124 Optimizing Optimization
in equity returns; (2) the time evolution of factor f j (^) , (^) t and of errors e i (^) , (^) t , and (3)
the comovements of the vector of the returns considering the skewness and
kurtosis of the joint distribution.
To deal with the third problem, we suggest employing a skewed copula with
heavy tails. A copula function C associated to random vector υ ( υ 1 , ... , υ (^) n )
is a probability distribution function on the n- dimensional hypercube, such
that:
Table 5.2 Estimated coefficients
MMM AA AXP T BAC BA CAT
Alpha 0.001% 0.017% 0.004% 0.010% 0.004% 0.005% 0.015%
Beta 1 0.528% 0.600% 0.660% 0.501% 0.522% 0.497% 0.552%
Beta 2 0.061% 0.175% 0.082% 0.189% 0.116% 0.090% 0.172%
Beta 3 0.079% 0.392% 0.137% 0.139% 0.273% 0.146% 0.212%
Beta 4 0.004% 0.122% 0.121% 0.166% 0.359% 0.031% 0.078%
Beta 5 0.182% 0.054% 0.035% 0.310% 0.017% 0.195% 0.080%
Beta 6 0.014% 0.063% 0.026% 0.252% 0.009% 0.031% 0.052%
Beta 7 0.057% 0.202% 0.019% 0.221% 0.074% 0.242% 0.032%
Beta 8 0.083% 0.020% 0.009% 0.068% 0.010% 0.185% 0.043%
Beta 9 0.118% 0.058% 0.028% 0.093% 0.000% 0.114% 0.117%
Beta 10 0.001% 0.194% 0.070% 0.020% 0.057% 0.152% 0.106%
Beta 11 0.118% 0.155% 0.010% 0.026% 0.052% 0.327% 0.148%
Beta 12 0.021% 0.069% 0.107% 0.026% 0.002% 0.004% 0.078%
Beta 13 0.069% 0.058% 0.076% 0.027% 0.093% 0.015% 0.020%
Beta 14 0.128% 0.193% 0.010% 0.033% 0.068% 0.105% 0.097%
HD IBM INTC JNJ JPM KFT MCD
Alpha 0.018% 0.003% 0.012% 0.006% 0.001% 0.001% 0.016%
Beta 1 0.585% 0.444% 0.670% 0.434% 0.646% 0.287% 0.374%
Beta 2 0.121% 0.067% 0.156% 0.324% 0.152% 0.188% 0.020%
Beta 3 0.158% 0.017% 0.050% 0.048% 0.251% 0.135% 0.010%
Beta 4 0.085% 0.184% 0.408% 0.032% 0.239% 0.036% 0.052%
Beta 5 0.172% 0.103% 0.057% 0.060% 0.129% 0.182% 0.196%
Beta 6 0.001% 0.043% 0.043% 0.142% 0.013% 0.330% 0.047%
Beta 7 0.049% 0.092% 0.267% 0.025% 0.076% 0.050% 0.223%
Beta 8 0.209% 0.036% 0.086% 0.025% 0.047% 0.399% 0.185%
Beta 9 0.056% 0.013% 0.045% 0.101% 0.009% 0.124% 0.475%
Beta 10 0.318% 0.040% 0.018% 0.045% 0.072% 0.216% 0.181%
Beta 11 0.191% 0.087% 0.000% 0.148% 0.017% 0.060% 0.027%
Beta 12 0.007% 0.183% 0.056% 0.205% 0.019% 0.150% 0.142%
Beta 13 0.196% 0.066% 0.110% 0.002% 0.007% 0.033% 0.066%
Beta 14 0.008% 0.156% 0.104% 0.183% 0.100% 0.001% 0.001%
Fyυ()( 11 ,...,ynnP y 1 ,..., yn)(( )( ))CP 1 y 1 ,...,Pnyn
υυ υ υ
CCF y(()υυ 1 1 ,...,F yn())n ,