Recovering the market risk premium from higher‐order moment risks

• Published date: 01 January 2021
• Author: George Chalamandaris,Leonidas S. Rompolis
• Date: 01 January 2021
• DOI: http://doi.org/10.1111/eufm.12287

Eur Financ Manag. 2021;27:147–186. wileyonlinelibrary.com/journal/eufm © 2020 John Wiley & Sons Ltd.

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DOI: 10.1111/eufm.12287

ORIGINAL ARTICLE

Recovering the market risk premium from

higher‐order moment risks

George Chalamandaris |Leonidas S. Rompolis

Department of Accounting and Finance,

Athens University of Economics and

Business, Athens, Greece

Correspondence

Leonidas S. Rompolis, Department of

Accounting and Finance, Athens

University of Economics and Business,

76 Patission Street, 10434 Athens, Greece.

Email: rompolis@aueb.gr

Funding information

Research Center of Athens University of

Economics and Business,

Grant/Award Numbers: EP‐3087‐01,

EP‐2237‐01, EP‐2259‐01

Abstract

We propose a consistent approach for the estimation

of the market risk premium. As a first step, we define

the broadest possible set of ex ante estimators from

the viewpoint of a power utility optimiser holding the

market portfolio. We then employ an evaluation

framework to optimise the parametrisation of the

methodology. We show that this theoretical

framework can still produce reasonable market risk

premium estimates, even when the representative

agent is not a power utility optimiser. Our results

show that the inclusion of higher‐order moment

risk premia improves the accuracy of the method.

KEYWORDS

ex ante market risk premium, physical cumulants, risk aversion

coefficient, risk‐neutral cumulants

JEL CLASSIFICATION

G12; G17; C51; C53

EUROPEAN

FINANCIAL MANAGEMENT

The authors would like to thank John Doukas (the editor) and an anonymous referee for very helpful comments and

suggestions that greatly improved the quality of the paper. We also thank Elias Tzavalis, Ciprian Necula, Thijs van der

Heijden, Alex Taylor, Ser‐Huang Poon, Alexandros Kostakis, Gianluca Fusai, Giannis Kyriakou, Michail Anthropelos,

Gikas Hardouvelis, Massimo Guidolin and seminar participants at Manchester Business School, Cass Business School,

the University of Piraeus (Department of Banking and Financial Management) and participants of the 2013 National

Conference of the FEBS, the 2014 Paris Financial Management Conference and the 2016 European Financial Man-

agement Conference for useful comments and suggestions. Lykourgos Alexiou provided excellent research assistance.

G. Chalamandaris acknowledges financial support from the Research Center of Athens University of Economics and

Business (EP‐3087‐01). L. S. Rompolis acknowledges financial support from the Research Center of Athens University

of Economics and Business (EP‐2237‐01 and EP‐2259‐01).

1|INTRODUCTION

The market risk premium (MRP) is a central quantity in various core fields of finance, including

asset pricing, portfolio allocation and risk management. Despite its undisputed importance, it is

also inherently unobserved by all interested parties. Most critically, according to a vast body of

empirical research (for references, see Martin, 2017), the MRP can be very volatile, subject to

both slow cyclical changes and abrupt shifts caused by unexpected shocks.

Indicative of the difficulty of capturing MRP dynamics is the controversial nature of the

results produced by the various estimation methods. The high dispersion of their time avera-

ges,

1

their markedly different responses to extraordinary events,

2

as well as their contradictory

statistical relations with specific observable time series

3

are all typical findings in the literature.

The main obstacle researchers face in this strand of literature is the lack of a visible benchmark

against which to compare different MRP estimates, even within the narrow context of same‐

family estimators. Equally debilitating, however, is the lack of a theoretically consistent em-

pirical framework that would allow for the testing of the MRP estimates' validity. Therefore, a

significant part of this controversy can be attributed to arbitrary modelling decisions that still

have a strong impact on the results, even though their legitimacy remains unproven.

4

It seems

reasonable, therefore, that a more consistent approach to precluding such ad hoc econometric

decisions would involve first defining a general set of estimators built on the same theoretical

set of hypotheses and then using the testing framework to determine the optimal para-

metrisation of the estimation method.

In this study, we propose an estimation approach that follows exactly this line of research:

First, we define an encompassing set of ex ante MRP estimators from the perspective of in-

vestors with power utility preferences holding the market portfolio. Second, we present an

evaluation framework to optimise the parametrisation of the methodology, given that it allows

us to compare MRP estimates from within a specific family of estimators. We apply this fra-

mework to the set of MRP estimators we define in this study.

We make several contributions to the literature in this paper. First, we derive a general

system of equations relating physical to risk‐neutral cumulants through the relative risk

aversion coefficient of a power utility investor that holds the market portfolio. This coefficient is

known to the literature as the projected relative risk aversion coefficient (PRRAC) (Rosenberg &

Engle, 2002), to distinguish it from the actual risk aversion of the representative agent. The

PRRAC can be interpreted as the relative risk aversion coefficient of a power utility investor

whose preferences best fit the preferences of the representative agent when expressed in terms

of the future states of the market portfolio. In effect, our methodology extends the MRP

1

A relatively recent review of equity risk premium models by Duarte and Rosa (2015) describes great dispersion of

mean estimates, ranging from

−1

to 14.5%. It is worth noting that these estimates do not constitute the outliers of

the literature. On the contrary, almost all well‐cited estimates outside of Duarte and Rosa's (2015)reviewarealso

inconsistent with each other. For example, one can easily find mean estimates around 2–3% Pastor et al. (2008)or

12% Santa‐Clara and Yan (2010).

2

For instance, Campbell and Thompson's (2008) valuation ratio‐based estimates of the MRP barely responded to the

Black Monday Crash of 1987, whereas option‐implied MRP estimates (Martin, 2017) exploded on the same day.

3

For instance, Greenwood and Shleifer (2014) indicate that valuation ratio‐based estimates of expected returns (ERs) are

negatively correlated with investor expectations retrieved from survey data, whereas other estimates (e.g. Martin, 2017)

are positively correlated with the same data.

4

For example, Duarte and Rosa (2015) documents that seemingly minor choices, such as the length of the estimation

window in historical means or the choice between expected and current earnings‐to‐price ratios in a valuation ratio

model, completely change the value of the estimated MRP.

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CHALAMANDARIS AND ROMPOLIS

estimator of Duan and Zhang (2014, DZ henceforth), which is driven mainly by the variance

risk premium, to a broad set of estimators that use additional higher‐order cumulant premia to

restrict the possible values of the MRP further. In that context, DZ's approach is only a nested

specification within the set of estimators that we examine.

The direction in which we extend the DZ estimator is motivated by recent theoretical and

empirical evidence indicating that higher‐order moment (or cumulant) risk premia determine the

time variability of the MRP (Sasaki, 2016). Other studies (Kozhan et al., 2013;Schneider,2015)also

indicate that the size of these higher‐order risk premia is related to risk aversion and business cycle

conditions. Thus, the set of MRP estimators defined in this study includes possibly better‐defined

specifications, which –because of their fit on higher‐order cumulant risk premia –are more likely to

benefit from this type of information, compared with that of DZ. Our empirical results confirm this

intuition. We find that the inclusion of skewness and kurtosis risk premia improves the quality of

the ex ante MRP estimates. At first glance, a notable difference between our estimates and those

obtained by the DZ specification is the average value of the MRP. Following DZ's approach, this is

close to 20% for our sample period. In contrast, the time‐series average of our estimate is around 7%,

which is substantially closer to the average equity premium reported in the literature.

Our second key contribution is that we explain why this theoretical framework can

produce reasonable MRP estimates, even though the representative agent is not a power

utility optimiser. To illustrate this argument, we conduct a lab‐controlled experiment,

assuming that real‐world dynamics are governed by the considerably more complex habit

formation model of Bekaert et al. (2009,BEXhenceforth).BycomparingtheknownMRP

produced by the BEX model with those we inferusingourmethodology,weshedlighton

the relative advantages of the extended MRP estimators versus those based solely on the

variance risk premium. Specifically, we show that we can still estimate the MRP with

accuracy, as long as the stochastic discount factor (SDF) of the representative agent can be

approximated by the SDF of the power utility expressed in terms of the market portfolio.

Moreover, we gain valuable insight from this experiment about the sometimes puzzling

dynamics of the power utility PRRAC, which has been documented but left mostly

unexplained by previous studies (Bliss & Panigirtzoglou, 2004;Duan&Zhang,2014). We

explicitly demonstrate that, in the context of the economy described by the BEX model, the

power utility PRRAC can be very different from the actual risk aversion of the re-

presentative model. For instance, we show that, when the actual risk aversion coefficient is

countercyclical, the PRRAC can well be cyclical and thus appear to decrease during periods

of distress. However, we also confirm that the accuracy of our end result (MRP estimates) is

robust to any such ‘counterintuitive’behaviour of the PRRAC.

5

Third, we propose a framework to evaluate the quality of the ex ante MRP estimates objec-

tively. The first pillar of this framework borrows from the forecasting literature and specifically

relies on tools used therein to rank competing forecasts. The second pillar is a portfolio allocation

exercise in the spirit of Kostakis et al. (2011) that allows us to evaluate the usefulness of our

estimates for the implementation of market‐timing strategies. The final pillar of this framework

5

Throughout the financial literature, we witness similar phenomena with the intermediate results of other highly

successful models, the most well known of which is probably the implied volatility of the Black–Scholes–Merton model.

Although this volatility estimate famously exhibits implausible dependence on the option's moneyness (volatility smile),

they are still the ‘wrong numbers’fed through the ‘wrong model’to produce the right answer. In this context, therefore,

we argue that our PRRAC estimates are the equivalent of the Black–Scholes–Merton‐implied volatilities, in the sense

that they still lead to reasonable MRP estimates, despite the lack of the PRRAC's interpretability as a measure of the

representative agent's risk aversion.

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