2 I- I- (1) 2 I- (2) 2 I- 1 [18] I- I-. 1 I- I- Jensen [11] I- FF 3 I- FF CAPM n ( i = 1,..., n) M t R i,t, i = 1,..., n R M,t ( ) R i,t = r i
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1 1 Idiosyncratic,, Idiosyncratic (I- ) I- 1 I- I- Jensen I- Fama-French 3 I- Fama-French 3 1 Fama-French (FF) 3 [6] (Capital Asset Pricing Model; CAPM [12, 15]) CAPM ( [2, 10, 14, 16]) [18] Idiosyncratic (I- ) [3, 4, 7] [5, 9] [7] [18] I- I- 1 I- (1 ) 2 2 I I-, [email protected], [email protected], [email protected]
![Idiosyncratic (I- ) [3, 4, 7] [5, 9] 1 2 3 1 [7] [18] I- I- 1 I- (1 ) 2 2 I- 2 2 3 3 I-, Email:](/docs-images/56/20085570/images/page_1.jpg "yuji@gssm.otsuka.tsukuba.ac.jp, Email: takaaki.yoshino.rc@jp.daiwacm.com, Email: tetsuro.saito.")
2 2 I- I- (1) 2 I- (2) 2 I- 1 [18] I- I-. 1 I- I- Jensen [11] I- FF 3 I- FF CAPM n ( i = 1,..., n) M t R i,t, i = 1,..., n R M,t ( ) R i,t = r i,t r f, R M,t = r M,t r f (2.1) R i,t = r i,t r i, R M,t = r M,t r M (2.2) 1. r i,t i r f r i := E [r i,t ], r M := E [r M,t ] i k C (m) 2 C (k) := Cov [ R i,t, RM,t k ] (2.1) 1 CAPM [12, 15] (2.3) r i r f = κ (1) ( r M r f ), κ (1) := C (1) Var [R M,t ] (2.4) [7] 3 r i r f = â 1 κ (1) + â 2κ (2) + + â mκ (m) + ĉ i (2.5) κ (k) := C (k) k = 1, 2,..., m (2.6) k+1], E [{R M,t E [R M,t ]} â k, k = 1,..., m ĉ i 4. (2.6) κ (2) i κ(3) [7]. [5, 9] ( ) 1 X, Y E [X] Var [X] Cov[X, Y ] X := E [X] X m X R m X. 2 r f (2.5) [8] 3 CAPM m. [7] m = 4 (2.5) m. 4 CAPM â 1 = r M r f, â k = 0 (k 2), ĉ i = 0.
![3) r i r f = κ (1) ( r M r f ), κ (1) := C (1) Var [R M,t ] (2.4) [7] 3 r i r f = â 1 κ (1) + â 2κ (2) + + â mκ (m) + ĉ i (2.5) κ (k) := C (k) k = 1, 2,..., m (2.](/docs-images/56/20085570/images/page_2.jpg "6) k+1], E [{R M,t E [R M,t ]} â k, k = 1,..., m ĉ i 4. (2.6) κ (2) i κ(3) [7]. [5, 9] ( ) 1 X, Y E [X] Var [X] Cov[X, Y ] X := E [X] X m X R m X. 2 r f 1 2. 3 (2.5) [8] 3 CAPM m.")
3 3 2.2 Idiosyncratic (I- ) [18] I- 1 (I- ) β (k) k I-. [ ] β (k) := Cov ε (k 1) i,t Var, RM,t k ], k = 1, 2, 3,... (2.7) [ R k M,t ε (0) i,t := R i,t ε (k) i,t, k 1 ε (k) i,t := ε(k 1) i,t β (k) Rk M,t, k = 1, 2, 3,... (2.8) I- β (1) β(1) = κ(1) β (1) R M,t R M,t ε (0) i,t := R i,t ε (1) i,t = R i,t β (1) ε (1) i,t β(1) R M,t ε (1) i,t 1 β(2) R2 M,t R2 M,t ε (2) i,t = ε(1) i,t β(2) R2 M,t ε(2) i,t k I- β (k) Rk M,t I- R i,t β (k) β(2) R2 M,t ε(k 1) i,t, k = 1,..., m [18] R M,t 1 ε(k) i,t k R i,t = β (1) R M,t + β (2) R2 M,t + + β (m 1) Rm 1 M,t + β (m) Rm M,t + ε (m) i,t (2.9) [ ] Cov ε (k) i,t, Rk M,t = 0, k = 1,..., m (2.9), C (k) (2.3) C (k) k I- C (k) = β(1) Cov [ R M,t, RM,t k ] [ (k 1) + + β Cov R k 1 M,t, Rk M,t ] + β (k) Var [ RM,t k ] I- [18] I- 1 2 I- 0 (2.10) β (k) = 0, k 2 (2.11) 1 2 I- [18] w i > 0, n wi = 1 i=1 I- k I- k (k = 1, 2,...) n i=1 n i=1 w i β (k) = 0 (k 2) (2.12) w i ε (k) i,t = 0 (k 1) (2.13) 5 w i > 0..
4 4 n i=1 w i β (1) = 1 (2.14) I- 2 I- ( ) 2 I- β (2) 2 y = cx2 ( c ) β (2) > 0 β (2) < 0 2 I- β (2) 3 I- β (3) 3 y = cx 3 β (3) > 0 β(3) < 0 I- y = cx k I- 2 2 I- 0 I- 0 0 I- ( ) 3 I- I- FF NEEDS- FinancialQUEST ,016 R i,t R M,t (2.1) ,016 I- 6 I- i R M,t β (1) (1 ) R2 M,t β(2) β (i), i = 1,..., 6 t 2 0 t 5% ( ) 2 I- 3 I (2.2) (2.1).
5 5 Table 3.1: β (k) ( 5% t ). k = 1 k = 2 k = 3 k = 4 k = 5 k = t t (%) (%) ( ) 2 50%, 3 65% 4 25% 5, 6 % 2 I % % % % % % % % % プラス マイナス 有 有 意 意 比 比 率 率 0.000% 5.000% % % % % % % 0.000% Fig. 3.1: 1 4 ( ) ( ) プラス 3 I- マイナス 有 意 有 比 意 率 比 率 I- 9. 1: 1978/10/3 1986/4/2, 2: 1986/4/3 1994/8/8, 3: 1994/8/9 2003/4/14, 4: 2003/4/ /12/30 Fig I- (1 4) 2 ( ) I- 4 3 I Fig I I I- 3.2 : FF I- I- 2 I- (= β (1) ), 10 ᾱ i := r i r f β (1) (r M,t r f ) (3.1) , 1017, 1100, Jensen [11].
6 % % % % % % % % % 0.000% 5.000% プラス 有 意 比 率 マイナス 有 意 比 率 2 I % % % % % % % % % 0.000% 5.000% プラス 有 意 比 率 マイナス 有 意 比 率 3 I Fig. 3.2: ( ) ( ) 1: 1978/10/3 1982/7/2, 2: 1982/7/3 1986/4/2, 3: 1986/4/3 1990/4/6, 4: 1990/4/9 1994/8/8 5: 1994/8/9 1998/12/4, 6: 1998/12/7 2003/4/14, 7: 2003/4/ /8/15, 8: 2007/8/ /12/30 I-. ᾱ i c 2 β (2) + + c m 1β (m 1) + c m β (m) (3.2) FF 3 SMB (Small Minus Big), HML (High Minus Low) (SMB and HML Factor Loadings; SMB-FL, HML-FL) SMB-FL, HML-FL 11. SMB-FL, HML-FL (= β (1) ) 12 SMB, HML α i,t := r i,t r f β (1) (r M,t r f ) (3.3) α i,t d 1 SMB t + d 2 HML t (3.4) i SMB, HML SMB-FL i = d 1, SMB-FL i, HML-FL i HML-FL i = d 2 ᾱ i c s SMB-FL i + c h HML-FL i (3.5) ᾱ i c 2 β (2) + + c m 1β (m 1) + c m β (m) + c ssmb-fl i + c h HML-FL i (3.6) Table 3.2 1,016 6 I- SMB-FL, HML-FL, Panel A 6 I- Panel B SMB-FL, HML-FL Panel C t p VIF (Variance Inflation Factor; ) Panel A C R 2, F F p Panel A I- 5% SMB-FL, HML-FL 2 3 I- SMB-FL HML-FL (2 3 ) I- SML-FL, HML-FL HML-FL SML-FL I- 11 SMB, HML B. 12 (3.3) α i,t CAPM. α i,t (= E [α i,t ]).
7 7 全 Panel 説 期 明 間 A: I- 共 変 のみ Table 係 変 数 2 次 I 共 変 動 3 次 I 共 変 動 4 次 I 共 変 動 5 次 I 共 変 動 6 次 I 共 変 動 SMB-FL HML-FL 定 数 項 3.2: 数 t 値 1.77E E E E E E R2/F 1.37E-01 値 /p 値 Panel 係 t 値 数 B: SMB, HMLファクターローディングのみ -2.58E E E R2/F 2.33E-01 値 /p 値 Panel 係 数 C: I- 共 -6.90E E E E-07 変 動 +SMB, HMLファクターローディング p t 値 < E E < E E <.0001 R2/F 2.84E-01 値 /p 値 p VIF 値 < < < <.0001 p VIF 値 < < <.0001 VIF <.0001 SMB-FL, HML-FL (3.4) SMB-FL, HML-FL Fig. 3.3 SMB-FL, HML-FL 5% / SMB-FL, HML-FL All Table SMB-FL 90% HML-FL 85% (3.3) α i,t (= ) SMB, HML SMB-FL, HML-FL SMB, HML (= ᾱ i ) SMB-FL % SMB-FL % % % % % % % % % プラス 有 意 比 率 マイナス 有 意 比 率 HML-FL ALL % % % % % % % % % プラス 有 意 比 率 マイナス 有 意 比 率 ALL Fig. 3.3: 1 4 (All) SMB-FL, HML-FL 3.3 (B/P) B/P ( ) 5 ( 1 5 ) Fig I- 5% 2 I- 3 I- 3 I- Fig. 3.5 B/P 1 5 5% 2 3 I- B/P 2 I-
8 8 プラス 有 意 比 率 マイナス 有 意 比 率 2 I % % % % % % ( 低 い) ( 5高 い) プラス 有 意 比 率 マイナス 有 意 比 率 3 I % % % % % % % % ( 低 い) ( 5高 い) Fig. 3.4: ( 1, 2,..., 5 ) 3 I- 2 I- 3 I- B/P 2 I % % % % % % プラス 有 意 比 率 マイナス 有 意 比 率 3 I % % % % % % % プラス 有 意 比 率 マイナス 有 意 比 率 ( 低 い) ( 高 い) ( 低 い) ( 高 い) Fig. 3.5: B/P ( 1, 2,..., 5 ) B/P 5 ( ) Table 3.3 ( 1 ) ( 5 ) 2 1 Panel A I- 2 I- 3 I- Panel A, C 4 I- Panel B, C SMB-FL 5% 5 2 I- 3 I- Panel A, C SMB-FL Panel B, C Panel B SMB-FL, HML-FL Table 3.2 Panel C SMB-FL, HML-FL Panel A I- Table 3.4 B/P B/P ( 1 ) ( 5 ) 1 Panel A Panel B Table 3.2 R 2 F Panel C SMB-FL Panel A 2 I- 1% 10% 3 5 I- R 2 5 Panel A I- 4 6 I- 1% Panel B SMB-FL, HML-FL SMB-FL 0.01% Panel C 4 6 I- 2 I- HML-FL 10% B/P B/P ( 1 ) ( 5 ) I- FF 3 FL
9 9 Panel 時 説 明 価 総 額 第 1 分 位 Table 3.3: 係 変 数 A: 数 I- 2 共 次 2.28E-05 I 変 共 変 動 のみ 3 次 1.62E E E-07 I 共 変 動 4 次 I 共 変 動 5 次 I 共 変 動 6 次 1.39E-07 I 共 変 動 SMB-FL HML-FL -2.16E-05 定 数 項 R2/F ( : 値 /p 値 1, : 5 ) t 値 E-01 Panel 係 t 値 数 B: SMB, HMLファクターローディングのみ -2.90E E E R2/F 9.20E-02 値 /p 値 Panel 係 数 C: I- 共 -1.93E-06 変 動 +SMB, 2.48E E E-07 HMLファクターローディング p t 値 E E < E E R2/F 2.62E-01 値 /p 値 Panel 時 説 明 価 A: 総 額 I- 共 第 変 5 分 位 係 変 数 2 次 I 共 変 動 のみ t 値 数 -6.45E E E 次 I 共 変 動 4 次 I 共 変 動 5 次 8.98E E-08 I 共 変 動 6 次 I 共 変 動 SMB-FL HML-FL 定 2.40E-04 数 項 R2/F 1.90E-01 値 /p 値 Panel 係 t 値 数 B: SMB, HMLファクターローディングのみ -2.19E E E R2/F 2.05E-02 値 /p 値 Panel 係 数 C: I- 共 -6.86E E E-07 変 動 +SMB, HMLファクターローディング p t 値 E E E E E-04 < R2/F 1.85E-01 値 /p 値 p VIF 値 < < < p VIF 値 < <.0001 VIF <.0001 p VIF 値 < <.0001 p VIF 値 < VIF <.0001 B/P Panel B/P 説 明 第 1 分 位 Table 3.4: 係 変 数 A: 数 I- 2 共 次 -2.42E E E E E-09 I 変 共 変 動 のみ 3 次 I 共 変 動 4 次 I 共 変 動 5 次 I 共 変 動 6 次 I 共 変 動 SMB-FL HML-FL 定 数 項 B/P 2.25E-04 R2/F ( : 値 /p 値 1, : 2 ) t 値 E-01 Panel 係 t 値 数 B: SMB, HMLファクターローディングのみ -2.52E E E R2/F 3.01E-01 値 /p 値 Panel 係 数 C: I- 共 -5.34E E E E E E-04 変 動 +SMB, HMLファクターローディング p t 値 < E E <.0001 R2/F 3.79E-01 値 /p 値 B/P Panel B/P 説 明 A: 第 I- 共 5 分 変 位 係 変 数 2 次 I 共 変 動 のみ t 値 数 -2.19E E E E 次 I 共 変 動 4 次 I 共 変 動 5 次 I 共 変 動 6 次 1.23E-07 I 共 変 動 SMB-FL HML-FL 定 2.24E-05 数 項 R2/F 1.35E-01 値 /p 値 Panel 係 t 値 数 B: SMB, HMLファクターローディングのみ -4.36E E E R2/F 1.73E-01 値 /p 値 Panel 係 数 C: I- 共 -2.27E E E E-08 変 動 +SMB, HMLファクターローディング p t 値 E E < E E R2/F 2.93E-01 値 /p 値 p VIF 値 < <.0001 p VIF 値 < < <.0001 VIF <.0001 p VIF 値 < p VIF 値 < <.0001 VIF <.0001
10 10 4 I- 1 I-. 3 I- 5 2 I- 3 I- 2 6 I- I- FF 3 SMB, HML (SMB-FL, HML-FL) I- 2 3 I- B/P 2 3 I- I- FF 3 FL SMB-FL, HML-FL I- [1] P. Artzner, F. Delbaen, J. M. Eber, and D. Heath, Coherent Measures of Risk, Mathematical Finance, 9(3), , [2] M.M. Carhart, On persistence in mutual fund performance, Journal of Finance, 52(1), 57 82, [3] R. Christie-David and M. Chaudhry, Coskewness and cokurtosis in futures markets, Journal of Empirical Finance 8, 55-81, [4] Y.P. Chung, H. Johnson and M. Schill, Asset pricing when returns are nonnormal: Fama-French factors vs. higher-order systematic comoments, Journal of Business, 79(2): , [5] R.F. Dittmar, Nonlinear pricing kernels, kurtosis preference, and evidence from the cross-section of equity returns, Journal of Finance 51, , [6] E.F. Fama and K.R. French, Common risk factors in the returns on stocks and bonds, Journal of Financial Economics, 33(1), 3 56, [7] H. Fang and T.-Y. Lai, Co-kurtosis and capital asset pricing, Financial Review 32, , [8] A. Kraus and R. Litzenberger, Skewness preference and the valuation of risk assets, Journal of Finance 31, , [9] C. Harvey and A. Siddique, Conditional skewness in asset pricing tests, Journal of Finance 55, , [10],, Fama-French,, 22, 3 23, [11] M.C. Jensen, The Performance of Mutual Funds in the Period , Journal of Finance 23, , [12] J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics, 13 37, [13] Y. Malevergne and D. Sornette, Extreme Financial Risks: From Dependence to Risk Management, Springer, [14] L. Pastor and R.F. Stambaugh, Liquidity risk and expected stock returns, Journal of Political Economy, 111(3), , 2003.
![Christie-David and M. Chaudhry, Coskewness and cokurtosis in futures markets, Journal of Empirical Finance 8, 55-81, 2001. [4] Y.P. Chung, H. Johnson and M.](/docs-images/56/20085570/images/page_10.jpg "Schill, Asset pricing when returns are nonnormal: Fa")
11 11 [15] W. Sharpe, Capital asset prices: a theory of market equilibrium under conditions of risk, Journal of Finance 19, , [16],,, UFJ, , [17],, VaR, / / , , [18],,, Idiosyncratic, 2011 JAFEE, 49 60, A (Generalized Capital Asset Pricing) [13] (2.6) [13] CAPM ρ r i r f = ρ ( w r ) r M r f, w i ρ (r w =w M ) i = 1, 2,..., n (A.1) w i, i = 1, 2,..., n 1 w R n i r R n i r i, i = 1,..., n w i, i = 1, 2,..., n [13] ρ i (i = 1, 2,...) 1 1 X, Y, c, ρ 13. 1) 1 (Positive homogeneity): ρ (λx) = λρ (X) 0, λ > 0 2) (Convexity): ρ (λx + (1 λ) Y ) λρ (X) + (1 λ) ρ (Y ), λ [0, 1] 3) (Translation invariance): ρ (X + c) = ρ (X), ρ (0) = 0 ρ ρ (X) = ( [ (X ]) 2 1/2 1/2 E X) = (Var [X]) (A.2) 1 ρ ( w r ) 1 1 = w i ρ (r w =w M ) 2ρ (w r) w =w = = w i E [ {w (r r) } 2 ] w =w 1 ρ (r M ) 1 2ρ 2 (r M ) E [ 2 { w (r r) } (r i r i ) ] w =w 1 ρ 2 (r M ) E [(r M r M ) (r i r i )] = Cov [r M, r i ] Var [r M ] (A.3) (A.1) CAPM ( [ X l X l := E X X l]) 1/l ρ(x) = j η j X j, η j > 0 (A.4) , [1] (Monotonicity; X Y ρ(x) ρ(y )) [17].
![1) w i, i = 1, 2,..., n 1 w R n i r R n i r i, i = 1,..., n w i, i = 1, 2,..., n [13] ρ i (i = 1, 2,...) 1 1 X, Y, c, ρ 13.](/docs-images/56/20085570/images/page_11.jpg "1) 1 (Positive homogeneity): ρ (λx) = λρ (X) 0, λ > 0 2) (Convexity): ρ (λx + (1 λ) Y ) λρ (X) + (1 λ) ρ (Y ), λ [0, 1] 3) (Translation invariance): ρ (X + c) = ρ (X), ρ (0) = 0 ρ ρ (X) = ( [ (X ]) 2")
12 12 1 (A.4) ρ w r [ j {w j E (r r) } ] j [ w i = w i = E j { w (r r) } ] j 1 w (ri r i ) =w w =w [ w =w ] = je (r M r M ) j 1 (r i r i ), j = 2, 4,... w r j j w i w =w = j r M j 1 j w r j w i w =w R i,t, R M,t (2.2) w r [ ] j E (r M r M ) j 1 (r i r i ) = w i w =w r M j 1 j (2.6) w r j w i w =w = r M j κ (j 1) = C(j 1) r M j 1 j (A.5) (A.6) r i r f = ρ ( w r ) r M r f w i ρ (r w =w M ) = = {j=2,4,...} ω l, l = 2, 4,... {j=2,4,...} η j w r j w i w =w r M r f {j=2,4,...} η j r M j ω j κ (j 1), i = 1, 2,..., n (A.7) ω l := η l r M l ( r M r f ) {j=2,4,...} η j r M j, l = 2, 4,... (A.8) B SMB/HML SMB/HML SMB/HML FF 3 [6] [10] 1. ( 1 ) 8 (Small) (Big) (B/P) 30% 70% B/P (Low), (Middle), (High) 2. Step 1 Small Low Small/Low Small Middle Small/Middle Small/Low, Small/Middle, Small/High, Big/Low, Big/Middle, Big/High 6 3. (a) (b) SMB (a) Small/Low, Small/Middle, Small/High 1/3 (b) Big/Low, Big/Middle, Big/High 1/3 4. (a) (b) HML (a) Small/High, Big/High 1/2 (b) Small/Low, Big/Low 1/2
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