# Credit Spreads, Leverage and Volatility: A Cointegration Approach

## Abstract

**:**

## 1. Introduction

- (a)
- A new structural model is developed along the lines of [1,5]. The model developed here is an extension of the Merton’s model in which the firm’s equity is priced as an n-fold compound call option instead of a vanilla call option; this allows to account for more than one debt maturing at only one future date, which surely is one of the most evident limitations of the Merton’s model.
- (b)
- A new estimation technique is implemented for those variables which structural models of default predict to be the drivers for the spreads. More specifically, a simple estimation which relies only on the joint calibration on the price of the equity and CDS spreads (and, indirectly, by the book value of the firm’s debt) is proposed to estimate the value of the firm’s assets alongside its volatility, which are both unobservable quantities.
- (c)
- Once the asset volatility and the market leverage are estimated, the goodness of these estimates is tested as their own ability to predict the one-period ahead CDS spreads. Different combinations of the model parameters are tested in order to obtain a satisfying calibration.
- (d)
- Finally, an econometric analysis of the determinants of the credit spreads is conducted using an error correction mechanism (ECM).

## 2. Literature Review

- Provide a novel and richer, though still tractable, structural model of default which removes one of the most stringent restriction of the Merton’s model, namely clustering the firm’s debt at one single point in time (theoretical);
- Develop a new estimation technique for some crucial unobservable variables, such as the value of the firm’s assets and volatility (methodological);
- Conduct a cointegration analysis on a large panel of US CDS spreads (using the estimated variables) to show that large part of the empirical failure of structural models is more apparent than real as it was largely due to an omitted variable problem (empirical).

## 3. The Model, Estimation Methodology and Data Description

#### 3.1. The Compound Option Model

#### 3.2. Estimation of the Unobservable Asset Value and Volatility

#### 3.3. Estimation of the Risk-Neutral Survival Probabilities

#### 3.4. Data Description and Aggregation Schemes of the Firms’ Capital Structures

`DD1Q`(long-term debt due in one year), that is ${F}_{1}=\mathtt{DD}\mathtt{1}\mathtt{Q}$. The remaining two bonds clustered at ${t}_{2}=5$ and ${t}_{3}=10$ are obtained from

`DLTQ`(long-term debt total), such that ${F}_{2}+{F}_{3}=w\xb7\mathtt{DLTQ}+(1-w)\xb7\mathtt{DLTQ}$. The weight is set as $w=1/3$, as motivated in the next section. This results in a sequence of outstanding debt, which is increasing with maturities. The choice of setting $n=3$ is considered optimal, as it is the smallest number of maturity dates needed in order to match both the level, slope and curvature of the term structure of the survival probabilities extracted from the CDSs. As a matter of fact, an effective calibration of the model should aim at reproducing the aforementioned term structure as accurately as possible.

## 4. Estimating the Cointegration

## 5. Discussion on the Main Results and Robustness Checks

#### Robustness Checks

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Stochastic Process Driving the Firm’s Equity

## Appendix B. The Delta of the Equity

**Theorem**

**A1.**

**Γ**positive definite and $\mathsf{{\rm Y}}\left(x\right)={\bigcap}_{i=1}^{k}\{{y}_{i}\in \mathbb{R}:{y}_{i}\le {d}_{i}\left(x\right)\}$, with $\mathbf{d}\left(x\right):{\mathbb{R}}_{+}\to {\mathbb{R}}^{k}$, ${d}_{i}\left(x\right)=\frac{lnx+{a}_{i}}{{b}_{i}}$ with ${a}_{i}\in \mathbb{R}$ and ${b}_{i}\in {\mathbb{R}}_{+}$. Then

**Proof.**

## Appendix C. The Vega of the Equity

**Theorem**

**A2.**

**Γ**positive definite and $\mathsf{{\rm Y}}\left(x\right)={\bigcap}_{i=1}^{k}\{{y}_{i}\in \mathbb{R}:{y}_{i}\le {d}_{i}\left(x\right)\}$, with $\mathbf{d}\left(x\right):{\mathbb{R}}_{+}\to {\mathbb{R}}^{k}$, ${d}_{i}\left(x\right)={b}_{i}x\pm \frac{{a}_{i}}{x}$ with ${a}_{i}:{\mathbb{R}}_{+}\to \mathbb{R}$ and ${b}_{i}\in {\mathbb{R}}_{+}$. Then

**Proof.**

## References

- Merton, R.C. On the pricing of corporate debt: The risk structure of interest rates. J. Financ.
**1974**, 29, 449–470. [Google Scholar] - Jones, E.P.; Mason, S.P.; Rosenfeld, E. Contingent Claims Analysis of Corporate Capital Structures: An Empirical Investigation. J. Financ.
**1983**, 39, 611–625. [Google Scholar] [CrossRef] - Huang, J.Z.; Huang, M. How Much of the Corporate-Treasury Yield Spread Is Due to Credit Risk? Rev. Asset Pricing Stud.
**2012**, 2, 153–202. [Google Scholar] [CrossRef] - Du, D.; Elkamhi, R.; Ericsson, J. Time-Varying Asset Volatility and the CreditSpread Puzzle. J. Financ.
**2019**, 74, 1841–1885. [Google Scholar] [CrossRef] - Geske, R. The Valuation of Corporate Liabilities as Compound Options. J. Financ. Quant. Anal.
**1977**, 12, 541–552. [Google Scholar] [CrossRef] - Collin-Dufresne, P.; Goldstein, R.S.; Martin, J.S. The Determinants of Credit Spread Changes. J. Financ.
**2001**, 56, 2177–2207. [Google Scholar] [CrossRef] - Cremers, K.J.M.; Driessen, J.; Meanhout, P.; Weinbaum, D. Individual Stock-Option Prices and Credit Spreads. J. Bank. Financ.
**2008**, 32, 2706–2715. [Google Scholar] [CrossRef] - Ericsson, J.; Jacobs, K.; Ovied, R. The Determinants of Credit Default Swap Premia. J. Financ. Quant. Anal.
**2009**, 44, 109–132. [Google Scholar] [CrossRef] - Zhang, B.Y.; Zhou, H.; Zhu, H. Explaining Credit Default Swap Spreads with the Equity Volatility and Jump Risks ofIndividual Firms. Rev. Financ. Stud.
**2009**, 22, 5099–5131. [Google Scholar] [CrossRef] - Engle, R.F.; Granger, C.W.J. Co-Integration and Error Correction: Representation, Estimation, and Testing. Econometrica
**1987**, 55, 251–276. [Google Scholar] [CrossRef] - Blanco, R.; Brennan, S.; Marsh, I.W. An Empirical Analysis of the Dynamic Relation between Investment-Grade Bonds and Credit Default Swaps. J. Financ.
**2005**, 60, 2255–2281. [Google Scholar] [CrossRef] - Eom, Y.H.; Helwege, J.; Huang, J.Z. Structural Models of Corporate Bond Pricing: An Empirical Analysis. Rev. Financ. Stud.
**2004**, 17, 499–544. [Google Scholar] [CrossRef] - Driessen, J. Is Default Event Risk Priced in Corporate Bonds? Rev. Financ. Stud.
**2005**, 18, 165–195. [Google Scholar] [CrossRef] - Longstaff, F.A.; Mithal, S.; Neis, E. Corporate Yield Spreads: Default Risk or Liquidity? New Evidence from the Credit Default Swap Market. J. Financ.
**2005**, 60, 2213–2253. [Google Scholar] [CrossRef] - Elton, E.J.; Gruber, M.J.; Agrawal, D.; Mann, C. Explaining the Rate Spread on Corporate Bonds. J. Financ.
**2001**, 56, 247–277. [Google Scholar] [CrossRef] - Campbell, J.Y.; Taksler, G.B. Equity Volatility and Corporate Bond Yields. J. Financ.
**2003**, 58, 2321–2349. [Google Scholar] [CrossRef] - Chen, L.; Collin-Dufresne, P.; Goldstein, R.S. On the Relation between the Credit Spread Puzzle and the Equity Premium Puzzle. Rev. Financ. Stud.
**2009**, 22, 3367–3409. [Google Scholar] [CrossRef] - Campbell, J.Y.; Cochrane, J.H. By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior. J. Political Econ.
**1999**, 107, 205–251. [Google Scholar] [CrossRef] - Leland, H.E. Corporate Debt Value, Bond Covenants, and Optimal Capital Structure. J. Financ.
**1994**, 49, 1213–1252. [Google Scholar] [CrossRef] - Heston, S.L. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Rev. Financ. Stud.
**1993**, 6, 327–343. [Google Scholar] [CrossRef] - Merton, R.C. On the pricing of contingent claims and the Modigliani-Miller theorem. J. Financ. Econ.
**1977**, 5, 241–249. [Google Scholar] [CrossRef] - Black, F.; Cox, J.C. Valuing corporate securities: Some effects of bond indenture provisions. J. Financ.
**1976**, 31, 351–367. [Google Scholar] [CrossRef] - Leland, H.E.; Toft, K.B. Optimal Capital Structure, Endogenous Bankruptcy, and the TermStructure of Credit Spreads. J. Financ.
**1996**, 51, 987–1019. [Google Scholar] [CrossRef] - Frey, R.; Sommer, D. The Generalization of the Geske-Formula for Compound Options to Stochastic InterestRates is Not Trivial—A Note. J. Appl. Probab.
**1998**, 35, 501–509. [Google Scholar] [CrossRef] - Geske, R. The Valuation of Compound Options. J. Financ. Econ.
**1979**, 7, 63–81. [Google Scholar] [CrossRef] - Geman, H.; Karoui, N.E.; Rochet, J.C. Change of numéraire, change of probability measure and option pricing. J. Appl. Probab.
**1995**, 32, 443–458. [Google Scholar] [CrossRef] - Brigo, D. Market Models for CDS Options and Callable Floaters. Risk Magazine, 2005. [Google Scholar]
- Brigo, D.; Mercurio, F. Interest Rate Models: Theory and Practice; Springer Finance: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Duffie, D.; Singleton, K.J. Modeling Term Structures of Defaultable Bonds. Rev. Financ. Stud.
**1999**, 12, 687–720. [Google Scholar] [CrossRef] - Litterman, R.; Scheinkman, J. Common Factors Affecting Bond Returns. J. Fixed Income
**1991**, 1, 54–61. [Google Scholar] [CrossRef] - Zhou, C. The term structure of credit spreads with jump risk. J. Bank. Financ.
**2001**, 25, 2015–2040. [Google Scholar] [CrossRef] - Pesaran, M.H.; Shin, K.; Smith, R.P. Pooled mean group estimation of dynamic heterogeneous panels. Pool. Mean Group Estim. Dyn. Heterog. Panels
**1999**, 94, 621–634. [Google Scholar] [CrossRef] - Longstaff, F.A.; Schwartz, E. A simple approach to valuing risky fixed and floating rate debt. J. Financ.
**1995**, 50, 789–821. [Google Scholar] [CrossRef] - Duffee, G.R. The relation between treasury yields and corporate bond yield spreads. J. Financ.
**1998**, 53, 2225–2241. [Google Scholar] [CrossRef] - Anderson, R.W.; Sundaresan, S. Design and Valuation of Debt Contracts. Rev. Financ. Stud.
**1996**, 9, 37–68. [Google Scholar] [CrossRef] - Mella-Barral, P.; Perraudin, W. Strategic Debt Service. J. Financ.
**1997**, 52, 531–556. [Google Scholar] [CrossRef] - Collin-Dufresne, P.; Goldstein, R.S. Do Credit Spreads Reflect Stationary Leverage Ratios? J. Financ.
**2001**, 56, 1929–1957. [Google Scholar] [CrossRef] - Carr, P.; Wu, L. Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions. J. Financ. Quant. Anal.
**2017**, 52, 2119–2156. [Google Scholar] [CrossRef] - Carr, P.; Wu, L. A Simple Robust Link Between American Puts and Credit Protection. Rev. Financ. Stud.
**2011**, 24, 473–505. [Google Scholar] [CrossRef] - Cox, J.C. The constant elasticity of variance option pricing model. J. Portf. Manag.
**1996**, 23, 15–17. [Google Scholar] [CrossRef] - Dupire, B. Pricing with a Smile. Risk
**1994**, 7, 18–20. [Google Scholar] - Henry-Labordère, P. Calibration of Local Stochastic Volatility Models to Market Smiles: A Monte-Carlo Approach. Risk Magazine, 23 October 2009; 16. [Google Scholar]
- Black, F.; Scholes, M. The Pricing of Options and Corporate Liabilities. J. Political Econ.
**1973**, 81, 637–654. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Continuation value (dashed and dotted line) and value (solid line) of the equity. Checking whether the continuation value of the equity ${S}^{\star}$ is greater than the face value of the bond F is equivalent to finding the value of the assets V greater than the default threshold $\overline{V}$.

**Figure 2.**Time series of the 5-year CDS spreads (

**top**), equity volatility (

**middle**) and financial leverage (

**bottom**) estimated as in (9) for four different companies: Capital One Financial (financials), Exelon (mining, energy and utilities), McDonald’s (retail, wholesale and services), and United Technologies (manufacturing). Visual inspection suggest non-stationarity and a strong comovement of the three variables. The non-stationarity of the time series is confirmed by unit root tests.

**Figure 3.**Cointegration equations (residuals of regression (11)) for four different companies: Capital One Financial (financials), Exelon (mining, energy and utilities), McDonald’s (retail, wholesale and services), and United Technologies (manufacturing). Visual inspection suggest stationarity, and therefore cointegration of CDS spreads, leverage, volatility and the treasury yield. Unit root tests confirm the stationarity of the residuals. (

**a**) Financials; (

**b**) mining, energy and utilities; (

**c**) retail, wholesale and services; (

**d**) manufacturing.

**Figure 4.**Scree plots for the first 10 PCs of the demeaned 1-, 5-, and 10-year CDS spread changes. The spread changes for different maturities have similar principal components and display the kink around the 3rd/4th component. Overall, the first component explains 25–35% of the total variance of the spread changes; the second component explains around 15%; the third component explains around 10%; the fourth component explains less then 10%. The first 10 PCs are able to explain 80% of the total variance for 1- and 10-year spread changes, and almost 90% of the total variance for the 5-year spread changes. However, the first four are able to explain only about 60% of the total variance.

**Table 1.**List of the selected companies (ticker) and their SIC code. The sample is further divided into four categories based on the industry/type or business: (a) financial companies; (b) mining, energy and utilities companies; (c) manufacturing; (d) retail, wholesale and services. Credit ratings are obtained from Compustat and the mode of the ratings over January 2013 to Decemeber 2017 are reported.

Ticker | SIC | Division | S&P Credit Rating |
---|---|---|---|

AAPL | 3663 | Manufacturing | AA+ |

ABT | 2834 | Manufacturing | A+ |

ALL | 6331 | Finance, Insurance and Real Estate | A− |

AMGN | 2836 | Manufacturing | A |

BA | 3721 | Manufacturing | A |

BAC | 6020 | Finance, Insurance and Real Estate | A− |

BMY | 2834 | Manufacturing | A+ |

C | 6199 | Finance, Insurance and Real Estate | BBB+ |

CAT | 3531 | Manufacturing | A |

CL | 2844 | Manufacturing | AA− |

CMCSA | 4841 | Transportation, Communications, Electric, Gas and Sanitary service | A− |

COF | 6141 | Finance, Insurance and Real Estate | BBB |

COP | 1311 | Mining | A |

COST | 5399 | Wholesale Trade | A+ |

CSCO | 3576 | Manufacturing | AA− |

CVS | 5912 | Retail Trade | BBB+ |

CVX | 2911 | Manufacturing | AA− |

DD | 2821 | Manufacturing | A− |

DIS | 4888 | Transportation, Communications, Electric, Gas and Sanitary service | A |

EMR | 3823 | Manufacturing | A |

EXC | 4911 | Transportation, Communications, Electric, Gas and Sanitary service | BBB |

F | 3711 | Manufacturing | BBB− |

FDX | 4513 | Transportation, Communications, Electric, Gas and Sanitary service | BBB |

GD | 3721 | Manufacturing | A+ |

GE | 4911 | Transportation, Communications, Electric, Gas and Sanitary service | AA+ |

HAL | 1389 | Mining | A |

HD | 5211 | Wholesale Trade | A |

IBM | 7370 | Services | AA− |

INTC | 3674 | Manufacturing | A+ |

JNJ | 2834 | Manufacturing | AAA |

JPM | 6020 | Finance, Insurance and Real Estate | A− |

KO | 2086 | Manufacturing | AA− |

LLY | 2834 | Manufacturing | AA− |

LOW | 5211 | Wholesale Trade | A− |

MCD | 5812 | Retail Trade | A |

MDT | 3845 | Manufacturing | A |

MMM | 2670 | Manufacturing | AA− |

MO | 2111 | Manufacturing | BBB+ |

MON | 5169 | Retail Trade | BBB+ |

MRK | 2834 | Manufacturing | AA |

MS | 6211 | Finance, Insurance and Real Estate | BBB+ |

MSFT | 7372 | Services | AAA |

ORCL | 7370 | Services | AA− |

OXY | 1311 | Mining | A |

PEP | 2080 | Manufacturing | A |

PFE | 2834 | Manufacturing | AA |

PG | 2840 | Manufacturing | AA− |

PM | 2111 | Manufacturing | A |

RTN | 3812 | Manufacturing | A |

SLB | 1389 | Mining | AA− |

SO | 4911 | Transportation, Communications, Electric, Gas and Sanitary service | A− |

SPG | 6798 | Finance, Insurance and Real Estate | A |

T | 4812 | Transportation, Communications, Electric, Gas and Sanitary service | BBB+ |

TGT | 5331 | Wholesale Trade | A |

TWX | 8748 | Services | BBB |

TXN | 3674 | Manufacturing | A+ |

UNH | 6324 | Finance, Insurance and Real Estate | A+ |

UNP | 4011 | Transportation, Communications, Electric, Gas and Sanitary service | A |

USB | 6020 | Finance, Insurance and Real Estate | A+ |

UTX | 3724 | Manufacturing | A− |

VZ | 4812 | Transportation, Communications, Electric, Gas and Sanitary service | BBB+ |

WFC | 6020 | Finance, Insurance and Real Estate | A |

WMT | 5331 | Retail Trade | AA |

XOM | 1311 | Mining | AAA |

**Table 2.**Pricing error of the CDS spread and risk-neutral probabilities of default for $w=1/2$. All the model-implied spreads are calculated setting the loss given default equal to 50%, 60% or 80%. This allows to jointly test for the effect of the aggregation scheme in the firm’s capital structure and on the selected value of LGD. The table reports the market CDS spread alongside those produced by the model (expressed in basis points) based on the estimates of the firm’s asset and volatility on the previous week. The results are clustered based on the maturity of the CDS contract (1, 5, and 10 years) and on the firm’s average leverage. A positive/negative pricing error ${\mathrm{CDS}}^{mrk}-{\mathrm{CDS}}^{model}$ indicates that the model under/overpredicts the level of the spread. This is reflected into the over/underprediction of the survival probabilities. Errors are also reported as percentages in brackets. For the probabilities of survival, only percentage errors are reported. Based on leverage, the number of companies in each bucket are ${N}_{low}=44$, ${N}_{med}=15$, ${N}_{high}=5$.

LGD | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{50}\mathbf{\%}$ | $\mathbf{60}\mathbf{\%}$ | $\mathbf{80}\mathbf{\%}$ | $\mathbf{50}\mathbf{\%}$ | $\mathbf{60}\mathbf{\%}$ | $\mathbf{80}\mathbf{\%}$ | $\mathbf{50}\mathbf{\%}$ | $\mathbf{60}\mathbf{\%}$ | $\mathbf{80}\mathbf{\%}$ | |||

LEV | CDS${}^{\mathit{mrk}}$ | CDS${}^{\mathit{model}}$ | CDS${}^{\mathit{mrk}}$ − CDS${}^{\mathit{model}}$($\mathbf{1}$ − CDS${}^{\mathit{model}}$/CDS${}^{\mathit{mrk}}$) | $\mathbf{1}\mathbf{-}{\mathbb{Q}}^{\mathit{model}}\mathbf{/}{\mathbb{Q}}^{\mathit{mrk}}$ | |||||||

1-year | $(0,0.25]$ | $7.68$ | $4.47$ | $4.12$ | $3.80$ | $\phantom{-}3.21$ $(42\%)$ | $\phantom{-}3.55$ $(46\%)$ | $\phantom{-}3.87$ $(50\%)$ | $-0.06\%$ | $-0.05\%$ | $-0.04\%$ |

$(0.25,1]$ | $15.55$ | $20.19$ | $20.02$ | $20.43$ | $-4.64$ $-(30\%)$ | $-4.47$ $-(29\%)$ | $-4.88$ $-(31\%)$ | $\phantom{-}0.09\%$ | $\phantom{-}0.08\%$ | $\phantom{-}0.06\%$ | |

$(1,\infty )$ | $28.85$ | $91.44$ | $86.25$ | $82.77$ | $-62.59$ $-(217\%)$ | $-57.40$ $-(199\%)$ | $-53.92$ $-(187\%)$ | $\phantom{-}1.19\%$ | $\phantom{-}0.93\%$ | $\phantom{-}0.66\%$ | |

5-year | $(0,0.25]$ | $35.20$ | $43.59$ | $41.51$ | $40.20$ | $-8.39$ $-(24\%)$ | $-6.31$ $-(18\%)$ | $-4.99$ $-(14\%)$ | $\phantom{-}0.70\%$ | $\phantom{-}0.50\%$ | $\phantom{-}0.31\%$ |

$(0.25,1]$ | $60.12$ | $107.84$ | $105.93$ | $106.59$ | $-47.71$ $-(79\%)$ | $-45.80$ $-(76\%)$ | $-46.47$ $-(77\%)$ | $\phantom{-}4.30\%$ | $\phantom{-}3.60\%$ | $\phantom{-}2.78\%$ | |

$(1,\infty )$ | $89.97$ | $192.52$ | $189.47$ | $189.75$ | $-102.55$ $-(114\%)$ | $-99.50$ $-(111\%)$ | $-99.78$ $-(111\%)$ | $\phantom{-}8.69\%$ | $\phantom{-}7.40\%$ | $\phantom{-}5.76\%$ | |

10-year | $(0,0.25]$ | $62.28$ | $63.28$ | $61.72$ | $61.82$ | $-1.00$ $-(2\%)$ | $\phantom{-}0.57$ $(1\%)$ | $\phantom{-}0.46$ $(1\%)$ | $-0.25\%$ | $-0.16\%$ | $-0.09\%$ |

$(0.25,1]$ | $93.55$ | $90.40$ | $87.45$ | $86.76$ | $\phantom{-}3.15$ $(3\%)$ | $\phantom{-}6.10$ $(7\%)$ | $\phantom{-}6.79$ $(7\%)$ | $-2.47\%$ | $-1.97\%$ | $-1.49\%$ | |

$(1,\infty )$ | $134.89$ | $121.84$ | $117.62$ | $115.07$ | $\phantom{-}13.05$ $(10\%)$ | $\phantom{-}17.27$ $(13\%)$ | $\phantom{-}19.82$ $(15\%)$ | $-6.97\%$ | $-5.52\%$ | $-4.15\%$ |

**Table 3.**Pricing error of the CDS spread and risk-neutral probabilities of default for $w=1/3$. All the model-implied spreads are calculated setting the loss given default equal to 50%, 60% or 80%. This allows to jointly test for the effect of the aggregation scheme in the firm’s capital structure and on the selected value of LGD. The table reports the market CDS spread alongside those produced by the model (expressed in basis points) based on the estimates of the firm’s asset and volatility on the previous week. The results are clustered based on the maturity of the CDS contract (1, 5, and 10 years) and on the firm’s average leverage. A positive/negative pricing error ${\mathrm{CDS}}^{mrk}-{\mathrm{CDS}}^{model}$ indicates that the model under/overpredicts the level of the spread. This is reflected into the over/underprediction of the survival probabilities. Errors are also reported as percentages in brackets. For the probabilities of survival, only percentage errors are reported. Based on leverage, the numbers of companies in each bucket are ${N}_{(0,0.25]}=44$, ${N}_{(0.25,1]}=15$, ${N}_{(1,+\infty )}=5$.

LGD | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{50}\mathbf{\%}$ | $\mathbf{60}\mathbf{\%}$ | $\mathbf{80}\mathbf{\%}$ | $\mathbf{50}\mathbf{\%}$ | $\mathbf{60}\mathbf{\%}$ | $\mathbf{80}\mathbf{\%}$ | $\mathbf{50}\mathbf{\%}$ | $\mathbf{60}\mathbf{\%}$ | $\mathbf{80}\mathbf{\%}$ | |||

LEV | CDS${}^{\mathit{mrk}}$ | CDS${}^{\mathit{model}}$ | CDS${}^{\mathit{mrk}}$ − CDS${}^{\mathit{model}}$($\mathbf{1}$ − CDS${}^{\mathit{model}}$/CDS${}^{\mathit{mrk}}$) | $\mathbf{1}\mathbf{-}{\mathbb{Q}}^{\mathit{model}}\mathbf{/}{\mathbb{Q}}^{\mathit{mrk}}$ | |||||||

1-year | $(0,0.25]$ | $7.68$ | $0.86$ | $0.67$ | $0.50$ | $\phantom{-}6.81$ $(89\%)$ | $\phantom{-}7.00$ $(91\%)$ | $\phantom{-}7.18$ $(93\%)$ | $-0.13\%$ | $-0.11\%$ | $-0.08\%$ |

$(0.25,1]$ | $15.55$ | $11.37$ | $10.79$ | $10.39$ | $\phantom{-}4.18$ $(27\%)$ | $\phantom{-}4.75$ $(31\%)$ | $\phantom{-}5.16$ $(33\%)$ | $-0.10\%$ | $-0.08\%$ | $-0.07\%$ | |

$(1,+\infty )$ | $28.85$ | $75.65$ | $70.27$ | $66.58$ | $-46.80$ $-(162\%)$ | $-41.41$ $-(144\%)$ | $-37.73$ $-(131\%)$ | $\phantom{-}0.89\%$ | $\phantom{-}0.67\%$ | $\phantom{-}0.46\%$ | |

5-year | $(0,0.25]$ | $35.20$ | $25.22$ | $23.06$ | $21.19$ | $\phantom{-}9.98$ $(28\%)$ | $\phantom{-}12.14$ $(34\%)$ | $\phantom{-}14.01$ $(40\%)$ | $-1.07\%$ | $-0.99\%$ | $-0.86\%$ |

$(0.25,1]$ | $60.12$ | $78.66$ | $75.47$ | $74.14$ | $-18.53$ $-(31\%)$ | $-15.35$ $-(26\%)$ | $-14.01$ $-(23\%)$ | $\phantom{-}1.45\%$ | $\phantom{-}1.12\%$ | $\phantom{-}0.77\%$ | |

$(1,+\infty )$ | $89.97$ | $162.23$ | $159.91$ | $162.48$ | $-72.26$ $-(80\%)$ | $-69.94$ $-(78\%)$ | $-72.51$ $-(81\%)$ | $\phantom{-}6.15\%$ | $\phantom{-}5.25\%$ | $\phantom{-}4.17\%$ | |

10-year | $(0,0.25]$ | $62.28$ | $64.91$ | $63.27$ | $63.04$ | $-2.62$ $-(4\%)$ | $-0.99$ $-(2\%)$ | $-0.76$ $-(1\%)$ | $0.40\%$ | $0.35\%$ | $0.27\%$ |

$(0.25,1]$ | $93.55$ | $95.76$ | $93.09$ | $92.90$ | $-2.21$ $-(2\%)$ | $\phantom{-}0.46$ $(0\%)$ | $\phantom{-}0.65$ $(1\%)$ | $-0.69\%$ | $-0.49\%$ | $-0.31\%$ | |

$(1,+\infty )$ | $134.89$ | $131.64$ | $127.62$ | $126.99$ | $\phantom{-}3.25$ $(2\%)$ | $\phantom{-}7.27$ $(5\%)$ | $\phantom{-}7.90$ $(6\%)$ | $-4.05\%$ | $-3.14\%$ | $-2.25\%$ |

**Table 4.**Pricing error of the CDS spread and risk-neutral probabilities of default for $w=2/3$. All the model-implied spreads are calculated setting the loss given default equal to 50%, 60% or 80%. This allows to jointly test for the effect of the aggregation scheme in the firm’s capital structure and on the selected value of LGD. The table reports the market CDS spread alongside those produced by the model (expressed in basis points) based on the estimates of the firm’s asset and volatility on the previous week. The results are clustered based on the maturity of the CDS contract (1, 5, and 10 years) and on the firm’s average leverage. A positive/negative pricing error ${\mathrm{CDS}}^{mrk}-{\mathrm{CDS}}^{model}$ indicates that the model under/overpredicts the level of the spread. This is reflected into the over/underprediction of the survival probabilities. Errors are also reported as percentages in brackets. For the probabilities of survival, only percentage errors are reported. Based on leverage, the number of companies in each bucket are ${N}_{(0,0.25]}=44$, ${N}_{(0.25,1]}=15$, ${N}_{(1,+\infty )}=5$.

LGD | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{50}\mathbf{\%}$ | $\mathbf{60}\mathbf{\%}$ | $\mathbf{80}\mathbf{\%}$ | $\mathbf{50}\mathbf{\%}$ | $\mathbf{60}\mathbf{\%}$ | $\mathbf{80}\mathbf{\%}$ | $\mathbf{50}\mathbf{\%}$ | $\mathbf{60}\mathbf{\%}$ | $\mathbf{80}\mathbf{\%}$ | |||

LEV | CDS${}^{\mathit{mrk}}$ | CDS${}^{\mathit{model}}$ | CDS${}^{\mathit{mrk}}$ − CDS${}^{\mathit{model}}$($\mathbf{1}$ − CDS${}^{\mathit{model}}$/CDS${}^{\mathit{mrk}}$) | $\mathbf{1}\mathbf{-}{\mathbb{Q}}^{\mathit{model}}\mathbf{/}{\mathbb{Q}}^{\mathit{mrk}}$ | |||||||

1-year | $(0,0.25]$ | $7.68$ | $16.57$ | $16.29$ | $16.53$ | $-8.89$ $(-116\%)$ | $-8.61$ $(-112\%)$ | $-8.85$ $(-115\%)$ | $\phantom{-}0.17\%$ | $\phantom{-}0.14\%$ | $\phantom{-}0.11\%$ |

(0.25, 1] | $15.55$ | $26.77$ | $26.60$ | $27.15$ | $-11.22$ $(-72\%)$ | $-11.05$ $(-71\%)$ | $-11.61$ $(-75\%)$ | $\phantom{-}0.21\%$ | $\phantom{-}0.18\%$ | $\phantom{-}0.14\%$ | |

$(1,+\infty )$ | $28.85$ | $94.77$ | $89.07$ | $83.35$ | $-65.92$ $(-228\%)$ | $-60.22$ $(-209\%)$ | $-54.49$ $(-189\%)$ | $\phantom{-}1.25\%$ | $\phantom{-}0.97\%$ | $\phantom{-}0.67\%$ | |

5-year | $(0,0.25]$ | $35.20$ | $63.77$ | $62.07$ | $62.05$ | $-28.57$ $(-81\%)$ | $-26.87$ $(-76\%)$ | $-26.84$ $(-76\%)$ | $\phantom{-}2.63\%$ | $\phantom{-}2.15\%$ | $\phantom{-}1.63\%$ |

$(0.25,1]$ | $60.12$ | $129.01$ | $125.68$ | $125.52$ | $-68.89$ $(-115\%)$ | $-65.55$ $(-109\%)$ | $-65.39$ $(-109\%)$ | $\phantom{-}6.17\%$ | $\phantom{-}5.10\%$ | $\phantom{-}3.90\%$ | |

$(1,+\infty )$ | $89.97$ | $196.16$ | $190.20$ | $186.96$ | $-106.19$ $(-118\%)$ | $-100.23$ $(-111\%)$ | $-96.99$ $(-108\%)$ | $\phantom{-}8.92\%$ | $\phantom{-}7.39\%$ | $\phantom{-}5.54\%$ | |

10-year | $(0,0.25]$ | $62.28$ | $60.77$ | $59.16$ | $58.83$ | $\phantom{-}1.51$ $(2\%)$ | $\phantom{-}3.12$ $(5\%)$ | $\phantom{-}3.45$ $(6\%)$ | $-1.35\%$ | $-1.07\%$ | $-0.80\%$ |

$(0.25,1]$ | $93.55$ | $84.65$ | $81.19$ | $79.41$ | $-22.37$ $(-24\%)$ | $-18.91$ $(-20\%)$ | $-17.13$ $(-18\%)$ | $-4.25\%$ | $-3.53\%$ | $-2.76\%$ | |

$(1,+\infty )$ | $134.89$ | $111.07$ | $106.77$ | $103.77$ | $-48.79$ $(-36\%)$ | $-44.49$ $(-33\%)$ | $-41.49$ $(-31\%)$ | $-9.47\%$ | $-7.61\%$ | $-5.80\%$ |

**Table 5.**Average absolute mean errors (expressed in basis points). Considering both all maturities and the 5-year maturity only, which is the most liquid, the error is smallest for the aggregation scheme $w=1/3$. Additionally, setting LGD = $0.5$ makes the pricing error smallest. As expected, the largest average pricing error is for the scheme $w=2/3$, which puts a lot of debt expiring in the short-term (which is unlikely to be for most of the companies). Reported figures are weighted averages in which the weights are the numbers of companies in each leverage bucket.

All Maturities | |||
---|---|---|---|

LGD | |||

w | 0.50 | 0.60 | 0.80 |

$1/2$ | $11.86$ | $24.51$ | $30.00$ |

$1/3$ | $9.58$ | $20.14$ | $25.86$ |

$2/3$ | $20.99$ | $44.80$ | $55.85$ |

5-year | |||

LGD | |||

$\mathit{w}$ | 0.50 | 0.60 | 0.80 |

$1/2$ | $24.96$ | $49.43$ | $59.25$ |

$1/3$ | $16.85$ | $37.66$ | $49.78$ |

$2/3$ | $44.08$ | $90.14$ | $110.80$ |

**Table 6.**ECM for 1-year CDS spreads. All the variables which structural models predict to influence the change in spreads are statistically significant and have the predicted signs. The loading on the cointegrating equation ($\epsilon $) is negative and statistically significant, thus confirming the existence of a long-term equilibrium to which spreads, volatility and leverage converge. This model constrains the long-run coefficient vector to be equal across panels while allowing for group-specific short-run and adjustment coefficients. The averaged short-run parameter estimates are reported.

1-Year CDS Spread | ||||
---|---|---|---|---|

Long-Run Equilibrium | ||||

Coefficient | t-Stat | p-Value | ||

VOL | $\phantom{-}0.0028$ | $\phantom{-}9.88$ | $0.000$ | *** |

LEV | $\phantom{-}0.0024$ | $\phantom{-}14.45$ | $0.000$ | *** |

Short-term adjustment | ||||

Coefficient | $\mathit{t}$-Stat | $\mathit{p}$-Value | ||

$\epsilon $ | $-0.1005$ | $-11.52$ | 0.000 | *** |

$\Delta $VOL | $\phantom{-}0.0074$ | $\phantom{-}6.29$ | $0.000$ | *** |

$\Delta $LEV | $\phantom{-}0.0036$ | $\phantom{-}5.23$ | $0.000$ | *** |

$\Delta $Level | $-0.0566$ | $-4.55$ | $0.000$ | *** |

$\Delta $Slope | $\phantom{-}0.0213$ | $\phantom{-}4.56$ | $0.000$ | *** |

$\Delta $Curvature | $\phantom{-}1.2409$ | $\phantom{-}3.77$ | $0.000$ | *** |

$\Delta ln$(S&P500) | $-0.0013$ | $-5.13$ | $0.000$ | *** |

$\Delta $Skew | $\phantom{-}5\times {10}^{-7}$ | $\phantom{-}2.37$ | $0.018$ | ** |

Constant | $-0.0001$ | $-9.66$ | $0.000$ | *** |

**Table 7.**ECM for 5-year CDS spreads. All the variables which structural models predict to influence the change in spreads are statistically significant and have the predicted signs. The loading on the cointegrating equation ($\epsilon $) is negative and statistically significant, thus confirming the existence of a long-term equilibrium to which spreads, volatility and leverage converge. This model constrains the long-run coefficient vector to be equal across panels while allowing for group-specific short-run and adjustment coefficients. The averaged short-run parameter estimates are reported.

5-Year CDS Spread | ||||
---|---|---|---|---|

Long-Run Equilibrium | ||||

Coefficient | t-Stat | p-Value | ||

VOL | $\phantom{-}0.0225$ | $\phantom{-}17.13$ | $0.000$ | *** |

LEV | $\phantom{-}0.0159$ | $\phantom{-}15.54$ | $0.000$ | *** |

Short-term adjustment | ||||

Coefficient | $\mathit{t}$-Stat | $\mathit{p}$-Value | ||

$\epsilon $ | $-0.0293$ | $-9.60$ | $0.000$ | *** |

$\Delta $VOL | $\phantom{-}0.0252$ | $\phantom{-}10.39$ | $0.000$ | *** |

$\Delta $LEV | $\phantom{-}0.0114$ | $\phantom{-}7.69$ | $0.000$ | *** |

$\Delta $Level | $-0.0975$ | $-4.90$ | $0.000$ | *** |

$\Delta $Slope | $\phantom{-}0.0346$ | $\phantom{-}3.42$ | $0.001$ | *** |

$\Delta $Curvature | $\phantom{-}1.8381$ | $\phantom{-}3.69$ | $0.000$ | *** |

$\Delta ln$(S&P500) | $-0.0025$ | $-5.71$ | $0.000$ | *** |

$\Delta $Skew | $-6\times {10}^{-7}$ | $-1.84$ | $0.065$ | * |

Constant | $-0.0003$ | $-9.75$ | $0.000$ | *** |

**Table 8.**ECM for 10-year CDS spreads. All the variables which structural models predict to influence the change in spreads are statistically significant and have the predicted signs. The loading on the cointegrating equation ($\epsilon $) is negative and statistically significant, thus confirming the existence of a long-term equilibrium to which spreads, volatility and leverage converge. This model constrains the long-run coefficient vector to be equal across panels while allowing for group-specific short-run and adjustment coefficients. The averaged short-run parameter estimates are reported.

10-Year CDS Spread | ||||
---|---|---|---|---|

Long-Run Equilibrium | ||||

Coefficient | t-Stat | p-Value | ||

VOL | $\phantom{-}0.0335$ | $\phantom{-}49.94$ | $0.000$ | *** |

LEV | $\phantom{-}0.0335$ | $\phantom{-}28.24$ | $0.000$ | *** |

Short-term adjustment | ||||

Coefficient | $\mathit{t}$-Stat | $\mathit{p}$-Value | ||

$\epsilon $ | $-0.0275$ | $-5.18$ | $0.000$ | *** |

$\Delta $VOL | $\phantom{-}0.0399$ | $\phantom{-}15.01$ | $0.000$ | *** |

$\Delta $LEV | $\phantom{-}0.0168$ | $\phantom{-}7.86$ | $0.000$ | *** |

$\Delta $Level | $-0.1129$ | $-4.95$ | $0.000$ | *** |

$\Delta $Slope | $\phantom{-}0.0242$ | $\phantom{-}2.31$ | $0.021$ | ** |

$\Delta $Curvature | $\phantom{-}2.2031$ | $\phantom{-}3.98$ | $0.000$ | *** |

$\Delta ln$(S&P500) | $-0.0027$ | $-5.41$ | $0.000$ | *** |

$\Delta $Skew | $-9\times {10}^{-7}$ | $-2.86$ | $0.004$ | *** |

Constant | $-0.0004$ | $-5.31$ | $0.000$ | *** |

**Table 9.**Average absolute mean errors (expressed in basis points) based on the results in [3]. There, the authors analyze the ability of structural models of default to reproduce observed credit spreads. They test a simple baseline model with and without stochastic interest rates ([33]), a model with endogenous default barrier ([23]), a model with strategic default ([35,36]), a model with mean-reverting leverage ratios ([37]), a model with countercyclical market risk premium, and a jump-diffusion model. A loss given default parameter of 48.69% is used by the authors for their calibration.

Structural Model | AAME |
---|---|

Baseline ([33]) | $89.49$ |

Baseline plus stochastic interest rates ([33]) | $105.67$ |

Endogenous default barrier ([23]) | $86.27$ |

Strategic default ([35,36]) | $76.89$ |

Mean-reverting leverage ratios ([37]) | $93.25$ |

Countercyclical market risk premium ([3]) | $83.19$ |

Jump-diffusion ([3]) | $84.78$ |

Compound option model | $9.58$ |

**Table 10.**Adjusted ${R}^{2}$s of the firm-specific time-series regressions in (13) (short-term adjustments). As shown by both the mean- and median-adjusted ${R}^{2}$, the explanatory power of the variables, which should affect credit spread changes as predicted by structural models, diminishes with the maturity of the spread.

1-Year | 5-Year | 10-Year | 1-Year | 5-Year | 10-Year | ||
---|---|---|---|---|---|---|---|

Ticker | adj–${\mathit{R}}^{2}$ | Ticker | adj–${\mathit{R}}^{2}$ | ||||

AAPL | 0.81 | 0.42 | 0.01 | LLY | 0.71 | 0.44 | 0.40 |

ABT | 0.86 | 0.36 | 0.24 | LOW | 0.76 | 0.74 | 0.07 |

ALL | 0.56 | 0.22 | 0.22 | MCD | 0.32 | 0.22 | 0.05 |

AMGN | 0.62 | 0.38 | 0.30 | MDT | 0.92 | 0.88 | 0.79 |

BA | 0.58 | 0.51 | 0.16 | MMM | 0.81 | 0.82 | 0.57 |

BAC | 0.49 | 0.26 | 0.21 | MO | 0.65 | 0.41 | 0.19 |

BMY | 0.71 | 0.29 | 0.15 | MON | 0.87 | 0.57 | 0.54 |

C | 0.45 | 0.27 | 0.23 | MRK | 0.81 | 0.77 | 0.35 |

CAT | 0.52 | 0.13 | 0.08 | MS | 0.63 | 0.53 | 0.50 |

CL | 0.85 | 0.74 | 0.20 | MSFT | 0.90 | 0.59 | 0.15 |

CMCSA | 0.57 | 0.20 | 0.13 | ORCL | 0.84 | 0.60 | 0.68 |

COF | 0.87 | 0.72 | 0.73 | OXY | 0.71 | 0.33 | 0.14 |

COP | 0.34 | 0.10 | 0.07 | PEP | 0.91 | 0.61 | 0.36 |

COST | 0.89 | 0.89 | 0.87 | PFE | 0.64 | 0.43 | 0.19 |

CSCO | 0.74 | 0.66 | 0.62 | PG | 0.86 | 0.82 | 0.20 |

CVS | 0.75 | 0.57 | 0.24 | PM | 0.86 | 0.74 | 0.33 |

CVX | 0.80 | 0.08 | 0.05 | RTN | 0.53 | 0.16 | 0.04 |

DD | 0.66 | 0.39 | 0.43 | SLB | 0.36 | 0.12 | 0.05 |

DIS | 0.77 | 0.46 | 0.32 | SO | 0.39 | 0.70 | 0.11 |

EMR | 0.52 | 0.65 | 0.36 | SPG | 0.29 | 0.10 | 0.08 |

EXC | 0.94 | 0.69 | 0.40 | T | 0.63 | 0.18 | 0.17 |

F | 0.55 | 0.36 | 0.31 | TGT | 0.80 | 0.70 | 0.41 |

FDX | 0.71 | 0.54 | 0.45 | TWX | 0.61 | 0.25 | 0.19 |

GD | 0.81 | 0.81 | 0.76 | TXN | 0.81 | 0.40 | 0.44 |

GE | 0.89 | 0.91 | 0.90 | UNH | 0.67 | 0.28 | 0.06 |

HAL | 0.22 | 0.09 | 0.10 | UNP | 0.54 | 0.18 | 0.06 |

HD | 0.79 | 0.55 | 0.33 | USB | 0.69 | 0.28 | 0.38 |

IBM | 0.59 | 0.29 | 0.24 | UTX | 0.76 | 0.40 | 0.17 |

INTC | 0.88 | 0.07 | 0.05 | VZ | 0.62 | 0.32 | 0.24 |

JNJ | 0.77 | 0.50 | 0.47 | WFC | 0.62 | 0.53 | 0.52 |

JPM | 0.57 | 0.41 | 0.39 | WMT | 0.71 | 0.33 | 0.09 |

KO | 0.73 | 0.80 | 0.57 | XOM | 0.83 | 0.35 | 0.20 |

1-year | 5-year | 10-year | |||||

adj–${R}^{2}$ | |||||||

Mean | 0.69 | 0.45 | 0.30 | ||||

Median | 0.71 | 0.41 | 0.24 | ||||

Min | 0.22 | 0.07 | 0.01 | ||||

Max | 0.94 | 0.91 | 0.90 |

**Table 11.**ECM for 1-year CDS spreads using the average implied volatility of put options instead of ${\sigma}_{S}$. Similar results are obtained; however, the implied volatility is significant only at the 10% significance level in the short-term adjustment equation. Additionally, $\Delta $LEV, $\Delta $Curvature and $\Delta $Skew have become insignificant, and $\Delta $Level is significant at the 10% significance level only.

1-Year CDS Spread | ||||
---|---|---|---|---|

Long-Run Equilibrium | ||||

Coefficient | t-Stat | p-Value | ||

IV | $\phantom{-}0.0008$ | $\phantom{-}3.77$ | $0.000$ | *** |

LEV | $\phantom{-}0.0025$ | $\phantom{-}12.18$ | $0.000$ | *** |

Short-term adjustment | ||||

Coefficient | $\mathit{t}$-Stat | $\mathit{p}$-Value | ||

$\epsilon $ | $-0.1063$ | $-9.79$ | 0.000 | *** |

$\Delta $IV | $\phantom{-}0.0002$ | $\phantom{-}1.83$ | $0.067$ | * |

$\Delta $LEV | $\phantom{-}0.0012$ | $\phantom{-}1.42$ | $0.156$ | |

$\Delta $Level | $-0.0390$ | $-1.90$ | $0.057$ | * |

$\Delta $Slope | $\phantom{-}0.0318$ | $\phantom{-}3.17$ | $0.002$ | *** |

$\Delta $Curvature | $\phantom{-}0.5474$ | $\phantom{-}1.41$ | $0.157$ | |

$\Delta ln$(S&P500) | $-0.0020$ | $-5.22$ | $0.000$ | *** |

$\Delta $Skew | $\phantom{-}4\times {10}^{-7}$ | $\phantom{-}1.35$ | $0.176$ | |

Constant | $\phantom{-}0.0001$ | $2.31$ | $0.000$ | *** |

**Table 12.**ECM for 5-year CDS spreads using the average implied volatility of put options instead of ${\sigma}_{S}$. Similar results are obtained; however, the implied volatility is not significant in the short-term adjustment equation. Additionally, $\Delta $LEV, $\Delta $Curvature and $\Delta $Skew are significant at the 10% significance level only, and $\Delta $Level is significant at the 5% significance level only.

5-Year CDS Spread | ||||
---|---|---|---|---|

Long-Run Equilibrium | ||||

Coefficient | t-Stat | p-Value | ||

IV | $\phantom{-}0.0147$ | $\phantom{-}8.67$ | $0.000$ | *** |

LEV | $\phantom{-}0.0038$ | $\phantom{-}7.18$ | $0.000$ | *** |

Short-term adjustment | ||||

Coefficient | $\mathit{t}$-Stat | $\mathit{p}$-Value | ||

$\epsilon $ | $-0.0360$ | $-10.68$ | $0.000$ | *** |

$\Delta $IV | $\phantom{-}0.0002$ | $\phantom{-}0.91$ | $0.361$ | |

$\Delta $LEV | $\phantom{-}0.0027$ | $\phantom{-}1.93$ | $0.053$ | * |

$\Delta $Level | $-0.1099$ | $-2.19$ | $0.028$ | ** |

$\Delta $Slope | $\phantom{-}0.0886$ | $\phantom{-}2.71$ | $0.007$ | *** |

$\Delta $Curvature | $\phantom{-}1.1884$ | $\phantom{-}1.74$ | $0.081$ | * |

$\Delta ln$(S&P500) | $-0.0045$ | $-5.87$ | $0.000$ | *** |

$\Delta $Skew | $-{1}^{-6}$ | $-1.90$ | $0.057$ | * |

Constant | $-{6}^{-6}$ | $-0.49$ | $0.623$ |

**Table 13.**ECM for 10-year CDS spreads using the average implied volatility of put options instead of ${\sigma}_{S}$. Similar results are obtained; however, neither the implied volatility nor leverage are significant in the short-term adjustment equation. Additionally, $\Delta $Level, $\Delta $Slope, $\Delta $Curvature and $\Delta $Skew are significant at the 5% significance level only.

10-Year CDS Spread | ||||
---|---|---|---|---|

Long-Run Equilibrium | ||||

Coefficient | t-Stat | p-Value | ||

IV | $\phantom{-}0.0312$ | $\phantom{-}11.43$ | $0.000$ | *** |

LEV | $\phantom{-}0.0285$ | $\phantom{-}17.22$ | $0.000$ | *** |

Short-term adjustment | ||||

Coefficient | $\mathit{t}$-Stat | $\mathit{p}$-Value | ||

$\epsilon $ | $-0.0277$ | $-5.82$ |