Appendices. Appendix 1 Buy ranges for each portfolio

Transcription

1 Appendices Appendix 1 Buy ranges for each portfolio 67

2 Appendix 2 Every recession declared by the NBER Source: The National Bureau of Economic Research 68

3 Appendix 3 Multifactor portfolio mathematics Mathematics based on Cochrane (1999, p ). 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!!!!

4 !"#!! =!!!" =!!!!!!!!! Or, writing out the sum of the matrix notation,!"#!! =! 1!!!!!!!!! 1!!! 7

5 Appendix 4 Descriptive statistics Big Growth 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,%), 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,%), 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

6 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,%), 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,%), 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

7 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

8 Appendix 5 Regression attributes Model 1: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: BigGrowth const <.1 *** dummy <.1 *** Mean dependent var.6947 S.D. dependent var.5447 Sum squared resid S.E. of regression R-squared.2752 Adjusted R-squared.2662 F(1, 982) P-value(F) 2.12e-7 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho Durbin-Watson Model 2: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: BigNeutral const <.1 *** dummy <.1 *** Mean dependent var.7739 S.D. dependent var Sum squared resid S.E. of regression.5654 R-squared.2613 Adjusted R-squared F(1, 982) P-value(F) 3.48e-7 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho Durbin-Watson Model 3: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: BigValue const <.1 *** dummy <.1 *** Mean dependent var.8685 S.D. dependent var.7264 Sum squared resid S.E. of regression.7192 R-squared.2778 Adjusted R-squared.2679 F(1, 982) P-value(F) 1.45e-7 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho Durbin-Watson

9 Model 4: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: SmallGrowth const <.1 *** dummy <.1 *** Mean dependent var.7168 S.D. dependent var.7788 Sum squared resid S.E. of regression R-squared Adjusted R-squared F(1, 982) P-value(F) 4.47e-7 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho.1488 Durbin-Watson Model 5: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: SmallNeutral const <.1 *** dummy <.1 *** Mean dependent var.9986 S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(1, 982) P-value(F) 1.64e-7 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho.1574 Durbin-Watson Model 6: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: SmallValue const <.1 *** dummy <.1 *** Mean dependent var.187 S.D. dependent var.893 Sum squared resid S.E. of regression.7997 R-squared Adjusted R-squared F(1, 982) P-value(F) 1.9e-6 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho Durbin-Watson

10 Model 7: OLS, using observations 1928:1-29:12 (T = 984) Dependent variable: LongBonds const *** dummy * Mean dependent var.4337 S.D. dependent var Sum squared resid S.E. of regression R-squared.375 Adjusted R-squared.269 F(1, 982) P-value(F) Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho Durbin-Watson

11 Appendix 6 Angles of the multifactor efficient frontier 77

12 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