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

Rational Pricing of Leveraged ETF Expense Ratios