β are found at the 10% significance level. In the case of zero transaction costs for 31 data series out of 51 the estimate of α is significantly positive at the 10% significance level. This number decreases to 21 (10, 4, 3, 2) if 0.10% (0.25, 0.50, 0.75, 1%) costs per trade are implemented. Table 4.15B shows the results of the CAPM estimation for the case that the best strategy in the training period is selected by the Sharpe ratio criterion. Now in the case of zero transaction costs for 33 data series it is found that the estimate of α is significantly positive at the 10% significance level. If transaction costs increase to 0.10% (0.25, 0.50, 0.75, 1%), then for 24 (11, 2, 2, 2) out of 51 data series the estimate of α is significantly positive. Hence, after correction for 1% transaction costs and risk it can be concluded, independently of the selection criterion used, that the best recursive optimizing and testing procedure shows no statistically significant out-of-sample forecasting power.
Selection criterion: mean return costs α<0 α>0 β<1 β>1 α>0 ∧ α>0 ∧ β<1 β>1 0% 1 31 35 2 25 0 0.10% 1 21 32 3 15 0 0.25% 1 10 34 4 8 0 0.50% 2 4 31 3 1 0 0.75% 3 3 29 4 1 1 1% 3 2 30 2 1 0 Selection criterion: Sharpe ratio costs α<0 α>0 β<1 β>1 α>0 ∧ α>0 ∧ β<1 β>1 0% 0 33 42 2 30 1 0.10% 0 24 39 1 21 0 0.25% 0 11 40 2 10 0 0.50% 0 2 36 2 1 0 0.75% 0 2 34 2 1 0 1% 0 2 35 2 1 0
Table 4.16: Summary: significance CAPM estimates for best out-of-sample testing procedure. For each transaction cost case, the table shows the number of data series for which significant estimates are found at the 10% significance level for the coefficients in the Sharpe-Lintner CAPM. Columns 1 and 2 show the number of data series for which the estimate of α is significantly negative and positive. Columns 3 and 4 show the number of data series for which the estimate of β is significantly smaller and larger than one. Column 5 shows the number of data series for which the estimate of α is significantly positive as well as the estimate of β is significantly smaller than one. Column 6 shows the number of data series for which the estimate of α is significantly positive as well as the estimate of β is significantly larger than one. Note that the number of data series analyzed is equal to 51 (50 stocks and the AEX-index).