Macroeconomic and Financial Determinants of the Volatility of. Corporate Bond Returns

Transcription

1 Macroeconomic and Financial Determinants of the Volatility of Corporate Bond Returns Belén Nieto University of Alicante, Spain Alfonso Novales University Complutense, Spain Gonzalo Rubio CEU Cardenal Herrera University, Spain This version: June 18, 2013 Abstract This paper analyzes the relationship between the volatility of corporate bond returns and standard financial and macroeconomic indicators reflecting the state of the economy. We employ the GARCH- MIDAS multiplicative two-component model of volatility that distinguishes the short-term dynamics from the long-run component of volatility. We show that recognizing the existence of a stochastic lowfrequency component captured by macroeconomic and financial indicators significantly improves the fit of the model to actual bond return data, relative to the constant long-run component embedded in a typical GARCH model. JEL classification: G12; C22; E44 Keywords: corporate bonds, volatility, low-frequency component, high-frequency component, macroeconomic indicators, financial indicators The authors acknowledge financial support from the Ministry of Science and Innovation through grant ECO (B. Nieto), and the Ministry of Economics and Competitiveness through grants ECO (G. Rubio), and ECO (A. Novales). Financial support from Generalitat Valenciana grant PrometeoII/2013/015 is also acknowledged. We thank seminar participants at University of Navarra for helpful comments on the article. We assume full responsibility for any remaining errors. 1

2 1. Introduction Volatility is the key variable in investment analysis, security pricing, and risk management. It is also the most important variable in option pricing, and it is heavily used in the regulation of financial institutions as the key input to estimate value-at-risk (VaR) measures. Hence, it is not surprising that volatility forecasting and the analysis of the determinants of volatility have become fundamental research issues in financial economics. Most of the empirical literature has focused so far on the behavior of stock market volatility. Contrary to previous research, this paper analyzes the macroeconomic and financial determinants of corporate bond return volatilities across six credit rating categories using the multiplicative two-component GARCH-MIDAS model of volatility recently proposed by Engle, Ghysels, and Sohn (2009). Although the typical persistence in stock market volatility is captured by the popular ARCH/GARCH-type models of Engle (1982) and Bollerslev (1986), the dynamics of volatility seems to be better characterized by the component model introduced by Engle and Lee (1999). Their proposal consists of two additive GARCH(1,1) components, one interpreted as a short-run or transitory component, and a second one identified as the long-run or trend component of volatility. 1 Recently, however, Engle and Rangel (2008) suggest a multiplicative component structure, the Spline-GARCH model, to accommodate non-stationarity features that are captured by the long run volatility component. Volatility is therefore a product of a slowly changing, low-frequency deterministic component picking up the non-stationary characteristic of the process, and a short-run/high frequency part described by a GARCH(1,1) process which mean-reverts to one. The deterministic component is supposed to be a function of macroeconomic variables, and hence volatility ends up being a combination of 1 Other relevant papers related to this approach are Chernov, Gallart, Ghysels, and Tauchen (2003), and Adrian and Rosenberg (2008). See Wang and Ghysels (2011) for a review from a statistical perspective. 2

3 macroeconomic effects and time series dynamics. Engle and Rangel (2008) apply this model to stock market volatilities across 50 countries and conclude that high volatility is explained by high inflation, slow output growth, high volatility of short-term interest rates, high volatility of production growth, and high inflation volatility. On top of this, econometric methods involving data sampled at different frequencies have been shown to be useful for forecasting volatility in equity assets as well as to explain the relationship between conditional variance and expected market returns, especially in comparison with the evidence available from the GARCH family. The mixed frequency approach to modeling and predicting volatility known as mixed data sampling (MIDAS hereafter) was introduced in a series of papers by Ghysels, Santa-Clara, and Valkanov (2003, 2005, 2006). The success of MIDAS lies in the additional statistical power that mixed data frequency regressions incorporate from using daily data in estimating conditional variances. In addition, MIDAS allows for a very flexible functional form for the weights to be applied to past squared returns to explain current volatility. 2 The insight of the MIDAS specification when combining different frequencies motivates Engle, Ghysels, and Sohn (2009) to modify the dynamics of low-frequency volatility methodology employed by Engle and Rangel (2008) under the Spline- GARCH model. They suggest interpreting the long-run/low-frequency volatility component in the spirit of MIDAS so that it can directly employ macroeconomic data sampled at lower frequency, while maintaining the mean reverting unit GARCH dynamics for the short-run component. This new class of models is called GARCH- MIDAS. 2 González, Nave, and Rubio (2012) also show the relevance of the weighting schemes of MIDAS when estimating conditional covariances as the cross products of portfolio returns and aggregate factor returns in the cross-sectional estimation of the market risk premium. 3

4 Relatively little work has been done to understand corporate bond volatility dynamics. This paper fills this gap by analyzing the macroeconomic and financial determinants of the volatility of corporate bond returns across six credit rating categories by applying the GARCH-MIDAS specification. This methodology allows us to disentangle the impact of macroeconomic conditions on corporate bond volatility from the short-run dynamics. To the best of our knowledge this is the first paper directly analyzing this issue. Regarding the behavior of corporate bonds, one of the most demanding issues refers to the credit spread puzzle. Huang and Huang (2003) show that structural models generate credit spreads much lower than the historical differences between Aaa and Baa corporate bonds. The two papers putting forward a potential solution of this puzzle consider the effects of macroeconomic conditions on corporate bond yields. Chen, Collin-Dufresne, and Goldstein (2009) use the Campbell and Cochrane (1999) model with habit preferences to show that time-varying risk aversion, together with a capital structure mechanism to match the countercyclical nature of defaults, can account for the high corporate bond spreads. In their model, investors are sensitive to the timing of defaults since high yield bond defaults are more likely to occur in recessions, when risk aversion is particularly high. Rather than modeling risk aversion, Chen (2010) employs time-varying consumption risk in the spirit of the long-run model of Bansal and Yaron (2004) to show that heteroskedastic long run aggregate consumption risk makes firms default more likely in recessions and generates a more disrupting environment for both stock and bond holders. Given the response of firms to macroeconomic conditions, Chen s model endogenously incorporates countercyclical fluctuations in risk prices, default probabilities, and default losses. This simultaneous co-movement generates the 4

5 large credit spreads that explain not only the credit spread puzzle but also the lowleverage ratios historically reported by firms. Therefore, macroeconomic conditions have been used to explain the level of credit spreads over time. However, none of these papers address the issue of how macroeconomic conditions affect corporate bonds volatility. Similarly, Rachwalski (2011) shows that corporate bond returns predict consumption growth and labor income growth even after controlling for equity returns. Additionally, the covariance between corporate bonds and stock returns predicts the stock market index. But, as the previous authors, Rachwalski (2011) does not discuss the relationship between corporate bond volatility and macroeconomic conditions. Finally, there are other papers analyzing the cross-sectional variation of corporate bond returns. Fama and French (1993) first show that default and term premia are priced factors in the corporate bond market. Gebhardt, Hvidkjaer, and Swaminathan (2005) show that default betas are significantly related to the cross-sectional variation of average bond returns. Furthermore, yield-to-maturity remains the only significant characteristic after controlling for default and term betas, suggesting that systematic risk factors are important for pricing corporate bonds. Lin, Wang, and Wu (2011) argue that market-wide liquidity risk is also a priced factor in the cross-section of corporate bonds as implied by their finding of a positive and significant relation between average bond returns and liquidity beta which is robust to including default and term betas. Acharya, Amihud, and Bharath (2013) also show that time-varying liquidity risk matters for corporate bonds, suggesting a flight-to-liquidity aspect in the corporate bond market on top of the well-known flight-to-quality. Using transaction-level data from 2003 to 2009, Bao, Pang, and Wang (2011) find that market-wide liquidity explains a substantial variation of credit yield spreads, and 5

6 that illiquidity is also priced in the cross-section of corporate bond returns. Once again, an aggregate liquidity factor is used to explain the time series behavior of yield spreads but not the volatility of corporate bonds. The work more closely related in spirit to our research is Cai and Jiang (2008) who show that, during the period, corporate bond excess return volatility is directly related to contemporaneous bond excess returns. They also argue that bond volatility is a significant predictor of corporate three-month and six-month future bond excess returns. More importantly, they decompose aggregate bond volatility into market, time-to-maturity, and rating components, to find that corporate bond volatility has both a slow-moving and a high-frequency component. They identify the lowfrequency component with the trend displayed by the rating volatility that shows a positive trend until 2002, followed by a declining pattern until However, they do not statistically decompose both components and they do not investigate the macroeconomic sources of the slow-moving pattern of volatilities. Moreover, their analysis is performed at the aggregate level rather than investigating the behavior of corporate bond volatilities throughout credit ratings. We find that for most financial or macroeconomic indicators, the recognition of a low frequency component of volatility in corporate bond returns significantly improves the likelihood of the GARCH specification, relative to a constant long-term component, with independence of the credit rating category. In particular, high volatility is explained by high levels of default premium, VIX, equity market-wide illiquidity, and low levels of the term premium. Similarly, high volatility of corporate bonds is related to slow growth of industrial production, consumption, and employment, high inflation and high volatility of consumption growth. 6

7 We also get evidence that volatility sensitivities to changes in financial or macroeconomic indicators are often monotonic in the rating of corporate bonds. For instance, high levels of default, VIX, market-wide illiquidity, and inflation have a much higher impact on CCC than in AAA bonds. Similarly, a low level of the term premium, and slow industrial production, consumption, and employment growth rates have a stronger effect on the volatility of BB and B bonds than on the volatility of AAA corporate bonds. However, they affect slightly less the volatility of CCC bonds than that of BB and B bonds. Interestingly, higher volatility of stockholders consumption growth anticipates a higher increase in the volatility of AAA bonds than in the volatility of CCC corporate bonds. All of these effects are generated by the low frequency component in the multiplicative specification we employ when modeling the daily behavior of corporate bond volatilities. This paper is organized as follows. Section 2 briefly describes the new class of multiplicative component models for asset volatility. Section 3 describes the data employed in the analysis, and section 4 reports and discusses our empirical results on the relationship between corporate bond volatility and macroeconomic and financial indicators. Section 5 further motivates the results analyzing the behavior of the model during normal/expansion and recession periods. Section 6 concludes. 2. The multiplicative GARCH-MIDAS two-component model of volatility Let us denote by r i, t the return of an asset on day i of any arbitrary period t which can be a month, quarter, year, etc. with N t days in each period. In our specific research we employ monthly data, so that t refers to months. Our macroeconomic and financial indicators will have a single t subindex since they are observed at monthly frequencies. We assume that daily returns follow a statistical structure given by, 7

8 where i, t ri,t η is an innovation with distribution ( 0, ) = µ + η, (1) i,t 2 i, t 2 t N σ with σ i, = τ t gi, t, where g i, t is the high-frequency component following a unit GARCH(1, 1) process, and τ t is the stochastic low-frequency component. 3 In particular, the return on day i in month t is written as, r i,t = µ + τ g ε, (2) t i,t i,t where ε i, t is a shock with distribution N(0,1) given the information available up to day ( i 1) of month t. As in Engle and Rangel (2008), the specification employed by Engle, Ghysels, and Sohn (2009) assumes that the volatility dynamics of the component g i, t is a daily GARCH(1, 1) process given by, where α + β < 1. g ( 1 ) ( r ) 2 i 1, t µ = α β + α + β g, (3) i, t i 1, t τ t On the other hand, the low-frequency (monthly) component τ t is assumed to respond to economic conditions over a relatively long period of time where these conditions are represented by either macroeconomic or financial indicators. The essence of the MIDAS approach is to consider data with different sampling frequency. In our case, we combine the monthly data for the macroeconomic and financial indicators with the daily data for the returns of corporate bonds across different credit ratings. Thus, in the spirit of MIDAS regression and filtering, the τ t component is assumed to be a 3 Note that in the original two-component model of Engel and Rangel (2008), the low frequency component is deterministic, while in this specification τ t is stochastic. This is the modification suggested by Engle, Ghysels and Sohn (2009). 8

9 smoothed measure of either the level or the volatility of the macroeconomic and financial indicators and it is written as, 4 t l Kl l k= 1 k ( ω1,l, 2,l ) X l,t k logτ = m + θ ϕ ω (4a) t v Kv v k= 1 k ( ω1,v, 2,v ) X v,t k logτ = m + θ ϕ ω, (4b) where X l and X v denote the level or the variance of the state variable indicators respectively. In this specification, the low-frequency component log( τ t ) varies from month to month but it stays the same for all days in a given month. This is the GARCH- MIDAS model with fixed time span indicator. 5 As in Engle, Ghysels, and Sohn (2009), we assume the beginning of period expectation of the high-frequency component, ( g ) expectation, E ( g ) 1 t 1 i, t = Et 1 i,t, to be equal to its unconditional. Therefore, the long-run component is given by, E 2 [( ri,t µ ) ] = τ t Et 1( gi,t ) = τ t t 1 (5) Finally, we assume a beta weighting scheme for expression (4a-4b), given by φ k ( ω, ω ) 1 2 ω k = k k 1 K K = K φ ω ω (, ) 1 2 ω (6) As discussed by Ghysels, Sinko, and Valkanov (2006), this beta-specification is very flexible, being able to accommodate increasing, decreasing or hump-shaped weighting schemes. 4 In the usual MIDAS approach, the low-frequency component is a smoothed measure of the realized variance of the asset itself. This can be easily introduced in the specification above. However, in this research, we will focus on the impact that either macroeconomic or financial indicators have on the future variance of asset returns. 5 It can be easily extended to allow for a rolling window structure, which we do not pursue in this paper. 9

10 The model can be estimated using log-likelihood techniques. For each credit rating, and for each financial and macroeconomic indicator, we estimate the set of parameters ( µ, α, β,m, θ, ω ω ) by, Θ = 1, 2 by maximizing the log-likelihood function given = K log L( { ri, t} 1,2,, ) = log e i= K N t t= 1 i= 1 2πτ t gi, t ( ri, t µ ) 2 T Nt t 1,2,, T 1 2τ t gi, t T Nt 1 1 = log 2π + log ( τ t gi, t ) ( ri, t µ ) τ g t= 1 i= 1 t i, t 2 (7) 3. The Data Corporate bond data comes from the daily yields on US corporate bonds reported in the files of Bank of America / Merrill Lynch from December 31, 1996 to January 31, 2012, for six credit bond rating classes: AAA, AA, A, BBB, BB, and CCC. Yields for the last day of each month for all credit ratings are displayed in Figure 1. Their evolution over time reflects a relatively parallel behavior, with the expected peaks during financial crises, especially for corporate bonds rated as BB, B, and CCC. Yields of CCC corporate bonds tend to be much higher than for other ratings, with an impressive overall high of almost 40% during November Our objective is to understand the behavior of corporate bond volatilities, which implies that we are specifically concerned with percentage changes in corporate bond prices. Given that we need daily corporate bond returns in order to estimate the multiplicative GARCH-MIDAS model, and that TRACE data is only available from 2002 onwards, we calculate the return on a given bond as, ( + yt+ 1) T ( 1+ y ), (8) T T P N 1 t y t log = log = log Pt N 1 y t + t+ 1 10

11 where P t is the price of a corporate bond at time t, y t is the yield, N is the nominal value of the bond, and T is the time to maturity. 6 Table 1.a displays descriptive statistics for corporate bond yields in our sample. High yield bonds present a relatively high standard deviation, positive skewness and excess kurtosis relative to high-rated bonds. Additionally, correlation coefficients are high for similar rating classes, and they decrease when we move to rating classes apart from the one we consider. The correlation between AAA and CCC bond returns is as low as Table 1.b contains similar information for corporate bond returns. Average returns present a decreasing pattern, due to the effect of ignoring coupon payments. However, this is not relevant for our study. What is really important is the behaviour of corporate bond return volatilities. The annualized volatility of CCC bonds is 37.9%, relative to a volatility of 7.6% for AAA bonds. Most of the volatility in corporate bonds comes from the percentage changes in prices and not from coupon payments. Indeed, the maximum and minimum annualized returns correspond to CCC bonds. In terms of returns, the BBB and BB categories have the highest negative skewness and excess kurtosis, and CCC bonds have more negative skewness and higher excess kurtosis than AAA bonds. The correlation patterns are very similar to the correlations reported for yields. The lowest correlations among all corporate bonds are those between the returns of the AAA/AA categories and the returns of B/CCC bonds. Yields for the 10-year government bond and Moody s Baa corporate bond series are obtained from the Federal Reserve Statistical Releases. We then compute two state variables based on these interest rates: a term structure slope (Term), computed as the 6 We obtain the average time-to-maturity, T, as the average maturity of the corporate bonds available in TRACE from 2002 to It turns out that the average time-to-maturity is 7.8 years. We keep this number fixed throughout our sample period. 11

12 difference between the 10-year government bond and the risk-free rate, and a default premium (Default), calculated as the difference between Moody s yield on Baa corporate bonds and the 10-year government bond yield. We collect these data on a monthly basis from January 1997 to January Daily data on VIX is obtained from January 1990 to July 2012 from the Chicago Board Options Exchange (CBOE). We employ the last day of the corresponding month to create a final monthly option-implied volatility series from January 1997 to January We use Pastor and Stambaugh (2003) measure of liquidity, which reflects the amount in which stock returns rebound upon high volume. Their measure is based on daily regressions of individual stock excess returns over the market return in a calendar month, em [ sign ( R j, t ) ] DVol j,t e j,t 1 em R j,t + 1 a + b R j,t + g + + =, (9) where R em j,t + 1 denotes the excess return of stock j over the market return. Pastor and Stambaugh aggregate the estimates of the g coefficient across stocks and scale it for growing dollar volume. They finally propose the innovations in this regression as the final measure of illiquidity. 7 The intuition is that high volume moves prices away equilibrium and they rebound the following day, which suggests that g is typically negative. We collect four alternative series of monthly macroeconomic growth, and the price deflator rate. We obtain nominal consumption expenditures on nondurable goods and services from Table of the National Income and Product Accounts (NIPA) available at the Bureau of Economic Analysis. Population data are from NIPA s Table 2.6 and the price deflator is computed using prices from NIPA s Table 2.8.4, with the 7 The monthly series are available in Lubos Pastor s web site. 12

13 year 2000 as its basis. All this information is used to construct monthly rates of growth of seasonally adjusted real per capita consumption expenditures on nondurable goods and services ( C) from January 1959 to September Monthly data for the industrial production index (IPI) are downloaded from the Federal Reserve, with series identifier G17/IP Major Industry Groups from January 1927 to January We also collect non-farm employment growth rate from the Bureau of Labor Statistics using the B tables of the seasonal adjusted employment situation release from January 1960 to October Finally, we use aggregate per capita stockholder consumption growth rate obtained from January 1960 to September Exploiting micro-level household consumption data, Malloy, Moskowitz, and Vissing-Jorgensen (2011) show that longrun stockholder consumption risk explains the cross-sectional variation in average stock returns better than the aggregate consumption risk obtained from nondurable goods and services. On top of that, they report plausible risk aversion estimates. They employ data from the CEX for the period March 1982 to November 2004 to extract consumption growth rates for stockholders, the wealthiest third of stockholders, and nonstockholders. In order to extend their available time period for these series, they construct factor-mimicking portfolios by projecting the stockholder consumption growth rate series from March 1982 to November 2004 on a set of instruments, and use the estimated coefficients to obtain a longer time series of instrumented stockholder consumption growth. They use a small growth portfolio (average of the two smallest size, two lowest BE/ME portfolios) from the 25 Fama-French portfolios, a large growth portfolio (average of the largest size, two lowest BE/ME portfolios), a small value portfolio (average of the two smallest size, two highest BE/ME portfolios), and a large value portfolio (average of the two largest size, two highest BE/ME portfolios) as instruments. Using the estimated coefficients of this regression, they generate a factor- 13

14 mimicking portfolio for stockholder consumption growth from July 1926 to November In this paper, we employ the reported estimated coefficients by Malloy, Moskowitz, and Vissing-Jorgensen (2011) to obtain a factor-mimicking portfolio with the same set of instruments for stockholder consumption from January 1960 to September Figure 2 displays aggregate per capita consumption, stockholder consumption, and employment annual growth rates from 1960 to The time series behaviour of aggregate consumption and employment growth rates are very similar and smoother than the stockholder consumption growth rate. However, the troughs and peaks of the three series tend to have the same time location. On the other hand, as expected, these peaks are much more pronounced for stockholder consumption growth than for either aggregate consumption or employment growth. Table 2 contains descriptive statistics for the financial and macroeconomic indicators used throughout this research. Annualized volatility is relatively high for industrial production and stockholder consumption growth, VIX, and especially marketwide liquidity. VIX is highly and positively correlated with the default premium and negatively correlated with macroeconomic variables, especially stockholder consumption growth. In terms of monthly growth, employment presents a higher correlation with industrial production and aggregate consumption than with stockholder consumption, and the default premium has a negative correlation with industrial production, consumption, and especially with employment growth. As preliminary evidence as well as to illustrate potentially interesting patterns between corporate bond returns and financial and macroeconomic indicators, we run simple regressions of monthly bond returns and volatility of bond returns on the described indicators. Table 3.a displays R -statistics for the contemporaneous regressions of bond returns on indicators and p-values for the F-test of global 2 14

15 significance. It also shows the sign of estimated effects when the p-values for the F-test of global significance are below As expected, we find low 2 R s except for the regressions of BB, B, and CCC returns on stockholder consumption. The 2 R for CCC corporate bond returns is surprisingly high and equal to Indeed, Figure 3.A displays the actual CCC bond returns and the fitted value from the contemporaneous regression with stockholder consumption. Both series closely resemble to each other. This is a striking finding that deserves future research. It may be interesting to check whether any of the instruments employed in deriving the factor-mimicking portfolio of stockholder consumption explain this high correlation. At this contemporaneous level, the regression of CCC returns on VIX also shows a relatively high 2 R of 9.7%. We also run dynamic regressions on the contemporaneous value and three lags of each financial and macroeconomic indicator, with the results shown in Table 3.b. There is an expected increasing pattern of the adjusted 2 R s of regressions of corporate bond returns on the default premium when we move from AAA to CCC bonds. 2 R s for BB, B, and CCC bonds are close to 0.24 in these regressions while being just 0.04 for the AAA bond. The behaviour with respect to the term premium is precisely the opposite. The higher 2 R s are reported for the AAA, AA, and A bonds. The presence of the contemporaneous observation in these dynamic regressions leads again to high 2 R s for BB, B, and CCC bonds with respect to stockholder consumption. When using VIX as an explanatory variable, 2 R s are essentially monotonically increasing with credit rating. The 2 R for BB and CCC bonds is 0.44 and 0.42 respectively, while being of just 0.05 for AAA bonds. In Figure 3.B we easily appreciate the improvement in the fit between CCC bond returns and VIX when we move from the contemporaneous to the dynamic regression. 15

16 Since we are interested on the volatility of bond returns, similar regressions explaining monthly volatility of corporate bond returns may be more useful as initial illustrations of our findings reported in the next section of the paper. At this point, monthly volatility is estimated as the average of daily squared returns during a given month and for each credit rated bonds. Tables 4.a and 4.b show estimates for the contemporaneous and the dynamic regressions, respectively. As before, adjusted 2 R s tend to be higher when we add three lags to the regressions. In particular, even though default premium and VIX are important explanatory variables of corporate bond volatility even contemporaneously, it is clear that the results improve when we allow for dynamics in the regressions. Industrial production and employment growth also seem to explain quite well the volatility of corporate bonds. Inflation shows explanatory power for the volatility for B and CCC rated credit bonds, while aggregate consumption has higher 2 R s for highly rated bonds, and stockholder consumption shows a slightly higher explanatory power for lower rated bonds. Figures 4.a and 4.b shows the good fit of these regressions for the case of the volatility of AAA and CCC bonds, respectively, when we use the default premium as a volatility indicator. 4. Estimates of GARCH-MIDAS Model of Corporate Bond Volatilities with Financial and Macroeconomic Indicators The GARCH-MIDAS model that we estimate next is given by expressions (2), (3), and (4a) or (4b) depending on whether we employ the level or the volatility of the financial and macroeconomic indicators, where the weights applied to past values of the indicators are given by the beta-weighting scheme in (6). For each credit rating, and for each financial and macroeconomic indicator we maximize the log-likelihood function given by (7). The number of lags in the long-run component, K and K, is different for l v 16

17 each corporate bond and indicator, and in all cases, we employ the lag that maximizes the log-likelihood function. The estimation combines the daily return data for corporate bonds with the monthly data for the financial and macroeconomic indicators. The empirical results are reported in Table 5.a through Table 5.k. Each panel corresponds to a particular financial or macroeconomic state variable and it contains the results for all credit rating categories. We report the estimated parameters given by the set ( µ, α, β,m, θ, ω ω ) Θ = 1, 2 with the standard errors in parentheses, the value of the log-likelihood function, and the likelihood ratio obtained by comparing the estimated model with the nested benchmark model given by the GARCH-MIDAS specification with constant long-run component. In brackets, below the likelihood ratio statistic, we report its p-value. In all cases, the α and β parameters of the short-run component given by the unit GARCH process are estimated with precision and present reasonable and similar values across all state variable indicators. The exception is the B rated bond for default premium, term premium, liquidity, and consumption volatility, for which we estimate a higher alpha relative to other bonds and indicators. The average estimated alpha (beta) for the AAA bond is slightly higher (lower) than for the CCC bond, and the persistence, measured as the sum of both parameters, is also slightly higher for the CCC bond. The estimated mean for the short-run dynamics given by µ is close to zero, as expected in daily return data. Estimated weights tend to vary across cases presenting different shapes ranging from monotonically decreasing to hump-shaped weights. For the later, the lag attaining the maximum value also varies across ratings and state variable indicators. It is also the case that they the weight parameters tend to be estimated with low precision. Given that 17

18 we take logs to estimate the long-run component, the mean of the long-run component, m, is by construction negative and it is estimated with precision. θ l and We are particularly interested in the slope parameters of the long-run components θ v. They indicate whether the past behavior of a financial or macroeconomic indicator anticipates either an increase or a decrease in the volatility of corporate bond returns. It turns out that θ is strongly significant in all cases, independently of the corporate bond or the state variable employed in estimation. This suggests that the recognition of the long-run component is a key factor for a better understanding of the behavior of the volatility of corporate bond returns. The θ estimates tend to have the expected sign. They are positive for the state variable indicators for which an increasing value implies a negative shock for the economy, and negative for those cases in which increasing values represent a positive shock. Overall, we find positive ˆθ l and ˆθ v for default, inflation, VIX, market-wide equity illiquidity, and consumption volatility, while a negative sign is obtained for term, industrial production, consumption, stockholder consumption, and employment growth. This implies that increasing values of default, VIX, illiquidity, inflation, and consumption volatility anticipate higher future volatility of corporate bond returns, while increasing values of production, consumption, or employment growth as well as the term premium, anticipate lower volatility in corporate bond returns. Moreover, although the signs of ˆθ l and ˆθ v are exactly what we expected, the relative impact of macroeconomic and financial indicators on volatility is quite different across credit rating categories. In order to appreciate this point, Figure 5.A and Figure 5.B display the θ estimates for state variable indicators that generate more and less volatility respectively across credit rating categories. For a better comparison, ˆθ l and ˆθ v are divided by the 18

19 cross-sectional standard deviation of the estimated valuesθ for a given indicator across credit ratings. Analyzing bad-news indicators, VIX seems to be the most relevant factor explaining corporate bond returns volatilities for all credit ratings, although the impact of VIX is especially large for the BB, B, and CCC categories. Similarly, the default premium presents an increasing impact when we move from AAA to CCC bonds. Together with VIX, they become the key factors generating return volatility for CCC bonds. Consumption volatility also has a similar, although smoother, increasing pattern across ratings, but it becomes less relevant for CCC bonds. On the other hand, stockholder consumption volatility seems to have a higher impact on the AAA category. Finally, inflation, and market-wide illiquidity become more important the lower is the credit rating class. Hence, in terms of junk-bonds, VIX, default, inflation, and illiquidity shocks in equity markets are the most relevant indicators of bond return volatility, while VIX and stockholder consumption volatility dominate the volatility of AAA bonds. On the good-news side, industrial production growth is practically in all cases the most relevant factor anticipating a reduction in corporate bond volatility. Consumption and employment growth are also macroeconomic state variables explaining a long-run reduction in volatility. It is interesting to point out that these macroeconomic indicators seem to have more impact on the BB and B categories than on junk-bonds. Indeed, the effects of production, consumption, and employment growth on the volatility of AAA and CCC are quite similar, but less than for the BB and B bonds. Finally, we must test the overall statistical significance of the model specification that incorporate a stochastic behavior for the long-run component of volatility relative to the specification in which the low-frequency component is assumed to be constant. Hence, the benchmark model is the GARCH-MIDAS model with constantτ. The last column of Table 5 provides the likelihood ratio test statistic with its corresponding p- 19

20 value. Out of the 77 possible cases analyzed, combining 11 state variable indicators and 7 credit rating categories, in 54 cases the test suggests a statistically significant improvement in fitting the data when incorporating the stochastic long-term volatility component. It also implies that a number of macroeconomic and financial indicators contain relevant information concerning future conditional volatility of corporate bond returns. Moreover, for some of the most relevant indicators, VIX, the term premium, industrial production, and consumption and employment growth we always reject the constantτ specification. The default premium, inflation, market-wide illiquidity, stockholders consumption growth and aggregate consumption volatility contain explanatory power on future volatility of bond returns for the lower rating classes (BB and beyond). The volatility of stockholder consumption seems to contain information on future volatility for high credit rating bonds.. 5. Interpreting the role of economic indicators in volatility estimation Since we have just shown that there is ample evidence of information content in macroeconomic and financial indicators on future bond returns volatility for all bond classes, the next step is to try to advance some intuition on this evidence. To that end, we split the sample between recession and normal/expansion periods to examine the different behavior of AAA and CCC bond return volatility in each sub-sample. We also consider macroeconomic and financial indicators separately, since they might contain different types of information on future volatility. The approach we follow is that having shown a generally better likelihood fit for the GARCH-MIDAS model, we take any significant departure from its implied volatility relative to the constant-τ volatility as an improvement in volatility estimation. 20

21 Figure 6 and 7 shows that differences between volatility estimates from the GARCH-MIDAS approach and the constant-τ approach become more important around the financial and economic crisis than during stable periods in high- as well as in lowrating bonds. This observation suggests comparing the volatility estimates from both modeling approaches in recession and normal/expansion times separately, taking into account that there are in our sample two recession periods according to the NBER chronology: March-November 2001 and a larger period, December 2007-June As shown in Figure 6.a, and for AAA bonds, the volatility estimated with macroeconomic indicators, industrial production, inflation, employment, consumption growth, and stockholder consumption growth is higher than the one estimated under a constant-τ specification over most of the sample. It is only in the last part of the second recession period that the constant-τ specification produces a higher volatility. Figure 6.b displays the differences generated by financial indicators. If we now compare Figure 6.a and 6.b, we note that volatility estimates from macroeconomic indicators are higher than the ones obtained from financial indicators, which fluctuate around the constant-τ volatility, staying close to it at all time periods. The important exception is the liquidity factor that behaves very similarly to macroeconomic indicators. Differences between volatility estimates and the constant-τ volatility generated by term, default, and VIX are relatively small except again at the end of the second recession period. As it was the case with macroeconomic indicators, in this specific period the volatilities obtained from financial indicators become noticeably lower than constant-τ volatility. In normal/expansion times the difference between the volatility estimated with macroeconomic indicators and the constant-τ volatility becomes particularly large. In such periods, the volatility estimates from financial indicators present small differences with respect to the constant-τ volatility, especially if we compare them with the 21

22 differences between the volatility estimates from macroeconomic indicators and the constant-τ volatility. We have previously shown a statistically significant improvement in likelihood fitting, and we also see that there are periods when they clearly depart from the volatility produced by the constant-τ specification. Volatility estimates from macroeconomic indicators for AAA bonds deviate from the benchmark constant-τ volatility in recession as well as in normal/expansion time periods, whereas volatility estimates from financial indicators differ from the benchmark volatility estimates just on recessions. Moving to low-rating bonds in Figures 7.a and 7.b the first thing to bear in mind is that, as expected, our data shows that the volatility of CCC bond returns is well higher than that of AAA bonds. This is the case when we consider volatility estimates from daily returns (7.6% for AAA bonds against 37.9% for CCC bonds in Table 1.b), as well as when we examine conditional volatilities from a constant-τ specification (6.61% average estimated volatility for AAA bonds against 20.13% for CCC bonds). Nevertheless, the GARCH-MIDAS models adjust well also to the higher volatilities of CCC bonds. As displayed in Figures 7.a and 7.b, over the whole sample, estimated volatility for CCC bonds from macroeconomic and financial indicators oscillate around the volatility from the constant-τ specification. If we examine recession periods, it is once again in the second part of the last recession period that we observe the larger differences between volatility from indicators and constant-τ volatility. During recession periods volatility estimates from financial indicators depart by a larger amount from the constant-τ volatility. This is especially clear for the term premium and the VIX index, which generate a noticeably lower volatility than the constant-τ specification towards the end of the second recession period. 22

23 Overall, macroeconomic indicators seem to do a good job at estimating volatility for high-rating bonds in recession as well as in normal times. For such bonds our estimates suggest an underestimation of volatility from the constant-τ specification over normal/expansion periods, for which a volatility model with macroeconomic indicators might be more appropriate. They are also consistent with an overestimation of volatility from the constant-τ model over recessions, when a model with financial indicators like the default premium or the VIX index should be preferred. For low-rating bonds, our GARCH-MIDAS volatility estimates generally suggest again that the constant-τ specification overestimates volatility during recessions. In such periods, differences between volatility estimates obtained with and without indicators are numerically large. On the other hand, over normal/expansion periods, differences are minor, being even lower than those obtained for the high-rating bonds, with no evidence of potential bias in standard volatility estimates from the constant-τ model. To examine whether this graphical evidence leads to statistically significant conclusions on the comparison between the characteristics of the time series of volatility estimated from both modeling strategies, we use a simple regression approach: where σ GM σ = α + β EXPANSION + u, (10) t t σ GM t is the volatility generated by the GARCH-MIDAS model, σ t is the volatility obtained under the constant-τ specification, and EXPANSION is a dummy variable that takes the value of 1 whenever month t does not belong to the NBER official recession dates, and zero otherwise. This implies that the intercept α is the average difference between the volatilities generated by both models over recessions, while the slope β indicates how the difference in estimated volatilities changes in normal/expansions times, relative to recession periods. The sum α + β is the average difference of volatilities over normal/expansion times. t t 23

24 Table 6 contains estimates of the intercept and the slope in each regression, as well as the p-values for the significance tests, obtained using variance estimates robust to the presence of heteroskedasticity and autocorrelation. In the interpretation that follows we will focus on coefficients estimated with a p-value below Coefficient estimates are consistent with our previous comments on the time behavior of volatility from both types of models over recession and expansion times. For AAA bonds, we obtain a positive intercept for macroeconomic indicators, reflecting the observation that the GARCH-MIDAS volatility is, on average, higher in recession periods than the constant-τ volatility estimate. For financial indicators, the intercept is negative when significant, meaning that average GARCH-MIDAS volatility is in this case lower than the constant-τ volatility. In normal/expansion times the volatility estimated with all macroeconomic indicators departs from the constant-τ volatility by an even higher amount, whereas the volatility estimated from financial indicators loses explanatory content relative to the constant-τ volatility estimate, as shown by a lower average difference in the volatilities from both models. For CCC bonds, volatility estimates from the GARCH-MIDAS approach are, on average, lower in recessions than the constant-τ volatility estimate, leading to negative intercept estimates. The exception is the default premium although the estimated coefficient is not statistically different from zero. Slope estimates have in all cases the opposite sign to the corresponding intercept, leading to a minor difference between the volatility estimates from both modeling approaches in normal/expansion times. Indeed, as shown in Table 6, a striking result is that in normal/expansion periods the estimated volatility from macroeconomic and financial indicators for CCC bonds is closer to the constant-τ volatility than for AAA bonds. Hence, our estimates suggest that the use of 24

25 indicators to estimate volatility is of interest for AAA bonds in any conditions, whereas in the case of CCC bonds, it is mostly interesting during recession periods. To better appreciate differences in estimated volatility, the left side of panel A in Table 7 shows the frequency of sample points in which the estimated GARCH-MIDAS volatility exceeds the volatility estimated from the constant-τ model by more than 10%, 5% or by any amount. The right side of panel A shows the frequency of points in which the opposite happens. We can see that for AAA bonds, the GARCH-MIDAS volatility estimated with macroeconomic indicators is almost always larger than the constant-τ volatility. This is consistent with our previous interpretation on the fact that the constant-τ specification tends to underestimate volatility for AAA bonds over normal/expansion periods, which stretch over most of the sample. Besides, we now see that the differences between both estimates are sizeable. Financial indicators provide contrary evidence when estimating volatility, since the GARCH-MIDAS volatility estimated for AAA bonds from the default premium and the VIX index is often below the volatility obtained from the constant-τ specification. This is again according to our previous observation on Figure 6. Table 7 also shows that the difference between constant-τ volatility and the volatility obtained from financial indicators is small, rarely surpassing 5%. Such cases should correspond mostly to recession periods, 8 for which we already saw above that the constant-τ model tends to overestimate volatility for AAA bonds. Hence, the statistics in this panel are consistent with the graphical evidence displayed above on the fact that macroeconomic indicators produce a large improvement in volatility estimation, raising the estimate relative to the use of a 8 Note that the percentage of cases where the difference between estimates exceeds 5% is close to the 15.5% of sample observations corresponding to recession periods. 25

26 benchmark constant-τ specification over most of the sample. The evidence in this panel also seems to be consistent with the benefit of using financial indicators to estimate volatility of high-rating bonds in recession periods, which extend over 15.5% of the sample. For CCC bonds the GARCH-MIDAS volatility estimated with macroeconomic indicators is very often below the constant-τ volatility. In a number of cases, the difference between volatility estimates is above 5%. We now see that over normal/expansion periods, the constant-τ model also seems to overestimate volatility although by a relatively small amount. This is again consistent with our suggestion on the graphical analysis above on the fact that the use of macroeconomic indicators for low-rating bonds did not seem to lead to an apparent difference in volatility estimates in normal/expansion periods. When using financial indicators for CCC bonds, each volatility estimate exceeds the other for about half of the sample periods. VIX and the term premium lead to some percentage of observations for which the difference between volatility estimates is larger than 5%. These observations should correspond to recession periods, for which we already concluded that financial indicators would lead to an improved volatility estimate below the constant-τ volatility estimate. How relevant are these differences in estimated volatility for risk management? Under the assumption of Normality of returns, the percent difference in estimated VaR would be equal to the percent difference in estimated standard deviation. Panel B in Table 7 displays the largest differences in VaR estimates observed over the whole sample. As we can see, these extreme differences can be very large in some cases, particularly when using macroeconomic indicators in the GARCH-MIDAS model, and even larger for CCC bonds than for AAA bonds. The reported number in this panel 26