P SS P SS P SS P SS P SS
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3 P SS
4 P SS P SS P SS P SS t P SS t t P SS
5 P SS P SS P SS P SS P SS P SS
6 P SS
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8 i t r i,t E t 1 [r i,t r f,t ] = γ M,t Cov t 1 (r i,t, r M,t ) + γ M 2,tCov t 1 ( ri,t, r 2 M,t ) r f,t t r M,t t γ M,t γ M 2,t t E t 1 [ ] Cov t 1 ( ) t 1 ( Cov t 1 ri,t, rm,t) 2 γ M 2,t
9 Cos i,t ( Cos i,t = Cov t 1 ri,t, rm,t) 2. β M 2,i,t E[r i,t r f,t ] = β M,i,t µ M,t + β M 2,i,tµ M 2,t µ M,t µ M 2,t β HS,i,t i [ ] E t 1 ϵi,t ϵ 2 M,t β HS,i,t = [ ] [ ] E t 1 ϵ 2i,t Et 1 ϵ 2M,t ϵ i,t = r i,t r f,t α i β M,i (r M,t r f,t ) i ϵ M,t = r M,t r f,t µ M
10 Cos i,t β M 2,i,t β HS,i,t r rm 2 r M rm 2 β HS,i,t β HS,i,t t t 60 t 1 t 12 t 1 β HS,i,t β M 2,i,t % 30% 0.78% 2.77% 0.23 β M β M
11 t 60 t 1 t r i,t rm,t 2 r i,t r 2 M,t α α MKT MKT SMB HML MOM MKT SMB HML RMW CMA α
12 t F (Cos i,k 12 k 1 ) = κ + F (Y i,k 24 k 13 ) θ + F (X i,k 13 ) ϕ + ε i,k 12 k 1, k = 25, 26,..., t i = 1,..., N k N k k Cos i,k 12 k 1 Cos i k 12 k 1 K Y Y i,k 24 k 13 β M i k 24 k 13 K X X i,k 13 i k 13 ε i,k 12 k 1 F (x i,t ) = Rank(x i,t) N t+1 x i,t x t Rank(x i,t ) N t x i,t x t N t K Y θ K X ϕ κ 25 t Cos Y X 12 t ˆκ ˆθ ˆϕ F (Cos i,t t+11 ) = ˆκ + F (Y i,t 12 t 1 ) ˆθ + F (X i,t 1 ) ˆϕ.
13 30% 30% P SS β M Cos Y F (CSt t+11,i )
14 P SS P SS 5.37% β HS Cos β M 2 P SS α 4.02% 0.36% 5.91% P SS P SS P SS MKT P SS P SS β M P SS P SS MKT rm 2 α
15 P SS α α θ ϕ ˆθ ˆϕ 5 th 95 th
16 t 12 t 2 t 12 t 7 t 1 P SS θ P SS P SS
17 P SS R 2 t R 2 N = 25 P SS MKT P SS t R 2 p R 2 R 2 P SS t R 2 R 2 R 2 t
18 P SS 5.55 t t 2.91 P SS β M P SS MKT r M,t r M,t MKT + r M,t r M,t MKT MKT + P SS MKT MKT + P SS P SS MKT SMB HML MOM MKT SMB HML RMW CMA HML MOM SMB R 2 R 2 P SS P SS P SS SMB R 2 SMB P SS 0.73 α P SS P SS SMB P SS
19 SMB MKT P SS HML MOM R 2 R 2 SMB P SS P SS SMB R 2 P SS R 2 p P SS SMB SMB P SS p R 2 SMB P SS MKT P SS HML RMW CMA R 2 MOM MKT P SS
20 p p P SS P SS
21 i 5 th 95 th
22 t 12 t 7 t 6 t 1 β M P SS t t t % 30%
23 30% 40% 30% 30% 30% MKT P SS MKT P SS HML MOM MKT P SS HML RMW CMA MKT P SS MKT P SS P SS MOM RMW HML α α α α α P SS α
24 P SS α P SS α α α MKT P SS
25 QSK(x t ) = (q 0.95(x t ) q 0.50 (x t )) (q 0.50 (x t ) q 0.05 (x t )) q 0.95 (x t ) q 0.05 (x t ) x t q 0.05 (x t ) q 0.50 (x t ) q 0.95 (x t ) 5 th 50 th 95 th x t QSK QSK QSK P SS MOM RMW α P SS α
26 α
27 i t d β M β M,i,t t+11 r i,td r f,td = α i,t t+11 + β M,i,t t+11 (r M,td r f,td ) + ϵ i,td. ϵ i,td Cos r i,td rm,t 2 d β M 2 β M 2,i,t t+11 r i,td r f,td = α i,t t+11 + β M,i,t t+11 (r M,td r f,td ) + β M 2,i,t t+11 (r M,td r f,td ) 2 + υ i,td. β HS ϵ i,td ϵ 2 M,t d ϵ M,td = r M,td 1 Td,t t+11 T d,t t+11 t d =1 r M,td T d,t t+11 t t + 11 ϵ 2 i,t d ϵ 2 M,t d r i,td r i,td υ i,td υ i,td
28 t 12 t 2 t 12 t 7 t 1
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33 βm βhs α α α Cos β M βhs Cos β M βhs P SS P SS βm βhs α 30% 30% Cos β M 2 βhs P SS P SS 10, 000 T % 1%
34 Cumulative Log-return P SS MKT 1$ P SS MKT
35 Regression parameters for coskewness rank prediction ˆθ ˆϕ
36 5 th 95 th β M Cos ˆθ ˆϕ th 95 th
37 P SS MKT P SS MKT MKT + R 2 R (5.27) ( 2.62) (5.27) ( 2.65) (6.08) ( 3.64) (2.62) (0.04) (0.01) (6.08) ( 3.77) (2.65) P SS β M β+ M (6.00) (2.47) ( 1.41) ( 0.56) (0.66) (0.76) (6.00) (2.70) ( 1.29) ( 0.33) (4.14) ( 1.26) (4.14) ( 1.27) (4.45) ( 2.29) (3.06) (0.00) (0.00) (4.45) ( 2.55) (2.91) P SS β M β+ M (2.87) (2.64) ( 0.72) ( 0.08) (1.00) (0.65) (2.87) (2.02) ( 0.75) (0.06) MKT P SS MKT MKT MKT + t R 2 p R 2 R 2 p R
38 P SS MKT P SS SMB HML MOM R 2 R (2.50) ( 1.12) (1.72) (2.64) (1.47) (2.50) ( 0.25) (2.01) (2.09) (1.38) P SS (2.67) ( 1.34) (2.03) (1.73) (2.65) (1.50) (0.92) (0.34) (2.67) ( 0.49) (0.96) ( 0.03) (1.98) (1.17) MKT P SS HML UMD (2.69) ( 1.34) (2.39) (2.65) (1.51) (0.93) (0.65) (2.69) ( 0.49) (1.95) (1.98) (1.18) (1.75) ( 0.76) (2.25) (0.04) (3.80) (1.75) ( 0.56) (2.11) (0.15) (1.66) P SS (1.65) ( 0.68) (0.82) (2.25) ( 0.12) (3.72) (0.78) (0.68) (1.65) ( 0.51) ( 0.41) (1.07) (0.03) (1.83) MKT P SS HML UMD (1.64) ( 0.64) (2.61) (0.50) (3.89) (0.99) (0.51) (1.64) ( 0.39) (1.77) (0.54) (1.33) MKT SMB HML MOM P SS t R 2 P SS p R 2 R 2 p R
39 P SS MKT P SS SMB HML RMW CMA R 2 R (4.18) ( 2.31) (2.04) (2.57) (1.03) (0.79) (4.18) ( 1.66) (3.34) (0.56) (1.10) ( 0.09) P SS (3.50) ( 1.60) (3.09) (2.11) (2.60) (1.84) (1.25) (0.00) (0.00) (3.50) ( 0.35) (2.87) ( 1.70) (0.41) (2.48) (0.79) MKT P SS HML RMW CMA (4.35) ( 2.29) (3.01) (2.57) (1.86) (1.00) (0.12) (0.00) (4.35) ( 1.34) (4.39) (0.62) (2.72) (0.33) (2.04) ( 0.90) (3.27) ( 1.66) (0.69) (1.27) (2.04) (0.00) (2.70) ( 2.48) (1.39) (2.21) P SS (1.85) ( 0.69) (1.14) (3.02) ( 0.71) (0.41) (1.45) (0.28) (0.44) (1.85) (0.32) (0.76) (0.65) ( 2.14) (1.32) (2.26) MKT P SS HML RMW CMA (1.72) ( 0.52) (2.33) ( 0.34) (0.06) (2.06) (0.75) (0.82) (1.72) (0.68) (2.40) ( 1.95) (1.04) (2.37) MKT SMB HML RMW CMA P SS t R 2 P SS p R 2 R 2 p R
40 p MKT P SS (0.80) (1.40) (2.37) (1.33) (0.64) (2.37) (2.34) ( 0.99) (2.65) (2.26) (1.73) (1.89) (2.52) (1.38) (2.00) (1.69) (1.56) (1.99) MKT P SS MKT P SS t
41 Regression parameters for idiosyncratic skewness rank prediction ˆθ ˆϕ
42 5 th 95 th β M Cos ˆθ ˆϕ th 95 th
43 Forward daily idiosyncratic skewness Predicted idiosyncratic skewness Lagged idiosyncratic skewness Lagged idiosyncratic volatility % 30%
44 α β MKT β P SS β HML β MOM β RMW β CMA R 2 MKT P SS (2.43) (45.06) (1.09) (2.17) (36.15) (3.59) (0.04) (24.83) (6.88) (1.11) (5.21) (9.46) MKT P SS HML MOM (0.99) (73.09) (1.51) (3.97) (1.57) (2.78) (48.68) (8.40) (6.03) (8.34) (1.78) (23.60) (13.61) (1.47) (7.53) (1.50) (3.13) (15.70) (0.59) (6.78) MKT P SS HML RMW CMA (1.31) (78.13) (1.10) (3.32) (6.25) (4.09) (0.46) (52.87) (5.84) (5.64) (0.40) (1.51) (0.23) (24.66) (4.65) (2.90) (2.98) (0.25) (0.46) (4.40) (4.28) (1.89) (3.96) (0.00) MKT P SS MKT P SS HML MOM MKT P SS RMW CMA α R 2 t T
45 α β MKT β P SS β HML β MOM β RMW β CMA R 2 MKT P SS (2.78) (78.43) (12.32) (0.43) (30.63) (0.75) (1.44) (24.22) (6.11) (1.85) (6.36) (7.63) MKT P SS HML MOM (0.04) (148.90) (16.97) (2.36) (8.80) (1.08) (52.78) (6.13) (7.96) (12.14) (0.36) (31.05) (14.01) (4.13) (9.65) (0.33) (6.70) (15.95) (2.79) (10.16) MKT P SS HML RMW CMA (0.50) (118.45) (8.47) (1.77) (6.66) (3.97) (0.71) (44.58) (0.92) (6.03) (3.12) (0.81) (0.93) (27.57) (4.60) (4.66) (3.67) (0.68) (0.76) (7.14) (5.21) (4.38) (4.21) (1.18) MKT P SS MKT P SS HML MOM MKT P SS RMW CMA α R 2 t T
46 α β MKT β P SS β HML β MOM β RMW β CMA R 2 MKT P SS (2.63) (42.80) (0.94) (2.15) (35.77) (3.80) (0.12) (24.44) (6.59) (1.25) (4.96) (9.03) MKT P SS HML MOM (0.83) (73.57) (1.52) (4.41) (2.37) (2.83) (48.49) (8.74) (5.91) (8.67) (1.76) (23.04) (13.21) (1.37) (7.70) (1.54) (2.79) (15.45) (0.81) (7.16) MKT P SS HML RMW CMA (1.34) (78.43) (1.51) (3.86) (6.70) (3.97) (0.50) (51.79) (5.97) (5.40) (0.37) (1.53) (0.20) (24.13) (4.44) (2.80) (3.05) (0.25) (0.43) (4.15) (3.98) (1.67) (4.08) (0.97) MKT P SS MKT P SS HML MOM MKT P SS RMW CMA α R 2 t T
47 α β MKT β P SS β HML β MOM β RMW β CMA R 2 MKT P SS (2.93) (77.17) (12.02) (0.02) (29.28) (1.41) (1.65) (23.13) (5.74) (2.05) (6.14) (7.29) MKT P SS HML MOM (0.02) (149.20) (16.87) (2.92) (9.07) (1.14) (53.31) (7.95) (6.70) (13.81) (0.36) (30.42) (13.63) (3.60) (9.89) (0.34) (6.40) (15.68) (2.29) (10.66) MKT P SS HML RMW CMA (0.51) (117.89) (8.51) (1.32) (6.91) (4.03) (0.78) (40.99) (1.37) (5.60) (3.58) (0.85) (0.94) (26.57) (4.12) (4.40) (4.05) (0.72) (0.77) (6.85) (4.78) (4.11) (4.57) (1.21) MKT P SS MKT P SS HML MOM MKT P SS RMW CMA α R 2 t T
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