SKEWNESS AND COSKEWNESS IN BOND RETURNS

• Published date: 01 June 2016
• DOI: http://doi.org/10.1111/jfir.12093
• Date: 01 June 2016
• Author: I‐Hsuan Ethan Chiang

SKEWNESS AND COSKEWNESS IN BOND RETURNS

I-Hsuan Ethan Chiang

University of North Carolina at Charlotte

Abstract

Bond skewness and coskewness (i.e., bond return comovement with market volatility)

are both time varying, with cross-sectional variation driven by maturity and credit rating.

Other things being equal, longer maturity bonds have lower skewness, and lower

coskewness with respect to the bond market index; lower quality bonds have lower

skewness, and higher coskewness with respect to the bond market index. Three-moment

bond alphas (which account for coskewness effects) are time varying and predictable by

market default spread. They are significantly different from, and often are closer to zero

than, two-moment alphas (which ignore coskewness effects).

JEL Classification: G12

I. Introduction

In this article, I explore properties of the third moments—skewness and coskewness (an

asset’s return comovement with market volatility)—in discrete holding-period bond

returns. I ask three empirical questions: Do bonds have significant skewness or

coskewness? What drives cross-sectional and time-series variations in bond skewness

and coskewness? How are bond alphas, after beta and coskewness risks are controlled,

related to bond attributes and macroeconomic variables?

Using panel regressions to separately analyze term premium (via a Treasury

bond data set) and default premium (via a corporate bond data set), I find that U.S. bond

returns display time-varying skewness and coskewness, whose cross-sectional variation

is driven by maturity and credit rating. Other bond attributes (including convexity and

current yield) and macroeconomic variables (e.g., short rate, term spread, default spread,

and interest rate volatility) also explain the time series and the cross-section of third

I am greatly indebted to my advisors: Wayne Ferson, Pierluigi Balduzzi, David Chapman, Edith Hotchkiss,

and Alan Marcus for their encouragement and helpful discussions. I am grateful to Harry Turtle (the associate

editor) and Scott Cederburg (the referee) for their numerous constructive suggestions and comments. This paper

has benefited from useful comments of Sirio Aramonte (discussant), Evangelos Aspiotis (discussant), Te-Feng

Chen (discussant), Ying Duan, Philip Dybvig, Roger Edelen, Richard Evans, Clifford Holderness, Kris Jacobs

(discussant), Edward Kane, Min Kim, Darren Kisgen, Chung-Ming Kuan, Justin Kuan, Xinxin Li, Wilson Liu

(discussant), David McLean, Fabio Moneta, Jeffrey Pontiff, Karl Snow (discussant), Philip Strahan, Hassan

Tehranian, Dengli Wang (discussant), and seminar participants at Academia Sinica, Boston College, University

of Connecticut, DePaul University, HKU, HKUST, University of Iowa, Southern Methodist University,

SUNY–Albany, SUNY–Buffalo, University of North Carolina at Charlotte, University of Waterloo, the Seventh

Trans-Atlantic Doctoral Conference at London Business School, and the annual meetings of the 2007 Southern

Finance Association, 2008 Washington Area Finance Association, 2008 Northern Finance Association, 2008

Financial Management Association, and 2012 Midwest Finance Association. I also thank Kenneth French for

providing access to his online data library. All remaining errors are solely mine.

The Journal of Financial Research Vol. XXXIX, No. 2 Pages 145–178 Summer 2016

145

© 2016 The Southern Finance Association and the Southwestern Finance Association

RAWLS COLLEGE OF BUSINESS, TEXAS TECH UNIVERSITY

PUBLISHED FOR THE SOUTHERN AND SOUTHWESTERN

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moments of bonds. When implementing a conditional three-moment two-index model on

the data sets, I find that three-moment bond alphas (which adjust for coskewness risk) are

time varying and predictable using the market default spread. They are significantly

different from, and often are closer to zero than, two-moment alphas (which ignore

coskewness effects).

This article makes the following contributions. First, from the portfolio

management and performance evaluation perspectives, I focus on the estimation of

skewness and coskewness in bond returns, whereas extant empirical research mainly

explores higher moments of stock returns. Second, both skewness and coskewness are

determined by three categories of predictive variables jointly: static bond attributes,

time-varying bond attributes, and time-varying macroeconomic variables. Very few

existing studies explore variations in the third moments, and those that do focus on time-

series aspects. Third, I provide a simple and unifying approach for portfolio managers

whose asset selectivity and market-timing ability are subject to evaluation. As the three

categories of predictive variables are observable in real time and the estimation

procedure is regression based, portfolio managers will find the alpha (representing asset

selectivity) and coskewness (resembling market-timing ability) estimators useful.

Fourth, I identify two bond attributes—maturity and credit rating—that drive the cross-

sectional variation in bond skewness and coskewness. These attributes enter various risk

measures in different ways.

Classical mean-variance portfolio analysis confines its focus to the first two

moments of returns, assuming that the distributions of asset returns are fully

characterized by the first two moments (as in a normal distribution) or assuming that

investors do not care about higher moments. The resulting equilibrium asset pricing

model is Sharpe’s (1964) and Lintner’s (1965) “two-moment”capital asset pricing

model (CAPM). However, returns are asymmetric empirically, and intuitively an

investor prefers a positively skewed payoff because it implies substantial chances of

large payoffs. Arditti (1967), Arrow (1971), Kraus and Litzenberger (1976), and Scott

and Horvath (1980) show that risk-averse investors with nonincreasing absolute risk

aversion (ARA) have a positive preference for portfolio skewness, and Kraus and

Litzenberger (1976) and Simkowitz and Beedles (1978) discuss portfolios featuring such

a preference.

1

Arditti (1967) relates stock returns to total skewness, and Barberis and

Huang (2008) provide theoretical justification using prospect theory. In the same vein,

Kraus and Litzenberger (1976, 1983) take a rational perspective and advocate a three-

moment CAPM, in which both covariance and coskewness drive cross-sectional

variation in expected asset returns. Kraus and Litzenberger (1976), Friend and

Westerfield (1980), Barone Adesi (1985), Sears and Wei (1988), Lim (1989), and Barone

Adesi, Gagliardini, and Urga (2004) find mixed empirical evidence for the three-moment

model using equity market data. Recent literature introduces conditioning information

1

A risk-averse investor’s preference for positive skewness can be formally motivated by a cubic

approximation of an expected utility function with nonincreasing ARA (see Arditti 1967; Arrow 1971). Although

Levy (1969) points out that the domain for wealth is restricted when a cubic approximation is imposed, Arditti

(1969) shows that nonincreasing ARA applies to general forms of utility functions and thus is not subject to Levy’s

critique.

146 The Journal of Financial Research

in this context. Harvey and Siddique (1999, 2000), Smith (2007), and Chang,

Christoffersen, and Jacobs (2013) all find that conditional coskewness matters in equity

returns. Patton (2004) finds conditional skewness matters in forming optimal stock

portfolios.

2

A growing literature (e.g., Fang and Lai 1997; Dittmar 2002; Ang, Chen, and

Xing 2006; Chung, Johnson, and Schill 2006) extends to even higher moments than

skewness, or to other comoments (Boguth 2010). However, the empirical literature on

skewness and coskewness has largely ignored bond returns.

In this article, I study the cross-sectional and time-series properties of skewness

and coskewness in discrete holding-period bond returns.

3

The cash-flow structure of

bonds motivates asymmetry in bond returns. Unlike stock prices, bond prices face an

upper bound: the maximum price of a bond is the sum of future promised cash flows, and

thus bond returns are skewed by nature. Furthermore, convexity, regime switching in

state variables, default risk, or underlying callability or putability may also create

asymmetry in bond returns. Therefore, it is important to examine whether discrete bond

returns have nontrivial higher moments and how they affect investment decisions.

The crucial empirical issue is what portfolio management content the third

moments in bond returns carry. Suppose an investor has a preference for positive

skewness. When he holds only one asset, obviously he cares about the asset’s asymmetry

measure on a stand-alone basis—the total skewness of that asset. But, when he holds

portfolios, he cares about an asset’s asymmetry measure on a portfolio basis—the

“coskewness”of that asset. Statistically, coskewness measures an asset’s marginal

contribution to portfolio skewness; thus, positive coskewness in an asset is desirable

because it makes the portfolio more positively skewed.

4

As coskewness is typically

formulated as the comovement of an asset’s return with portfolio volatility, an economic

interpretation of an asset’s coskewness is its ability to hedge against shocks to portfolio

volatility. Consider that when the stock market becomes more volatile and distressed,

investors tend to sell their equity holdings and buy Treasury bonds. This typical “flight to

quality”behavior is reflected in Treasury bonds’positive coskewness with respect to the

stock market index. However, riskier bonds do not have Treasury bonds’distinct

insurance feature and behave like most stocks when the stock market is distressed,

exhibiting negative coskewness with respect to the stock market index. My empirical

evidence shows that Treasury bonds have positive coskewness with respect to the stock

market index, whereas corporate bonds have negative coskewness with respect to the

stock market index. With the possible sign differences, the third-moment asset pricing

and portfolio choice implications in fixed-income securities have the potential to be

richer than those in stocks, as empirically most stocks have negative skewness and

negative coskewness with respect to the stock market index.

2

Skewness in other financial instruments has been studied in the literature, including foreign exchange (Peir

o

1999), futures (Eastman and Lucey 2008), and lottery tickets (Bhattacharyya and Garrett 2008).

3

The amount of attention this article pays to third moments has a theoretical justification: Kraus and

Litzenberger (1983) show that individual preferences for positive skewness can be aggregated so that the

representative investor has a similar preference; however, preference for kurtosis cannot be aggregated.

4

The coskewness formulation in this article is motivated by Barone Adesi, Gagliardini, and Urga (2004); see

Section II for a discussion on alternative coskewness measures.

Skewness and Coskewness 147