Active Portfolio Management, Positive Jensen-Jarrow Alpha, and Zero Sets of CAPM

TL;DR

This paper demonstrates conditions under which positive alpha can exist in active portfolio management, challenging prior claims of its illusionary nature by analyzing zero sets of CAPM and systemic market events.

Contribution

It introduces new conditions for positive alpha existence, linking systemic events, local time on zero sets, and hedging strategies, contrasting with Jarrow's earlier results.

Findings

Positive alpha exists when asset volatility is greater than zero.

Systemic events like market crashes occur almost surely in finite time.

Hedging strategies can be based on exotic options or swaptions related to systemic events.

Abstract

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 that assumption may well have probability measure zero of occurrence as well. Technically, the existence of [Tanaka] local time on a zero set for CAPM 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.