Market Excess Returns, Variance and the Third Cumulant

• Author: Huimin Zhao,Eric C. Chang,Jin E. Zhang
• DOI: http://doi.org/10.1111/irfi.12234
• Published date: 01 September 2020
• Date: 01 September 2020

Market Excess Returns, Variance

and the Third Cumulant*

JIN E. ZHANG

†

,ERIC C. CHANG

‡

AND HUIMIN ZHAO

§

†

Department of Accountancy and Finance, Otago Business School, University of

Otago, Dunedin, New Zealand

‡

Faculty of Business and Economics, The University of Hong Kong, Pokfulam Road,

Hong Kong and

§

Sun Yat-Sen Business School, Sun Yat-Sen University, Guangzhou, China

ABSTRACT

In this paper, we develop an equilibrium asset pricing model for market

excess returns, variance and the third cumulant by using a jump-diffusion

process with stochastic variance and jump intensity in Cox et al. (1985) pro-

duction economy. Empirical evidence with the S&P 500 index and options

from January, 1996 to December, 2005 strongly supports our model predic-

tion that the lower the third cumulant, the higher the market excess returns.

Consistent with existing literature, the theoretical mean–variance relation is

supported only by regressions on risk-neutral variance. We further demon-

strate empirically that the third cumulant explains significantly the variance

risk premium.

JEL Codes: G12; G13

Accepted: 8 August 2018

I. INTRODUCTION

The third cumulant or skewness has been shown to be an important factor that

drives future cross-sectional stock returns. Chang et al. (2013) demonstrate that

stocks with high exposure to innovations in implied market skewness exhibit

low returns on average. Conrad et al. (2013) find more ex ante negatively (posi-

tively) skewed returns yield subsequent higher (lower) returns. However, the

research on studying the importance of the third cumulant in time-series

* We are grateful to Jamie Alcock (our AFM discussant), Hendrik Bessembinder, Leonid Kogan, Roni

Michaely, Neng Wang and seminar participants at 2015 Asian Finance Association Annual Meeting

and 2015 Auckland Finance Meeting (AFM) for helpful comments and suggestions. Jin E. Zhang has

been supported by an establishment grant from the University of Otago and the National Natural

Science Foundation of China grant (Project No. 71771199). Eric C. Chang gratefully acknowledges

financial support from the General Grant Council (GRF-HKU 17500617) and the Chung Hon-Dak

Professorship. Huimin Zhao has been supported by the National Natural Science Foundation of

China (Project No.71303265 and 71573289) and the Innovative Research Group Project of National

Natural Science Foundation of China (No. 71721001).

© 2018 International Review of Finance Ltd. 2018

International Review of Finance, 20:3, 2020: pp. 605–637

DOI: 10.1111/irfi.12234

market excess returns is sparse. Guo et al. (2015) document that realized jump

risk, especially that associated with bad or negative jumps, is a significant deter-

minant of conditional equity premium. This paper contributes the literature by

developing an equilibrium model for market excess returns, variance and the

third cumulant, and by testing the model empirically.

Following Cox et al. (1985), Dumas (1989), Vasicek (2005, 2013) and Zhang

et al. (2012), we establish our equilibrium asset pricing model in a production

economy.

1

The production variable or the price of a market portfolio follows a

jump-diffusion process with stochastic variance and stochastic jump intensity.

2

Solving the optimal control problem of one representative investor with the

constant relative risk aversion (CRRA) utility function gives us an equilibrium

condition for the instantaneous equity premium. Our setup is an extension of

Zhang et al. (2012) from constant variance and jump intensity to stochastic

ones. The dynamic setting allows us to study the time-series relationship

between market excess returns and risk that is measured not only by variance

1 Financial economists build general equilibrium models to explain the value and its time varia-

tions of equity premium, risk-free rate and variance risk premium. There are three ways to

establish an equilibrium model. The first approach is consumption-based asset pricing that is

built on a pure exchange economy, in which capital is completely illiquid. With one or multi-

ple stochastic aggregate endowments (dividends), investors maximize their expected utilities

by choosing an optimal level of consumption at each period. The literature started with Lucas

(1978) and Breeden (1979), followed by many others. The most recent works on one repre-

sentative investor is Branger et al. (2016), on heterogeneous investors is Bhamra and Uppal

(2014), and on multiple endowments is Martin (2013a). The second approach is production-

based asset pricing that is built on a production economy, in which capital reallocation while

feasible, is often costly. The model is able to capture the impact of capital illiquidity on equi-

librium resource allocation between consumption and production. The literature started with

Lucas and Prescott (1971) and Hayashi (1982), followed by Jermann (1998), Tallarini (2000)

and Boldrin et al. (2001) in discrete-time settings. Recently, Pingyck and Wang (2013) study

the economic and policy consequences of catastrophes with a jump-diffusion capital stock

process. Eberly and Wang (2011) present a two-sector general equilibrium model. The third

approach is based on a pure production economy, in which capital is perfectly liquid and can

be reallocated frictionlessly. Having a production technology to grow its commodity stochas-

tically and with a fixed amount of initial endowment, investors choose an optimal level of

consumption at each period and leaves the rest in the production to grow for the future con-

sumption. The literature started with Cox et al. (1985), followed by Dumas (1989), Vasicek

(2005, 2013) and Zhang et al. (2012). We choose to use the third approach in order to address

our main issue with a parsimonious model.

2 Anderson et al. (2002) find that any reasonably descriptive continuous-time model for equity-

index returns must allow for discrete jumps as well as stochastic volatility with a pronounced

negative relationship between return and volatility innovations. In their model, the jump

intensity is assumed to be a linear function of instantaneous variance. Huang and Wu (2004)

find that to better capture the behavior of the S&P 500 index options, we need to incorporate

a high frequency jump component in the return process and generate stochastic volatilities

from two different sources, the jump component and the diffusion component. Their focus is

on different specifications of time-changed Lévy processes. In our model, the stochastic jump

intensity is designed to be a new factor. As shown in Section 2, this new factor is the main

driver of the uncertainty in the third cumulant. It also enters explicitly into the total variance

together with instantaneous diffusive variance in the spirit of Huang and Wu (2004).

© 2018 International Review of Finance Ltd. 2018606

International Review of Finance

but also by the third cumulant. Recently, Martin (2013b) has developed a the-

ory of consumption-based asset pricing with higher cumulants by using a

cumulant-generating function technique. However, it is not possible to study a

time-series relationship between risk and return using his model because its

return distribution is static. To the best of our knowledge, our equilibrium

model is the first one that is capable of capturing the dynamic relationship

between market excess returns and higher cumulants.

With some further analysis, we establish a relationship between term market

excess return and term variance and term third cumulant, where the term vari-

ance is an aggregate effect between mean variance and mean jump intensity

during the period, and the term third cumulant is proportional to the mean

jump intensity. Our theoretical model has the following testable predictions:

the higher (lower) the variance (third cumulant), the higher the market excess returns.

Our theory is further tested empirically by using S&P 500 index and options

data from January 4, 1996 to December 30, 2005.

The research on testing the mean–variance relationship of a market portfolio

is controversial. Campbell (1987), and Glosten et al. (1993) document that con-

ditional volatility and risk premium are negatively related, contrary to eco-

nomic theory, while Turner et al. (1989), and Harrison and Zhang (1999) find a

positive relation between them. Using a VAR technique, Brandt and Kang

(2004) document that the conditional correlation between mean and volatility

is negative and the unconditional correlation is positive. Recently, Guo and

Whitelaw (2006) have found that market returns are positively related to

implied volatilities. Banerjee et al. (2007) also document that implied volatility

of the market has predictive power for future return on portfolios, even control-

ling with Fama and French risk factors. Consistent with the existing literature,

we find that the standard theoretical mean–variance relationship is supported

by regressions on risk-neutral variance based on monthly and quarterly returns,

but not supported by those on physical variance.

Our theory on the relation between market excess returns and the third

cumulant is strongly supported by empirical evidence. Combining the third

central moments as another risk factor with variance, we find that all of the

inter-temporal and contemporaneous relations between market excess returns

and the third cumulant are significantly negative for quarterly, semi-annual

and annual return regressions. Even combined with other predicting variables,

such as P/D ratio, the default spread and the consumption-wealth ratio (CAY),

the coefficients of the ex-post, risk-neutral and contemporaneous third cumu-

lant of market returns are negatively significant. Our research shows that the

third cumulant should be included as a measure of risk in addition to variance

when investigating the risk–return relation.

Guo et al. (2015) focus on the econometrical estimation of realized jumps and

study their relationship with conditional equity premium. Their simulation-

based theoretical result is built in a discrete time setting with Epstein-Zin prefer-

ences. The source of frictions that yields the results is the difference between up-

and downside jumps, which is consistent with the third cumulant studied in this

© 2018 International Review of Finance Ltd. 2018 607

MER, Variance and the Third Cumulant