting and Studying Risk Measure in Hedge Funds Hironori Kato The Graduate University for Advanced Studies This paper provides empirical analyses of value at risk and expected shortfall calculated by various models which assume i.i.d. with the hedge fund s losses. We use hedge funds monthly returns, which include before and after the subprime crisis of We use the normal distribution as the hedge fund s loss distribution and also Gaussian mixture, Johnson distribution, which can express negative skewness and large kurtosis of the hedge fund s return. We conclude that the ratios rejecting risk measure models for the global-macro are low. On the other hand, all risk measures for the relative-value are mostly rejected. Key words: Hedge fund, risk management, value at risk, expected shortfall.

バリュー・アット・リスクのリスク指標としての妥当性について ― 理論的サーベイによる期待ショートフォールとの比較分析―

バリュー・アット・リスクのリスク指標としての妥当性について ― 理論的サーベイによる期待ショートフォールとの比較分析― aaaab aabab VaR VaRArtzner et al. VaR VaR VaR Artzner et al.var VaR VaR VaR ρ XY ρ (X+Y ) ρ(x) + ρ(y ) XY ρ VaRArtzner et al.1999basak and Shapiro1999Danielsson2000Rootzén and Klüppelberg VaR VaR VaRVaR