  This paper studies the general relationship between the gearing ratio of a
Leveraged ETF and its corresponding expense ratio, viz., the investment
management fees that are charged for the provision of this levered financial
service. It must not be possible for an investor to combine two or more LETFs
in such a way that his (continuously-rebalanced) LETF portfolio can match the
gearing ratio of a given, professionally managed product and, at the same time,
enjoy lower weighted-average expenses than the existing LETF. Given a finite
set of LETFs that exist in the marketplace, I give necessary and sufficient
conditions for these products to be undominated in the price-gearing plane. In
a beautiful application of the duality theorem of linear programming, I prove a
kind of two-fund theorem for LETFs: given a target gearing ratio for the
investor, the cheapest way to achieve it is to combine (uniquely) the two
nearest undominated LETF products that bracket it on the leverage axis. This
also happens to be the implementation that has the lowest annual turnover. For
the writer's enjoyment, we supply a second proof of the Main Theorem on LETFs
that is based on Carath\'eodory's theorem in convex geometry. Thus, say, a
triple-leveraged ("UltraPro") exchange-traded product should never be mixed
with cash, if the investor is able to trade in the underlying index. In terms
of financial innovation, our two-fund theorem for LETFs implies that the
introduction of new, undominated 2.5x products would increase the welfare of
all investors whose preferred gearing ratios lie between 2x ("Ultra") and 3x
("UltraPro"). Similarly for a 1.5x product.
