Understanding the Role of VIX in Explaining Movements in Credit Spreads

Transcription

1 Understanding the Role of VIX in Explaining Movements in Credit Spreads Xuan Che Nikunj Kapadia 1 First Version: April 3, University of Massachusetts, Amherst. Preliminary; comments are welcome. Please address correspondence to Nikunj Kapadia Isenberg School of Management, University of Massachusetts, Amherst, MA nkapadia@som.umass.edu.

2 Abstract Why does the VIX and market return explain changes in credit spreads? Existing literature suggests these factors proxy for macroeconomic risk. In this paper, we investigate an alternative hypothesis that the VIX in its role as a fear index impacts intermediary and arbitrageur capital, impacting spreads and resulting in decreased market integration across credit and equity markets. We document that hedging credit default swaps in the equity markets is surprisingly ineffective. On average, hedging reduces the RMSE reduces by 10% and the VaR by only 12%. However, a passive hedge kept in place over a period as long as a month is (multifold) more effective than dynamic daily hedging. We demonstrate that the VIX and market returns predict both the RMSE as well as the improvement in hedging effectiveness that occurs over time. Our results suggest that frictionless structural models of credit risk are of limited use in explaining changes in credit spreads because factors which are excluded from the pricing kernel have significant impact on credit spreads. Keywords: Hedge Ratio, credit default swap JEL classification: G12, G14

3 In the uproar over AIG, the most important lesson has been ignored. AIG failed because it sold large amounts of credit default swaps without properly offsetting or covering their positions. George Soros (Wall Street Journal, March 24, 2009) 1 Introduction Arguably, the remarkable success of contingent claim pricing theory is because even though dynamic replication of the contingent claim is not possible in practice, market participants agree about certain restrictions placed by the theory on the pricing kernel. For example, it is now well accepted that the equity option is not replicable, and therefore the Black-Scholes formula cannot possibly price an option correctly. Yet, there is little debate on the primary restriction the theory places on the factors that should not enter into the pricing kernel. Although factors like the Fama-French factors enter the pricing kernel of the underlying asset (and determine the stock return), these factors are deemed irrelevant for determining the option price after controlling for their impact on the stock price. Thus, option price changes are closely related to stock price changes, and the Black-Scholes hedge ratio is a first order approximation of the true hedge ratio. Indeed, the academic literature (and practitioners) have used the hedge ratios recommended by the Black-Scholes formula (with ad hoc adjustments to the volatility) for hedging at a daily frequency. 1 In this context, a result first documented in Collin-Dufresne, Goldstein and Martin (2001) poses a significant challenge to contingent claim theory. Their finding that changes in credit spreads are poorly explained has received considerable attention in the literature. A number of papers, including Blanco, Brennan, and Marsh (2005), Cremers, Driessen, Maenhout, and Weinbaum (2008), Ericsson, Jacobs, and Oviedo (2009), and Zhang, Zhou, and Zhu (2009) have focused on finding a complete set of factors that should be in the pricing kernel, and therefore explain credit spread changes. Our focus is, however, the opposite. We are concerned with factors that impact credit spreads, but should not be in the pricing kernel. The finding we find puzzling is that credit spreads are explained by the S&P 500 return 1 Consistent with the implication of contingent claim theory, dynamic hedging of index options at a daily interval reduces hedging errors than if a passive hedge is in place over a longer horizon. In their Table 9, Bakshi, Cao and Chen (2000) document that hedging errors for both at- and away- from the money index options increase by over 90% when the hedge is rebalanced once every two days instead at a daily frequency. 1

4 and the VIX index after controlling for the underlying stock return. These two variables are jointly significant after controlling for macroeconomic variables such as term spread and the 10-year treasury yield (Collin-Dufresne, Goldstein and Martin, 2001), and firm specific covariates. They are economically significant. In our dataset, the median R-square of a regression of changes in credit default swap spreads on the S&P 500 return and change in VIX is 22% - almost exactly the same as the regression on the underlying equity return and changes in equity volatility and leverage. Should the S&P 500 return and VIX be included in the contingent claim pricing model? If not, why do they explain credit spreads? The rationale provided in the literature for including the S&P 500 return and VIX is that they proxy for macroeconomic covariates. Chen, Collin-Dufresne and Goldstein (2009) make a persuasive case that macroeconomic variables are relevant to pricing credit as the default boundary may co-vary with the business cycle. 2 However, it is not clear whether the market return and the VIX (as opposed to, say, the yield curve) truly proxy for the macroeconomy. In particular, the interpretation of the VIX as a macro covariate differs from its more common interpretation as a fear index. In this paper, we examine the role of the market return and VIX using the latter interpretation, and consider the implications for structural models. There are good reasons to view the impact of the market variables, especially the VIX, in terms of market fears. There is increasing recent evidence that amount of deployed capital in the market has pricing effects (e.g., Duffie, 2010), and that when market fears are high, both arbitrageurs and market makers reduce capital. Brunnermeier, Nagel and Pedersen (2008) document that returns on currency carry trades are related to the VIX, and that an increase in the VIX coincides with reductions in speculator carry positions. Nagel (2011) documents that returns on short-term risk-reversal strategies are highly correlated with the VIX, consistent with the hypothesis that market making activity is highly correlated with market fears. In considering the role of the VIX, we focus on the effectiveness with which credit default swaps can be hedged in the equity market. As we discussed earlier, we do not expect structural models of credit risk to accurately estimate the credit spread, and there is sufficient evidence that they do not (Eom, Helwege and Huang, 2004). However, we do expect structural models to provide a robust hedge ratio, and dynamic hedging to be effective. 2 Chen (2010), Bhamra, Kuehn, and Strebulaev (2010a, 2010b) and Hackbarth, Miao, and Morellec (2006) provide structural models and discuss the role of macroeconomy in determining the pricing kernel. 2

5 Indeed, the effectiveness of hedging allows us to directly confront the two contrasting views of the VIX. Macro expectations of the business cycle do not, excepting crisis situations, fluctuate significantly from one day to another. Market fears, on the other hand, do fluctuate significantly on a daily basis (as is evident from the high daily correlation between the market return and the VIX index). Under the former interpretation, dynamic hedging credit default swaps in the equity market should be effective in reducing the risk of the hedged portfolio (as in other contingent claim markets). Under the latter interpretation, dynamic hedging will be ineffective, on average, and be more ineffective when market returns are lower and market fears are higher. We conduct our empirical analysis on a sample of 207 single name credit default swaps over the period 2001 to The period ranges from the beginning of the credit default swap market to the Big Bang in April of 2009, when contract specifications for North American credit default swaps were standardized. Our first finding is that credit default swaps are poorly hedged in equity markets. Across our entire sample, depending on the model used to construct the hedge ratio, the root mean square error (RMSE) of a portfolio of credit default swaps hedged by the stock of the firm ranges from about $16,500 to $17,500 a day for a CDS notional of $10 million. In comparison, the RMSE of the unhedged CDS portfolio is about $18,000 a day. That is, on average, hedging a portfolio of credit default swaps in the equity market reduces daily volatility by only about 10%. Disconcertingly, hedging sometimes increases volatility; the Merton model hedge ratios result in greater volatility for investment grade firms. The finding is robust. First, it holds across sub-samples of rating classes, for both above investment grade firms as well as below investment grade firms. Second, it holds over subperiods. The best hedging performance is in the financial crisis period of , when correlations across all asset classes increase. But even in this period, the reduction in daily RMSE only 12%. Finally, hedging with equity is only slightly more effective in reducing the tail risk as it is in reducing volatility; the maximum reduction in the Value at Risk over the entire sample period is 12%. Hedging ineffectiveness is not because of model risk associated with the estimation of the hedge ratio. We use four different specifications used to determine the hedge ratio, including the hedge ratio from the classic zero-coupon Merton model and an in-sample regression-based estimates of the sensitivity of spread changes to the equity. Our results are similar across all hedge ratios. Indeed, consistent with the finding of Schaefer and 3

6 Strebulaev (2008), Merton model hedge ratios are not statistically different from the insample empirically estimated hedge ratios. Consequently, the effectiveness of the hedge ratio in reducing RMSE is about the same across all four models. Having documented that hedging is unusually ineffective in credit markets, we next consider the specific role played by VIX and macroeconomic factors in determining the RMSE. If hedging errors occur because changes in macro expectations, the RMSE will be correlated with contemporaneous changes in the variables proxying for macro expectations. If hedging errors are because of limited funding liquidity, then hedging errors should be predictable by factors correlated with funding liquidity. Consistent with the implications of Hackbarth, Miao, and Morellec (2006), Chen, Collin- Dufresne and Goldstein (2009), Chen (2010), and Bhamra, Kuehn, and Strebulaev (2010a, 2010b), we find that contemporaneous changes in the yield curve are correlated with the RMSE. Thus, changes in macro expectations do play a role in determining hedging errors. However, only factors related to funding liquidity, the VIX and S&P return, have predictive power. The traditional macro variables constructed of the yield curve do not predict hedging errors. Thus, the evidence suggests that the VIX and S&P return play a role that is apart from the role played by macro factors in determining changes in credit spreads. To consider whether hedging ineffectiveness is consistent with short term fears in the market, we test the implication of slow-moving capital noted in Duffie (2010). He observes that the defining characteristic of limited intermediary or investment capital is a price reversal over a short period. Nagel (2011) also focuses on price reversals in his investigation. In our setting, a price reversal implies that a passive hedge kept in place over a short period may be more effective than dynamic hedging, contrary to the implication of contingent pricing theory. We, indeed, find that a passive hedge over periods as long as two months is more effective than rebalancing the hedge on a daily basis. Hedging effectiveness increases for all hedge ratios with the greatest improvement at 31% - a three-fold increase over daily dynamic hedging. Notably, the hedge ratio that is the most effective is the one estimated from a univariate regression of credit spreads on the underlying stock return, without controlling for the other variables, including the VIX and market return. The hedging improvement over time provides our final and most direct test. If the VIX is associated with a reduced deployment of capital, then the price reversal and hedging improvement is predictable. Moreover, when more capital that is withdrawn from deployment, the hedging errors will persist for a longer period of time. Thus, the improvement in 4

7 hedging effectiveness over a fixed interval of time will be reduced. This is precisely what we find. The increase in hedging effectiveness is predictable with the VIX, and the higher the VIX, the lower is the improvement in hedging effectiveness over the following two weeks. In related literature, Schaefer and Strebulaev (2008) consider whether the Merton model provides a robust hedge ratio, and conclude that the hedge ratio cannot be differentiated from the empirically observed sensitivity. We corroborate their finding, and, in addition, investigate whether the hedge ratio is effective in practice. Our results suggest that structural models of credit risk are less useful than envisaged simply because of limited liquidity in the market. Finally, a number of papers use the effectiveness of dynamic hedging to evaluate models. Li and Zhao (2007) evaluate term structure models by considering how they hedge interest rate derivatives, while Bakshi, Cao and Chen (1997) evaluate option pricing models by considering how effectively they help to hedge index options. Our focus and application is different not just because we focus on CDS markets, but we use hedging effectiveness to evaluate contingent claim models more broadly. The rest of the paper is as follows. Section 2 discusses the determination of the hedge ratio. Section 3 describes our data and also describes the pricing model used to market the spread to market. Sections 4 present the empirical results. Section 5 provides illustrations on the results. The last section offers brief conclusions. 2 Hedging in Structural Models of Credit Risk 2.1 Hedge Ratio Let A i,t be the value of assets of a firm i with equity value of S i,t. The firm has outstanding debt in the form of a zero-coupon bond of face value F, maturity T, and market value of B i,t. From the absence of arbitrage, A i,t = S i,t + B i,t. (1) In the Merton one factor model, equity and debt prices are impacted only by changes in the firm value. From equation (1), S i,t A i,t + B i,t A i,t = 1. (2) 5

8 Following Schaefer and Strebulaev (2008), define the hedge ratio for the bond, δi,t b, as the amount of equity required to hedge the bond. From (1) and (2), δi,t b = B i,t/ A i,t S i,t, (3) S i,t / A i,t B i,t ( ) ( ) 1 1 = 1 1 (4) i,t L i,t where L i,t is the firm leverage, defined as the market value of debt over the market value of the asset, and i,t is the sensitivity of equity to the firm value. In the Merton (1974) model, is the delta of a European call option with the firm as the underlying asset. In a wide class of models, including Merton (1974), is strictly bounded by 1 prior to maturity of the debt. It follows from (4) that δi,t b is strictly positive, and a long position in the bond can be hedged by shorting the stock. 2.2 Merton Model Hedge Ratios The spread, cs i,t, over the riskfree rate r f t is equal to cs i,t = 1 T ln(f/b i,t) r f t. From equation (4), it follows that the sensitivity of the spread to the equity of the firm is, cs i,t S i,t /S i,t = 1 T ( 1 i,t 1)( 1 L i,t 1). (5) From the Merton model (suppressing the dependency on A t ), i,t = N(d 1 (K i,t, T )), where N( ) is the cumulative normal distribution, and d 1 (K i, T ) = ln(a i,t/k i ) + (r f t y i + σ 2 i /2)T σ i T, (6) where y and σ are the constant dividend yield and the asset volatility, respectively, and K is the default threshold. In the classical Merton model, K is equal to the face value of the bond F. For the classical Merton model, we define the hedge ratio for the credit default swap in 6

9 the Merton model hedge ratio as δ m i,t, δ m i,t = CDS i,t S i,t /S i,t, (7) = cs i,t S i,t /S i,t D i,t, (8) where D i,t is defined as the CDS duration, the dollar change in the value of the swap for a one bps spread change. D i,t is determined by the pricing model used to mark the swap to market; we defer discussion on the mark to market model to a later section. In addition, to the classical Merton model hedge ratio, we also construct a hedge ratio from the Merton model extended to price a coupon bond. Duffie (1999) demonstrates that under reasonable assumptions, the spread on a coupon bond priced at par is equal to the CDS spread. Thus, it may be more accurate to compute the hedge ratio from the extended Merton model. Consider a bond B i,t, t 0, of face value F, maturity T and an annual coupon c (as a fraction of the face value) payable semi-annually on dates T n, n = 1, 2.., 2T. If the bond defaults on a coupon date, T n, then the holder of the bond receives either the firm value or a constant fraction, w, of the contracted cash flow on that date, whichever is less. The firm defaults if the firm value at T n is below a known threshold K i. Under these assumptions, treating this coupon bond as a portfolio of zero coupon bonds as in Longstaff and Schwartz (1995), Eom, Helwege and Huang (2004) provide the value of the coupon bond from the extended Merton model as, B i,0 = where, 2T n=1 e rtn E Q 0 [ ( F 1 [n=2t ] + c ) ( ( 1 [Ai,Tn K] + min wf 1 [n=2t ] + c ) ) ], A i,tn 1 [Ai,Tn <K], 2 2 (9) E Q 0 E Q 0 [ 1[Ai,Tn Ki] ] [ min(u, Ai,Tn )1 [Ai,Tn <Ki] ] = N (d 2 (K i, T n )), = A i,0 e yitn N( d 1 (u, T n )) + u [N(d 2 (u, T n )) N(d 2 (K i, T n ))], where N( ) is the cumulative standard normal distribution, r is the constant riskfree interest rate and d 2 (X i, T n ) = d 1 (X i, T n ) σ i Tn, 7

10 and d 1 has been defined earlier in equation (6). Define c(a i,t ) as the coupon rate that results in the bond being priced at par for a given set of parameters and firm value A i,t. The sensitivity c(a i,t )/ A i,t can be computed numerically. From c(a i,t )/ A i,t, we can estimate the sensitivity of c(a i,t ) to S i,t as, c(a i,t ) S i,t /S i,t = c(a i,t)/ A i,t S i,t / A i,t S i,t. (10) Given the equivalence of the spread on the bond priced at par and the CDS spread, the sensitivity of c(a i,t ) to the stock will equal to the sensitivity of the CDS spread to the stock. Therefore, defining the extended Merton model hedge ratio for firm i as δi,t m, we get the hedge ratio as, δ m i,t = c S i,t /S i,t D i,t, (11) where, as in equation (8), D i,t is the duration of the credit default swap. 2.3 Empirical Hedge Ratio When a single factor model does not determine relative pricing of equity and credit, then hedging credit in the equity market is no longer an act of undertaking an arbitrage as in the Merton model, but a means of reducing the variance of the hedged portfolio. As in the early literature on the hedging of derivatives with basis risk (e.g. Figlewski, 1984), the optimal hedge ratio under a linearity assumption can be computed from a regression of the change in CDS spread against the stock return. Let the empirical hedge ratio, δi,t e, for the credit default swap be defined as the dollar amount of stock required to hedge one CDS contract. Consider the linear regression of the change in CDS spread, CDS i,t = CDS i,t CDS i,t 1 on the stock return, CDS i,t = α i + β i r i,t + ẽ i,t. (12) The slope coefficient, β i, is the sensitivity of the CDS spread to changes in the stock price. To compute the hedge ratio, we convert β i into a dollar sensitivity as follows, δ e i,t = β i D i,t. (13) 8

11 The specification of equation (12) can be extended to include other variables, CDS i,t = α i + β i r i,t + γ X t + ẽ i,t, (14) where X t are additional variables that might impact the CDS spread. In line with previous research (e.g., Collin-Dufresne, Goldstein and Martin, 2001), we include firm-specific variables (changes in leverage and equity volatility), index equity and option market variables (S&P 500 return, change in VIX) and interest rate market variables (changes in 10-year Treasury rate and slope of the yield curve). Below, we will estimate and use the empirical hedge ratio both in-sample and outof-sample (the latter through a rolling regression). Although the in-sample hedge ratio cannot be used in practice, it will serve as a useful benchmark to understand the potential effectiveness of hedging credit risk in the equity markets. 2.4 Hedging Effectiveness As in Figlewski and Green (1999), we assume that the objective of the financial institution is to minimize the daily volatility of its hedged CDS portfolio position. Suppose that at time t, the market maker hedges a portfolio of CDS contracts of N t names with the underlying equity, and make no additional trade until t + κ. At t + κ, the position is closed out and the hedging error over the κ-day period at time t is computed as, e t (κ) = 1 N t N t ( )] t+κ [( 1) c (CV i,t+κ CV i,t ) + ( 1) c δ i,t (P i,t+κ + Div i,τ )/P i,t 1. i=1 τ=t where c {1, 2} denotes whether the position holds the CDS short (c = 1) or long (c = 2). We alternate daily between long and short positions to minimize the impact of non-linearity on the hedge. CV i,t CV(CDS i,t ) is the cash settlement, or mark to market, value of the swap. δ i,t is the dollar amount of equity of firm i required to hedge one CDS contract at time t, computed either from empirically observed credit-equity sensitivity or from the Merton model as discussed in previous subsections. P i,t is the stock price at time t and Div i,t is the cash dividend received at time t. Our focus is on understanding hedging effectiveness when the position is dynamically hedged on a daily basis, κ = 1. We will also consider cumulative hedging errors over longer horizons corresponding to κ > 1 in a later section. (15) 9

12 Following Bertsimas, Kogan and Lo (2000), we use root-mean-squared error (RMSE) as the summary statistic for hedging errors over a period t τ to t. For a given κ, the RMSE is defined as RMSE (τ) t (κ) = 1 τ κ t κ+1 s=t τ e 2 s(κ). (16) The RMSE is equal to the standard deviation when the mean hedging error is zero. For comparison, we also compute the RMSE of the portfolio when the swap is not hedged, δ i,t 0, and denote it as RMSE. We define our measure of hedging effectiveness as the reduction in the RMSE as a result of hedging, E (τ) t (κ) = 1 RMSE (τ) t (κ)/ RMSE (τ) t (κ) (17) When the hedge works perfectly, then E = 1. If E is negative, it implies that hedging increases volatility relative to the unhedged position. In general, we expect the hedge to be imperfect, and E to be bounded between zero and one. 3 Data and Implementation 3.1 Data We get our data on credit default swap spreads (CDS) from Markit. Specifically, we use daily observations for five-year credit default swap on senior, unsecured debt of non-financial firms. The 341 obligors that enter our sample are components of the Dow Jones CDX North America Investment Grade (CDX.NA.IG), the Dow Jones CDX North America High Yield (CDX.NA.HY), CDX North America Crossover (CDX.NA.XO), over 2001 to Our sample ends on March 31st, 2009, right before the Big Bang of April 2000 which changed pricing conventions in the CDS market. Of the 341 firms, 34 firms could not be matched to CRSP and Compustat, 92 firms were delisted during this period, and 8 firms have less than a year of data, leaving us with a final sample of 207 firms. Of these, 112 obligors have an average rating of investment grade (AAA, AA, A, and BBB), and the remaining 95 obligors are below investment grade (BB, B, and CCC). The data of CDS spread has different DocClauses, including MM, MR, XR, CR, for each firm. For below investment 10

13 grade firms, we use XR; for investment grade firms, we use MR or XR, if MR is missing. Stock prices and outstanding number of shares are adjusted for stock dividends and splits, and obtained from CRSP. We take into account cash dividends. Accounting information required to compute leverage and face value of debt is collected from Compustat Quarterly file. If the debt data is missing, we use the closest available quarterly data in the same year, or else use the annual data. Interest rate data including Treasury, LIBOR and swap rates are from the Federal Reserve. Panel A of Table 1 reports the summary statistics of our sample. Across rating classes, the average CDS spread increases from 52.9 bps for A rated firms to bps for B rated firms. As would be expected, higher rated firms are larger and have lower leverage. Market capitalization ranges from 46.8 billion dollars to 3.9 billion dollars, and leverage from 0.14 to 0.38 across the rating classes. Panel B reports the summary statistics of daily changes in CDS spread. The average daily spread change for the whole sample is 0.48 bps. Daily spread changes are positive on average in line with the overall increase in credit risk over this period. Spread changes for lower rated classes are larger and more volatile. 3.2 Merton Model Implementation To estimate the asset volatility and firm value, we follow Duffie, Saita and Wang (2007) to jointly solve for the asset value and volatility by iterating the following two equations over a rolling one-year period, S i,t = A i,t N[d 1 ] K i e rt N[d 2 ], σ i = stdev [ln A i,t ln A i,t 1 ]. As is now standard, the default threshold, K i, is defined as the sum of the firm s book value of short-term debt [Compustat item 45] and half of its long term debt [Compustat item 51] in the previous quarter. The 1-year Treasury rate is used as the riskfree rate. If the number of observations in the rolling window is less than 30, then the volatility and asset value are set as missing. Panel A of Table 1 also reports the average asset volatilities estimated through the above procedure. The average asset volatility over our sample period is 31%, ranging from 28% to 35%. As would be expected, the higher rated firms have lower asset volatility. 11

14 3.3 CDS Pricing Model We use the ISDA CDS Standard model to mark the credit default swap to market. Documentation of the model as well as the source code for the model is available at The provider of our CDS quotes, Markit, maintains an implementation (Markit Converter) as does Bloomberg ( CDSW ). Using the model, we define the duration D i,t as the average change in the mark to market value for a ± 1 bps change in spread as follows, D i,t = 1 2 ( CV(CDS i,t + 1) CV(CDS i,t ) + CV(CDS i,t ) CV(CDS i,t 1) ). (18) The change in the value of the CDS will depend on the level of the spread. Figure 1 plots the relation between the duration and the level of the spread. For example, when the spread is below 250 bps, a one bps change in an investment grade swap initially at par results in a change in the settlement value of the CDS of between $4,000 and $4,500 for a notional of $10 mm. The computed duration, D i,t is used to estimate the hedge ratio in equations (8), (11), and (13). 4 Results 4.1 Hedge Ratio We begin by reporting the estimated hedge ratios for each of the four specifications discussed earlier, viz., the classical and extended Merton models, and the empirical hedge ratio from univariate and multivariate specifications, respectively Merton Model Hedge Ratio On each date, t, we compute the hedge ratio for the rating class as follows. First, we compute spreads and sensitivities on a firm-by-firm basis for the classical and extended Merton models. Then, for each day, we calculate the average spread and sensitivity across all firms within each rating class. The duration is now computed at the average spread for the rating class, and the hedge ratio is set equal to the product of the duration and average sensitivity for each of the rating classes. The CDS of each firm is hedged using the hedge ratio ratio of its rating class. Using the hedge ratio for the rating class as opposed to 12

15 firm-specific hedge ratios reduces estimation noise. When we instead use the firm-specific sensitivity and spread to compute the hedge, we find that the RMSE is higher and the hedge is less effective. Table 2 reports the time-series mean and standard deviation of the spreads, sensitivities and hedge ratios for each rating category. First, consider the estimate of the spreads from the Merton model. Both the classical and extended Merton model underestimate the actual spreads. Consistent with theory, the extended Merton model provides estimates of spreads that are closer to the observed spreads. For example, for BBB rated firms, the observed average CDS spread is 98 bps. In comparison, the average spread from the classical and extended Merton model is 36 bps and 50 bps, respectively. Nevertheless, both models underestimate the spread, consistent with Eom, Helwege and Huang (2004), who also document in their Table 3 that the Merton model underestimates bond spreads by about 50%. The pricing error in our calibration of classical Merton ranges from -47% to -71% across the rating classes, and from -31% to -56% for the extended Merton. Next, consider the sensitivity of the spread to the stock return, and corresponding hedge ratios. The average sensitivity of the spread to equity computed with the extended Merton model is higher than that from the classical model. Across all firms, the average sensitivity of the spread for a one percent stock return is 0.69 bps and 0.81 bps for the Merton and extended Merton models, respectively. The difference in sensitivities impacts the estimated hedge ratios. The corresponding hedge ratios, the amount of equity required to hedge one CDS contract of $10 million notional, are $282,865 and $335,525 for the classical and extended Merton models, respectively. The sensitivity of the spread to the stock return increases monotonically as the rating declines. For example, the hedge ratio for BB firms is about two-fold that for BBB firms Empirical Hedge Ratio Next, we estimate the slope coefficient β in the univariate specification of equation (12) and the multivariate specification of (14), respectively. We estimate β through a panel regression allowing for firm effects using weekly changes in CDS spreads and stock return. β estimated from a weekly regression results in lower hedging errors than if estimated from daily or monthly spread changes. Table 3 reports the results. Across all rating classes, the coefficient on the stock return is negative and statistically significant at the highest levels. 13

16 For the univariate model, the sensitivity of spread changes to the stock return ranges from 0.70 bps for firms rated A to 4.22 bps for firms rated B. For the multivariate model, excepting firms rated B or below, the magnitude of the sensitivity is lower. For example, the slope coefficient of rating BBB is 0.63 bps for the multivariate regression, only half of the 1.29 bps for the univariate specification. Following the procedure in Merton hedge ratio, the duration at the average CDS spread level for each rating class on each day is used to calculate the empirical hedge ratios. Table 3 also reports the time-series mean for the duration and hedge ratio for each rating class. The average amount of equity required to hedge a CDS contract of notional $10 million is $1,196,485 using the sensitivity from the univariate specification, and $1,158,299 for the multivariate specification. Consistent with the sensitivities previously estimated from the Merton models, the absolute magnitude of the sensitivity increases monotonically as the rating of the firm declines. Although the Merton model consistently underestimates the level of the spread, Schaefer and Strebulaev (2008) find that the sensitivities from the Merton model are not significantly different from those estimated from empirically observed bond spreads. We implement a similar regression test by running the following panel regression in rating j for our data, CDS i,t = α j + λ j β M j,t r i,t + ẽ i,t. (19) where βj,t M is the average sensitivity for all firms in rating j at time t calculated from the Merton model. If the Merton model correctly estimates the sensitivity of the CDS spread to equity, then λ j would be equal to one. Panel B reports the results of the regression. we find that the hypothesis, λ j = 1, is not rejected for any rating class for the extended Merton model. The classical Merton model fares slightly worse, rejecting the hypothesis for firms of rating class B and below. Overall, our results are consistent with Schaefer and Strebulaev (2008) that the empirically estimated sensitivities are close to those estimated from the Merton models. That is, there does not appear to be significant model risk in the estimation of the hedge ratios. 4.2 Hedging Effectiveness Table 4 reports RMSE of the daily hedging errors under each of the four hedge ratios. In computing the RMSE, to avoid a negative bias, we do not include a day if the CDS spread has does not change. The RMSE of the unhedged position serves as a benchmark 14

17 to calculate hedging effectiveness E. The mean hedging error in each case is close to zero, so the RMSE can also be interpreted as the volatility of the market maker s daily P&L RMSE across Models As shown in panel A of Table 4, the average RMSE across the entire sample ranges from $16,544 to $17,497 across the four hedge ratios, with the lowest (highest) hedging error arising from the hedge ratio estimated from the empirical multivariate (classical Merton) specification. How well does the Merton model perform relative to the empirical multivariate hedge ratio? The RMSE for the extended Merton model over the entire sample is $17,353, only about 5% higher than the RMSE from the multivariate specification. Interestingly, the RMSE from the classical Merton model is also about the same as that for the extended Merton model differing by less than 1% even though the extended Merton model hedge ratios are closer to the multivariate empirical hedge ratio. Overall, on average, both the Merton models are about as useful for hedging purposes, and their hedging effectiveness is close to that of the empirically estimated hedge ratios. Across rating classes, we see wider differences across the models. The hedge ratio from the multivariate empirical model performs better for higher rating classes than either of the hedge ratios from the Merton models, but its performance deteriorates for the lowest rating class. The empirical hedge ratio results in 9% lower volatility than the Merton model hedge ratios for investment grade firms, but results in 3% higher volatility for firms rated B or below. Sub-period results reported in Panels B to D are consistent with conclusions based on the entire sample period. Except for the financial crisis period of , the Merton model hedge ratios perform about the same, or even better, than the empirically estimated hedge ratios. Moreover, the Merton model always dominates the empirically estimated hedge ratios for the riskiest firms. Overall, our first set of findings supports Schaefer and Strebulaev s conclusion that Merton model hedge ratios are not different from those estimated from data. On average, across the entire sample, Merton model hedge ratios result in RMSE of about the same order of magnitude as the empirical hedge ratio, and within sub-samples, often improves upon the latter. 15

18 4.2.2 Hedging effectiveness across models To evaluate the hedging effectiveness, E, Table 4 reports the RMSE of the unhedged portfolio in the first column. Surprisingly, as can be observed from Panel A, the RMSE of the unhedged portfolio of $18,341 over the entire sample is not much different from that of the best hedged position. Across the entire sample, hedging using the multivariate hedge ratio reduces RMSE by only 9.8%, while the two Merton model hedge ratios reduce the volatility by around 5%. Results across rating classes in Panel A are consistent with the overall sample. Although hedge ratios tends to perform better for lower rated firms than higher rated for all the four models, the best E through hedging is less than 10%. Sub-period results reported in Panels B to D provide consistent results. The best E of 12% (using the multivariate hedge ratio) occurs in the the period corresponding to the most volatility CDS markets. In the least volatile period corresponding to , the RMSE reduces at most by 6%. Indeed, hedging often increases volatility in sub-samples. For example, across the entire sample period, the Merton model hedge ratios increase volatility for above investment grade firms. In Panel A, the RMSE for the hedged portfolio for A rated firms is about 6% higher than the unhedged portfolio when the extended Merton model hedge ratio is used to construct the hedge. In summary, consistently across sub-samples and sub-periods, the equity hedge is of limited effectiveness in reducing the volatility of the CDS portfolio across all models. At best, across sub-samples, the reduction in volatility is 12%. At worst, hedging increases volatility of the CDS portfolio compared with leaving the portfolio unhedged. 4.3 Out of Sample Empirical Hedge Ratio The empirically estimated sensitivities that we used to construct the hedge cannot be used in practice as these were estimated within the sample period. How do out-of-sample hedge ratios perform? To check, we estimate the hedge ratio from a rolling regression over the previous one year. Using the previous one year not only allows us to check the out-of-sample usefulness of the empirical hedge ratio, but also allows a fairer comparison with the Merton model. The results, reported in Table 5, are not encouraging. Hedging reduces RMSE across 16

19 all firms by only 5.38% - less than the reduction observed when the hedge ratio was estimated in-sample. Moreover, there is considerable variation in hedging effectiveness of the sensitivities estimated from the one-year rolling regressions. While the hedge ratio from multivariate regression reduces the volatility by 10% in the subperiod , it results in 32% higher RMSE than the unhedged portfolio for subperiod In comparison, the Merton model hedge ratios provide a more consistent hedging performance across sub-samples. 4.4 Alternative Measure of Risk The RMSE may not be the appropriate risk measure, especially when the distribution of hedging errors does not follow a normal distribution. In Figure 2, we plot the distribution of the hedging error from Section 4.2. The distribution of hedging error is not normally distributed. Both the distribution of the unhedged and hedged CDS portfolio have fat tails. The Kolmogorov-Smirnoff test rejects the hypothesis of a normal distribution at the 5% significance level. As an alternative to volatility, we consider the Value at Risk (VaR). Investigating whether hedging can reduce the VaR is important especially as regulators often set capital requirements based on this measure. We construct the VaR at the 99 % by averaging the absolute hedging error at the 0.5% percentile and 99.5% percentile. The daily VaR for the unhedged portfolio is $73,000. Hedging in the equity market is slightly more effective at the tail. Across the entire sample, using the in-sample empirical hedge ratio estimated from the multivariate specification reduces the VaR by 12% as opposed to the 9.8% reduction in volatility. The hedge ratios from the Merton model have less effectiveness than the empirical hedge ratio with the VaR reducing by only about 2.5%. Hedging effectiveness using the Merton model hedge ratios varies across sub-periods, ranging from 2.5% to 9%. Hedging effectiveness also varies across rating classes. Consistent with our previous observations with respect to the RMSE, the Merton model hedge ratios are most effective for the riskiest firms, but can increase the VaR for the higher rated firms. Thus, for example in Panel A, while the extended Merton model reduces the VaR by 5% for BB rated firms, it increases the VaR by about the same amount for BBB rated firms. In summary, the conclusions using VaR are consistent with those using the RMSE. Hedging credit risk in the equity markets is of limited effectiveness across the entire sample, 17

20 although the Merton model performs creditably in comparison with the in-sample estimated empirical sensitivity. 5 The Role of VIX Our results indicate that hedging in the equity market is of limited effectiveness in reducing the volatility or VaR of a CDS market-maker. The ineffectiveness of the equity hedge is not because of model risk as the performance of the Merton model is about the same order of magnitude as the in-sample estimated hedge ratio, and is often even superior. We now investigate the role of the VIX in determining hedging effectiveness. 5.1 Are Hedging Errors Predictable? Hedging errors may be explained by two alternative hypotheses. First, hedging errors may occur because over the holding period, changes in macro expectations result in relative re-pricing of equity and credit risk. Hackbarth, Miao, and Morellec (2006), Chen, Collin- Dufresne and Goldstein (2009), Chen (2010), and Bhamra, Kuehn, and Strebulaev (2010a, 2010b) provide models that demonstrate why changes in the business cycle can impact credit spreads. Second, hedging errors may be because of limited liquidity in the market. If hedging errors are because of the macroeconomy, then hedging errors should be contemporaneously related to changes in the macro variables. If hedging errors are because of limited funding liquidity, then hedging errors should be predictable by factors that are correlated with the funding liquidity. We first investigate whether the RMSE of a portfolio of hedged credit default swaps can be explain by contemporaneous changes in systematic and macro variables. We estimate a time-series regression of the RMSE of the portfolio of all firms at a monthly frequency as follows, log RMSE t = α + β 0 log RMSE t 1 + +β 1 V IX t + β 2 r m,t + β 3 SMB t + β 4 HML t + β 5 slope t + β 6 r 10 t + ẽ t. (20) where RMSE in month t is computed as in equation 16 using daily hedging errors. V IX t is the change in VIX index over the month; r m,t, SMB t, HML t are market return and 18

21 returns on the Fama-French factors in month t; slope t and rt 10 are the change in the 10-year Treasury rate and slope of yield curve, respectively, over the month. We include a lagged term to control for autocorrelation. Panel A of Table 7 reports the results over the entire sample period and the pre-crisis period ( ). The lagged RMSE and the set of macroeconomic and systematic variables together explain 81.9% (72.3%) variation in RM SE for the full period (pre-crisis period). Of the variables, the FF factors are not significant (although the HML factor is weakly significant at the 10% level for pre-crisis period). The RMSE is significantly related to the VIX index and the S& P return. The sign of the coefficient on the VIX (S& P return) is positive (negative), indicating that an increase in the VIX and a decrease in S& P return increases the RMSE. Importantly, the coefficient on the slope of the yield curve is significant, indicating that a steepening of the yield curve results in greater hedging errors. The 10-year Treasury bill is significant over the entire sample period that includes the financial crisis, but not significant over the pre-crisis period. The significance of the yield curve is consistent with the literature that indicates macro expectations of the business cycle impact credit spreads. Default boundaries may be especially sensitive to a perceived reduction in economy-wide growth rates, and this may lead to a re-pricing of credit risk when the slope of the term structure steepens. It is, however, unclear that such an explanation would hold for the VIX. The VIX is especially sensitive to market fears over short horizon, and market fears are correlated with funding liquidity. If VIX impacts hedging errors through this channel, then hedging errors should be predictable with the level of the VIX. To check this, we estimate the following regression of the RMSE on the 1-month lag of the above market and macroeconomic variables, log RMSE t = α + β 0 log RMSE t 1 + +β 1 V IX t 1 + β 2 r m,t 1 + β 3 SMB t 1 + β 4 HML t 1 + β 5 slope t 1 + β 6 r 10 t 1 + ẽ t. (21) Regression results are reported in Panel B of table 7. As can be observed, the VIX index and the S& P return are both significant at the highest levels. However, neither of the variables based on the yield curve are significant. Of the Fama-French factors, HM L is significant over the entire sample period, Overall, regression results show that, as expected, macroeconomic and market variables 19

22 are contemporaneously related to the hedging errors. However, only factors related to funding liquidity such as VIX and S&P return have predictive power. The traditional macro variables constructed of the yield curve cannot predict hedging errors. Thus, the evidence suggests that the VIX and S&P return play a role that is apart from the role played by macro factors in determining changes in credit spreads. 5.2 VIX and Market Fear How does the VIX affect hedging effectiveness and credit spreads in terms of market fears? Motivated by Brunnermeier, Nagel and and Pedersen (2008) and Nagel (2012), we relate market fears to arbitrage activity across credit and equity markets. When market fears are high, both arbitrageurs and market makers could reduce arbitrage capital deployed in the credit default swap market rather than take advantage of pricing discrepancies, or provide liquidity to the market. Slow-moving capital does not replace the reduction in capital deployed by existing institutions. Duffie (2010) provides a comprehensive discussion of the possible reasons why capital may be slow moving. Kapadia and Pu (2012) also document pricing discrepancies between the equity and credit markets which reduce over time, consistent with the notion that slow moving capital prevents instantaneous arbitrage. We explore the hypothesis in the following two steps. First, we provide evidence on slowmoving capital that arbitrages across credit and equity market. Second, we provide direct evidence on the role of the VIX in slowing down market integration between equity and credit markets Evidence on slow-moving capital Duffie (2010) observes that the defining characteristic of limited intermediary or investment capital is a price reversal over a short period. Nagel (2011) also focuses on price reversals in his investigation. In our setting, a price reversal implies that a passive hedge kept in place over a short period may be more effective than dynamic hedging. To investigate, we consider cumulative hedging errors over horizons longer than one day, that is e t (κ) when κ > 1. Results for horizons corresponding to κ {5, 10, 25, 50} are reported in table 8. As would be expected, the magnitude of RMSE increases with horizon. The rate of increase is slightly greater than κ. For example, the RMSE of the unhedged portfolio over a 50 20

23 (25) business day horizon is 3.32 (2.42) times that of the RMSE over 5 business days. Next, with regard to hedging effectiveness, each of the four hedge ratios are effective in reducing the RMSE. For example, the hedging effectiveness E using the extended Merton model is around 17% for a horizon of 5 business days, and 21% for the 10-day horizon. The hedge ratio from the classical Merton model also shows similar magnitudes of effectiveness. Thus, a passive hedge held over a period of time as long as two months significantly improves hedging performance compared with a hedge position that is rebalanced every day. Much of the improvement in hedging effectiveness occurs over a short period. Panel A of Figure 6 plots the RMSE of the cumulative hedging error from one to 50 days. The gap between RMSE of unhedged portoflio and that of the hedged increases with horizon. Panel B of Figure 6 plots hedging effectiveness as a function of horizon. There is a steep improvement in E from 1-day to 10-days over all hedge ratios, and then a slower improvement for longer horizons upto 50 days. The improvement over the longer period is limited to the two empirical hedge ratios. Significantly, the hedge ratio from univariate regression provide the best performance now, even though it works relatively poorly in daily hedging. Overall, the results indicate that the passive equity hedge is more effective over longer horizons. This is remarkable because standard textbook theory indicates that a passive hedge worsens with time because of gamma risk. The implication of the finding is that the lack of short-term market integration on hedging is larger than the impact of a stale hedge ratio, and suggests that slow-moving capital has an important role to play in integrating equity credit markets Role of VIX in determining market integration When less arbitrage capital is deployed in the market, relative equity-credit market pricing errors will persist for a longer time and therefore the improvement in hedging effectiveness over a fixed interval of time (or the speed of the improvement with horizons) will be reduced. If the VIX is associated with a reduced deployment of capital in credit default swaps market, then the speed of price reversal and hedging improvement with horizons is predictable. We test the hypothesis by relating the speed of improvement in hedging effectiveness with horizon, which is notated as υ, to the VIX in a panel regression as follows, υ j,t = α + β 1 V IX t 1 + β 2 r m,t 1 + β 3 SMB t 1 + β 4 HML t 1 + I j + ẽ j,t, (22) 21

24 where υ j,t is hedging improvement of rating portfolio j over the days of month t. The independent variables now include S&P 500 return, VIX, and the Fama-French factors. We include these factors as previous results demonstrated that these variables predicted the RMSE. Clustered standard errors are used to calculate t stat. We measure υ t for each rating class in following two ways. First, we measure the improvement in hedging effectiveness as the difference between hedging effectiveness computed using 15-day cumulative hedging errors, and the hedging effectiveness computed from daily hedging errors, E t (15) E t (1). Second, we estimate the regression, E t (κ) = α + ηκ + ẽ κ, where κ {1, 3, 5, 10, 15, 20}. The slope coefficient η is our second measure. Regression results are reported in table 9. Over the full period, the coefficient on the VIX is consistently significant for both measures indicating that the improvement in hedging effectiveness is predictable by the VIX. The negative sign indicates that a higher VIX predicts a slower improvement in hedging effectiveness over a particular horizon. Only the VIX is consistently significant across both specifications, although the market return and Fama-French factors show occasional significance, suggesting that these factors may have a secondary role to play. In summary, we find that the increase in hedging effectiveness is predictable with the VIX, and the higher the VIX, the lower is the improvement in hedging effectiveness over a given period. Our results are consistent with the interpretation of VIX as a fear index, as in times of market stress, integration of equity-credit markets occurs at a slow rate. 6 Conclusion We examine the role of the VIX in explaining credit spreads from the perspective of the effectiveness with which credit default swaps can be hedged in the equity market. Our surprising finding is that hedging in the equity markets is of limited effectiveness in reducing the volatility of a CDS portfolio at a daily frequency. Over our entire sample, hedging in the equity market reduce daily volatility about 10%. Moreover, in sub-samples, hedging can increase the RMSE. The lack of effectiveness is not because of model risk as the Merton model hedge ratios are similar to the hedge ratios based on the empirical observed sensitivity of CDS spread to stock return. Instead, we document that hedging errors in portfolios of credit default swaps are predictable with the VIX index. Moreover, we also demonstrate 22