The Cumulant Risk Premium

70 Pages Posted: 19 Jan 2023 Last revised: 11 Oct 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 leveraged ETFs. We show that the risk premium on these ETFs reflects the difference between physical and risk-neutral cumulants, which
we call the cumulant risk premium (CRP). We show that the CRP is different from zero across
asset classes (equities, bonds, commodities, currencies, and volatility) and is large in times of
stress. We illustrate that highly leveraged strategies are extremely exposed to higher-order cumulants. Our results have implications for hedge funds, factor models, momentum strategies,
and options.

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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