Appendices Appendix 1 Buy ranges for each portfolio 67
Appendix 2 Every recession declared by the NBER Source: The National Bureau of Economic Research 68
Appendix 3 Multifactor portfolio mathematics Mathematics based on Cochrane (1999, p. 74-75). The optimization problem is: min var(!! ) =!!" subject to!!!! = µμ!!1 = 1!!! =!! The Lagrangian is 1!!!"!!!!!!!!!! 1 1!!!!!!! 2 The first order condition with respect to w give! =!!!!!! + 1!! +!!! =!!!!" Where! =! 1!! =!!!!!!! =! 1!! Plugging this value of w into the constraint equations!!! =!, We get!!!!!" =!! =!!!!!!!! =!!!!!!!!! The portfolio variance is then 69!!!!
!"#!! =!!!" =!!!!!!!!! Or, writing out the sum of the matrix notation,!"#!! =! 1!!!!!!!!! 1!!! 7
Appendix 4 Descriptive statistics Big Growth 2 18 16 14 12 1 8 6 4 2 BigGrowth Mean,69 Standard Error,17 Median,18 Mode -,62 Standard Deviation,544 Sample Variance,3 Kurtosis 5,623 Skewness -,6981 Range,6118 Minimum -,336 Maximum,2812 Sum 6,8358 Largest(1),2812 Smallest(1) -,336 Confidence Level(95,%),34 3 25 2 15 1 5 Big Neutral BigNeutral Mean,77 Standard Error,18 Median,123 Mode,25 Standard Deviation,573 Sample Variance,33 Kurtosis 1,6318 Skewness,112 Range,7417 Minimum -,3256 Maximum,4161 Sum 7,6148 Largest(1),4161 Smallest(1) -,3256 Confidence Level(95,%),36 3 25 2 15 1 5 Big Value BigValue Mean,87 Standard Error,23 Median,13 Mode,44 Standard Deviation,721 Sample Variance,52 Kurtosis 1,4579 Skewness,641 Range,973 Minimum -,4363 Maximum,5339 Sum 8,5465 Largest(1),5339 Smallest(1) -,4363 Confidence Level(95,%),45 71
25 2 15 1 5 Small Growth SmallGrowth Mean,72 Standard Error,25 Median,119 Mode,32 Standard Deviation,771 Sample Variance,59 Kurtosis 5,874 Skewness -,915 Range,8873 Minimum -,397 Maximum,4966 Sum 7,534 Largest(1),4966 Smallest(1) -,397 Confidence Level(95,%),48 2 18 16 14 12 1 8 6 4 2 Small Neutral SmallNeutral Mean,1 Standard Error,22 Median,156 Mode -,99 Standard Deviation,695 Sample Variance,48 Kurtosis 8,7396 Skewness,118 Range,866 Minimum -,3689 Maximum,4971 Sum 9,8265 Largest(1),4971 Smallest(1) -,3689 Confidence Level(95,%),43 3 25 2 15 1 5 Small Value SmallValue Mean,19 Standard Error,26 Median,153 Mode,55 Standard Deviation,89 Sample Variance,65 Kurtosis 9,619 Skewness,3748 Range 1,99 Minimum -,411 Maximum,5989 Sum 1,6963 Largest(1),5989 Smallest(1) -,411 Confidence Level(95,%),51 72
3 25 2 15 1 5 Long- term bonds LongBonds Mean,43 Standard Error,8 Median,3 Mode,21 Standard Deviation,238 Sample Variance,6 Kurtosis 4,8744 Skewness,3795 Range,261 Minimum -,1192 Maximum,1418 Sum 4,2672 Largest(1),1418 Smallest(1) -,1192 Confidence Level(95,%),15 73
Appendix 5 Regression attributes Model 1: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: BigGrowth const.112651.1972 5.9267 <.1 *** dummy -.22844.437179-5.2253 <.1 *** Mean dependent var.6947 S.D. dependent var.5447 Sum squared resid 2.83168 S.E. of regression.53693 R-squared.2752 Adjusted R-squared.2662 F(1, 982) 27.349 P-value(F) 2.12e-7 Log-likelihood 1482.443 Akaike criterion -296.885 Schwarz criterion -2951.12 Hannan-Quinn -2957.164 rho.73659 Durbin-Watson 1.852584 Model 2: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: BigNeutral const.1223.2149 6.97 <.1 *** dummy -.236179.46357-5.134 <.1 *** Mean dependent var.7739 S.D. dependent var.57263 Sum squared resid 3.139217 S.E. of regression.5654 R-squared.2613 Adjusted R-squared.25112 F(1, 982) 26.3255 P-value(F) 3.48e-7 Log-likelihood 1431.69 Akaike criterion -2859.219 Schwarz criterion -2849.436 Hannan-Quinn -2855.498 rho.115698 Durbin-Watson 1.768495 Model 3: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: BigValue const.144813.251662 5.7543 <.1 *** dummy -.3662.57884-5.2971 <.1 *** Mean dependent var.8685 S.D. dependent var.7264 Sum squared resid 4.96366 S.E. of regression.7192 R-squared.2778 Adjusted R-squared.2679 F(1, 982) 28.5966 P-value(F) 1.45e-7 Log-likelihood 126.249 Akaike criterion -248.498 Schwarz criterion -2398.714 Hannan-Quinn -244.776 rho.114715 Durbin-Watson 1.77181 74
Model 4: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: SmallGrowth const.131228.26957 4.8692 <.1 *** dummy -.31524.619885-5.82 <.1 *** Mean dependent var.7168 S.D. dependent var.7788 Sum squared resid 5.691864 S.E. of regression.76133 R-squared.25626 Adjusted R-squared.24634 F(1, 982) 25.82646 P-value(F) 4.47e-7 Log-likelihood 1138.838 Akaike criterion -2273.676 Schwarz criterion -2263.892 Hannan-Quinn -2269.954 rho.1488 Durbin-Watson 1.7336 Model 5: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: SmallNeutral const.155532.242789 6.461 <.1 *** dummy -.29458.558431-5.2739 <.1 *** Mean dependent var.9986 S.D. dependent var.69514 Sum squared resid 4.619248 S.E. of regression.68585 R-squared.27543 Adjusted R-squared.26553 F(1, 982) 27.81354 P-value(F) 1.64e-7 Log-likelihood 1241.57 Akaike criterion -2479.141 Schwarz criterion -2469.357 Hannan-Quinn -2475.419 rho.1574 Durbin-Watson 1.697922 Model 6: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: SmallValue const.16975.28391 5.9725 <.1 *** dummy -.319392.651129-4.952 <.1 *** Mean dependent var.187 S.D. dependent var.893 Sum squared resid 6.2894 S.E. of regression.7997 R-squared.23916 Adjusted R-squared.22922 F(1, 982) 24.61 P-value(F) 1.9e-6 Log-likelihood 19.451 Akaike criterion -2176.92 Schwarz criterion -2167.119 Hannan-Quinn -2173.181 rho.155256 Durbin-Watson 1.68984 75
Model 7: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: LongBonds const.363812.84643 4.3278.2 *** dummy.369494.193354 1.911.563 * Mean dependent var.4337 S.D. dependent var.23779 Sum squared resid.55378 S.E. of regression.23747 R-squared.375 Adjusted R-squared.269 F(1, 982) 3.651816 P-value(F).56299 Log-likelihood 2285.21 Akaike criterion -4566.421 Schwarz criterion -4556.637 Hannan-Quinn -4562.699 rho.24539 Durbin-Watson 1.943834 76
Appendix 6 Angles of the multifactor efficient frontier 77
Appendix 7 Calculation of two-funds!"#$h!!"#$%& = Example of 6/4 in market/risk-free 1!"#$!"#$%&'(,6 = 1 (,6,2) 2,,18! With a new value of the return variance in the fall of 28 (,7%) 1 (,6,2) 2,,7! =,4 (!!! )!! Equation 28 Now, 4% is invested in the market portfolio and 96% in the risk-free asset,6 = 1 2,!,2,7!! =,61 The expected return would have to increase of 61% to maintain the 6/4 in market/risk-free. 78