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Nothing else is requiredWhat may have appeared sensible in 1991 may not be so sensible in 2022. Now there are all sorts of esoteric high cost indexes and active ETFs having costs as low or sometimes lower than passive ETFs (e.g. the second and fourth lowest cost ultrashort term ETFs in M*'s table are actively managed - ICSH and VUSB).If "active" and "passive" management styles are defined in sensible ways, it must be the case that
(1) before costs, the return on the average actively managed dollar will equal the return on the average passively managed dollar and
(2) after costs, the return on the average actively managed dollar will be less than the return on the average passively managed dollar.
These assertions will hold for any time period. Moreover, they depend only on the laws of addition, subtraction, multiplication and division. Nothing else is required.
But Many Vanguard managed fund quietly out perform many Vanguard ETFsBasic Math
William Sharpe’s arithmetic of active management is admirably direct: “After costs, the return on the actively managed dollar will be less than the average passively managed dollar,” wrote the good professor in 1991. “These assertions will hold for any time period. Moreover, they depend only on the laws of addition, subtraction, multiplication, and division. Nothing else is required.”
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