We present conditions under which positive alpha exists in the realm of active portfolio management– in contrast to the controversial result in (Jarrow, 2010, pg. 20) which implicates delegated portfolio management by surmising that positive alphas are illusionary. Specifically, we show that the critical assumption used in (Jarrow, 2010, pg. 20), to derive the illusionary alpha result, is based on a zero set for CAPM with Lebesgue measure zero. So conclusions based on the assumption may well have probability measure zero of occurrence. Technically, the existence of [Tanaka] local time on that set implies existence of positive alphas. In fact, we show that positive alpha exists under the same scenarios of ”perpetual event swap” and ”market systemic event” Jarrow (2010) used to formulate the illusionary positive alpha result. First, we prove that as long as asset price volatility is greater than zero, systemic events like market crash will occur in finite time almost surely. Thus creating an opportunity to hedge against that event. Second, we find that Jarrow’s ”false positive alpha” variable constitutes portfolio manager reward for trading strategy. For instance, we show that positive alpha exists if portfolio managers develop hedging strategies based on either (1) an exotic [barrier] option on the underlying asset–with barrier hitting time motivated by the ”market systemic” event, or (2) a swaption strategy for the implied interest rate risk inherent in Jarrow’s triumvirate of riskless rate of return, factor sensitivity exposure, and constant risk premium for a perpetual event swap.

## Discussion

Godfrey, if the market is essentially complete and equilibrium characterizes asset prices, I'm not sure whether "alpha return" would not be illusionary in that instances... moreover, in continuous time, I'm not sure whether the cost of maintaining your "hedged" trading strategy would not lead to "vanishing alpha" or a ruin a la LTCM!