BIS Working PapersNo 1128 The cumulant risk premium by Albert (Pete) S. Kyle, Karamfil Todorov Monetary and Economic Department October 2023 JEL classification: G1, G12, G13, G23 Keywords: Cumulants, leverage, ETF, factor models, VIX, momentum, options BIS Working Papers are written by members of the Monetary and Economic Department of the Bank for International Settlements, and from time to time by other economists, and are published by the Bank. The papers are on subjects of topical interest and are technical in character. The views expressed in them are those of their authors and not necessarily the views of the BIS. This publication is available on the BIS website (www.bis.org). © Bank for International Settlements 2023. All rights reserved. Brief excerpts may be reproduced or translated provided the source is stated. ISSN 1020-0959 (print) ISSN 1682-7678 (online) The Cumulant Risk Premium⋆Albert S. (Pete) Kylea, Karamfil TodorovbaUniversity of Maryland, Robert H. Smith School of Business, askyle4@umd.edubBank for International Settlements, karamfil.todorov@bis.orgAbstractWe develop a novel methodology to measure the risk premium of higher-order cumulants (closelyrelated to the moments of a distribution) based on leveraged ETFs. We show that the risk pre-mium on these ETFs reflects the difference between physical and risk-neutral cumulants, whichwe call the cumulant risk premium (CRP). We show that the CRP is different from zero acrossasset classes (equities, bonds, commodities, currencies, and volatility) and is large in times ofstress. We illustrate that highly leveraged strategies are extremely exposed to higher-order cu-mulants. Our results have implications for hedge funds, factor models, momentum strategies,and options.Keywords:Cumulants, leverage, ETF, factor models, VIX, momentum, optionsJEL classification:G1, G12, G13, G23⋆The views expressed herein are those of the authors and do not necessarily reflect the views of the Bank forInternational Settlements. We are grateful to Steven Heston, Semyon Malamud, Ian Martin, Andreas Schrimpf,Tobias Sichert, Nancy Xu (discussant), and seminar participants at the AFA, BIS, and University of Lausanne forhelpful comments and suggestions. 1. IntroductionMany episodes of market turbulence, including the March 2020 COVID-19 crisis, show thatasset returns are not normally distributed and that higher-order moments play an importantrole in financial markets. How can we measure the risk premium of higher-order momentsacross asset classes in a tractable way? The classic approach in the existing literature is to useoption portfolios (e.g., Bakshi et al. (2003), Schneider and Zechner (2020)). However, the prob-lem with implementing this approach in practice is that options are often unavailable for allstrikes and are illiquid, especially for out-of-money (OTM) strikes and for less liquid assets thanequities. With average bid-ask spreads above 74%,1it is hard to infer higher-order momentsfrom options because option prices are measured very imprecisely. In this paper, we develop anovel methodology to quantify the risk premium of higher-order moments based on leveragedETFs, which are much more liquid than options with average bid-ask spreads of only 0.27%,more than274 timessmaller than those of options. We implement our new methodology acrossseveral asset classes: equities, bonds, commodities, currencies, and volatility (VIX).Leveraged ETFs (which we also label “constant-beta assets”) are assets that maintain a con-stant leverageβwith respect to a given benchmark index: e.g, a double-leveraged ETF (β=2)on the S&P 500 index should deliver twice the performance of the index on a given day. In or-der to maintain a constantβ, these ETFs need to rebalance when the index moves (see Chengand Madhavan (2009) and Todorov (2019)).2We show that this dynamic rebalancing by ETFsexposes them to higher-order moments. Thus, by observing the returns on leveraged ETFs, wecan quantify the risk premium of higher-order moments on the index.We use cumulants to measure the risk premium of higher-order moments in a tractableway. Cumulants are convenient to summarise the main characteristics of a given distributionfunction, but are more intuitive to work with compared to non-central moments. Cumulantsare also more convenient to use in the case of log-returns that appear over multiple periods in1Based on the average across equity indexes, Treasuries, currencies, commodities, for the period 2006–2020and across strikes, data from IVolatility.2For example, if the index rises, a double-leveraged ETF makes money and becomes less leveraged if it doesnot rebalance. To maintain the leverage constant, the ETF then needs to lever up and buy more of the index.1 our setting, and to model linear combinations of random variables. Cornish and Fisher (1938)describe cumulants in a general setting, whereas Martin (2013) is