Do Vanguard's Managed Funds Beat Its Index Funds? Looking for Prescient Stock and Style Selection Draft 1

Is John Bogle right that Vanguardís index and managed funds are comparable and by his example that investors should hold both? Each Vanguard managed fund produces the same return as the tracking basket of Vanguardís indexed funds plus a differential, a. If a, is positive the managed fund is superior to the tracking index. Managed funds that are closely tracked by their tracking basket are beaten by their baskets. Those that are not, beat them. We also find that during the period considered: aís in one period predict aís in the next period, and managers make prescient style choices.