The cumulant risk premium

BIS Working Papers  |  No 1128  | 
09 October 2023



Many episodes of market turbulence, including the Covid-19 crisis in March 2020, show that asset returns are not normally distributed and that higher-order moments play an important role in financial markets. This raises two questions: (1) What are the implications of higher-order moments for standard finance models? (2) How can the risk premium of higher-order moments across asset classes be measured in a tractable way? In this paper, we show that leveraged ETFs can be used as an alternative to options for measuring the risk of higher-order moments, and we quantify this risk across equities, bonds, commodities, currencies and volatility. To quantify the exposure to higher-order moments in a tractable way, we use the concept of cumulants, which are close relatives to the moments of a given distribution.


We develop a novel methodology for measuring the risk premium of higher-order moments based on leveraged ETFs. The methodology presents an alternative way to measure the risk of higher-order moments by using liquid ETFs instead of illiquid option portfolios, as widely used in the existing literature. Our paper illustrates that higher leverage involves increasing exposure to higher-order moments, in contrast to a common misperception that this risk declines with the number of higher-order terms. These results have implications for hedge funds and anyone who uses leverage. Our findings have also implications for portfolio theory, momentum strategies, factor models and option pricing.


First, we find that the risk premium on any asset is the sum of a linear term in an asset's leverage, and a non-linear term, which depends on the difference between higher-order physical and risk-neutral cumulants of the factor. We call the sum of these differences the cumulant risk premium (CRP). The returns on highly leveraged strategies are dominated by the non-linear term, and are thus highly sensitive to the CRP. Second, we find that the average CRP is –7.4% annualised across assets and is more than 100% of the factor risk premium. Third, we show that momentum strategies are exposed to cumulants. Fourth, we find that liquidity provision to leveraged strategies is exposed to higher-order even cumulants and show that a strategy to mimic liquidity provision earns large Sharpe ratios in some assets. Finally, we develop a global index of market stress based on cumulants. The index is easy to compute in real time and takes the prospective of a liquidity provider who is exposed to cumulants across asset classes.


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.

JEL classification: G1, G12, G13, G23

Keywords: : cumulants, leverage, ETF, factor models, VIX, momentum, options