The term structure of systematic and idiosyncratic risk

• Published date: 01 April 2019
• Date: 01 April 2019
• DOI: http://doi.org/10.1002/fut.21985
• Author: Marcel Prokopczuk,Chardin Wese Simen,Fabian Hollstein

Received: 26 June 2018

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Revised: 5 November 2018

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Accepted: 5 November 2018

DOI: 10.1002/fut.21985

RESEARCH ARTICLE

The term structure of systematic and idiosyncratic risk

Fabian Hollstein

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Marcel Prokopczuk

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Chardin Wese Simen

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1

School of Economics and Management,

Leibniz University of Hannover,

Hannover, Germany

2

ICMA Centre, Henley Business School,

University of Reading, Reading, UK

Correspondence

Fabian Hollstein, School of Economics

and Management, Leibniz University

of Hannover, Koenigsworther Platz 1,

30167 Hannover, Germany.

Email: hollstein@fmt.uni-hannover.de

Abstract

We study the term structure of variance (total risk), systematic, and

idiosyncratic risk. Consistent with the expectations hypothesis, we find that,

for the entire market, the slope of the term structure of variance is mainly

informative about the path of future variance. Thus, there is little indication of a

time‐varying term premium. Turning the focus to individual stocks, we cannot

reject the expectations hypothesis for systematic variance, but we strongly reject

it for idiosyncratic variance. Our results are robust to jumps and potential

statistical biases.

KEYWORDS

expectations hypothesis, idiosyncratic risk, implied correlation, model‐free option‐implied variance,

options, systematic risk, term structure

JEL CLASSIFICATION

G11, G12, G17

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INTRODUCTION

Recent studies document the predictive power of the term structure of risk‐related variables. For example, Koijen,

Moskowitz, Pedersen, and Vrugt (2017) and Vasquez (2016) show that the spread related to option‐implied volatilit ies of

different maturities predicts future option returns. At the same time, security exchanges are increasingly disseminating

information about the term structure of option‐implied volatility. For instance, the Chicago Board Options Exchange

(CBOE) now publishesinformation not only about the popular 1‐month volatility index(VIX) but also about the 3‐month

VIX and the option‐implied correlationof various maturities. The academic and professional interest in theterm structure

of risk raises several questions: What does the term structure of variance tell us about future developments? Are there

differences in the term structures of market and stock option prices? Do the term structures of systematic and

idiosyncratic risk behave differently? In particular, does the term structure encode information about the future path of

the variable of interest or does it instead reflect variations in a possible term premium?

Understanding whether there is a time‐varying term premium or not is important in many situations. For asset

managers, knowledge about the term premium is essential for strategies that take positions in the long‐term variance

and roll over short positions in the short‐term variance. If the term premium varies over time in a predictable fashion,

investors could exploit this. Contrarily, it is important to know whether it is cheaper to hedge against variance increases

by buying a long‐term variance swap contract or rolling over short‐term variance swaps. Understanding the differences

in the term structures of systematic and idiosyncratic variance can help asset managers decide how to hedge individual

stock variance. Answers to the above questions are also important for risk managers who need an estimate of future

stock or market variance. To the extent that there is a time‐varying premium in the term structure of variance, the

implied forward variance will be a noisy proxy for the expected future implied variance. Thus, a risk manager would

need to purge the implied forward variance from the time‐varying term premium.

J Futures Markets. 2019;39:435–460. wileyonlinelibrary.com/journal/fut © 2018 Wiley Periodicals, Inc.

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This paper analyzes the term structures of total, systematic, and idiosyncratic variance. We formally derive testable

predictions of the expectations hypothesis. The expectations hypothesis states that the spread between the current long‐

term estimate of these risk measures and the current short‐term estimate of risk is mainly informative about future

developments in short‐term risk. Our derivation points to a relationship between the term structure of equity index

options prices and that of the option prices on the underlying equities.

We use a large options data set to empirically test the expectations hypothesis. Our results suggest that the expectations

hypothesis generally cannot be rejected for the term structures of the option‐implied variance of the market as well as for

systematic stock variance. Thus, there is little indication of a time‐varying term premium associated with systematic risk. As

a consequence, the slope of each term structure is informative about investors’expectations of future short‐term (systematic)

variance. As opposed to that, we typically detect a negative term premium in the term structure of option‐implied

idiosyncratic variance. Additionally, we also cannot reject the expectations hypothesis in the term structure of option‐implied

correlation. These results are robust to the presence of jumps in the underlying price process, as well as potential statistical

biases in our tests. We thus conclude that overall the expectations hypothesis provides a good description of the term

structure of market option prices, but not to the extent that it accounts for idiosyncratic variance.

Our work extends the literature on the term structure of variance and volatility.

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Campa and Chang (1995) and

DellaCorte, Sarno, and Tsiakas (2011) study the term structure of foreign exchange variance and volatility, respectively.

Johnson (2016) and Mixon (2007) extend these studies to the term structure of equity index implied variance. Our work is

related to a study by Heynen, Kemna, and Vorst (1994), who focus on the term structure of the index and individual equity

option‐implied volatility. Taken together, the above studies reach conflicting conclusions. These range from a rejection of (an

implication of) the expectations hypothesis (Della Corte et al., 2011; Johnson, 2016) to mixed results (Mixon, 2007) and not

being able to reject the expectations hypothesis for the term structure of variance (Campa & Chang, 1995; Heynen et al.,

1994). Our study is different in several important aspects. First, unlike Campa and Chang (1995), Heynen et al. (1994), and

Mixon (2007), we study the model‐free option‐implied variance, which makes our results immune to potential

misspecification of a specific option pricing model used. That is, we avoid performing a joint test of correct option pricing

model specification and the expectations hypothesis. Second, we extend the work of Johnson (2016) and Mixon (2007), who

focus on the market index only. Because our derivation points to the link between the option‐implied variances of the index

and the individual equities, we study the term structure of the option‐implied variance of individual equities. Third, and

most importantly, motivated by partly differential results on the market and individual stocks, we decompose the term

structure of option‐implied variance into systematic and idiosyncratic variance.

Our paper is also related to Clara (2018), who uses the slope of the term structure of beta to predict the cross‐section

of excess stock returns. Our focus is very different. Our main interest is to understand the term structure of option‐

implied variance. We use forward‐looking betas to decompose the option‐implied variance of each stock into systematic

and idiosyncratic parts and test the expectations hypothesis for both components.

Feunou, Fontaine, Taamouti, and Tédongap (2014) show that the principal components from the option‐implied variance

termstructurehavepredictivepowerforbondandequityreturns. Our results indicate that factors capturing the slope of the

term structure in the market level are related to expectations about the future variance and may help rationalize these findings.

We also add to the literature on the term structure of option‐implied correlation. Faria and Kosowski (2016) study

the term structure of option‐implied correlation. However, they make no attempt to test the expectations hypothesis.

Our study carries implications for asset pricing and risk management in general, and the design of trading strategies

in particular. From an asset pricing standpoint, our findings imply, though do not directly test, that the cross‐sectional

strategies of Koijen et al. (2017) and Vasquez (2016) mainly sort on the expected path of the future short‐term option‐

implied volatility, rather than a related term premium. Thus, our results suggest that these studies capture a risk

premium associated with cross‐sectional differences in expectations about future short‐term risk. Furthermore, our

finding that the implied systematic variance term structure mainly reflects expectations about future short‐term

systematic variance can be used for risk management purposes. Finally, the results presented in this study reveal that a

trading strategy that buys the long‐term option‐implied variance and sells the future short‐term option‐implied variance

is not profitable on average at the market level but yields substantial negative returns when applied to individual stocks.

The remainder of this paper is organized as follows. In Section 2, we introduce the data and the methodology for the

estimation of the option‐implied quantities. In Section 3, we derive the theoretical relationship between option‐implied

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Ait‐Sahalia, Karaman, and Mancini (2018) extend the work of Egloff, Leippold, and Wu (2010), modeling the term structure of variance swaps in a continuous time setup. Further papers that model

the term structure of variance swap rates include Amengual and Xiu (2018), Andries, Eisenbach, Schmalz, and Wang (2015), Dew‐Becker, Giglio, Le, and Rodriguez (2017), and Filipović, Gourier, and

Mancini (2016).

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