I- Fama-French 3, Idiosyncratic (I- ) I- ( ) 1 I- I- I- 1 I- I- Jensen Fama-French 3 SMB-FL, HML-FL I- Fama-French 3 I- Fama-MacBeth Fama-MacBeth I- SMB-FL, HML-FL Fama-MacBeth 2, 3, 5 I- HML-FL 1 Fama-French (FF) 3 [7] (Capital Asset Pricing Model; CAPM [15, 18]) CAPM, Email: yuji@gssm.otsuka.tsukuba.ac.jp, Email: takaaki.yoshino@daiwa.co.jp, Email: tetsuro.saito@daiwa.co.jp 1
( [2, 14, 17, 19]) [20, 21] Idiosyncratic (I- ) I- FF 3 [3, 4, 10] [6, 11] 1 2 3 1 [10] [20] I- I- 1 I- (1 ) 2 2 I- 2 2 3 3 I- I- I- (1) 2 I- (2) 2 I- 1, (3) I- I- [21] I- I- 1 I- 7 I- FF 3 SMB, HML (SMB-FL, HML-FL) I- SMB-FL, HML-FL Fama-MacBeth [8, 9] I- 2
2 2.1 CAPM n ( i = 1,..., n) M t R, i = 1,..., n R M,t ( ) R = r r f, R M,t = r M,t r f (2.1) R = r r i, R M,t = r M,t r M (2.2) 1. r i r f r i := E [r ], r M := E [r M,t ] i k C (k) 2 ] C (k) [R := Cov, RM,t k (2.3) (2.1) 1 CAPM [15, 18] r i r f = κ (1) ( r M r f ), κ (1) := C (1) Var [R M,t ] (2.4) [10] (2.4) r i r f 1 3. [10] κ (k) := C (k) k = 1, 2,..., (2.5) E [{R M,t E [R M,t ]} k+1], κ (2) i κ(3) [6, 11] ( ) 1 t = 1, 2,..., T r, r M,t ( i.i.d.) X, Y X = E [X] Var [X] Cov[X, Y ] X m X R m X. 2 r f 1 2. 3 [13] 3 CAPM. 1. 3
2.2 I- [20, 21] I- I- 1 (I- ) β (k) k I-. [ ] ε (0) β (k) := Cov ε (k 1) Var, RM,t k ], k = 1, 2, 3,... (2.6) [ R k M,t := R ε (k), k 1 ε (k) := ε (k 1) β (k) Rk M,t, k = 1, 2, 3,... (2.7) I- β (1) β(1) = κ (1) β(1) R M,t R M,t ε (0) := R ε (1) = R β (1) R M,t ε (1) β (1) R M,t 4. ε (1) 1 β (2) R2 M,t R2 M,t ε (2) RM,t k β(2) R2 M,t 1 ε(2) = ε (1) β(2) ε(k) ε(k 1) R2 M,t k k I- β(k) (2.6), (2.7) I- 1 R k I- k [20, 21] R = β (1) R M,t + β (2) R2 M,t + + β (k 1) Rk 1 M,t + β(k) Rk M,t + ε (k) (2.8) [ ] Cov ε (j), Rj M,t = 0, j = 1,..., k (2.8) (2.3) ] [ ] ] C (k) = β(1) [R Cov M,t, RM,t k + + β (k 1) Cov R k 1 M,t, Rk M,t + β (k) [R Var M,t k (2.9) (2.9) I- I- 2.3 I- [20, 21] I- 1 2 I- 0 β (k) = 0, k 2 (2.10) 4 2 X, Y X Y E [XY ] = 0. 4
1 2 I- [20, 21] w i > 0, n wi = 1 i=1 I- k 5 2 2 I- k (k = 1, 2,...) n i=1 n i=1 w i β (k) = 0 (k 2) (2.11) w i ε (k) = 0 (k 1) (2.12) n i=1 w i β (1) = 1 (2.13) I- I- 2 2 I- 0 I- [20, 21] R, R M,t I- 1 I- I- 6. [21] I- 10 I- 1 I- 2 I- 3 I-. 5% ( ) 2 50%, 3 65% 5 w i > 0.. 6 I- R M,t. 5
4 I- 25% 5 % 3 I- 2 I- 2 I- 3 I- B/P B/P 2 I- 3 I- 3 I- B/P I- [21] I- I- I- Fama-French 3 [7] I- Fama-French 3 3 3.1 I- ᾱ i := r i r f β (1) ( r M,t r f ) (3.1) I- 7. ᾱ i c 2 β (2) + + c m 1β (m 1) + c m β (m) (3.2) m I- α := r r f β (1) (r M,t r f ) (3.3) 7 2 I- (= β (1) ), (3.1) (3.1) Jensen [12]. 6
CAPM α (= E [α ]) (2.1) R, R M,t α = ε (1) (3.2) A [16] CAPM I- I- (A.13) I- 3 I- ( ) 4 3 I- I- (A.13) I- ᾱ ( α ) 2 m I- 3.2 SMB HML FF 3 FF 3 SMB (Small Minus Big), HML (High Minus Low) 8,. 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 8 SMB/HML FF 3 [7] [14]. 7
(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 I- FF 3 (3.3) α α d 1 SMB t + d 2 HML t (3.4) SMB t HML t t SMB, HML (3.4) FF 3 r r f β 1 (r M,t r f ) + β 2 SMB t + β 3 HML t β 1 CAPM ( 1 I- β (1) ) (3.4) CAPM ( α ) SMB, HML (Factor Loading; FL) d 1, d 2 0 CAPM SMB, HML d 1, d 2 SMB, HML d 1, d 2 SMB, HML I- FF 3 SMB, HML (SMB-FL, HML-FL) SMB-FL, HML-FL (= β (1) ) α SMB, HML (3.4) i SMB, HML SMB-FL i = d 1, HML-FL i = d 2 SMB-FL i, HML-FL i 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) I- (3.2) 8
3.3 1 9 NEEDS-FinancialQUEST 1978 10 3 2013 3 29 10 2013 3 29 10 1,121 R R M,t (2.1) 11. I- i R M,t β (1) (1 ) SMB-FL, HML-FL R 2 M,t β(2) 1 α (3.4) i SMB-FL i, HML-FL i k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 0.839 0.582 11.083 17.582 49.929 209.227 2244.184 0.235 1.098 13.398 61.850 239.618 1535.606 7159.634 t 40.666 1.075 2.104 0.415 0.097 0.106 0.084 t 13.503 2.087 2.476 1.385 0.645 0.567 0.327 (%) 100.000 9.991 59.054 19.893 1.517 0.268 0.089 (%) 0.000 40.856 6.601 6.334 0.089 0.268 0.000 Table 3.1: β (k). 5% t. Table 3.1 1,121 I- 7 Table 3.1 1 β (i), i = 1,..., 7 t 2 0 t 5% ( ) 2 I- 3 I- ( ) 2 50%, 3 65% 4 9 33 4. 10 http://www.nikkei.co.jp/needs/services/fq.html., 1978 10 2. 8861. 11. (2.2) (2.1). 9
25% 5 % 12. I- SMB-FL HML-FL Table 3.2 Table 3.1 I- SMB-FL, HML-FL, Panel A 7 I- Panel B SMB-FL, HML-FL Panel C t p Panel A C, F F p Panel A I- 2 I- 10% 3 4, 6 7 I- 1% I- 5 I- 5 I- VIF 7.06 VIF (A.13) I- 3 7 5 Panel A: I- : 0.107, F : 23.441, p : 0.0001 2 3 4 5 6 7 SMB-FL HML-FL 1.35e-05 1.90e-06 1.53e-06 1.05e-08 3.21e-08 7.02e-09 1.50e-04 t 1.418 2.436 4.884 0.146 2.410 3.179 14.337 p 0.078 0.007 <.0001 0.442 0.008 0.001 <.0001 Panel B: SMB-FL, HML-FL : 0.191, F : 133.624, p : 0.0001 2.68e-04 4.81e-06 2.51e-04 t 14.858 0.098 15.280 p <.0001 0.461 <.0001 Panel C: I- + SMB-FL, HML-FL : 0.211, F : 38.440, p : 0.0001 5.83e-06 1.36e-07 5.06e-07 1.81e-07 1.75e-08 1.43e-09 2.60e-04 1.85e-05 2.46e-04 t 0.692 0.185 1.574 2.666 1.377 0.643 10.873 0.383 14.758 p 0.245 0.427 0.058 0.004 0.084 0.260 <.0001 0.351 <.0001 Table 3.2:. I-. SMB-FL, HML-FL I- 5 I- 5 I- 1% I- 3 7 I- I- 0 SMB-FL HML-FL 12 5 I- 7 I-. 10
I- Table 3.3 I- 1 2 3 4 5 6 7 SMB-FL HML-FL 1 1.000 0.147 0.209 0.265 0.198 0.242 0.173 0.070 0.054 2 0.138 1.000 0.022 0.015 0.106 0.034 0.006 0.263 0.127 3 0.238 0.023 1.000 0.041 0.317 0.151 0.174 0.217 0.131 4 0.268 0.012 0.079 1.000 0.351 0.874 0.128 0.431 0.067 5 0.207 0.104 0.406 0.378 1.000 0.405 0.851 0.029 0.119 6 0.239 0.002 0.210 0.881 0.481 1.000 0.199 0.289 0.073 7 0.181 0.006 0.268 0.202 0.881 0.344 1.000 0.233 0.092 SMB-FL 0.040 0.232 0.237 0.434 0.035 0.269 0.217 1.000 0.023 HML-FL 0.107 0.136 0.233 0.047 0.135 0.063 0.112 0.059 1.000 Table 3.3:. SMB-FL, HML-FL HML-FL SMB-FL SMB-FL, HML-FL (3.4) SMB-FL, HML-FL 13 Fig. 3.1 SMB- FL, HML-FL 5% / SMB-FL, HML-FL All Table 3.2 1 4 4 4 1: 1978/10/03-1986/07/08, 2: 1986/07/09-1995/03/23 3: 1995/03/24-2004/03/18, 4: 2004/03/19-2013/03/29 4 14. SMB-FL 90% HML-FL 85% (3.3) α (= ) SMB, HML SMB-FL HML-FL HML (= ᾱ i ) 13 (3.4) SMB t HML t. 14 1 855, 1034, 1106, 1447. 11
0.000% プラス有意比率 マイナス有意比率 プラス有意比率 マイナス有意比率 100.000% 90.000% 90.000% 80.000% 80.000% 70.000% 70.000% 60.000% 60.000% 50.000% 50.000% 40.000% 40.000% 30.000% 30.000% 20.000% 20.000% 10.000% 10.000% 0.000% 1 2 3 4 ALL 1 2 3 4 ALL SMB-FL HML-FL Fig. 3.1: 4 1 4 (All) SMB-FL( ), HML- FL( ) 3.4 (B/P) B/P ( ) 5 ( 1 5 ) Table 3.4 ( 1 ) ( 5 ) 2 Panel A I- 1 I- 5 5 I- 1 SMB-FL, HML-FL 1 SMB-FL 1% HML-FL 10% 5 F 5 Panel C SMB-FL, HML-FL Panel A I- 1 Panel C SMB-FL, HML-FL I- 4 I- 10% 6 I- 1% Panel A, Panel B Table 3.5 B/P B/P ( 1 ) 12
1 Panel A: I- : 0.0824, F : 4.339, p : 0.001 2 3 4 5 6 7 SMB FL HML FL 3.25e-05 6.80e-07 2.64e-06 1.43e-07 1.03e-07 7.66e-09 1.28e-05 t 1.374 0.342 2.902 0.748 2.760 1.220 0.301 p 0.085 0.366 0.002 0.228 0.003 0.112 0.382 Panel B: SMB FL, HML FL : 0.108, F : 14.531, p : 0.0001 2.94e-04 2.68e-04 9.36e-05 t 5.592 1.517 1.330 p <.0001 0.065 0.093 Panel C: I + SMB FL, HML FL : 0.195, F : 7.759, p : 0.0001 3.59e-06 7.27e-07 1.51e-06 1.78e-08 9.53e-08 3.46e-09 3.74e-04 3.31e-04 1.75e-04 t 0.182 0.395 1.612 0.091 2.768 0.558 4.863 2.024 2.616 p 0.428 0.347 0.054 0.464 0.003 0.289 <.0001 0.022 0.005 5 Panel A: I : 0.180, F : 9.163, p : 0.0001 2 3 4 5 6 7 SMB FL HML FL 5.62e-05 2.77e-06 2.80e-07 1.48e-07 3.77e-08 8.38e-09 2.28e-04 t 3.546 1.653 0.535 1.129 1.560 1.867 13.686 p 0.000 0.050 0.297 0.130 0.060 0.032 <.0001 Panel B: SMB FL, HML FL : 0.0141, F : 2.595, p : 0.077 4.62e-06 1.01e-04 2.34e-04 t 0.088 1.158 9.413 p 0.465 0.124 <.0001 Panel C: I + SMB FL, HML FL : 0.173, F : 6.835, p : 0.0001 5.73e-05 2.67e-06 2.09e-07 1.61e-07 3.96e-08 8.81e-09 2.11e-05 1.13e-06 2.29e-04 t 3.420 1.839 0.413 1.125 1.718 1.781 0.363 0.016 9.922 p 0.000 0.034 0.340 0.131 0.044 0.038 0.358 0.494 <.0001 Table 3.4:. I. ( : 1, : 5 ) 13
( 5 ) Panel C 1 5 5 5 HML-FL 1% 4 6 I- 1% 1 5 7 I- 5% ( p 1% ) 1 5 SMB-FL 2 I- SMB-FL, HML-FL 5 10% B/P 1 Panel A: I : 0.142, F : 7.153, p : 0.0001 2 3 4 5 6 7 SMB FL HML FL 2.32e-07 2.44e-06 1.39e-06 1.73e-07 3.40e-08 5.15e-09 2.23e-04 t 0.009 1.270 1.932 0.855 0.910 0.737 10.577 p 0.496 0.103 0.027 0.197 0.182 0.231 <.0001 Panel B: SMB FL, HML FL : 0.211, F : 30.840, p : 0.0001 2.80e-04 5.17e-05 3.14e-04 t 7.430 0.415 13.914 p <.0001 0.339 <.0001 Panel C: I + SMB FL, HML FL : 0.261, F : 10.831, p : 0.0001 2.65e-05 1.45e-06 2.23e-07 4.93e-07 4.42e-08 1.84e-08 3.03e-04 7.19e-06 3.03e-04 t 1.147 0.790 0.274 2.349 1.212 2.229 4.739 0.058 13.383 p 0.126 0.215 0.392 0.010 0.113 0.013 <.0001 0.477 <.0001 B/P 5 Panel A: I : 0.144, F : 7.265, p : 0.0001 2 3 4 5 6 7 SMB FL HML FL 5.80e-06 2.42e-06 3.26e-06 2.88e-07 1.26e-07 8.88e-09 6.16e-05 t 0.285 1.350 3.586 1.455 3.424 1.633 2.218 p 0.388 0.089 0.000 0.074 0.000 0.052 0.014 Panel B: SMB FL, HML FL : 0.203, F : 29.482, p : 0.0001 4.78e-04 3.15e-04 2.00e-04 t 6.023 2.189 2.753 p <.0001 0.015 0.003 Panel C: I + SMB FL, HML FL : 0.300, F : 12.926, p : 0.0001 2.23e-05 2.12e-06 2.11e-06 1.84e-07 1.12e-07 2.56e-09 4.21e-04 4.08e-04 1.60e-04 t 1.353 1.279 2.628 0.920 3.592 0.479 5.849 2.955 2.250 p 0.089 0.101 0.005 0.179 0.000 0.316 <.0001 0.002 0.013 Table 3.5: B/P. I. ( : 1, : 5 ) 14
3.5 Table 3.6 1 5 B/P 1 B/P 5 I- + 2 SMB-FL, HML-FL SMB-FL, HML-FL Panel C 2 3 4 5 6 7 SMB-FL HML-FL + ( ) ( ) + (+ ) + ( ) 1 + ( ) + (+ ) ( ) + (+ ) 5 ( ) ( ) ( ) + (+ ) B/P 1 ( ) (+ ) ( ) B/P 5 ( ) ( ) + + (+ ) ( ) + (+ ) Table 3.6:. +,,, 1%, 5%, 10%. 3 I- 4 I- 6 7 I- 2 5 I- I- I- 2 5 I- SMB-FL SMB-FL 4 Fama-MacBeth Fama-MacBeth ( ). 15 I- SMB-FL, HML-FL (250 ) 15 Fama-MacBeth [8, 9] [5]. Fama-MacBeth. 15
1 Fama-MacBeth I- 4.1 3.3 (1978 10 3 2013 3 29 ) I- 250 7 I- SMB, HML 8 1 ( ) SMB-FL, HML-FL 250 I- SMB-FL, HML-FL 3.3 1 16 I- SMB-FL, HML-FL 1 I- SMB-FL, HML-FL 1 1,752. Fama-MacBeth [8, 9] t I- 4.2 Table 4.1 Fama-MacBeth Panel A 1 7 I Fama-MacBeth 1 3 t t p 4 7, 5% 5% 10% 10%. 17 Panel B SMB-FL, HML-FL Panel C SMB-FL, HML-FL Panel D 16 t, t+1 s r t+1,..., r t+s s i=1 (1 + rt+i) [14]. 17 1,752. 16
1 2 3 4 5 6 7 SMB-FL HML-FL Panel A: I- ( ): 6.018% ( ) 9.465e-05 4.093e-05 4.325e-05 1.231e-05 1.899e-05 4.895e-06 1.783e-06 7.947e-04 t 0.161 1.524 2.437 0.629 2.439 0.985 1.500 1.833 p 0.436 0.064 0.007 0.265 0.007 0.162 0.067 0.033 5% ( ) 36.701% 21.861% 16.895% 17.751% 16.210% 15.468% 15.696% 39.269% 5% ( ) 40.183% 22.774% 16.667% 15.925% 16.610% 17.009% 16.495% 34.018% 10% ( ) 39.098% 26.598% 21.861% 23.116% 22.032% 22.146% 21.804% 41.724% 10% ( ) 42.580% 29.053% 23.345% 22.489% 22.089% 23.231% 22.603% 36.587% Panel B: I- + SMB-FL, HML-FL ( ): 8.739% ( ) 3.248e-04 4.758e-05 3.133e-05 4.269e-06 1.074e-05 1.570e-06 7.575e-07 9.568e-05 8.467e-04 4.400e-04 t 0.583 2.091 2.296 0.330 1.665 0.475 0.895 0.231 2.668 1.163 p 0.280 0.018 0.011 0.371 0.048 0.317 0.185 0.409 0.004 0.122 5% ( ) 37.386% 15.183% 11.073% 13.756% 11.587% 11.587% 11.416% 32.934% 31.336% 34.703% 5% ( ) 37.614% 17.295% 12.900% 11.815% 11.758% 12.785% 12.671% 36.016% 27.511% 30.879% 10% ( ) 39.897% 20.491% 17.409% 20.662% 18.094% 17.409% 18.779% 36.016% 35.788% 39.384% 10% ( ) 40.868% 22.945% 18.493% 18.607% 17.580% 18.836% 18.037% 39.269% 31.621% 34.589% Panel C: (1 I- ) + SMB-FL, HML-FL ( ): 7.757% ( ) 2.364e-04 3.842e-05 8.568e-04 5.364e-04 t 0.420 0.092 2.645 1.365 p 0.337 0.463 0.004 0.086 5% ( ) 37.900% 36.244% 33.733% 36.358% 5% ( ) 38.642% 38.470% 29.566% 32.078% 10% ( ) 40.068% 39.041% 37.500% 39.612% 10% ( ) 40.639% 41.381% 33.276% 34.989% Panel D: (1 I- ) ( ): 3.786% ( ) 2.390e-04 9.376e-04 t 0.396 1.953 p 0.346 0.026 5% ( ) 37.443% 40.811% 5% ( ) 41.096% 35.445% 10% ( ) 39.783% 43.550% 10% ( ) 43.779% 38.299% Table 4.1: Fama MacBeth. I-. 17
1 I- Panel A Panel D Panel A Panel B Fama-MacBeth [8] 5% 36.7% 40.2% 80% ( ) 2 I- Panel A 2 7 I- 10% 3 5 I- 1% Panel B SMB-FL, HML-FL 2 3 5 I- 5% I- 2 3 I- 5 I- 7 Panel A 5 I- 5 I- 3 I- 2 I- SMB-FL, HML-FL Panel C SMB-FL, HML-FL 3 HML-FL 1% HML-FL [8] B/P 18. HML-FL. (Table 3.2 ) HML-FL Fama-MacBeth SMB-FL [14] I- SMB-FL, HML-FL Fama-MacBeth 18 [8] Fama-MacBeth B/P B/P. B/P HML-FL HML-FL B/P HML-FL [8] B/P. 18
2 I- I- 5 I- 1 I- 7 I- 5 I- FF 3 SMB, HML (SMB-FL, HML-FL) 2 3 I- 5 I- SMB-FL, HML-FL 2 3 I- I- FF 3 FL I- Fama-MacBeth Fama- MacBeth 1 I- I- SMB-FL, HML-FL 2, 3, 5 I- I- HML-FL I- I- [1] P. Artzner, F. Delbaen, J. M. Eber, and D. Heath, Coherent Measures of Risk, Mathematical Finance, 9(3), 203-228, 1999. [2] M.M. Carhart, On persistence in mutual fund performance, Journal of Finance, 52(1), 57 82, 1997. 19
[3] R. 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. Schill, Asset pricing when returns are nonnormal: Fama-French factors vs. higher-order systematic comoments, Journal of Business, 79(2), 923 940, 2006. [5] J.H. Cochrane, Asset Pricing, Revised Edition, Princeton University Press, 2005. [6] R.F. Dittmar, Nonlinear pricing kernels, kurtosis preference, and evidence from the cross-section of equity returns, Journal of Finance 51, 369 403, 2002. [7] 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, 1993. [8] E.F. Fama and K.R. French, The cross-section of expected stock returns, Journal of Finance, 47(2), 427 465, 1992. [9] E.F. Fama and J.D. MacBeth, Risk, Return, and Equilibrium: Empirical Tests, The Journal of Political Economy, 81(3), 607 636, 1973. [10] H. Fang and T.-Y. Lai, Co-kurtosis and capital asset pricing, Financial Review 32, 293 307, 1997. [11] C. Harvey and A. Siddique, Conditional skewness in asset pricing tests, Journal of Finance 55, 1263 1295, 2000. [12] M.C. Jensen, The Performance of Mutual Funds in the Period 1945-1964, Journal of Finance 23, 389 416, 1968. [13] A. Kraus and R. Litzenberger, Skewness preference and the valuation of risk assets, Journal of Finance 31, 1085 1100, 1976. [14],, Fama-French,, 22, 3 23, 2007. [15] 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, 1965. [16] Y. Malevergne and D. Sornette, Extreme Financial Risks: From Dependence to Risk Management, Springer, 2006. [17] L. Pastor and R.F. Stambaugh, Liquidity risk and expected stock returns, Journal of Political Economy, 111(3), 642 685, 2003. 20
[18] W. Sharpe, Capital asset prices: a theory of market equilibrium under conditions of risk, Journal of Finance 19, 425 442, 1964. [19],,, UFJ, 407 430, 2008. [20],,, Idiosyncratic, 2011 JAFEE, 49 60, 2011. [21],,,, JAFEE 12, pp. 168 19, 2013. [22],, VaR, / /2001.12, 95 131, 2001. A (Generalized Capital Asset Pricing) [16] I- [16] CAPM ρ r i r f = ρ ( w r ) r M r f, i = 1, 2,..., n (A.1) w i ρ (r w =w M,t ) w i, i = 1, 2,..., n 1 w R n i r R n i r, i = 1,..., n w R n [16] ρ i (i = 1, 2,...) 1 1 X, Y, c, ρ 19. 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 19 1 3, [1] (Monotonicity; X Y ρ(x) ρ(y )) [22]. 21
( [ (X ]) 2 1/2 ρ ρ (X) := E X) 1 ρ ( w r ) 1 w i ρ (r w =w M,t ) ( ) ( [ { ]) 2 1/2 ( [ 1/2 ρ w r = E w (r r)} = Var w r]) (A.2) = = = 1 2ρ (w r) w =w 1 [ 2ρ 2 (r M,t ) E 2 1 [ { } ] 2 E w (r r) w i } { w (r r) ρ 2 (r M,t ) E [(r M,t r M,t ) (r r i )] = Cov [r M,t, r ] Var [r M,t ] ] w (r r i ) =w w =w 1 ρ (r M,t ) (A.3) (A.3) (A.1) CAPM ρ ρ (X) := j 1 η j X X j, η j > 0 (A.4) j j X X l := ( [ E X l]) 1/l, l = 1, 2,..., (A.5) ρ 1 η 1 = 1, η j = 0 (j 2) (A.4) η 2 = 1, η j = 0 (j 2) (A.4) ρ R, R M,t (2.2) ρ ρ ( ) ρ w w r := η j (r r), j η j > 0 (A.6) {j=2,4,..., ˆm} ˆm w (r r) [ j {w j E (r r) } ] j w i = w i w =w [ w =w { j 1 = E j w (r r)} (r r i )] [ ] w =w = je R j 1 M,t R, j = 2, 4,..., ˆm w (r r) j j w i = j R M,t j 1 j w =w w (r r) j w i w =w 22
κ (j 1) = C(j 1) [ E R j M,t ] = C(j 1) R M,t j j, j = 2, 4,..., ˆm R, R M,t (2.2) w (r r) [ ] j E R j 1 M,t R = w i w =w R M,t j 1 j ( ) Cov R j 1 M,t, R = R M,t j 1 j C (j 1) = R M,t j 1 j = R M,t j κ (j 1) r i r f = ρ ( w r ) r M r f w i ρ (r w =w M,t ) = w (r r) j η j w i {j=2,4,..., ˆm} w =w = ( r M r f ) ω j κ (j 1) {j=2,4,..., ˆm} r M r f {j=2,4,..., ˆm} η j R M,t j (A.7) (A.8) (A.9) ω l, l = 2, 4,..., ˆm ω l := η l R M,t l {j=2,4,..., ˆm} η j R M,t j, l = 2, 4,..., ˆm (A.10) η l > 0 ω l > 0 (2.9) κ (j 1) = C(j 1) [ E R j M,t ω 2 + ω 4 + + ω ˆm = 1 j 1 ] = l=1 [ ] E R j 1+l ϕ (j) M,t l := [ ], l = 1,..., j 1 E R j M,t ϕ (j) l β (l), j = 2, 4,..., ˆm (A.11) 23
(A.9) r i r f = ( r M r f ) = ( r M r f ) {j=2,4,..., ˆm} {j=2,4,..., ˆm} ω j κ (j 1) j 1 ω j l=1 ϕ l β (l) (A.12) I- r i r f r i r f = ( r M r f ) β(1) + + {j=4,6,..., ˆm} {j=6,8,..., ˆm} ω j ϕ (j) β (2) 2 ω j ϕ (j) 5 + {j=4,6,..., ˆm} ω j ϕ (j) β (3) 3 + β (5) + + ω ˆmϕ ( ˆm) ˆm 2) ˆm 2 β( + ω ˆm ϕ ( ˆm) ˆm 1 {j=6,8..., ˆm} ˆm 1) β( ω j ϕ (j) β (4) 4 (A.13) 4 2 I- 24