Credit Default Swap Spreads and Variance Risk Premia * Hao Wang † Hao Zhou ‡ Yi Zhou § First Draft: August 2009 This Version: November 2009 Abstract We find that variance risk premium, defined as the spread between the option- implied and expected variances, has a prominent explaining power for the credit default swap spreads at individual firm level. Such a predictability cannot be crowded out by those of the market and firm level credit risk factors identified in previous research. We demonstrate that the strong predictability of the implied variance for credit spreads is mostly explained by both variance premium and expected variance. Our finding sug- gests that the variance risk premium is a relatively clean measure of a firm’s exposure to macroeconomic uncertainty or systematic variance risk, while the option-implied and expected variances may be more contaminated by the idiosyncratic variance risk. Such a result is consistent with a setting that the firm’s asset is exposed to the priced time-varying systematic variance risk. JEL Classification: G12, G13, G14. Keywords: Variance Risk Premia, Credit Default Swap Spreads, Option-implied Vari- ance, Expected Variance, Realized Variance. * We would like to thank Darrell Duffie, Bing Han, William Megginson, George Tauchen, and Hong Yan for helpful discussions. The authors acknowledge the generous financial support from Global Association of Risk Professionals (GARP) and Center for Hedge Fund Research (CHFR) at Imperial College London. † Hao Wang, Assistant Professor, Tsinghua University, School of Economics and Management, 318 Weilun Building, Beijing 100084, China, E-mail: [email protected], Tel: 86 10-62797482. ‡ Hao Zhou, Senior Economist, Federal Reserve Board, Risk Analysis Section, Washington, DC 20551, USA, E-mail: [email protected], Tel: 1 202-452-3360. § Yi Zhou, Assistant Professor, The University of Oklahoma, Michael F. Price College of Business, Finance Division, 307 West Brooks Street, Adams Hall 250, Norman, OK 73069, USA, E-mail: [email protected], Tel: 1 405-325-1135.
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Credit Default Swap Spreads and Variance Risk Premia∗
Hao Wang† Hao Zhou‡ Yi Zhou§
First Draft: August 2009This Version: November 2009
Abstract
We find that variance risk premium, defined as the spread between the option-implied and expected variances, has a prominent explaining power for the credit defaultswap spreads at individual firm level. Such a predictability cannot be crowded out bythose of the market and firm level credit risk factors identified in previous research. Wedemonstrate that the strong predictability of the implied variance for credit spreads ismostly explained by both variance premium and expected variance. Our finding sug-gests that the variance risk premium is a relatively clean measure of a firm’s exposureto macroeconomic uncertainty or systematic variance risk, while the option-impliedand expected variances may be more contaminated by the idiosyncratic variance risk.Such a result is consistent with a setting that the firm’s asset is exposed to the pricedtime-varying systematic variance risk.
∗We would like to thank Darrell Duffie, Bing Han, William Megginson, George Tauchen, and Hong Yanfor helpful discussions. The authors acknowledge the generous financial support from Global Association ofRisk Professionals (GARP) and Center for Hedge Fund Research (CHFR) at Imperial College London.
†Hao Wang, Assistant Professor, Tsinghua University, School of Economics and Management, 318 WeilunBuilding, Beijing 100084, China, E-mail: [email protected], Tel: 86 10-62797482.
‡Hao Zhou, Senior Economist, Federal Reserve Board, Risk Analysis Section, Washington, DC 20551,USA, E-mail: [email protected], Tel: 1 202-452-3360.
§Yi Zhou, Assistant Professor, The University of Oklahoma, Michael F. Price College of Business, FinanceDivision, 307 West Brooks Street, Adams Hall 250, Norman, OK 73069, USA, E-mail: [email protected],Tel: 1 405-325-1135.
Credit Default Swap Spreads and Variance Risk Premia
Abstract
We find that variance risk premium, defined as the spread between the option-implied and
expected variances, has a prominent explaining power for the credit default swap spreads at
individual firm level. Such a predictability cannot be crowded out by those of the market and
firm level credit risk factors identified in previous research. We demonstrate that the strong
predictability of the implied variance for credit spreads is mostly explained by both variance
premium and expected variance. Our finding suggests that the variance risk premium is a
relatively clean measure of a firm’s exposure to macroeconomic uncertainty or systematic
variance risk, while the option-implied and expected variances may be more contaminated
by the idiosyncratic variance risk. Such a result is consistent with a setting that the firm’s
asset is exposed to the priced time-varying systematic variance risk.
It has long been recognized in literature that a critical component of systematic risk could
be absent in the existing credit risk models (Jones, Mason, and Rosenfeld, 1984; Elton,
Gruber, Agrawal, and Mann, 2001; Collin-Dufresne, Goldstein, and Martin, 2001; Huang
and Huang, 2003). As evidenced by the recent credit crisis, a possible missing economic
uncertainty or systematic variance risk factor plays a prominent role in affecting credit risk-
related security prices. To investigate this important issue in an effort to shed light on
improving the credit risk modeling, we construct individual firm’s variance risk premium
that captures the firm’s exposure to macroeconomic uncertainty or systematic variance risk,
in the context of Bollerslev, Tauchen, and Zhou (2009b) and Drechsler and Yaron (2008),
and carry out an extensive investigation on the cross-sectional relationships between credit
spreads and variance risk premia.1
Previous research presents empirical evidence that implied variance is always informa-
tionally more efficient than realized variance in predicting credit spreads (see, Cao, Yu, and
Zhong, 2006; Berndt, Lookman, and Obreja, 2006; Carr and Wu, 2008a; Buraschi, Trojani,
and Vedolin, 2009, among others). However, it remains unclear to what extent the pre-
dictability comes from better informational efficiency of option market, from risk premium
changes, or from expected variance changes. By decomposing option-implied variance, we
provide a clear economic interpretation as to where the predictability exactly comes from.
In particular, we investigate quantitatively to what extent the predicting power of implied
variance is due to risk premium or variance risk changes.
There is a great deal of debate as for whether the idiosyncratic volatility risk is priced
positively, negatively, or not priced at all in the cross-section of stock returns (see, Ang,
1The systematic risk implication is also consistent with recent research that recognizes the linkage amongmacro-economic conditions, equity risk premia and credit risk pricing (see, e.g., David, 2008; Bhamra, Kuhn,and Strebulaev, 2009; Chen, Collin-Dufresne, and Goldstein, 2009; Chen, 2009). However, such a equilibrium-structural approach that explains mostly the “representative” firms has limited explaining power for thecross-section of individual firm’s credit spreads with respect to economic uncertainty risk.
1
Hodrick, Xing, and Zhang, 2009; Cao and Han, 2009; Fu, 2009; Goyal and Saretto, 2009;
Huang, Liu, Rhee, and Zhang, 2009, among others). It is possible that these conflicting
findings are due to the model specific ways of separating systematic versus idiosyncratic
variance risks and caused by potential model misspecifications. For explaining credit spreads,
conceptually the variance risk premium variable will isolate only the systematic variance risk
which must be priced in all risky assets. This is because, by construction, the risk neutral
and objective expectations of firms’ idiosyncratic variance risk should cancel out with each
other.
We demonstrate empirically that VRP has a significant predicting power for credit de-
fault swap (CDS) spreads at the individual firm level. Such a predictability complements
that of the macro-economic and firm specific credit risk determinants suggested by existing
theoretical models and previous empirical evidence. In addition, VRP, as a firm-level varia-
in capturing the market systematic risk to predict CDS spreads. Moreover, the predicting
power of VRP for credit spreads increases as the credit quality of the CDS contracts deteri-
orates. This is an important finding in that the VRP may capture the systematic variance
risk exposures of the underlying firm asset dynamics and therefore explains a large chunk
of systematic credit spread variations. The VRP constructed with implied variance from
at-the-money put option has a stronger predicting power than alternatives from call options
or different moneyness.
In an effort to answer the important question as to where the predicting power of option-
implied variance on credit spreads comes from, we decompose implied variance into VRP and
expected variance. Although all three explanatory variables have almost the same statistical
significance in predicting the positive risk premia in CDS spreads, we find that both VRP
and expected variance can substitute most of the explaining power of the implied variance,
especially for the CDS spreads of 1-3 year maturities. Nevertheless, VRP and expected
variance are two equally important components in terms of predicting credit spreads, sug-
2
gesting that there may be a two-factor structure of the implied variance dynamics behind
the superior explaining power for CDS spreads. We also find that the predicting power of
VRP for credit spreads at the firm level decreases gradually as time passes by, which is
qualitatively similar to the aggregate level evidence that the predictive power of VRP for
credit spreads peaks at the short end (Zhou, 2009). Such a result is also consistent with
the findings in literature that the option implied risk premia are successful in explaining the
large credit spread variations (Cremers, Driessen, and Maenhout, 2008a; Coval, Jurek, and
Stafford, 2009).
Finally, to understand why VRP is an ideal measure for firms’ exposures to systematic
variance or economic uncertainty risk, we argue that the risk-neutral expectation and physical
expectation of the total variance only differ for priced systematic component, but remain the
same for non-priced idiosyncratic component. As supported in our empirical analysis, the
first principle component of VRP across all firms explains 80% of the total variation, while
that of implied variance only explains 37%. Therefore the informational efficiency of implied
variance may be largely due to credit and option markets comovement, while the economic
interpretation of implied variance in terms of explaining credit spreads resides largely on
VRP, which captures the systematic risk premium changes. Campbell and Taksler (2003),
Cremers, Driessen, Maenhout, and Weinbaum (2008b), Ericsson, Jacobs, and Oviedo (2004),
and Zhang, Zhou, and Zhu (2009) also examine the role of firm-level volatility risk in the
determination of bond and CDS spreads. However, we emphasize the nature of VRP as
isolating the firm’s exposure to the economic uncertainty or systematic variance risk factor.
The rest of the paper will be organized as the following: Section 2 introduces the VRP
measure and our empirical methodology; followed by a description of data sources and sum-
mary statistics in Section 3; then Section 4 presents empirical findings of VRP with respect
to predicting CDS spreads; and Section 5 concludes.
3
2 Variance Risk Premia and Empirical Methodology
In this section, we introduce the construction of the variance risk premium (VRP) measure for
individual firms, following the recent literature in defining the market variance risk premium
as a difference between the option-implied return variance and forecasted realized variance.
Then we outline our empirical strategy for explaining the credit spreads of individual firms,
using such a firm specific variance risk premium variable, together with key market and firm
control variables—noticeably the short rate and leverage ratio.
2.1 Constructing the VRP Measure for Individual Firms
We start our construction of the VRP measure for individual firms with estimating the
option-implied variance which, denoted by IVt, represents the market’s risk-neutral expec-
tation of the return variation between time t and t + 1, conditional on time t information,
which can be expressed as
IVt = EQt [Variance(t, t + 1)]. (1)
It has been established in literature that for highly liquid market indices, like the S&P
500, a model-free formula for computing the option-implied variance can be implemented
easily, even though only a limited number of strikes being available (Carr and Madan, 1998;
Britten-Jones and Neuberger, 2000; Jiang and Tian, 2005; Bollerslev, Gibson, and Zhou,
2009a). However, for individual firms, the liquidity of options of different moneyness and
maturities is still a big concern. Therefore we follow the existing literature to focus on the
implied variance from out-of-the-money put options (see, e.g., Cao, Yu, and Zhong, 2006,
and the references therein).
Our implied variances are obtained from OptionMetrics, which employs the Cox-Ross-
Rubinstein (CRR) binomial tree model and a curve-fitting approach to build the volatility
surface across strikes and maturities for individual entities (Cox, Ross, and Rubinstein,
1979).2 With the volatility surface, we are able to compute a panel of option-implied vari-
2We may alternatively construct our own measures of implied volatilities using option portfolios. However,
4
ances based on out-of-the-money, at-the-money and in-the-money put and call options.
In order to define the realized variance that we use in estimating the expected variance,
let pt denote the logarithmic price of equity. The realized variance over the discrete t− 1 to
t time interval may be measured as:
RVt ≡n∑
j=1
[pt−1+ j
n
− pt−1+ j−1
n(∆)
]2−→ Variance(t − 1, t), (2)
where the convergence relies on n → ∞; i.e., an increasing number of within period price
observations. As demonstrated in the literature (see, e.g., Andersen, Bollerslev, Diebold, and
Ebens, 2001a; Barndorff-Nielsen and Shephard, 2002), this “model-free” realized variance
measure based on high-frequency intraday data can provide much more accurate ex-post
observations of the unobserved return variance than those based on daily returns.
For a monthly horizon and monthly data frequency, where IVt is the end-of-month risk-
neutral expected variance for the next month, and RVt is the realized variance of the current
month, we adopt a linear forecast of the objective or statistical expectation of the return
variance as RVt+1 = α + βIVt + γRVt + ǫt+1, and the expected variance is simply the time t
forecast of realized variance from t to t + 1 based estimated projection coefficients,
EVt ≡ R̂V t+1 = α̂ + β̂IVt + γ̂RVt. (3)
Implied variance from options market is a better or more efficient forecast for expected
variance than the historical variance based on low-frequency daily return data (Jiang and
Tian, 2005). While realized variance based on high-frequency data also has remarkable
forecasting power for future realized variance (Andersen, Bollerslev, Diebold, and Labys,
2001b). Putting these two together, a joint forecast model with one lag of implied variance
and one lag of realized variances seems to do a decent job in forecasting expected variances
(Drechsler and Yaron, 2008).
given the close similarity between the methodologies, self-constructed implied variance could of no significantdifference compared to that computed based on the volatility surface provided by OptionMetrics.
5
The variance risk premium, or VRP, underlying our key empirical findings is defined as
the difference between the ex-ante risk-neutral expectation and the objective expectation of
future return variation over the [t, t + 1] time interval,
V RPt ≡ IVt − EVt. (4)
Such a construct at the market level has been shown to possess remarkable capability in
forecasting the short-run movements in credit spread indices, with short term interest rate as
a control for fundamental economic condition (Zhou, 2009). Here we are going to investigate
in detail how VRP of individual firms can help us to understand the firms’ credit risk premia.
2.2 Empirical Implementation Strategy
We examine the relationship between credit spreads and variance risk premium (VRP) in the
presence of market and firm level credit risk determinants suggested by theory and empirical
evidence. We will focus on monthly data to avoid picking up the market microstructure
induced high frequency comovements. For spreads and implied variance, they are just the
last available end-of-month daily observations. Because missing dates and stale quotes sig-
nify that daily or even weekly data quality is not reliable, and if we just ignore the daily
missing values, we will artificially introduce some serial dependent measurement errors in the
dependent variable—CDS spread. Monthly data will give us more conservative but reliable
results and is typically of the shortest horizon, compared to quarterly or annual data, for
picking up the low frequency risk premia movements.
As suggested by the structural form credit risk models (Merton, 1974), leverage is the
most important credit risk determinant—all else being equal, a firm with higher leverage has
a higher likelihood of default. The market leverage ratio, denoted by LEVi,t, is computed as
the book value of debt over the sum of the book value of debt and market value of equity.
Moreover, structural models predict that risk-free interest rates negatively influence the
credit spread (Longstaff and Schwartz, 1995). Under the risk-neutral measure, high interest
rates lead the firm’s underlying asset value to grow at a higher rate, reducing the likelihood
6
of financial distress. Therefore, CDS spreads should be determined by the leverage ratio of
the underlying firm and the risk-free spot rate. We define the risk-free rate variable to be
the one-year swap yield, denoted by rt.
Empirical research also shows that in practice, CDS spreads contain compensation for
non-default risks as well as risk premia which may be difficult to identify without the ag-
gregate macro variables. Henceforth, we will not limit our analysis to the traditional the-
oretically motivated regressors but augment our set of variables by the following market
variables: (1) the S&P 500 return, denoted by S&Pt to proxy for the overall state of the
economy; (2) the option-implied variance based on the S&P 500 index options denoted by
V IX2t ; (3) Moody’s default premium slope, denoted by DPSt, is computed as Baa yield
spread minus Aaa yield spread to capture the default risk premium in the corporate bond
market; and (4) the difference of five-year swap rate and five-year Treasury rate, denoted by
STSt, as a proxy for fixed income market illiquidity, which is expected to increase with CDS
spreads. As indicated by the recent empirical research, e.g., in Collin-Dufresne, Goldstein,
and Martin (2001) and Longstaff, Pan, Pedersen, and Singleton (2007), VIX turns out to be
a lead explanatory variable for both corporate and sovereign credit spreads, similar to short
rate and leverage ratio. To be comparable with the variance risk premium definition and
consistent with the popular variance swap contract, we use VIX-squared on monthly basis
as an additive systemic risk measurement.
For firm characteristic variables, besides market leverage, we include the following con-
trols: (1) the natural logarithm of sales, denoted by SALEi,t. As a proxy for firm size,
SALE should influence CDS spread analogously—a larger firm will attract more investor
attention, hence have more liquid credit default swap spreads with lower spreads, all else
being equal; (2) asset turnover, denoted by ATOi,t, is computed as sales divided by total
assets; (3) return on assets, denoted by ROAi,t, is computed as earnings divided by total
assets; (4) price-earnings ratio denoted by PEi,t; and (5) market-to-book ratio, denoted by
MBi,t. Firm asset turnover, market-book ratio and return on assets are all expected to be
7
negatively related to CDS spreads, in the notion that firms of high profitability and future
growth tend to have lower credit risk. Price-earnings ratio may have two opposite effects on
CDS spreads: on the one hand, high price-earnings ratio implies high future asset growth
reducing the likelihood of financial distress and credit risk. On the other hand, high growth
firms tend to have high return volatilities that increase credit risk.
Given the nature of our panel data, we adopt the robust standard error approach of
Petersen (2009) to account for both firm and time effects in large panel data sets. Therefore,
the above discussions suggest the following regression equation
Uncertainty,” Working Paper, Federal Reserve Board .
Zhu, Haibin (2006), “An Empirical Comparison of Credit Spreads Between the Bond Market
and the Credit Default Swap Market,” Bank of International Settlements, Working Paper .
24
Table 1: Descriptive statistics - CDS spreads, VRP, IV, EV and RV
This table presents the summary statistics� average across the 540 �rms� of the �ve-year CDS spreads and our benchmark VRP measure (Panel A),together with implied, realized, and expected variances (Panel B). The VRP is computed as the spread between 30-day out-of-the-money put option-impliedvariance and high-frequency expected variance. The average Moody�s and S&P ratings of the CDS reference entities range between AAA and CCC. Thenumbers of �rms in each rating category are reported in the second column in Panel (A). AR(1) denotes autocorrelation with one lag.
Panel A: The means of the statistics of CDS spreads and VRP across individual �rms
CDS Spread VRPRating Number of Firms Mean SD Skewness Kurtosis AR(1) Mean SD Skewness Kurtosis AR(1)
Table 2: Summary statistics - the market and firm characteristic variables
This table reports the descriptive statistics of the market and �rm level control variables. For �rm characteristics,we report the averages of the statistics across 540 �rms. The averages of one-year swap rate is 3:37%. The averagedefault premium slope (Moody�s Baa rateminus Aaa rate) is 0:99% and the average liquidity measure (�ve-year swaprate minus constant maturity Treasury rate) is 0:54%. The average market leverage ratio is 40% with a standarddeviation of 6%. AR(1) denotes autocorrelation with one lag.
This table reports the regression results of �ve-year CDS spreads on the benchmark VRP computed with the30-day out-of-the-money put option-implied variance (monthly) minus 30-day high-frequency expected variance(monthly). Regression (1) is for the univariate regression; regression (2) shows the relationship between CDS spreadsand VRP in the presence of VIX (monthly squared in percentage); regression (3) further includes �rm leverage ratio;and regression (4) includes all other control variables. We adjust two-dimensional (�rm and time) clustered standarderrors in the regressions as in Petersen (2009).
Table 4: VRP versus implied variance and expected variance
This table compares the predictability of VRP on CDS spreads to that of implied and expected variances fordi¤erent CDS maturities. Panel A reports the univariate regression results for VRP, implied and expected variances.Panel B reports the regression results of CDS spreads on each pairs of VRP, implied and expected variances re-spectively. We adjust two-dimensional (�rm and time) clustered standard errors in the regressions as in Petersen(2009).
Table 5: Principal Component Analyses of CDS Spreads, VRP, IV and EV
This table reports the principal component analyses of CDS spreads, VRP, implied and expected variances. We select �rms with 48 monthly observationsstarting in January 2004. The sample contains 217 �rms. VRP is explained mostly by �rst three components (92.8% cumulatively), whereas IV and EV aredriven marginally by several components. Robustness checks with various samples show that sample selection does not change the results qualitatively.
This table reports the regression results of CDS spreads on VRPs computed with expected and realized variances.For each VRP measure, we report the results of both univariate regression and the regression with all control variablesincluded. The results demonstrate consistent and signi�cant predictability of VRP on credit spreads. We adjusttwo-dimensional (�rm and time) clustered standard errors in the regressions as in Petersen (2009).
Table 7: CDS spreads and VRPs of different implied variances
This table reports the regression results of CDS spreads on VRPs computed from di¤erent measures of im-plied variances. Besides the benchmark out-of-the-money (OTM) put option-implied variance, we use implied vari-ances computed from at-the-money (ATM) and in-the-money (ITM) put options, together with out-of-the-money(OTM), at-the-money (ATM) and in-the-money (ITM) call options. All VRP measures display consistently signif-icant predictability on CDS spreads in the presence of market and �rm level credit risk determinants. We adjusttwo-dimensional (�rm and time) clustered standard errors in the regressions as in Petersen (2009).
VRP constructed with implied variance fromPut option Call option
Table 8: CDS spreads and VRP, IV, EV and RV by CDS rating
This table reports the regression results of CDS spreads on VRP, implied and expected variances for three sub-samples: AAA-A, BBB, and BB-CCC. The �rst sub-sample contains CDS of AAA, AA and AA grades. The secondcontains CDS of BBB grade. The third sub-sample contains CDS of speculative grades ranging between BB and CCC.The three sub-samples contain 7; 666, 10; 055 and 5; 378 �rm-month observations respectively. Two-dimensional (�rmand time) clustered standard errors in the regressions are adjusted as in Petersen (2009). To be focused, we omitreporting the multivariate regression results for the control variables, given their similarity to those reported in Table6 and 7.
Rating Univariate Regression Regression with All Control VariablesVariable Group Coe¢ cient T -statistic Adjusted R2 Coe¢ cient T -statistic Adjusted R2
Table 9: CDS spreads and lagged VRPs (the inter-temporal effect)
This table reports the regression results of CDS spreads on lagged VRPs from zero month to twelve months. Panel A reports that, in univariate regressions,both the coe¢ cient values and t-statistics decrease from 1.91 and 8.19 to 1.14 and 4.16 respectively as the lag increases from zero to twelve months. The R2sdecrease from 0.24 to 0.09. Panel B reports the results of the regressions with VIX and �rm leverage ratio included. Same inter-temporal pattern is observed.We adjust two-dimensional (�rm and time) clustered standard errors in the regressions as in Petersen (2009).