The Cumulant Risk Premium

60 Pages Posted: 19 Jan 2023

See all articles by Albert S. Kyle

Albert S. Kyle

University of Maryland

Karamfil Todorov

Bank for International Settlements

Date Written: December 30, 2022


We develop a novel methodology to measure the risk premium of higher-order cumulants (closely related to the moments of a distribution) based on assets satisfying a single-factor setting. We show that single-factor linear pricing works only if the difference between physical and risk-neutral cumulants, which we call the cumulant risk premium (CRP), is zero. To illustrate our approach empirically, we study leveraged ETFs, which are assets with constant betas and exposure to a single factor only. We show that the CRP is different from zero across asset classes: equities, bonds, commodities, currencies, and volatility. We quantify the even-order CRP by developing a simple strategy of shorting ETFs with opposite betas. The strategy mimics liquidity provision, earns Sharpe ratios above one, and can be used as a simple gauge of global market stress in real time. Our results have implications not only for factor models but also for portfolio theory, momentum strategies, option pricing, hedge funds, and leverage in general.

Keywords: Cumulants, leverage, ETF, CAPM, factor models, VIX

JEL Classification: G1, G12, G13, G23

Suggested Citation

Kyle, Albert (Pete) S. and Todorov, Karamfil, The Cumulant Risk Premium (December 30, 2022). Available at SSRN: or

Albert (Pete) S. Kyle

University of Maryland ( email )

College Park
College Park, MD 20742
United States

Karamfil Todorov (Contact Author)

Bank for International Settlements ( email )

Centralbahnplatz 2
Basel, Basel-Stadt 4002

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