CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
CREDIT DEFAULT SWAPS AND EQUITY VOLATILITY:
THEORETICAL MODELLING AND MARKET EVIDENCE
Giorgio CONSIGLI
E-mail: [email protected]
DMSIA Università di Bergamo and UniCredit Banca Mobiliare
ABSTRACT - The joint behaviour of equity premiums and credit spreads on securities issued by the same company provides a direct statistical evidence of the degree of efficiency of
equity and fixed income markets, whose participants are expected in the long term to provide
a common risk assessment.
Increasing interest in the financial industry is attracted both for financial engineering and
trading purposes, by the potentials offered by contracts with equity and fixed income components.
Increased liquidity in the credit default swaps (CDS) market, on the other hand, provides new
grounds for fixed income analysis based on the statistical study of theoretical versus actual
spread movements.
In the paper we analyse the statistical relationship between CDS spreads, stocks implied
volatility and theoretical spreads generated by an application of Merton seminal structural
default model. A measure of price discrepancies is also proposed, based on the difference
between theoretical and actual spread behaviours, leading to an application of relative value
analysis in the fixed income market.
We also test the dependence of credit spreads behaviour on the volatility of the associated
equity returns and the relationship over time between theoretical and observed CDS spreads.
The analysis is applied to six large corporations and back-tested over the 2002-2003 period.
Ford and General Motors are considered for the sector of Automotives, Deutsche Telekom
and France Telecom for Telecommunications, Endesa and RWE for the sector of Utilities.
We show that implied volatility movements drive significant spread movements: both theoretical and actual spreads follow volatility patterns. Furthermore, the impact on the theoretical spreads provides an accurate proxy of forthcoming movements in actual spreads. This
general results applies in particular to companies with a sufficient degree of risk (i.e. leverage), while poor evidence can be collected for companies relatively free of default risk.
KEYWORDS – Credit risk, Structural default model, Relative value analysis, Credit default
swaps, implied volatility.
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
1
INTRODUCTION
Increasing liquidity in the credit default swaps (CDS) market provides a much cleaner environment in which to study the statistical properties of credit spreads.
In what follows we extend an approach originally conceived for credit risk modelling to assess company-specific mis-pricing indicators. The analysis is built on the comparison of data
generated by an application of Merton’s seminal structural default [Merton, 1977] model
against CDS spreads (on the pricing of CDS see [Duffie, 1999] and [Hull and White, 2000]).
We test the dependence of credit spreads on the volatility of the associated equity returns and
the relationship over time between theoretically generated spreads and actual CDS spreads.
The general aim of the article is to test the potential of the structural approach to default risk
estimation as reference model for relative valuation of fixed income securities. Two such applications are also briefly described.
We support our conclusions relying on the following statistical analysis:
We clarify in a static framework, how theoretical spreads are implied out using the
Merton approach, and the general structure of the approach;
We introduce a measure of price discrepancy based on the relationship between theoretical and actual market spreads, described by the spreads quoted in the Credit Default Swap market, and
Test the opportunities offered by this approach in the construction of strategies in the
fixed income market. In terms of cheap-dear analysis and convergence trades effectiveness.
Observe that the developed theoretical model is not intended to provide accurate fair valuation of market spreads. There is already extensive evidence of a structural mis-valuation of
market spreads by this approach [Campbell and Taksler, 2003], [Cooper and Davydenko,
2003]. Rather we test whether normalised spreads, i.e. spreads adjusted by their mean and
volatility can be used for relative valuation purposes.
The analysis has been applied to six large corporations and back-tested over the 2002-2003
period. Ford and General Motors are considered for the Automotive sector, Deutsche Telekom and France Telecom for Telecommunications, Endesa and RWE for the sector of Utilities.
For each such company we apply Merton’s approach – see § 2 for details – to derive theoretical implied credit spreads, using as inputs, among others, the equity implied volatility,
and compare the resulting values with observable CDS spreads.
Results are extremely interesting. In particular, according to our model, implied volatility
movements drive significant spread movements and both theoretical and actual spreads follow these volatility patterns. Furthermore, the impact on the theoretical spreads provides an
accurate proxy of forthcoming movements in actual spreads. These general results apply
mostly to companies with a sufficiently high degree leverage while the relationships are
weaker for companies relatively free of default risk.
In the normalised set-up theoretical spreads show higher volatility than actual spreads and
provide alternatively upper and lower bounds to actual spread movements: this evidence is
used to introduce a measure of mis-pricing moving around 0 with actual spread movements
following mis-pricing indications.
The article structure is as follows. In section 2 we analyse the assumptions and main features
of Merton’s structural default model [Merton, 1974]. In section 3, we study the statistical interaction between theoretical and market spreads and analyse their relationship with equity
implied volatility. Finally in section 4 the potentials offered by this framework in terms of
relative value trading are tested.
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CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
2
THE MERTON STRUCTURAL DEFAULT MODEL
High and increasing leverage, measured by the debt-to-equity ratio implies increasing interest payments, potentially lowering earnings expectations and reducing expected returns on
the equity investment. Typically if a company needs to increase financial resources to meet
short-term obligations both the cost of debt and the cost of equity capital will increase penalising equity and bondholders alike.
The process, as is well known, does not need to be symmetric. Increasing earnings expectations are likely to induce an increase of the expected return on the equity investment and a
decrease of the cost of equity capital, while the cost of debt remains the same.
Merton’s method is generally considered to provide a coherent way to extract information on
theoretical credit spreads and default probabilities from information about the company’s
leverage, the company’s value and the volatility of the company’s value. Stock price volatility can be regarded as the crucial variable of the approach.
The option-pricing corporate default model exploits the analogy between holding the equity
of a company and going long a call option whose underlying is the company’s total asset
value. The strike price is given by the nominal value of the outstanding debt.
Figure 2.1
Equity holding as a long Call option
Call
price
C(t)
C(t)
V < D(T), E < 0
(default)
K=D(T)
V > D(T), E > 0
V = D+E
Asset value
In this framework the Black and Scholes option formula [Black and Scholes, 1973] can be used
to derive a credit spread/default probability from the company’s equity value and volatility
with a maturity given by the maturity of the outstanding debt.
Just to expand on this point a company’s assets can be funded through a combination of equity and debt. If the assets fall in value below the level of the debt, in principal, creditors
have the right to claim the nominal value of their credit. In this situation, on the other hand,
equity holders do not benefit before all debt-holders are paid off.
The asymmetry is reflected in the resulting payoff patterns: equal to a European short-put option for a debt-holder and a long-call option for an equity-holder. This is the evidence used
by Merton to propose an option-valuation framework: from a given debt-equity structure
and assets value volatility, the likelihood of a company’s default can be derived introducing
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CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
as strike price of the option (put or call), the outstanding nominal value of the debt (see figure 2.1).
2.1
The mathematical model
Suppose a very simple liability structure formed by equity and a zero-coupon bond issue maturing at time T . Let Vt and Et denote the value of the company’s assets and, respectively,
equity at time t ∈ [0, T ] , D the amount of debt to be repaid at time T and σ V and σ E the
volatility of the assets value and the equity value, respectively. To make clear the concept of
default put forward by Merton, consider the following: If VT < D , then is rational for the
company to default on the debt at time T . In this case the value of the equity is 0. If instead
VT > D , the company should make the repayment at time T and the value of the equity at
this time is VT − D . The resulting payoff for the firm’s equity at
T is thus
ET = max(VT − D,0) .
This shows that the equity can be treated as a call option on the value of the assets with a
strike price equal to the repayment required on the debt. We can thus apply the option pricing
formula, from which the value of the equity today comes out the well-known Black and
Scholes formula:
E 0 = V0 N (d1 ) − De − rT N (d 2 )
(2.1.1)
In (2.1.1): N is the cumulative standard normal distribution and r is a constant risk-free interest rate quoted at the current date for the debt maturity T. The value of the equity is decomposed in a weighted portfolio long the assets and short the discounted debt value. The
weights are determined by the two percentiles d1 = 1 / σ V T * ln(V0 / D) + (r + σ V2 / 2)T and
[
]
d 2 = d1 − σ V T .
The risk-neutral probability that the company will default on the debt at maturity is now
N (−d 2 ) : this is the probability that at option expiry, the assets value will be below the debt
value. To calculate this value we need V0 and σ V . The point is that neither of these is ob-
servable. However, we can observe E 0 and estimate σ E . In our implementation we use the
implied volatility estimated in the option market.
From Ito’s lemma we have
σ E E 0 = N (d1 )σ V V0
(2.1.2)
Equations (2.1.1) and (2.1.2) provide a pair of simultaneous equations that need to be solved
for V0 and σ V . The resulting current value of the debt is V0 − E 0 .
From the debt value, the implied yield to maturity can be computed and, given the prevailing
risk-free rate, the corresponding credit spread for that maturity calculated by subtraction, allowing comparison with the observable credit spread in the market.
V −D
The distance to default, given by 0
, can at this point be computed (see on the releV0σ V
vance of the D2D the remarks in [Kealhofer, 2003a and 2003b]).
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CDS and equity volatility: theoretical modelling and … _Giorgio CONSIGLI
The variable V0σ V is approximately equal to the size of one standard deviation move in the
current assets value. The distance to default is thus an approximate measure of the number
of standard deviations between the current value of the assets and the amount owed at maturity.
The system given by equations (2.1.1) and (2.1.2) is solved through the following fixed-point
procedure.
We first rearrange the system to obtain more numerically stable functions (the function on
the right of (2.1.1), being a difference between two quantities of same sign and similar size,
may indeed lead to numerical errors):
V0 =
De − rT N (d 2 ) + E 0
N (d1 )
and
σV =
(2.1.3)
σ E E0
De
− rT
(2.1.4).
N (d 2 ) + E0
Which is a fixed-point equation of type (V0 , σ V ) = G (V0 , σ V ) .
To solve numerically this system we simply compute the iterations G n of function G from
an arbitrary (but sensible) starting point, until our error function, given by
V 0n − G 1 (V 0n , σ Vn ) + σ Vn − G 2 (V 0n , σ Vn ) became less than 10^-6, or until we reach iteration number 1000. In the latter case the function returned an error.
The solution of the system allows the definition of the current debt value as:
D0 = V 0 − E 0
(2.1.5)
from which the implied credit spread for the given maturity and the recovery rate associated
with the given default probability, can be computed:
π 0,T = 10000( Ln(
D(T )
) / T − r0f,T )
D(0)

D(0)
RecRate = 1 − 1 −
f
 D(T )e − r0,T T


 / (N (−d ))
2



(2.1.6)
(2.1.7)
Equation (2.1.6) defines the credit spread in basis points.
In § 3 we analyse the relationship between the spread computed in (2.1.6) – model generated,
theoretical – with the 5Y CDS spreads observed in the market over a two-year period.
2.2
Model summary
Here below the information flows underlying the model implementation are displayed in order to summarise the model dependencies.
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CDS and equity volatility: theoretical modelling and … _Giorgio CONSIGLI
Several key financial variables are combined to derive an estimate of the current firm value
and thus, by difference with the company market cap, the implied debt value needed to estimate the default probability and the recovery rate.
Figure 2.2.1
Structural model INPUT-OUTPUT info
INPUT VARIABLES
INFO
OUTPUT & DEFAULT
Debt maturity,
approximated T
Firm value V(t)
Option
Pricing
function
inversion
Risk free
rate (t,T)
d1
Debt Nominal
amount D(T)
d2
Firm volatility
Debt value D(t)
(T-t) Implied Credit
Spread
Log return
volatility
(T-t) Default
Probability
Stock price S(t)
Equity value E(t)
No of issued
stocks
Recovery rate
Distance to default
(D2D)
Several simplifications are needed in order to apply in practice this framework (see [Black
and Scholes, 1973], [Geske, 1977], [Merton, 1974]):
• The company’s liability structure needs to be particularly simple. Only stocks and bonds
are present. Changes in assets value need to be fully reflected in corresponding changes
of equity and bond prices.
• All the debt matures at one point in time: this is the exercise time of the embedded option.
• The value of the equities and the assets follows a stochastic evolution described by a
geometric Brownian motion.
• The market of corporate securities is free of arbitrage.
Input variables of Merton’s pricing model are: the stock price volatility and market value,
the current nominal debt, the risk-free return for the average maturity of outstanding debt.
The outputs are the company market value and its volatility, the debt market value and the
associated default information. Namely the default probability and the recovery value – the
value of the debt recovered upon default – the implied credit spread for the assumed (average) maturity and the so-called distance-to-default (D2D) a measure of the number of standard deviations of the asset value away from default.
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CDS and equity volatility: theoretical modelling and … _Giorgio CONSIGLI
Theoretical spreads are derived within Merton’s approach by first computing the current debt
value as a difference between the company’s total value and the market value of the equity
capital and then by subtracting from the resulting implied yield the risk free rate.
2.3
Important theoretical relationships
The model allows a direct comparison between several key corporate variables. As clearly
described in figure 2.2.1 and the model mathematical structure, Merton’s approach can be
regarded as a mapping between five input variables, known at the current date and the company value and default information. The only input variable that typically requires statistical
estimation is the stock return volatility: this not the case if the implied volatility is used instead. This is indeed the volatility measure considered here below.
We complete the description of the model by focusing on a set of relationships that clarify
the assumptions underlying the proposed approach. Namely, between:
Stock price volatility and company value volatility
Credit spread and stock price and volatility
Implied bond price and stock price and volatility
Implied stock price and credit spread and default probability
Stock price, credit spread and leverage.
The first relationship is behind the structural concept of default: this is assumed to occur
when the asset value falls below the outstanding debt: the higher the firm volatility the higher
the probability of default and the greater the option premium to gain protection. Observe that
the very same reasoning is behind an increase of the spread to be paid to buy default protection via a CDS. Given the company value and the equity value we can by difference estimate
the debt current implied value and thus the credit spread and implied bond price. Finally we
are interested to understand the critical relationship between the leverage and the stock price
and credit spread.
2.3.1 Equity and firm volatility
Equation (2.1.4) clarifies the relationship between equity and company volatility: we present
below for a range of stock volatility values the estimated company volatility for a representative highly levered company.
Figure 2.3.1.1
Stock and Company volatility
Company value volatility as a function of stock return volatility
1.60000
1.40000
asset volatility
1.20000
1.00000
0.80000
0.60000
0.40000
0.20000
0.00000
0
0.2
0.4
0.6
0.8
1
stock volatility
1.2
1.4
1.6
1.8
2
sigmaE
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
sigmaV
0.05820
0.11640
0.17460
0.23281
0.29126
0.35073
0.41228
0.47693
0.54542
0.61831
0.69593
0.77845
0.86589
0.95812
1.05493
1.15599
1.26090
1.36922
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CDS and equity volatility: theoretical modelling and … _Giorgio CONSIGLI
As the equity volatility increases so does the firm value volatility: the rate of increase is exponential and for very high implied volatility inputs the company volatility increases more
than proportionally, leading to a sudden reduction of the distance to default and respectively
an increase of default probability.
2.3.2 Implied credit spread and bond price versus stock price and volatility
The spread defines the excess return – premium -- required for an investment in a risky debt
instrument. The described model does provide a unique insight into the relationship between
cost of debt and cost of equity capital. Consider the sensitivity of the implied spread to the
equity volatility: the risk associated with an equity investment in the given company.
Figure 2.3.2.1 shows a 3D plot whose entries are the implied spread, the stock price and its
volatility.
Figure 2.3.2.1
Credit spread, stock price and volatility
Implied 1Y Credit Spread
1800
1600
1400
1200
Spread in bps
1000
800
600
400
200
0.5
11.6
10.7
9.8
8
0.85
8.9
Stock price
7.1
6.2
5.3
1.2
4.4
3.5
0
Equity
return
volatility
The model assumes an exponential relationship between credit spreads and equity volatility:
the lower the price the stronger the volatility effect. Conversely the spread decreases for
higher stock prices as expected.
A similar impact is found on the bond price implicit to such theoretical spread: the relationship can again be implemented for given stock price and volatility by implying out the price
of one bond maturing at T from the estimated debt theoretical value. The bond price will decrease as the stock volatility increases and for increasing stock price a smaller reduction is
recorded.
The option formula can be inverted in order to recover for given implied spread and default
probability what the (implied) stock price should be. In figure 2.3.2.2 we report on the same
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CDS and equity volatility: theoretical modelling and … _Giorgio CONSIGLI
plot this interesting relationship: for increasing implied spread, thus increasing default probability and decreasing debt value, the stock price will tend to decrease. The sensitivity to the
implied spread, maintaining the default probability fixed, is almost zero for low default values and becomes slightly negative for high default values.
Figure 2.3.2.2
Implied stock price and credit spread and default probability
Implied Equity price
14.000000
12.000000
10.000000
8.000000
stock price
6.000000
4.000000
330
290
250
210
170
130
90
50
0.41
0.455
10
0.32
0.365
0.23
0.275
0.14
1Y default probability
0.185
0.05
0.095
0.000000
0.005
2.000000
1Y implied credit spread
In what follows we are interested in the first set of relationships from Figure 2.3.2.1, analysing the behaviour of the theoretical spread as a function of the stock price volatility.
The second set of relationships can support the analysis of theoretical versus actual stock
price movements in periods of stable volatility and is typical of equity research.
2.3.3
Stock price, credit spread and leverage
A relevant relationship, worth further analysis, is the one, for fixed debt value and equity
volatility, between stock price, leverage and implied spreads. Stock price movements induce
on one hand changes in market capitalization and for given debt value, corresponding
changes in the leverage measured by the ratio between D(t ) and E (t ) , and on the other hand
changes in the credit spreads as explained above. At any point in time, the stock price reflects
a given price yield: the dynamics of this yield relative to the credit spread is the fundamental
driver of the leverage behavior.
For fixed risk free rate and time to maturity we can map the implied spread on the leverage
for the current stock price and construct the corresponding theoretical leverage for the given
credit spread and stock price pair.
This is reported in figure 2.3.3.1.
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CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
Figure 2.3.3.1
Leverage as a function of credit spread and stock price
Leverage, credit spreads and stock price
for the XYZ company
for fixed risk free rate, nominal debt and maturity
1.6
1.4
1.2
1
0.8 leverage
0.6
3.5
4.4
5.3
6.2
7.1
8
10
190
100
370
280
550
730
0
460
credit spread
0.2
640
1000
910
820
0.4
stock price
The inputs are: the credit spread and the stock price for the X and Y axes, the risk free rate
and the outstanding debt which, as the number of issued equities, are kept fixed. We can in
this way recover the leverage as a ratio between the debt value associated with that risk free
rate and the corresponding spread and the equity value induced by the corresponding stock
price, given the number of equities.
The figure shows that as the stock price increases for constant credit spread the leverage
tends accordingly to decrease at an exponential rate. Similarly for fixed stock price the leverage tends to decrease for increasing credit spread, but at a low linear rate. This behavior is
consistent with a generally recognized evidence in market practice: the strong correlation between credit spreads, equity volatility and yields during periods of increasing market uncertainty and spread tightening, and the tendency of equity yields to react slower to turns during
periods of spread easing.
Questions such as the impact on the cost of equity and debt capital associated with a leverage
increase can in this framework receive an answer.
3
STOCK PRICE VOLATILITY, THEORETICAL AND CDS SPREADS
Merton’s approach provides a unique insight into the theoretical relationship between equity
value and volatility and default probability and credit spreads. A recognised drawback of the
approach, however, is that due to the necessary simplifying assumptions and in absence of
specific additional corporate information, the approach leads to a severe underestimation of
observable credit spreads and inaccurate assessment of the company default probability. Several approaches have been proposed to overcome this shortcoming (see [Campbell and Tak-
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CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
Taksler, 2003],[Geske, 1977],[Kealhofer, 2003a]) but no clear progress has yet been
achieved.
The ability of the structural approach to track continuously changing credit exposures, however, has been widely assessed, leading to a widespread adoption of this modelling technique
(see [Consigli, 2004],[Hull et al, 2004],[Kealhofer, 20003a and 2003b]). The method is robust with respect to credit risk ranking and as shown below can also be fruitfully applied to
relative valuation purposes.
It is worth recalling that the model described in section 2 provides the canonical reference for
the development of credit risk models based on this structural approach: the only required
market variable is the company’s equity. KMV, the US company recently acquired by
Moodys, is the most noticeable example of a successful story of application of this approach.
Here the behaviour of the credit spread and the distance to default are turned into a default
probability and rating group transition probabilities by introducing a set of class boundaries,
whose crossing is associated with a rating transition. The following figure describes the result of an estimation procedure of this type on a representative highly levered company.
Figure 3.1
An application of the structural approach to credit risk modelling
Firm XYZ Value pdf
1.2E-11
BBB
default
A
AA
AAA
1E-11
frequency
8E-12
6E-12
4E-12
2E-12
2,
76
2,
62
3,
63
0.
08
1Y
D(T
)
V(0)
30
,3
88
,8
59
,9
30
.9
0
58
,0
15
,0
96
,2
31
.7
1
85
,6
41
,3
32
,5
32
.5
3
11
3,
26
7,
56
8,
83
3.
34
14
0,
89
3,
80
5,
13
4.
16
16
8,
52
0,
04
1,
43
4.
97
19
6,
14
6,
27
7,
73
5.
79
22
3,
77
2,
51
4,
03
6.
60
25
1,
39
8,
75
0,
33
7.
42
0
Company market value
t=0
The company value pdf at the 1Y horizon is generated by an application of the lognormal
model. The boundaries for rating transitions are externally imposed to fit exogenous transition probabilities. Each sample path of the company value has an associated current debt
value and thus a given credit spread. This is the theoretical spread we focus on the article.
The assessment of these risk factors can be repeated for very many companies to derive a
credit portfolio risk profile.
Here next we follow a different route and rather than aiming at the valuation of portfolios default probabilities, we assess the dependence of market and theoretical spreads from the equity implied volatility expressed by the option market.
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CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
3.1
Market and theoretical spread dynamics
We develop a case study for the European telecom sector, focussing on the two market leaders, namely Deutsche Telekom and France Telecom. While limiting the analysis to those two
companies, the collected evidences have been tested for a larger sample including companies
in the Automotive and Utility sectors, as shown in Appendix.
Table 3.1.1 reports the correlation coefficients between actual and theoretical daily spreads
within each sector. The sample period is Jan.2002-Dec.2003 and we consider daily observations.
theor / actual
FORD
GM
DT
FT
ENDESA
RWE
FORD
0.7051
0.6968
GM
0.6069
0.5926
DT
FT
0.8707
0.7795
ENDESA
RWE
0.7966
0.7703
0.9157
0.2470
0.6893
0.7092
Table 3.1.1 Correlation between actual and theoretical spread values
The figures in the table support the introduction of normalised credit spreads constructed by
subtracting from actual values the sample means and dividing by the statistical standard deviation. As z-scores they measure the deviation from the mean of daily values per unit volatility. These transformations leave the correlations unaltered.
For each company we have collected the required data input from January 1st 2002 to December 22nd 2003, to compute the theoretical spreads from Merton’s model. The maturity of
the companies’ outstanding debt is assumed to be 5Y, in order to generate spread information
comparable with the 5y CDS quotes.
These theoretical spread approximations, jointly with the implied volatility, are used to test:
The dependence of actual spreads on stock price volatility
The correlation between theoretical and actual spreads
The differences between companies belonging to the same sector of activity
The possibility to derive a measure of mis-pricing and as a consequence
Trading indications within and across sectors.
The following figure reports the dynamics of theoretical (left) and actual (right) 5Y spreads
for the Telecom sector. Dots are used to highlight market quotes for hedges actually completed. Thin lines indicate instead theoretical spreads.
As previously pointed out, together with the figures in appendix the figure shows that the hierarchy – creditworthiness – across theoretical spreads is preserved compared with market
quotes.
Theoretical spreads are generally lower than actual spreads. When the equity volatility increases, however, theoretical spreads overall show a much higher volatility than the spreads
expressed by the CDS market.
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CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
Figure 3.1.1
Theoretical (left) and actual (right) 5Y spreads in the Telecom sector – 2002-2003
4495
800
Deutsche (black) & France (grey) TLC
700
3495
600
2995
500
2495
400
1995
300
1495
200
995
100
495
-5
Jan-02
3.2
5YCDS (diamonds)
theoretical 5Y spreads (lines)
3995
0
Apr-02
Jul-02
Sep-02
Jan-03
Apr-03
Jul-03
Oct-03
Equity volatility and market spreads during 2002-2003
The role played by the equity volatility in explaining the probability of default in Merton’s
model has been analysed in § 2. We intend now to assess the relevance of such influence.
CDS spreads are driven in general by several factors: stock price volatility is recognised to
play a relevant role, together with other variables, such as the credit rating and several accounting ratios.
The strength of the relationship between stock price volatility and actual CDS spreads can
thus vary over time and may in theory become negligible under specific conditions: the evidence we collect is instead that stock price volatility is a fundamental driver of spread
movements, both theoretical and actual.
Furthermore, the calculated movements in theoretical spreads provide a robust approximation of forthcoming market spreads variations.
The late summer and early autumn of year 2002 represent epochal changes of the Telecom
market structure and spread tightening was widespread.
The last year, instead, has witnessed a stable decrease of credit spreads with substantial
movements of funds from the still unstable equity market to the bond market and Telecom’s
have been among the beneficiaries.
The case of France Telecom is remarkable: figure 3.2.1 shows the very strong relationship
between the equity implied volatility and the theoretical spreads during periods of increased
volatility and, outside these periods, a close approximation of actual market movements by
theoretical spreads.
13
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
Figure 3.2.1
Deutsche Telekom and France Telecom
theoretical and actual spreads (left axis) versus equity implied volatility (right axis)
Deutsche Tlk
Theor (solid line) and Actual (diamonds) 5Y Normalised spreads
vs Equity volatility (dotted line with squares)
4
0.90
Normalised 5Y spreads
0.70
2
0.60
1
0.50
0
0.40
0.30
-1
0.20
-2
-3
Jan-02
Equity implied volatility
0.80
3
0.10
Apr-02
Jul-02
Sep-02
Dec-02
Mar-03
Jun-03
Sep-03
0.00
Dec-03
5
1.40
4
1.20
3
1.00
2
0.80
1
0.60
0
0.40
-1
0.20
-2
Jan-02
Apr-02
Jul-02
Sep-02
Dec-02
Mar-03
Jun-03
Sep-03
Equity implied volatility
Normalised 5Y spreads
France Tlc
Act (diamonds) and Theor (solid line) 5Y Spreads
vs Equity volatility (dotted line with squares)
0.00
Dec-03
Given the start-of-the-period theoretical spreads, changes of implied volatility do provide an
extremely accurate approximation of theoretical spread variations. These in turn drive normalized actual spreads.
Summarizing, the two figures show:
14
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
High correlation between volatility patterns for the two companies and corresponding
High correlation between volatility trends and theoretical spreads
A significant impact of volatility spikes on theoretical spread movements
The tendency of CDS spreads to move tightly around the corresponding normalized
theoretical spreads
During the last 12 months the decrease of implied volatility observed for both companies
has been paralleled by a stabilization of theoretical spreads around the mean and by a
convergence of actual spreads between the two companies.
3.3
CDS-Implied volatility lead-lag correlations and Granger causality
In table 3.1.1 we have reported daily correlation coefficients between theoretical and actual
spreads for each company and within sectors. The evidence of strong infra-sector correlation
has also emerged.
In this section we focus on the relationship between implied equity volatility and CDS
spreads in order to test the presence of a leading effect of the former on the latter and if so
the evidence of a Granger causality effect1 (see [Gourieroux and Monfort, 1997]).
The presence of a leading effect from, equity volatility to CDS spreads has far reaching practical consequences: to assess such a possibility we extend the correlation analysis in order to
estimate lead-lag correlations based on a –3, +3 day interval lags.
Due to lack of statistics on Endesa and RWE, whose CDS market as mention is not sufficiently liquid yet, the analysis is limited to the Automotive and Telecom sectors.
Table 3.3.1 displays the correlations between daily changes of implied volatility and CDS
spreads, for the four considered companies.
For t = 1,2,..., T − 3, we report in the first row the estimated correlation between volatility
changes recorded on day t and CDS spread changes on day t+3, in the second the correlation
between the former on day t and the latter on day t+2 and so on. Fort each company in bold
the maximum positive probability is highlight.
ImpVol,CDS
-3
-2
-1
0
1
2
3
FORD
0.0369
0.0716
0.0593
-0.0399
0.0502
-0.0908
0.0386
GM
-0.0735
0.0115
0.0617
0.0268
-0.0534
0.0468
0.0434
DT
-0.0048
0.0247
0.0269
0.1338
-0.0647
-0.0062
0.0224
FT
0.0111
-0.0106
0.0425
0.1326
-0.0479
0.0907
-0.0130
Table 3.3.1 Lead-lag correlation between daily variations of implied volatility and CDS spreads
In two out of four cases, namely Ford and GM, the figures support the hypothesis of a moderate leading effect of implied vol on CDS spreads. For Telecoms changes of implied volatility recorded the previous, two or three days before do not provide conclusive evidence on
current variations of CDS spreads.
In general no decisive indications on a leading effect of implied volatility on credit spreads
come from those results.
1
Granger causality studies the causal dependence of one variable from another. Positive lead correlation tends
typically to be related with the presence for Granger causality, but as shown in the section not necessarely.
15
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
In order to collect more evidence on a possible causality direction from implied volatility to
CDS spreads, we have developed a Granger causality test on the two variables. For the general theory on causality and exogeneity we refer again to [Gourieroux and Monfort, 1997].
We have tested both a lag 1 and lag 2 model of causality for the four companies: lag 1
Granger causality implies a causal effect on current spreads coming from the previous day
implied volatility. This hypothesis is rejected – the null hypothesis accepted – depending on
the value of a likelihood ratio, constructed as follows:
The numerator is defined by the sum of squared residuals for a complete linear regression
model containing lagged implied volatilies – to the second order in case of lag 2 causality
testing and to the first order for lag 1 causality – and
The denominator is determined by the sum of squared residuals in a simple autoregressive model with no exogenous variables considered.
As shown in tables 3.3.2 in three out of four cases the null hypothesis – implied volatility
does not Granger-cause CDS spreads – is rejected at the 95% confidence level. Quite remarkably the R square is equal to 1 or near 1 in all estimated models and the p-value is 0.
In the table: mod1 refers to the complete model and mod2 to the simple autoregressive model
with no exogenous variables.
LAG1
const
FORD
mod1
mod2
GM
mod1
mod2
DT
mod1
mod2
FT
mod1
mod2
y-1
impvol-1
STATS
R^2
p
GRANGER CAUSALITY
S1
F(1,T-3)
NULL
HYPOTH
-0.0008
0.0001
0.9479
0.9949
0.0047
1
1
6025.1
11602
0
0
10.3384 0.001480409
reject
0
0.0001
0.9635
0.9941
0.0025
1
1
3096.7
6060.6
0
0
6.1834
0.013570708
reject
-0.0008
0
0.8726
0.9942
0.0051
1
1
14624
26703
0
0
24.34
1.50248E-06
reject
0.0001
0.0001
0.9938
0.9879
-0.0004
1
1
13399
26883
0
0
0.2327
0.629965036
accept
LAG2
const
FORD
mod1
mod2
GM
mod1
mod2
DT
mod1
mod2
FT
mod1
mod2
F
y-1
y-2
impvol-1 impvol-2
STATS
R^2
F
p
GRANGER CAUSALITY
S1
Chi_sq(2)
NULL
HYPOTH
-0.0008
0.0001
1.029
1.0752
-0.085
-0.0813
0.0041
0.008
1
1
2998
5794
0
0
10.5088
0.0052245
reject
0.0001
0.0001
1.0296
1.058
-0.0662
-0.0651
0.0035
-0.011
1
1
1547
3029
0
0
7.1476
0.0280491
reject
-0.001
0
0.8297
0.9659
0.0191
0.0291
0.0038
0.0022
1
1
7448.8
13303
0
0
31.395
1.523E-07
reject
0.0002
0.0001
0.6056
0.6278
0.3963
0.3581
0.0031
-0.0041
1
1
7647.1
15131
0
0
4.6836
0.0961544
accept
Table 3.3.2 Granger causality between implied volatility and CDS spreads
The null hypothesis is rejected in the LAG1 model by the F(1,T-3) statistics – where 1 and T3 (here equal to 240) are the degrees of freedom for the F distribution— in presence of a likelihood ratio S1 greater than 3.087 at the 5% confidence level. It is rejected in the LAG2
model at the same confidence interval by the Chi-square(2) statistics when S1 exceeds 5.91.
16
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
In both models there is a strong evidence of a causality effect for DT and Ford, still present
for GM, while there is no evidence of causality for FT.
A final judgement on the stability of this causality effect requires a more extended statistical
back testing for a longer data horizon possibly considering moving time windows. This is left
to future research.
As a summary it is possible to conclude that at least during year 2003, implied volatility has
played a crucial role in determining spread levels and variations. Other variables such as risk
free rates, leverage and stock prices playing a less crucial role.
In conclusion of this section a set of general results that have proven true for all companies
considered in the study can also be sketched. In particular:
Movements of implied volatility (up-down) are positively correlated with movements
(up-down) of theoretical and actual spreads and tend to have a leading effect on these,
Changes of actual spreads, when delayed, can be accurately inferred from changes of
theoretical spreads,
Actual spreads tend to fluctuate around the more volatile theoretical spreads,
The difference between normalised spreads, both theoretical and actual, shows a cyclical
behaviour around zero.
From the first point we can infer that equity and fixed income investors rely on common information sets and that higher (lower) risk perception in the equity market is rapidly transferred into wider (lower) spreads in the bond market.
From the second and the third points we can conclude that in the normalised world the proposed theoretical model provides a good benchmark for actual spread movements (even if no
fair pricing appears possible).
Finally the fourth point is exploited in the mis-pricing analysis developed below.
4.
RELATIVE VALUE ANALYSIS
The results presented support the evidence of a strong correlation between implied volatility
and theoretical spreads and thereafter between theoretical and market spreads. Based on our
pricing approach, we can thus conclude that stock price volatility, an input to Merton option
model, represents the fundamental state variable of the model.
In this set-up the difference between CDS and theoretical spreads should indicate a possible
over/under valuation of market spreads relative to theoretical values. Furthermore, if this is
true, such a difference should fluctuate around an equilibrium – long term value of 0.
Figure 4.1 shows the behaviour of that difference for Deutsche and France Tlc during the
2002-2003 period.
17
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
Figure 4.1
Actual minus theoretical normalised spreads for the Telecom sector,
Jan.2002—Dec.2003
DTlc (black) & FTlc (grey) [Actual - Theoretical] 5Y Spreads
3
Normalised spread difference
2
ACTUAL > THEOR
1
0
-1
ACTUAL < THEOR
-2
-3
Sep-01
Dec-01
Mar-02
May-02
Aug-02
Nov-02
Feb-03
May-03
Jul-03
Oct-03
Jan-04
The figure, as anticipated by the previous statistical analysis, shows that Telecoms present
strong infra-sector correlations and there is a common credit factor driving the spreads of the
two corporations.
Due to the random evolution of the normalised spreads, we see that the mis-pricing measure
does change rapidly with occasional outliers, with sign changes even over short time periods.
In order to avoid this variability we identify through a 3rd degree polynomial a persistent
trend of the spreads difference and use this approximation as measure of price discrepancy.
We consider two applications of the price discrepancy measure:
Cheap-dear analysis that provides indication on optimal times to enter and exit the market
Convergence trade analysis that allows the definition of possible long-short positions to
exploit possible spreads convergence.
4.1 Mis-pricing measure and cheap-dear analysis
If the implied volatility is low (in relative terms) and the mis-pricing measure positive, than
we should expect a movement of CDS spreads towards the theoretical spreads.
Conversely if volatility is high, the theoretical spreads should be high and we should expect
an increase of market spreads. Consider DT as an example.
18
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
Figure 4.1.1
Mispricing measure vs (A) implied volatility and (B) CDS spreads.
Deutsche Tlk Jan.2002-Dec.2003
Deutsche Tlk
Implied volatility vs Mispricing measure
1.5
0.90
ACTUAL > THEOR
1
0.80
0.60
0
0.50
-0.5
0.40
0.30
-1
0.20
ACTUAL < THEOR
-1.5
-2
Jan-02
Implied vol
Mispricing measure
0.70
0.5
0.10
Apr-02
Jul-02
Sep-02
Dec-02
Mar-03
Jun-03
Sep-03
0.00
Dec-03
Deutsche Tlc 5Y CDS vs Mispricing measure
500
1.5
450
1
400
0.5
5YCDS
350
300
0
250
-0.5
200
150
-1
100
50
0
Jan-02
-1.5
ACTUAL < THEOR
Apr-02
Jul-02
Sep-02
Dec-02
Mar-03
Jun-03
Sep-03
[actual - theor] normalised spreads
ACTUAL > THEOR
-2
Dec-03
On both figures the zero equilibrium level (solid grey line) on the mispricing axis corresponds to the point where actual spreads are equal to theoretical values. The smooth black
line defines the current tendency of the mis-pricing condition.
Plot A displays the dynamics, relative to these two conditions, of the implied volatility (right
Y axis). During the first part of year 2002 and then again since April 2003, implied vol is
relatively low and, accordingly, theoretical spreads are low. Since the beginning of year 2003
there has been a general trend towards lower volatility coefficients. In March 2003 the mispricing measure becomes negative highlighting an underlying negative gap between theoretical and actual spreads: the latter, given the volatility pattern, is now expected to converge
towards the former.
A similar analysis holds for France Telecom.
19
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
Figure 4.1.2
Mispricing measure vs implied volatility (A) and CDS actual spreads (B).
France Tlc Jan.2002-Dec.2003
France Tlc
Implied volatility vs Mispricing measure
4
1.40
3
actual > theor
1.20
1.00
1
0.80
0
0.60
-1
-2
0.40
actual < theor
0.20
-3
-4
Jan-02
Implied vol
Mispricing measure
2
Apr-02
Jul-02
Sep-02
Dec-02
Mar-03
Jun-03
Sep-03
0.00
Dec-03
France Tlc
5YCDS vs Mispricing measure
3
600
actual > theor
2
5YCDS
500
1
400
0
300
200
-1
-2
actual < theor
100
0
Feb-02
-3
[actual - theor] normalised spreads
4
700
-4
May-02
Aug-02
Oct-02
Feb-03
May-03
Aug-03
Oct-03
From September 2002 CDS spreads have started decreasing very rapidly after a period of instability. The descent started at a stage in which the mis-pricing measure was already negative: it stayed slightly negative during the rest of the year and then in the current year. In this
period theoretical spreads are down, CDS spreads are down and below the theoretical value
in normalized terms and volatility is low. It is not possible to derive here a general rule: right
now CDS spreads appear at the current volatility level, fairly mispriced and an increase can
be expected during the next few weeks.
20
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
4.2 Trading rules
In § 4.1 and the appendix, we show that CDS market movements can to a certain extent, be
accurately foreseen by considering jointly the current level of stock price volatility and the
current difference between market and theoretical spreads.
We test now whether by introducing buy-sell indications in correspondence of deviations
from the 0 level of the mis-pricing measure and still conditioning on the behaviour of the ration between CDS spread and equity volatility, we can enter and leave the market at the right
time.
We consider France Telecom and limit the analysis to the year 2002. We assume that the investment decision relies on a 6-month history of actual CDS spreads, mispricing indicator
and CDS/volatility ratio. At the beginning of the year the investor has no position on Ford
and has collected the above data history.
The average CDS/volatility ratio, in green, has been computed from the original series
through a linear fit. The mispricing indicator in black (relative to the red 0 line: a positive
indicator indicates actual > theoretical normalised spreads) has instead been derived through
a 3rd degree polynomial fit of the original series. These are compared with actual CDS
spreads (left axis).
Figure 4.2.1 includes two plots: the first related to the market situation as at June 2002; the
second as at mid August 2003.
Figure 4.2.1
Buy – Sell indications for FT credits, year 2003
France TLC Trading rule 6 month history as at 15.8.2002
France TLC Trading rule 6 month history as at 30.6.2002
4
700
3
BUY SIGNAL
400
2.5
2
1.5
300
1
200
spread
difference
3
500
CDS spreads
CDS/volatility
500
2
400
SELL
SIGNAL
300
1
0
200
mispricing indicator
100
0.5
0
-0.5
08
/0
15 1/0
/0 2
22 1/0
/01 2
29 /0
/0 2
05 1/0
/0 2
12 2/0
/0 2
19 2/0
/02 2
26 /0
/0 2
05 2/0
/0 2
12 3/0
/0 2
19 3/0
/03 2
26 /0
/03 2
04 /0
/0 2
11 4/0
/0 2
18 4/0
/0 2
25 4/0
/04 2
03 /0
/0 2
10 5/0
/0 2
17 5/0
/0 2
24 5/0
/0 2
31 5/0
/0 2
07 5/0
/0 2
14 6/0
/0 2
21 6/0
/0 2
28 6/0
/06 2
/0
2
0
time
100
actual < theoretical
0
-1
-2
19
/0
26 2/0
/0 2
05 2/0
/0 2
12 3/0
/0 2
19 3/0
/0 2
26 3/0
/0 2
04 3/0
/0 2
11 4/0
/0 2
18 4/0
/0 2
25 4/0
/0 2
03 4/0
/0 2
10 5/0
/0 2
17 5/0
/0 2
24 5/0
/0 2
31 5/0
/0 2
07 5/0
/0 2
14 6/0
/0 2
21 6/0
/0 2
28 6/0
/0 2
05 6/0
/0 2
12 7/0
/0 2
19 7/0
/0 2
26 7/0
/0 2
02 7/0
/0 2
09 8/0
/0 2
8/
02
CDS spreads
4
risk premium
600
mispricing, risk premium
600
5
700
3.5
CDS spreads
BUY
SIGNAL
800
4.5
mispricing, risk premium
800
time
The first plot shows the market evolution over the 6 months preceding June: over the last 4
weeks CDS spreads increased by more than 250bp diverging significantly from the average
CDS/volatility ratio and driving up the mispricing indicator. The BUY signal is thus generated.
During the following months CDS spreads and stock market volatility start to decline, the
latter faster than the former: as a result we observe a decrease of theoretical spreads and a
convergence towards 0 of the mispricing indicator.
21
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
At the beginning of July the indicator becomes negative (actual spreads < theoretical
spreads) and actual spreads stabilise: a SELL signal is now generated and a spread difference
of 180 bps locked in.
4.3 Infra-sector relative values
We consider in this final section the potential of infra-sector relative value analysis based on
essentially two sources of information: the differences between normalised theoretical
spreads and between CDS market spreads for two different companies. The test is canonical:
we consider whether significant differences between theoretical spreads drive the movements
of market spreads for pair of companies.
Let’s consider Deutsche and France telecom: France Telecom being originally perceived by
the market to be riskier than Deutsche Telekom during year 2002.
Figure 4.3.1
Telecoms spread convergence analysis over 2002-2003
3
200
actual spread difference
4
2
100
1
0
0
-1
-100
-2
-200
-300
Jan-02
-3
Apr-02
Jul-02
Sep-02
Dec-02
Mar-03
Jun-03
Sep-03
Normalised theoretical spread difference
300
[Deutsche Tlc - France Tlc]
Actual (diamonds) vs Theoretical (solide line) Spreads difference
-4
Dec-03
As before, the theoretical spread difference remains around 0 and since October 2002 moves
remarkably in line with market spread differentials. In March 2003 the two credits are priced
the same and over the summer the small difference in theoretical spreads converges to 0 as
well.
A convergence trade [long France Tlc, short Deutsche Tlk] might have been constructed with
profit generated during the period in which France telecom recovered.
5.
CONCLUSIONS
In this report we have analysed in detail the interaction between stock price volatility, CDS
market spreads and theoretical credit spreads for six representative companies in the automotive, telecom and utility sectors.
22
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
The stock price volatility implied out in the option market turns out to be a major driver of
spread movements, both theoretical (not surprisingly, given the adopted model) and actual.
The strong correlation between theoretical and actual spreads has been used to derive a riskadjusted mispricing measure upon which market trading can be based.
The Merton model has been shown to provide a good benchmark model, despite of its limited ability to capture spreads’ fair values. We have pointed out that this generally recognised
drawback does depend on the limited impact on theoretical spreads of changes in the liability
structure. The model tends to undervalue the impact of changes in the leverage on the company default probability. This is instead a key variable in explaining market spread differences.
Observe however that the theoretical model has been shown to provide a coherent and robust
reference model in absence of actively traded debt.
A major result of the study we have conducted can be found in the identification and the
measurement of the key role of stock market volatility to define the risk source of credits and
thus, the opportunity to rely on this input variable to assess forthcoming market movements.
This result has been substantiated with a preliminary study on the presence of a causality effect from volatility patterns and CDS spreads: both in our setting estimated in the derivative
market.
Acknowledgements
The results presented in this study come from a project run at UniCredit Banca Mobiliare
Credit desk and the data have all being provided by the bank. The article has a companion
similar publication published as UBM research by the bank and included in the bibliography.
I acknowledge the fruitful collaboration with Daniele Palumbo, David Keeble and Giorgio
Frascella from UBM on this project.
23
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
BIBLIOGRAPHY
[1]
BLACK F. and SCHOLES M., The pricing of options and corporate liabilities, The
Journal of Political Economy 81, 637-654, 1973.
[2]
CAMPBELL J.Y and TAKSLER G.B., Equity volatility and Corporate Bond Yields,
The Journal of Finance, LVIII.6, 2321-2349, Dec.2003.
[3]
CONSIGLI G., 5Y Credit Default Swaps and theoretical spread dynamics: from pricing to relative value analysis, UniCredit Banca Mobiliare Special Issue n.6, contact
[email protected], 1-33, March 2004.
[4]
COOPER I.A and DAVYDENKO S.A., Using Yield Spreads to Estimate Expected Returns on Debt and Equity, Research Paper London Business school, contact author Ian
Cooper [email protected], 1-33, Dec.2003.
[5]
DUFFIE D., Credit Swap Valuation, Financial Analysts Journal, 55, 73-89, 1999.
[6]
GESKE R., The Valuation of Corporate Liabilities as Compound Options, Journal of
Financial and Quantitative Analysis, 5, 541-552, 1977.
[7]
GOURIEROUX C. and MONFORT A., Time Series and Dynamic Models, Themes in
Modern Econometrics, Cambridge University Press, 1997.
[8]
HULL J. and A.WHITE, Valuing Credit Default Swaps I: No Counterparty Default
Risk, Journal of Derivatives, 8, 29-40, 2000.
[9]
HULL J. NELKEN I and A.WHITE, Merton’s model, Credit Risk, and Volatility
Skews, Research Series University of Toronto, contact John Hull,
[email protected] 1-38, 2004.
[10] KEALHOFER S., Quantifying Credit Risk I: Default Prediction, Financial Analysts
Journal, 59, 30-44, 1 (2003a).
[11] KEALHOFER S., Quantifying Credit Risk II: Debt Valuation, Financial Analysts
Journal, 59, 78-92, 3 (2003b).
[12] MERTON R.C., On the pricing of corporate debt: The risk structure of interest rates,
Journal of Finance, 29, 449-470, 1974.
24
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
APPENDIX. STATISTICAL ANALYSIS FOR AUTOMOTIVES AND UTILITIES
Here below, we report without comment, the results collected for Automotives and Utilities
on the analysis described above for the Telecom sector. We refer to the article for comments
on the displayed dynamics.
Figure A.1
Theoretical (left) and actual (right) 5Y spreads
Automotive and Utility sectors – 2002-2003
700
1250
600
500
1000
400
750
300
500
200
250
100
0
Jan-02
5YCDS (diamonds)
theoretical 5Y spreads (lines)
FORD (grey) & GM (black)
1500
0
Apr-02
Jul-02
Oct-02
Jan-03
Apr-03
Jul-03
Oct-03
ENDESA (black) & RWE (grey)
195
250
200
155
135
150
115
95
100
75
55
35
5YCDS (diamonds)
theoretical 5Y spreads (lines)
175
50
15
-5
Jan-02
0
Apr-02
Jul-02
Oct-02
Jan-03
Apr-03
Jul-03
Oct-03
25
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
Figure A.2
Ford and GM
theoretical and actual spreads (left axis) versus equity implied volatility (right axis)
GM5YCDS(diamonds) andtheoretical (solidline) spreads
vsequityimpliedvolatility(dottedline)
FORD5YCDS(diamonds) andtheoretical (lines) spreads
vsequityimpliedvolatility(dottedwithsquares)
10
0.90
1.10
8
0.80
6
0.70
4
0.60
2
0.50
0
0.40
-2
0.30
-4
0.20
-6
Jan-02 Apr-02 Jul-02 Sep-02 Jan-03 Apr-03 Jul-03 Oct-03
0.10
6
0.90
0.80
4
0.70
2
0.60
0.50
0
equity implied vol
1.00
0.40
-2
0.30
-4
Jan-02 Apr-02 Jul-02 Sep-02 Jan-03 Apr-03 Jul-03 Oct-03
0.20
5Y nomalised spreads
5Y normalised spreads
8
1.20
equity return implied volatility
10
Figure A.3
Endesa and RWE
theoretical and actual spreads (left axis) versus equity implied volatility (right axis)
RWE
NormalisedActual (diamonds) andTheor (lines) 5YSpreads
vsEquityvol (dottedwithsquares)
0.3
0
0.2
4
0.1
0
Jul-02
Oct-02 Jan-03 Apr-03
Jul-03
Oct-03
0.4
3
2
0.3
1
0
-1
-1
0.5
0.2
equity volatility
1
normalised spreads
0.5
0.4
0.6
6
5
2
-2
Jan-02 Apr-02
7
0.6
3
normalised spreads
0.7
equity volatility
4
ENDESA
NormalisedTheor (lines) andActual (diamonds) 5YSpreads
vsEquityvol (dottedlineswithsquares)
0.1
-2
-3
Jan-02 Apr-02 Jul-02 Oct-02 Jan-03 Apr-03 Jul-03 Oct-03
0
26
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
Figure A.4
Actual minus theoretical normalised spreads for the Automotive and Utility sectors
Jan.2002—Dec.2003
FORD (diamonds grey) & GM (squared black)
[Actual - Theoretical] 5Y Normalised Spreads
4
Normalised spread difference
3
2
ACTUAL > THEOR
1
0
-1
-2
-3
-4
ACTUAL < THEOR
-5
-6
-7
Oct-01
Jan-02 Mar-02 Jun-02 Aug-02 Oct-02
Endesa (black) & RWE (grey) [Actual - Theoretical] 5Y Spreads
3
Normalised spread difference
2
Jan-03 Mar-03 Jun-03 Aug-03 Nov-03 Jan-04
ACTUAL > THEOR
1
0
-1
-2
ACTUAL < THEOR
-3
-4
Nov-01
Feb-02
May-02
Sep-02
Dec-02
Mar-03
Jun-03
Oct-03
Jan-04
Apr-04
27
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
Figure A.5
Mispricing measure vs implied volatility and CDS spreads. Jan 2002-Dec 2003
FORD
700
1.20
6
600
ACTUAL > THEOR
0.60
-2
0.40
Oct-02
Jan-03
Apr-03
Jul-03
0
Jan-02
6
2
Apr-03
Jul-03
Oct-03
0
400
0.60
350
4
2
300
250
0
200
0.30
150
0.20
100
-2
-4
50
0.10
-6
Jan-02 Apr-02 Jul-02 Sep-02 Jan-03 Apr-03 Jul-03 Oct-03
6
450
0.70
0.40
ACTUAL<THEOR
500
0.80
0.50
-2
0
Jan-02 Apr-02 Jul-02 Sep-02 Jan-03 Apr-03 Jul-03 Oct-03
0.00
2
0.6
250
actual >theor
2
actual >theor
0.5
200
1
-0.5
0.2
actual <theor
0.1
-1.5
0.5
150
5YCDS
0.3
implied vol
0
0
100
-0.5
-1
50
actual <theor
-2
Jan-02 Apr-02 Jul-02 Sep-02 Jan-03 Apr-03 Jul-03 Oct-03
0
1.5
1
0.4
0.5
-6
ENDESA
5YCDSvsMispricingmeasure
EndesaImpliedvolatilityvsmispricingmeasure
mispricing measure
Oct-02 Jan-03
Implied vol
5YCDS
Mispricing measure
4
-1
Jul-02
GM
5YCDSvs Mispricing measure
0.90
ACTUAL>THEOR
1.5
-6
Apr-02
Oct-03
GM
Impliedvolatilityvs Mispricingmeasure
-4
-4
ACTUAL<
THEOR
[actual - theoretical] normalised spreads
Jul-02
-2
200
0.00
Apr-02
0
300
[actual - theor] normalised spreads
-6
Jan-02
400
100
0.20
ACTUAL < THEOR
2
5YCDS spread
0
-4
ACTUAL>THEOR
500
0.80
2
Implied vol (line)
Mispricing measure
4
1.00
4
Mispricing measure
FORD
6
0
Jan-02 Apr-02 Jul-02 Sep-02 Jan-03 Apr-03 Jul-03 Oct-03
-1.5
-2
time
28
CDS and equity volatility: theoretical modelling and …_Giorgio CONSIGLI
RWE
Implied volatility vs mispricing measure
RWE
5YCDSvsMispricingmeasure
0.6
120
0.5
100
2
1
0.4
80
1
0
0.3
60
0
-1
0.2
40
-1
-2
0.1
20
-3
Jan-02
0
0
Jan-02 Apr-02 Jul-02 Oct-02 Jan-03 Apr-03 Jul-03 Oct-03
mispricing measure
actual <theor
Apr-02
Jul-02
Sep-02
Jan-03
Apr-03
Jul-03
Oct-03
&
actual >theor
actual <theor
3
-2
[actual - theor] normalised spreads
actual >theor
2
implied vol
5YCDS
3
-3
& &
29