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Understanding the Role of VIX in Explaining
Movements in Credit Spreads
Xuan Che Nikunj Kapadia 1
First Version: April 3, 2012
1University of Massachusetts, Amherst. Preliminary; comments are welcome. Please addresscorrespondence to Nikunj Kapadia Isenberg School of Management, University of Massachusetts,Amherst, MA 01003. Email: [email protected].
Abstract
Why does the VIX and market return explain changes in credit spreads? Existing litera-
ture 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
“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 return1Consistent 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 indexoptions increase by over 90% when the hedge is rebalanced once every two days instead at a daily frequency.
1
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 de-
fault 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 suffi-
cient 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.2Chen (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
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, fluc-
tuate 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 mar-
ket 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 2009. 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 sub-
periods. The best hedging performance is in the financial crisis period of 2008-09, 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
Strebulaev (2008), Merton model hedge ratios are not statistically different from the in-
sample 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 con-
sider 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 deploy-
ment, the hedging errors will persist for a longer period of time. Thus, the improvement in
4
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 struc-
tural 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 Ai,t be the value of assets of a firm i with equity value of Si,t. The firm has outstanding
debt in the form of a zero-coupon bond of face value F , maturity T , and market value of
Bi,t. From the absence of arbitrage,
Ai,t = Si,t +Bi,t. (1)
In the Merton one factor model, equity and debt prices are impacted only by changes
in the firm value. From equation (1),
∂Si,t∂Ai,t
+∂Bi,t∂Ai,t
= 1. (2)
5
Following Schaefer and Strebulaev (2008), define the hedge ratio for the bond, δbi,t, as the
amount of equity required to hedge the bond. From (1) and (2),
δbi,t =∂Bi,t/∂Ai,t∂Si,t/∂Ai,t
Si,tBi,t
, (3)
=(
1∆i,t− 1)(
1Li,t− 1)
(4)
where Li,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 δbi,t 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, csi,t, over the riskfree rate rft is equal to csi,t = 1T ln(F/Bi,t)−rft . From equation
(4), it follows that the sensitivity of the spread to the equity of the firm is,
∂csi,t∂Si,t/Si,t
= − 1T
(1
∆i,t− 1)(
1Li,t− 1). (5)
From the Merton model (suppressing the dependency on At), ∆i,t = N(d1(Ki,t, T )), where
N(·) is the cumulative normal distribution, and
d1(Ki, T ) =ln(Ai,t/Ki) + (rft − yi + σ2
i /2)Tσi√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
the Merton model hedge ratio as δmi,t,
δmi,t =∂CDSi,t∂Si,t/Si,t
, (7)
=∂csi,t
∂Si,t/Si,tDi,t, (8)
where Di,t is defined as the CDS “duration”, the dollar change in the value of the swap for
a one bps spread change. Di,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 Bi,t, t ≡ 0, of face value F , maturity T and an annual coupon c (asa fraction of the face value) payable semi-annually on dates Tn, n = 1, 2.., 2T . If the bonddefaults on a coupon date, Tn, then the holder of the bond receives either the firm value ora constant fraction, w, of the contracted cash flow on that date, whichever is less. The firmdefaults if the firm value at Tn is below a known threshold Ki. 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 theextended Merton model as,
Bi,0 =2T∑
n=1
e−rTnEQ0
[F(1[n=2T ] +
c
2
)1[Ai,Tn≥K] + min
(wF
(1[n=2T ] +
c
2
), Ai,Tn
)1[Ai,Tn <K]
],
(9)
where,
EQ0
[1[Ai,Tn≥Ki]
]= N (d2(Ki, Tn)) ,
EQ0
[min(u,Ai,Tn)1[Ai,Tn <Ki]
]= Ai,0e
−yiTnN(−d1(u, Tn)) + u [N(d2(u, Tn))−N(d2(Ki, Tn))] ,
where N(·) is the cumulative standard normal distribution, r is the constant riskfree interest
rate and
d2(Xi, Tn) = d1(Xi, Tn)− σi√Tn,
7
and d1 has been defined earlier in equation (6).
Define c(Ai,t) as the coupon rate that results in the bond being priced at par for a
given set of parameters and firm value Ai,t. The sensitivity ∂c(Ai,t)/∂Ai,t can be computed
numerically. From ∂c(Ai,t)/∂Ai,t, we can estimate the sensitivity of c(Ai,t) to Si,t as,
∂c(Ai,t)∂Si,t/Si,t
=∂c(Ai,t)/∂Ai,t∂Si,t/∂Ai,t
Si,t. (10)
Given the equivalence of the spread on the bond priced at par and the CDS spread, the
sensitivity of c(Ai,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 δmi,t, we get
the hedge ratio as,
δmi,t =∂c
∂Si,t/Si,tDi,t, (11)
where, as in equation (8), Di,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, δei,t, 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, ∆CDSi,t = CDSi,t − CDSi,t−1 on the stock return,
∆CDSi,t = αi + βiri,t + ei,t. (12)
The slope coefficient, βi, is the sensitivity of the CDS spread to changes in the stock price.
To compute the hedge ratio, we convert βi into a dollar sensitivity as follows,
δei,t = βiDi,t. (13)
8
The specification of equation (12) can be extended to include other variables,
∆CDSi,t = αi + βiri,t + γ Xt + ei,t, (14)
where Xt 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 vari-
ables (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 out-
of-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 Nt 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,
et(κ) =1Nt
Nt∑i=1
[(−1)c (CVi,t+κ − CVi,t) + (−1)cδi,t
((Pi,t+κ +
t+κ∑τ=t
Divi,τ )/Pi,t − 1
)].
(15)
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. CVi,t ≡ CV(CDSi,t) is the cash settlement, or mark to market, value of the
swap. δi,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. Pi,t is the stock price at time t and Divi,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.
9
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) =
√√√√ 1τ − κ
t−κ+1∑s=t−τ
e2s(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,
δi,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 2009. 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
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 infor-
mation 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 605.5 bps for 6 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,
Si,t = Ai,tN [d1]−Kie−rTN [d2],
σi = stdev [lnAi,t − lnAi,t−1] .
As is now standard, the default threshold, Ki, 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
3.3 CDS Pricing Model
We use the ISDA CDS Standard model to mark the credit default swap to market. Docu-
mentation of the model as well as the source code for the model is available at www.cdsmodel.com.
The provider of our CDS quotes, Markit, maintains an implementation (Markit Converter)
as does Bloomberg (“CDSW”). Using the model, we define the duration Di,t as the average
change in the mark to market value for a ± 1 bps change in spread as follows,
Di,t =12
(|CV(CDSi,t + 1)− CV(CDSi,t)|+ |CV(CDSi,t)− CV(CDSi,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, Di,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.
4.1.1 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
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.
4.1.2 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
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,
∆CDSi,t = αj + λjβMj,tri,t + ei,t. (19)
where βMj,t 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 λj would be equal to one. Panel B reports the results of the regression.
we find that the hypothesis, λj = 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
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.
4.2.1 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 2008-09, 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
4.2.2 Hedging effectiveness across models
To evaluate the hedging effectiveness, E, Table 4 reports the RMSE of the unhedged port-
folio 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 2008-09 - the period corresponding to the most
volatility CDS markets. In the least volatile period corresponding to 2004-07, 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
all firms by only 5.38% - less than the reduction observed when the hedge ratio was esti-
mated 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 2008-2009, it results
in 32% higher RMSE than the unhedged portfolio for subperiod 2002-2003. In compari-
son, 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
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,
logRMSEt = α+ β0 logRMSEt−1 + +β1∆V IXt + β2rm,t
+ β3SMBt + β4HMLt + β5∆slopet + β6∆r10t + et. (20)
where RMSE in month t is computed as in equation 16 using daily hedging errors. ∆V IXt
is the change in VIX index over the month; rm,t, SMBt, HMLt are market return and
18
returns on the Fama-French factors in month t; ∆slopet and ∆r10t 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 (2001- 2007). The lagged RMSE and the set of macroeconomic and systematic
variables together explain 81.9% (72.3%) variation in RMSE 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 follow-
ing regression of the RMSE on the 1-month lag of the above market and macroeconomic
variables,
logRMSEt = α+ β0 logRMSEt−1 + +β1V IXt−1 + β2rm,t−1
+ β3SMBt−1 + β4HMLt−1 + β5slopet−1 + β6r10t−1 + et. (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, HML is
significant over the entire sample period,
Overall, regression results show that, as expected, macroeconomic and market variables
19
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 slow-
moving 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.
5.2.1 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 et(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
(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.
5.2.2 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,
υj,t = α+ β1V IXt−1 + β2rm,t−1 + β3SMBt−1 + β4HMLt−1 + Ij + ej,t, (22)
21
where υj,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 υt for each rating class in following two ways. First, we measure the im-
provement 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, Et(15) − Et(1). Second, we estimate the regression, Et(κ) = α + ηκ + eκ,
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
that the VIX explains the rate at which equity-credit markets get integrated over time. In
summary, our findings suggests that the VIX plays a role consistent with its role as a “fear
index” in explaining credit spreads. In times of market stress, there is less liquidity in the
markets, and this results in lower equity-market integration.
What is the implication of our results for structural models of credit risk? Structural
models not only indicate which variables are important for pricing credit risk but, as im-
portantly, indicate the set of variables that should not enter the pricing kernel. However,
the restriction on the factors entering the pricing kernel depends on whether credit risk can
be dynamically hedged. Our results provide evidence in that the VIX is important for ex-
plaining both credit spread changes and hedging ineffectiveness because of limited arbitrage
activity rather than as a proxy for the business cycle. Thus, a (frictionless) contingent claim
model can at best do a limited job of explaining credit spreads in practice. Future research
should consider how an explicit role for funding liquidity be included in pricing models.
23
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26
Table 1: Summary Statistics
This table reports the summary statistics of five important variables for 207 firms overJanuary 2001 to March 2009. Panel A reports the mean and standard deviation of CDSspread, market cap, asset volatility and leverage. CDS Spread is given in basis points.Market Cap (billion dollars) is the product of the stock price and the outstanding numberof shares. Asset volatility is computed as noted in text of paper. Leverage is defined as theratio of the book value of debt (debt in current liabilities plus long term debt) to the sumof the book value of debt and market equity value. Panel B reports summary statistics ofdaily changes in CDS spread in basis points. The time-series averages of these variables arecalculated first for each firm, and then the statistics in the cross-section are reported foreach rating class and the whole sample. “All” refers to summary statistics of the variablesacross all firms in our portfolio.
Panel A: Summary Statistics of the SampleCDS (bps) Market Cap (B $) Asset Volatility Leverage
Rating N Mean SD Mean SD Mean SD Mean SD> A 33 52.9 22.7 46.8 46.8 0.28 0.06 0.14 0.09BBB 79 98.1 51.9 15.6 17.4 0.28 0.07 0.20 0.11BB 56 251.4 132.0 6.1 5.5 0.34 0.10 0.31 0.186 B 39 605.5 475.5 3.9 3.9 0.35 0.09 0.38 0.15ALL 207 228.0 293.2 15.8 25.9 0.31 0.09 0.25 0.16
Panel B: Summary Statistics of Daily Changes in CDS Spread (bps)Rating Mean SD Skew Kurtosis P95 P5> A 0.05 0.09 2.13 4.88 0.32 -0.03BBB 0.10 0.22 1.65 4.78 0.55 -0.14BB 0.28 0.77 3.80 20.15 1.64 -0.336 B 1.88 3.58 3.45 14.06 10.38 -0.33ALL 0.48 1.73 7.43 68.93 2.77 -0.19
27
Table 2: Merton Model Hedge Ratio
This table reports summary statistics of spreads, sensitivities and hedge ratios calculatedusing the Merton models for each rating class and the whole sample. Spread (bps) is thecredit spread computed with Merton model. Sensitivity is the spread change per unit ofstock return which is computed from equation (5) for the classic Merton, and from equation(10) for the extended Merton. The hedge ratio (dollars) is the amount of equity required tohedge one CDS contract of notional of $10 mm, and is equal to the product of the sensitivityand duration. The time-series mean and standard deviation are reported here.
Rating Classical Merton Extended MertonMean SD Mean SD
Spread (bps)>A 15.2 27.9 23.4 35.3BBB 35.8 47.9 50.2 59.8BB 131.9 158.0 172.6 187.16B 191.4 219.8 263.4 264.8All 69.4 87.5 95.7 105.7
Sensitivity>A 0.0021 0.0033 0.0030 0.0041BBB 0.0042 0.0048 0.0055 0.0057BB 0.0116 0.0106 0.0128 0.00976B 0.0159 0.0088 0.0165 0.0112All 0.0069 0.0062 0.0081 0.0058
Hedge Ratio ($)>A 96,390 148,487 133,201 184,748BBB 184,943 206,094 240,845 244,362BB 460,051 399,958 515,300 365,0816B 551,403 332,553 592,144 265,388All 282,865 229,134 335,525 221,346
28
Table 3: Empirical Hedge Ratio
This table reports the estimate of the empirical hedge ratio, δej,t = βjDj,t. βj is the slopecoefficient estimated from a panel regression of change in CDS spread on stock return,specified in equation (12) and equation (14), respectively, for rating j. Fixed effect isallowed here. The duration Dj,t is computed at the average observed CDS spread of ratingj on each date. Panel A reports the time-series average of duration ($) and hedge ratios($). Panel B reports the estimate of the slope coefficient λj in hedge ratio regression (19)for rating j. The null hypothesis is λj = 1, which means Merton hedge ratio is in line withthose empirically observed. The t-statistics for λj are with respect to the difference fromunity and are calculated with the clustered standard error.
Panel A: Empirical Hedge RatioUnivariate Multivariate
Rating Duration |β| Hedge Ratio ($) |β| Hedge Ratio ($)> A 4,490 0.0070 314,311 0.0018 80,823BBB 4,411 0.0129 569,070 0.0063 277,918BB 4,127 0.0252 1,039,982 0.0205 846,0176 B 3,750 0.0422 1,582,637 0.0431 1,616,390ALL 4,243 0.0282 1,196,485 0.0273 1,158,299
Panel B: |λ| in Hedge Ratio RegressionClassical Merton Extended Merton
Rating Coefficient t-stat Coefficient t-stat> A 0.97 -0.16 1.16 0.68BBB 0.94 -0.70 1.02 0.21BB 1.22 0.68 1.13 0.416 B 1.49 2.05** 1.10 0.47ALL 1.37 2.18** 1.10 0.61
29
Table 4: Comparisons of Hedging Effectiveness - RMSE
This table reports the RMSE of the hedging errors under four hedge ratios during the wholesample period and three subperiods. An equally weighted CDS portfolio is formed acrosseach rating class and the whole sample, respectively, and then is hedged dynamically inequity market. The four hedge ratios include two empirical ratios from a univariate andmultivariate regression, respectively, and two theoretical hedge ratios from classical andextended Merton models. The position is rebalanced on a daily basis. We also reportRMSE of the unhedged portfolio ( “No Hedge”) for comparison. Hedging effectiveness E isthe reduction of RMSE compared to that of unhedged position. The number in RMSE isin dollar term and that in E is in percentage term.
Empirical MertonRating No Hedge Univariate Multivariate Classical ExtendedRating RMSE RMSE E RMSE E RMSE E RMSE E
Panel A: Whole period 2001-2009> A 8,896 9,002 -1.19 8,681 2.42 9,070 -1.95 9,463 -6.37BBB 13,286 13,261 0.19 12,551 5.53 13,372 -0.65 13,756 -3.54BB 35,473 34,092 3.89 33,251 6.26 34,214 3.55 33,724 4.936 B 57,479 53,609 6.73 53,784 6.43 52,726 8.27 52,514 8.64ALL 18,341 17,285 5.76 16,544 9.80 17,497 4.60 17,353 5.39
Panel B: Subperiod 2001-2003> A 10,164 10,946 -7.69 10,083 0.80 10,450 -2.81 10,954 -7.77BBB 14,953 16,116 -7.78 14,768 1.23 15,104 -1.02 15,556 -4.03BB 55,361 52,809 4.61 52,082 5.92 50,847 8.15 50,950 7.976 B 79,432 73,343 7.67 73,538 7.42 71,993 9.36 72,269 9.02ALL 18,002 17,532 2.61 16,603 7.77 16,594 7.82 16,786 6.76
Panel C: Subperiod 2004-2007> A 2,199 2,916 -32.61 2,157 1.89 2,189 0.46 2,182 0.74BBB 4,056 5,178 -27.64 4,029 0.66 3,955 2.49 3,946 2.72BB 12,440 13,273 -6.69 12,437 0.02 11,988 3.63 11,910 4.266 B 26,197 26,597 -1.53 26,770 -2.19 25,175 3.90 24,937 4.81ALL 8,486 8,927 -5.19 8,175 3.66 8,090 4.67 8,002 5.70
Panel D: Subperiod 2008-2009> A 16,487 15,489 6.05 15,885 3.65 16,727 -1.45 17,432 -5.73BBB 24,389 21,990 9.84 22,091 9.42 24,548 -0.65 25,293 -3.71BB 43,143 41,016 4.93 39,297 8.91 46,546 -7.89 44,279 -2.636 B 77,283 71,142 7.95 71,357 7.67 71,320 7.72 70,095 9.30ALL 34,811 31,402 9.79 30,582 12.15 33,829 2.82 33,226 4.55
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Table 5: Out-of-The-Sample Hedging under Rolling Regression
This table reports RMSE ($) of hedging errors and hedging effectiveness E using two out-of-the-sample empirical hedge ratios. The β used to calculate hedge ratio is estimated fromweekly rolling regression, in which rolling window is 1 year. Since the rolling window startsfrom January 2001, the first date to compute hedge ratio is January 1st, 2002. Panel Areports the RMSE ($) of hedging errors across the whole sample. Panel B - D reports thatfor three subperiods. For the convenience of comparison, we also report the RMSE of ”NoHedge” and that using Merton hedge ratios. Hedging effectiveness E is the reduction ofRMSE compared to that of unhedged position. The number in RMSE is in dollar term andthat in E is in percentage term.
Out-of-sample Empirical MertonRating No Hedge Univariate Multivariate Classical ExtendedRating RMSE RMSE E RMSE E RMSE E RMSE E
Panel A: Whole period 2002-2009> A 7,779 7,874 -1.21 7,839 -0.76 7,915 -1.75 8,298 -6.68BBB 12,431 13,075 -5.19 11,936 3.98 12,312 0.96 12,605 -1.40BB 23,469 22,864 2.58 23,329 0.60 24,746 -5.44 23,921 -1.926 B 44,983 44,651 0.74 53,013 -17.85 42,852 4.74 42,356 5.84ALL 17,102 16,420 3.98 16,182 5.38 16,601 2.92 16,373 4.26
Panel B: Sub-period 2002-2003> A 5,604 7,343 -31.03 6,730 -20.08 5,953 -6.22 6,609 -17.93BBB 12,544 17,210 -37.20 13,453 -7.24 11,701 6.72 11,764 6.22BB 23,888 26,304 -10.11 25,742 -7.76 24,050 -0.68 24,561 -2.826 B 52,814 67,084 -27.02 100,252 -89.82 54,867 -3.89 55,097 -4.32ALL 10,670 14,376 -34.73 14,072 -31.89 10,251 3.93 10,495 1.64
Panel C: Sub-period 2004-2007> A 2,203 2,220 -0.81 2,270 -3.05 2,193 0.46 2,187 0.73BBB 4,059 3,979 1.96 4,020 0.95 3,958 2.49 3,949 2.70BB 12,121 11,338 6.46 11,576 4.50 11,674 3.69 11,605 4.266 B 25,946 24,645 5.01 26,698 -2.90 24,965 3.78 24,752 4.60ALL 8,409 7,686 8.60 7,952 5.44 8,018 4.65 7,935 5.64
Panel D: Sub-period 2008-2009> A 16,495 15,959 3.25 16,144 2.13 16,708 -1.29 17,412 -5.56BBB 24,398 23,036 5.58 22,490 7.82 24,596 -0.81 25,348 -3.89BB 42,868 40,953 4.47 42,247 1.45 46,565 -8.62 44,270 -3.276 B 77,133 71,247 7.63 74,647 3.22 71,380 7.46 70,096 9.12ALL 34,714 31,899 8.11 31,185 10.17 33,847 2.50 33,228 4.28
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Table 6: Comparisons of Hedging Effectiveness - Value at Risk
This table reports the Value at Risk (VaR) at 99% confidence interval of the hedging errorsand its hedging effectiveness E under four hedge ratios for the whole sample and each ratingclass. VaR ($) is measured as the average of the absolute value of hedging errors at 99.5 and0.5 percentile. E (%) is measured as the reduction of VaR compared to that of unhedgedposition. The number in VaR is in dollar term and that in E is in percentage term.
Empirical MertonRating No Hedge Univariate Multivariate Classical ExtendedRating VaR VaR E VaR E VaR E VaR E
Panel A: Whole period 2001-2009> A 39,157 39,270 -0.29 37,739 3.62 40,355 -3.06 40,445 -3.29BBB 57,613 56,553 1.84 57,537 0.13 62,076 -7.75 60,265 -4.60BB 145,181 131,645 9.32 129,543 10.77 140,454 3.26 137,499 5.296 B 247,185 222,850 9.85 223,442 9.61 229,259 7.25 245,243 0.79ALL 72,953 67,747 7.14 64,156 12.06 71,156 2.46 71,196 2.41
Panel B: Subperiod 2001-2003> A 43,761 48,174 -10.08 44,167 -0.93 46,810 -6.97 48,886 -11.71BBB 65,126 66,939 -2.78 64,949 0.27 66,756 -2.50 69,107 -6.11BB 235,651 187,262 20.53 191,561 18.71 203,804 13.51 197,797 16.066 B 313,827 290,607 7.40 291,341 7.17 283,931 9.53 284,350 9.39ALL 63,327 57,737 8.83 57,156 9.74 58,816 7.12 57,616 9.02
Panel C: Subperiod 2004-2007> A 8,114 9,338 -15.09 7,931 2.25 8,084 0.37 8,075 0.47BBB 14,139 16,015 -13.27 13,829 2.19 13,981 1.12 13,937 1.42BB 44,663 37,526 15.98 37,078 16.98 42,949 3.84 42,168 5.596 B 82,462 67,636 17.98 68,630 16.77 76,378 7.38 75,253 8.74ALL 32,018 29,016 9.38 27,494 14.13 30,658 4.25 30,160 5.80
Panel D: Subperiod 2008-2009> A 56,290 55,626 1.18 54,873 2.52 51,578 8.37 52,302 7.09BBB 85,471 72,589 15.07 71,931 15.84 75,246 11.96 77,000 9.91BB 145,550 132,971 8.64 128,654 11.61 155,653 -6.94 148,033 -1.716 B 278,060 240,416 13.54 239,387 13.91 231,550 16.73 247,140 11.12ALL 114,136 104,790 8.19 105,168 7.86 105,337 7.71 106,897 6.34
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Table 7: RMSE and market variables
This table reports the results of time series regressions of the RMSE over a set of non-equitymarket variables. Panel A reports the results of a time-series regression of change in logRMSE of the portfolio of all firms on its lagged term and a set of contemporaneous marketvariables on a monthly frequency as follows:
logRMSEt = α+ β0 logRMSEt−1 + β1∆V IXt + β2rm,t
+ β3SMBt + β4HMLt + β5∆slopet + β6∆r10t + et.
Panel B reports the results of a predictive time-series regression as follows:
logRMSEt = α+ β0 logRMSEt−1 + β1V IXt−1 + β2rm,t−1
+ β3SMBt−1 + β4HMLt−1 + β5slopet−1 + β6r10t−1 + et.
The dependent variable RMSEt is calculated with daily hedging errors in month t. Themarket variables is defined as follows, rm is the return on the S&P 500 index; SMB andHML, respectively, are the Fama-French Small Minus Big and High Minus Low factors;slope is the monthly change in slope of the yield curve, where the slope is defined as the10-year Treasury rate minus 1-year Treasury rate; r10 is monthly change in 10-year Treasuryrate. The hedge ratio to calculate hedging errors is constructed from univariate regression.Standard errors are computed using Newey-West with three lags. N is the number ofobservations in the regression. The results are reported for both full period and pre-crisisperiod (2002-2007).
Panel A: Contemporaneous Panel B: Prediction
Full Period Pre-crisis Period Full Period Pre-crisis Period
RMSEt−1 0.7247 0.6979 RMSEt−1 0.4765 0.2687(14.24)∗∗∗ (9.44)∗∗∗ (5.37)∗∗∗ (2.33)∗∗
∆V IXt 0.9406 1.4831 V IXt−1 1.1982 1.5786(3.36)∗∗∗ (2.86)∗∗∗ (4.10)∗∗∗ (4.97)∗∗∗
rm,t -1.4667 -1.2151 rm,t−1 -0.779 -0.8697(−4.91)∗∗∗ (−2.65)∗∗∗ (−2.52)∗∗∗ (−2.16)∗∗
SMBt 0.5788 0.6021 SMBt−1 0.7608 0.3185(1.25) (1.17) (1.62) (0.71)
HMLt 0.6156 1.0671 HMLt−1 2.23 1.5231(1.58) (1.89)∗ (3.91)∗∗∗ (1.69)∗
∆slopet 19.43 17.8354 slopet−1 -0.7997 1.0452(4.82)∗∗∗ (2.85)∗∗∗ (-0.62) (0.69)
∆r10t -8.37 -2.058 r10
t−1 -0.3269 8.0747(−2.00)∗∗ (-0.31) (-0.10) (1.76)∗
Intercept 1.1054 1.2033 Intercept 1.8949 2.2514(5.41)∗∗∗ (4.08)∗∗∗ (6.23)∗∗∗ (6.05)∗∗∗
R2 0.819 0.7225 R2 0.789 0.7136N 92 77 N 92 77Durbin-Watson 2.46 2.38 Durbin-Watson 2.15 1.96
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Table 8: Hedging Effectiveness Over Longer Time Horizons
For each rating class and the whole sample, the table reports the RMSE and hedgingeffectiveness E of the cumulative hedging errors of four hedge ratios over 5, 10, 25 and 50business days, respectively. Hedging effectiveness E is the reduction of RMSE comparedto that of unhedged position. The number in RMSE is in dollar term and that in E is inpercentage term.
Empirical MertonRating No Hedge Univariate Multivariate Classical ExtendedRating RMSE RMSE E RMSE E RMSE E RMSE E
Panel A: 5 business days> A 37,190 32,738 11.97 35,805 3.72 33,951 8.71 33,142 10.88BBB 42,839 37,254 13.04 39,104 8.72 38,617 9.86 38,295 10.61BB 101,946 83,938 17.66 85,316 16.31 86,890 14.77 85,761 15.886 B 147,933 117,447 20.61 117,370 20.66 122,518 17.18 124,058 16.14ALL 60,608 48,058 20.71 49,103 18.98 50,570 16.56 50,164 17.23
Panel B: 10 Business days> A 40,389 35,617 11.81 38,751 4.05 37,007 8.37 36,568 9.46BBB 64,138 54,180 15.53 57,974 9.61 57,151 10.89 56,151 12.45BB 139,177 107,739 22.59 110,982 20.26 112,730 19.00 111,517 19.876 B 222,432 166,384 25.20 165,877 25.43 175,554 21.08 180,891 18.68ALL 92,834 70,042 24.55 72,928 21.44 73,770 20.54 73,349 20.99
Panel C: 25 Business days> A 62,140 53,118 14.52 59,272 4.61 58,121 6.47 57,027 8.23BBB 100,919 84,491 16.28 91,142 9.69 92,486 8.36 90,429 10.39BB 226,202 167,999 25.73 174,959 22.65 182,268 19.42 180,117 20.376 B 362,331 260,831 28.01 259,697 28.33 291,105 19.66 294,428 18.74ALL 146,519 107,172 26.85 113,250 22.71 119,936 18.14 118,067 19.42
Panel C: 50 Business days> A 90,866 74,449 18.07 86,026 5.33 85,943 5.42 83,889 7.68BBB 138,446 110,346 20.30 122,498 11.52 128,297 7.33 124,701 9.93BB 320,795 228,295 28.83 240,687 24.97 265,381 17.27 259,789 19.026 B 479,803 307,965 35.81 306,166 36.19 381,041 20.58 377,348 21.35ALL 201,249 137,900 31.48 148,229 26.35 170,118 15.47 165,590 17.72
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Table 9: Improvement of Hedging Effectiveness over Horizon and Market-Wide Factors
This table reports the result of a panel regression across rating classes of the improvement ofhedging effectiveness over four market variables. The regression specification is as follows:
υj,t = α+ β1V IXt−1 + β2rm,t−1 + β3SMBt−1 + β4HMLt−1 + Ij + ej,t, (23)
where υj,t is the speed of improvement of hedging effectiveness over time of rating j in montht. The independent variables include V IX, rm, which is the return on the S&P 500 index,and SMB and HML, which, respectively, are the Fama-French Small Minus Big and HighMinus Low factors. υ is measured in two ways; 1. difference between Em(15) and Em(1),whose result is reported in “E(15)− E(1)” column; 2. we estimate the slope coefficient inthe regression of hedging effectiveness Em(κ) on horizon κ, where κ ∈ {1, 3, 5, 10, 15, 20}.Its result is reported in “slope of E(κ)” column. Standard errors are clustered to computet stats. Fixed effect is allowed here. N is the number of observations in the regression. Theresults are reported for both full period and pre-crisis period (2001 - 2007).
Full Period Pre-crisis Period(1) (2) (1) (2)
E(15)− E(1) slope of E(κ) E(15)− E(1) slope of E(κ)V IX -1.4893 -0.0667 -1.9352 -0.1195
(−7.08)∗∗∗ (−5.46)∗∗∗ (−5.35)∗∗∗ (−10.87)∗∗∗
rm 0.5534 -0.0123 1.3259 -0.0446(5.49)∗∗∗ -0.33 (6.79)∗∗∗ -0.68
SMB -1.7540 0.0321 -1.3061 0.0436(-1.79) (0.67) (-1.17) (0.93)
HML 0.2089 -0.0441 2.2367 -0.0571(0.25) (-1.14) (2.26)∗∗ (-0.63)
Intercept 0.7377 0.0269 0.8163 0.0363(16.71)∗∗∗ (11.06)∗∗∗ (11.33)∗∗∗ (15.05)∗∗∗
Firm effect Yes Yes Yes YesTime effect No No No NoR2 0.078 0.056 0.080 0.071N 359 366 303 306
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0 200 400 600 800 1000 1200 1400 1600 1800 20002000
2500
3000
3500
4000
4500
CDS spread (bps)
Dur
atio
n (d
olla
r)
Plot of Duration on different spread levels
Figure 1: Duration of different spread levels
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-0.3 -0.2 -0.1 0 0.1 0.20
10
20
30
40
50
60
Daily hedging error (million dollars)
Den
sity
No hedge
Univariateregression
-0.3 -0.2 -0.1 0 0.1 0.20
10
20
30
40
50
60
Daily hedging error (million dollars)
Den
sity
No hedge
Multivariateregression
-0.3 -0.2 -0.1 0 0.1 0.20
10
20
30
40
50
60
70
Daily hedging error (million dollars)
Den
sity
No hedge
ClassicalMerton
-0.3 -0.2 -0.1 0 0.1 0.20
10
20
30
40
50
60
70
Daily hedging error (million dollars)
Den
sity
No hedge
ExtendedMerton
Figure 2: Distributions of Daily Hedging Errors
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0 5 10 15 20 25 30 35 40 45 500
50
100
150
200
250
Business days
Tho
usan
d do
llars
RMSE of the whole sample
No HedgeUnivariateMultivariateClassical MertonExtended Merton
0 5 10 15 20 25 30 35 40 45 500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Business days
Hed
ging
effe
ctiv
enes
s
Hedging effectiveness of the whole sample
UnivariateMultivariateClassical MertonExtended Merton
Figure 3: RMSE and hedging effectiveness of four hedge ratios over longer time horizons
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