Master’s Degree
in Economics and Finance
Final Thesis
Corporate CDS spreads determinants from the
Eurozone crisis to COVID-19 pandemic:
A Bayesian Markov switching model
Supervisors:
Prof. Andrea Berardi
Prof. Roberto Casarin
Graduand:
Giacomo Bulfone
Matriculation number
875979
Abstract
A deep understanding of the CDS spreads determinants is crucial for both policy makers interested in preserving the stability of the financial system and of financial insiders interested in managing credit and financial
risks. The literature is mainly focused on the pre-subprime crisis, and
either consider linear models with a large number of covariates or non-linear models, such as regime Markov switching models, with a small number of explanatory variables and two regimes only. The aim of this thesis is to investigate the determinants of the European iTraxx corporate index considering a large set of explanatory variables (both macroeconomic and financial) within a Markov switching model framework. The focus is on the post 2007-2009 crisis and more precisely on the period from October 2011 to April 2020 which includes the sovereign debt crisis, the Brexit referendum and the recent COVID-19 pandemic events. The dataset includes financial and economic variables usually employed in CDS spreads analysis and some new explanatory variables such as changes in the Baltic Dry Index and WTI crude oil returns. The analysis is conducted in two steps. First a multivariate
regression model is estimated via OLS method. Second, stability tests
are also used to detect structural breaks in the linear relationship and to motivate the use of non-linear models. Finally, the in-sample and out-of-sample analysis of the forecasting performances of different Markov switching models has been performed. The empirical results suggest that: more than 2 regimes are significant to model CDS spreads; the impact of the covariates varies across regimes; and that both the economic activity index and the crude oil returns have some predictive power for changes in the iTraxx index. âĂŃ
Contents
1 Introduction 4
2 Data 12
2.1 Explanatory variables . . . 12
2.2 Preliminary data analysis . . . 16
2.3 Preliminary analysis of the variance . . . 16
3 Multiple regression models 19 3.1 Regression models results discussion . . . 23
4 Markov switching technique 26 4.1 Markov chain process . . . 27
4.2 Hamilton filter . . . 29
4.3 Gibbs sampler . . . 32
5 Markov switching regression models 35 5.1 2-regime Markov switching regression . . . 35
5.2 2-regimes Markov switching regression results discussion . . . 38
5.3 3-regimes Markov switching regression . . . 40
5.4 3-regimes Markov switching regression results discussion . . . 40
6 Conclusions 45 A 2-regime Markov switching posterior distributions 48 A.1 Model (33) . . . 48
A.2 Model (34) . . . 51
B 3-regime Markov switching posterior distributions 54 B.1 Model (35) . . . 54
1
Introduction
The credit default swap (CDS) is the most used derivate to hedge credit risk. Its origin dates back to mid 90s. In 1994, JP Morgan created CDSs, in order to reduce credit risk exposure allowing to extend its loan capability. The CDS was written with the European Bank of Reconstruction and Development (EBRD) and the aim of the deal was to allow the bank to offer a line of credit of USD 5 billions to Exxon, maintaining balance sheet flexibility (Vogel et al. (2013)). Over the years, CDSs have become more and more complex instruments, and saw a steady increase both in volumes and notional outstanding, reaching the peak of USD 60 trillions prior to the subprime crisis.
In 2008, the insurance giant AIG was one of the main CDSs seller. As stated by Dino Kos on Financial Times (ft.com), the use of those derivatives at that time was to concentrate risk rather than disperse it, pushing AIG on the brink of collapse as the subprime bubble burst.
Given their primary role in the 2007-2009 crisis and the fact that they are traded only on OTC markets1, causing a lack of transparency, has forced the regulator to a
standardization of the market, introducing CCPs in order to erase the counterparty risk (Aldasoro and Ehlser (2018)).
These changes were introduced for European markets in 2009, with the small-bang protocol. Two main features were introduced: the standardized coupon rates and the quoting conventions. The former asserts that European corporate CDS trade with fixed coupons of 25, 50, 100 and 1000 bps, while European sovereign CDS trade with 25 bps for entities with tighter spreads and with 100 bps for the ones with higher spreads. Similarly, the quoting conventions imply the introduction of standardized coupon dates for European CDS. Therefore, for any given year, the coupon dates are fixed on March 20th, June 20th, September 20th and December 20th. Lastly, the protocol also brings a standardized settlement procedure.
1Over-the-counter (OTC) derivative market is the location where banks, large financial
institutions and fund managers are the main participants. It is less regulated than the exchange traded market and the trade can either be presented to a central counterparty (CCP) or cleared bilaterally (Options, futures and other derivatives, p. 3-6).
My work focuses on the European iTraxx corporate index. The Markit iTraxx are a family of Asian, European and Emerging Market credit default swap indexes. The iTraxx group was formed by the merger in 2004 of JP Morgan and Morgan Stanley Trac-X indexes and the ones created by Deutsche Bank, ABN Amro and IBoxx (i.e. iBoxx CDS indexes). In November 2007, Markit acquired both CDX and iTraxx and by 2011 Markit owned and managed most of the CDS indexes. Their aim is to transfer risk in a more efficient way rather than using multiple single-name CDSs. The Markit iTraxx Europe Main index2 trades with maturities
of 3, 5, 7 and 10 years. It is an equally-weighted index composed by the 125 most liquid listed companies, with weight 0.8 for each constituent firm’s CDS (icmagroup.org). Every 6 months3 the index is updated, rolling out the firms that
either defaulted/merged or do not respect the liquidity parameter, including new ones.
A single-name CDS is a contract in which two parties are involved. On one
hand, the protection buyer, who pays periodic fixed payments to the counterparty until the CDS expires due to a credit event or maturity. On the other hand, the protection buyer, who receives the fixed payments and, as the credit event occurs, must buy the bonds owned by the counterparty at their face value. The fixed spread represents the compensation for the insurance in case of a credit event. Conversely, multi-name CDSs, which consider multiple entities, are contracts that include CDS indexes, basket products and CDS tranches. In this case the credit event of a single reference entity does not terminate the contract. So, a CDS is basically an insurance contract against the risk of default of the underlying reference entity. The total value of the bonds that can be sold is known as notional principal and the settlement can be physical, if the protection buyer delivers the bonds to the protection seller, or cash settlement in which just the difference between the face value and the value of the bonds at the time of default is paid. A key aspect is the aforementioned credit event, which is usually defined as either bankruptcy or restructuring of debt.
CDS are mainly used as an hedging tool. According to Vogel et al. (2005), with
2also known as "the Main"
the evolution of credit markets, CDS spreads have become a valuable indicator of the creditworthiness of the reference entity, as well as a barometer of health for the broad credit markets, since these information are generated from banks’ trading and hedging actions. Indeed, an increase in CDS spreads can be interpreted as an higher demand for credit protection, as financial institutions perceive a higher risk of these loans.
Comparing CDS spreads with credit ratings, it turns out that the former is a more reactive and accurate assessment of credit conditions than rating agencies. This is supported by the fact that CDS are continuously traded, making them a real time indicator, while ratings tend to be slow to change. In addition, during the subprime crisis, rating agencies have been severely criticized for their unethical behaviour, losing credibility.
Also CDS market has been criticized. Along with the aforementioned OTC
nature of this instrument, multiple papers have studied the CDS market efficiency (Byström (2005,2006), Avino (2014), Sensoy (2017)). The main findings are that several European CDS markets, which are the credit risk benchmarks in Europe, present a significant autocorrelated structure and a time-varying market efficiency. Byström (2005) analyses separately seven Europe iTraxx sub-indexes (i.e. industrials, autos, energy, TMT, consumers, senior financials and sub-ordinated financials), finding positive autocorrelation and that some firm-specific information is embedded first in stock prices respect to CDS prices. In Byström (2006), the author, considering the same sub-indexes, tries to exploit this inefficiency applying a simple trading strategy, which generates profits before considering transaction costs. In a similar way, Avino and Nneji (2014) consider only the financial senior and non-financial sectors. They based their trading strategy on predictions of both linear and non-linear models, generating similar results to the ones in Byström (2006). In addition, they find that autocorrelation is significant only during low volatility periods, while in volatile ones it ceases to exist. Moreover, Sensoy et al. (2017) analyses several sovereign CDS markets employing a permutation entropy approach in combination with an independence test, highlighting the fact that they have different degrees of time-varying efficiency, and that efficiency also changes
depending on the region to which the index refers.
The CDS spreads and credit determinants have been widely studied in literature. There is evidence of an arbitrage link between CDS spreads and credit spreads (Blanco et al. (2005), Duffie (1999)).
Blanco et al. (2005) discover the evidence of cointegration for all the U.S. firms
where CDS and bonds market priced risk equally on average. Conversely, in
Europe there is little evidence of cointegration for the entities involved in the research. Nevertheless, the cointegration test partially failed due to possibly different contract specifications between Europe and U.S. (i.e. CTD option). In the light of the reasons given above, a fully understanding of CDS spreads determinants are in the interest of both financial actors and policy makers. The former interested in an efficient assessment of risks, while the latter in keeping financial stability in the economy. Moreover, CDS spreads have become a powerful price discovery tool (Blanco et al. (2005), Schreiber et al. (2012), Lee at al. (2018)). Their findings are united by the fact that CDS market leads stock and bond markets, and also equity market volatility.
There is a large literature on the pricing of credit risk, divided in structural and reduced form models. The main difference between them is the information set used. Therefore, if firm-specific information is used, the model considered
is structural. On the other hand, reduced form models assume less detailed
information about the firm, and the estimation of the model relies on market data (Jarrow and Protter (2004)).
The first structural model, which has been extended in several different directions, is given by Merton (1974) (see also Black & Scholes (1973)). The model assumes that the dynamics of a firm’s value can be described by means of stochastic differential equations and finds that the risk premium depends on three variables: the firm’s volatility, the present value of the financial leverage and the risk-free rate (used in the calculus of the present value).
The structural approach has been criticized, among others, by Collin-Dufresne et al. (2001). The authors compare the explained variance by two multiple regression
models, one estimated just considering firm-specific variables and the other adding macroeconomic factors. The result is that the second one outperforms the first one in terms of credit spread explained. Nevertheless, most of the variation is due to a common unknown systematic factor.
Other related works are Abid e Naifar (2006), Zhang et al. (2009), Ericsson et al. (2009), Annaert et al. (2010), Kajurova et al. (2014) and Fu et al. (2020). Abid e Naifar (2006) estimate a multiple regression model finding that macroeconomic variables have a greater explanatory power than firm-specific ones. Similar results are achieved by Fu et al. (2020), Ericsson et al. (2009) and Blanco et al. (2005). In particular, the last one find that macroeconomic variables, such as interest rates, term structure, equity market returns and market volatility, have a larger and immediate impact on credit spreads, while CDS spreads are more sensitive to firm-specific variables. However, considering the long-run arbitrage-based equivalence, they are equally sensitive to both macro and firm-specific factors. In any case, theoretical determinants, which are used in structural models, must be considered in order to analyse credit risk.
Zhang et al. (2009) analyse changes in single-name CDS spreads considering both jump risk effects and firm-specific variables, explaining an additional 14% to 18% of spread variation.
Kajurova et al. (2014) analyses the CDS spreads determinants applying multiple regression models in three different time periods (pre, during and post subprime crisis). Their result is that explanatory power of the considered variables tend to change upon the economic conditions. Similarly, Annaert at al. (2010) show by applying a rolling multiple regression model that explanatory variables’ sign of bank CDS spreads changes in relation with the economic conditions, and so, for efficient policy decisions, policy makers should not rely only on financial institutions’ spreads to monitor their credit risk, but also on liquidity and business cycle factors.
To my best knowledge, Alexander and Kaeck (2008) were the first to analyse daily corporate CDS spread determinants of different iTraxx sub-indexes in a regime-dependent framework. They first develop a multiple regression analysis
considering level and slope of term structure, volatility of the market, equity
returns and a lagged value of iTraxx sub-indexes. Then, assessing the high
instability of the estimated parameters, they extend the regression model with a 2-state Markov switching regression. Their conclusion is that the influence of theoretical determinants have a regime dependent behaviour and that the unknown systematic factor found in Collin-Dufresne et al. (2001) is due to the regime specific behaviour of CDS spreads.
Other works that study CDS spreads in which a Markov switching model is employed are Riedel et al. (2013), Avino and Nneji (2014), Chan and Marsden
(2014), Ma et. Al (2018), Guidolin et al. (2019) and Sabkha et al. (2019).
The mentioned above works analyse both sovereign and corporate CDS spreads.
Sovereign spreads are analysed by: Riedel et al. (2013), which apply a
Markov switching model to estimate the credit cycle and study the daily spread determinants in emerging sovereign debt markets, finding that credit spreads are characterized by a varying influence of the spread determinants. Ma et al. (2018) use the Markov regime switching approach to examine the relationship of short-term sovereign CDS spreads considering county-specific and global variables, finding that significance of the considered variables changes according to the underlying regime. Sabhka et al. (2019) analyse the non-linear relationship between oil shocks and sovereign CDS spreads. They find that during the high volatility regime sovereign debt is sensitive to these shocks. I will extend their analysis to the corporate sector, trying to see if oil shocks could also influence corporate CDS spreads, in particular during volatile regimes.
In the empirical analysis on corporate CDS spreads comprises the aforementioned work of Avino and Nneji (2014), in which the MS model is compared with other linear models in order to predict changes in iTraxx index. They found that linear models have a better predictive power than non-linear models. Chan and Marsden (2014) study CDX spreads determinants in a Markov switching framework for both investment-grade and high-yield companies. They use a large set of explanatory variables, both macroeconomic and firm-specific ones, finding that regressor exhibit a regime dependent behaviour, and that monetary policy is influent in both
Markov switching vector error correction model to capture that the adjustment mechanism between corporate and CDS spreads may be characterized by a non-linear relationship.
In my thesis, I try to conjugate the line of research about CDS and credit spreads determinants started by Collin-Dufresne et al. (2001), with the regime switching technique introduced by Alexander and Kaeck (2008). The work is related to the Alexander and Kaeck (2008) paper, as I consider daily observations and the European iTraxx corporate index in a Markov switching framework, and to Collin-Dufresne et al. (2001), among others, as I will also enlarge the number of CDS spread determinants including macroeconomic factors. I consider explanatory variables derived from both structural and reduced form models and introduce novel variables in the corporate sector analysis (i.e. BDI changes and WTI crude oil returns), in the context of a regime switching framework. In this regard, my contribution to the literature is threefold: i) I update the Alexander and Kaeck (2008) paper in a new time range after the 2007-2009 crisis; ii) I extend the Collin-Dufresne et al. analysis by applying a Markov switching regression model in the evidence of a non-linear behaviour of CDS spreads; iii) I try to introduce a third regime, which has never been analysed before. The rationale behind the new regime is to divide the time series in normal volatility, high volatility and extreme volatility which characterizes periods of economic and financial distress.
The extreme volatility regime is supported by the empirical evidence offered by the flourish literature about the economic impact of COVID-19 pandemic (Baker et al. (2020), Gormsen and Koijen (2020), Onali (2020)). Gormsen and Koijen (2020) analyse the stock and bond market returns of both U.S. and Europe. The result is that equity indexes returns have dropped more than 30% in March. The same decline happen for the 30-year maturity Bund, signalling an increase in investors’ risk aversion (see also FSB (2020)). A similar result is achieved by Baker at al. (2020), where the authors make a comparison between the past disease (i.e. Spanish flu, Ebola), finding that COVID-19 pandemic has caused a stock market impact never seen before, with levels of volatility comparable to the ones of 2007-2009 crisis. Onali (2020) studies the VIX index in relation to the number of deaths, finding that deaths impacted VIX positively and Dow Jones returns negatively.
The reminder of this work is organized as follows: in Section 2 the data is discussed in detail and some preliminary analysis are shown; Section 3 reports the estimation results of the multiple regression model; Section 4 explains in detail the theoretical functioning of the Markov switching model; in Section 5 the Markov switching regression model results are discussed and Section 6 concludes.
2
Data
2.1
Explanatory variables
In this section, I give a detailed explanation of each variable considered and its expected relationship with CDS spreads.
The first three variables are derived from structural models (Merton (1974)) and are: firm’s debt-to-equity ratio, firm’s operation volatility and risk-free interest rate. In the structural models, the default happen when the firm’s value falls under a certain threshold. The rationale behind financial leverage is that as it increases, exceeding the optimal value, the firm’s value starts to decrease due to the presence of bankruptcy costs. So, unsustainable levels of debt translate into higher levels of financial leverage, increasing the firm’s probability of the default since the value of firm will approach the default barrier. Vice versa, upside movements of the company stock price, reduce the debt-to-equity ratio and at the same time the probability of default. Since daily financial leverages are not available, a feasible proxy is the company stock price. Therefore, it is straightforward to see the negative expected relationship between CDS spreads and stock returns. Alexander and Kaeck (2008) compose an equally-weighted portfolio considering stocks of the companies included in the iTraxx index. Due to lack of data, I cannot do the same for my sample period. The equity component is proxied by the two main European stock indexes: the Euro Stoxx 50 and the MSCI Europe. The first one is composed by 50 Eurozone blue chip companies, in which the same criterion of the iTraxx index for including or removing components is applied, and is calculated considering the free-float market capitalization of each firm (stoxx.com); the second one considers 437 companies both large and mid cap. It is calculated following the same methodology of the previous index (msci.com).
The firm’s volatility is an unobservable variable. Anyway, can be proxied by the equity historical volatility. Volatility is positively related to CDS spreads, since as it increases, it makes easier for the company’s value to hit the default barrier (Merton (1974)). Nevertheless, Benkert (2004) has empirically showed in his paper
that implied volatility has higher efficiency in explaining CDS premia of a large set of international companies than historical volatility. A possible explanation is related to the fact that implied volatility is a forward-looking measure affecting CDS spreads more than historical volatility. Consequently, I will consider changes in the VStoxx index, which is composed by implied volatility of options that have the Euro Stoxx 50 index as underlying asset.
The risk-free rate is another unobservable variable. According to Alexander and Kaeck (2008), I consider the Euro swap rates with maturities from 1 to 30 years instead of government bond yields as proxy for risk-free rate. Houweling and Vorst (2005) have empirically demonstrated that by the end of 90s the market used different interest rates to price derivatives rather than Treasury bills yields. They achieve this conclusion comparing the result of different credit risk models for credit default swaps, in which the ones estimated using swap rate outperformed the ones in which government bond yield was used. Moreover, Euro swap rates have the advantage of no-short selling constraint, high liquidity and are not influenced by special tax regimes. A negative relationship is expected between level of interest rates and credit default swap spreads, since higher levels are associated with economic expansion periods, signalling a shift of investor’s risk aversion towards risky assets. Vice versa, lower levels imply an increase in risk aversion and so are usually seen during recession periods. A theoretical argument is given by Duffie (1999), who finds that an increase in risk-free interest rate, lowers the risk-neutral default probability. For what concerns the slope of the yield curve, the relationship is uncertain. Collin-Dufresne et al. (2001) interpret a positive slope as an expected increase in interest rates and so a negative relationship with CDS spreads. Chan and Marsden (2014) add a further interpretation, consisting on the fact that a steeper slope can forecast an economic environment with rising inflation due to a tightened monetary policy, which could lead to less liquidity in the bond market and so a widening in credit spreads. This argument predicts a positive relationship with CDS spreads for the arbitrage link with bonds spreads, even though the authors have found that the sign of the slope coefficient is regime dependent. Moreover, I apply a principal component analysis (PCA) to the differenced term structure. According to theory, the first principal component represents changes
in the level of interest rates, so upward and downward shifts in the yield curve, while the second principal component is associated to changes in the slope of the yield curve. The advantage of using a PCA is twofold: I do not have to consider arbitrary maturities in the term structure to compute the slope and permits to reduce multicollinearity (Alexander and Kaeck (2008), Statistics and data analysis for financial engineering, p. 443-452).
Following the findings of Byström (2005,2006) and Avino and Nneji (2014), in which they discover a significant positive autocorrelation structure in the iTraxx index, I consider a lagged value of iTraxx spreads, in order to see if autocorrelation is still present in a more mature CDS market. I can guess a positive sign for the coefficient, but it will depend, since the range of time on which is based the previous analysis is different from mine.
As suggested by Annaert et al. (2010) and Collin-Dufresne et al. (2001), I also consider macroeconomic and liquidity indicators. Collin-Dufresne et al. (2001) use the difference between of the 10-year swap index and the 10-year Treasuries. If investor’s risk-aversion increase pushing them to buy safe assets, their yield consequently decrease and the difference increases. Therefore this shift from risky to safe assets, entails that liquidity dries up in the corporate sector in favour of the government bond segment. For my work, I consider changes between the 10-year Euro swap rate and the 10-year Bund yield. The coefficient for this variable is therefore expected to affect positively CDS spreads.
The first macroeconomic indicator is represented by WTI crude oil returns. Oil shocks are a common macroeconomic indicator used by analysts to assess the economy health. Sabhka et al. (2019) find that in high volatility regimes, sovereign credit volatility becomes high sensitive to these shocks. I want to extend their analysis to the corporate sector including it among my explanatory variables. Historically, high oil prices are seen during economic expansion periods, while low prices characterize periods of recession, since oil prices are driven by the law of supply and demand. Therefore, returns in crude oil prices should affect CDS spreads negatively.
Table 1: Explanatory variables with expected parameter’s sign
Explanatory Variable Abbrevation Expected sign
First principal component P C1
-Second principal component P C2
-Changes in VStoxx index ∆V +
Equity index returns ES50ret and M SCIret
-Changes in bond market liquidity ∆liq +
WTI crude oil returns W T Iret
-Changes in Baltic Dry Index ∆BDI
-Lagged value of iTraxx spreads ∆iT raxxt−1 + or
-(BDI). The index was created in 1985 and is measured by the intersection between the demand of shipping capacity and the supply of dry bulk carriers. It refers mainly to the shipping of dry raw materials (steel, concrete, food and so on), so it does not consider oil, across oceans and is seen by financial analysts as an efficient indicator of economic activity. The index has two main advantages: it is completely devoid to speculation and it is one of the few macroeconomic variables with daily observations. Lin et al. (2019) analyse the BDI spillover effects, finding that the BDI drives movements of equity, commodity and currency markets. For this reason the index can also drive other financial markets when major events occur in the U.S or in relevant emerging markets such as China, which is characterized by an high demand of raw materials. The authors conclude that BDI is a useful indicator during financial turmoil periods. Since the COVID-19 pandemic has expanded from China, causing a total blockage of both transport and production and according to Lin et al. (2019) findings, the Baltic Dry Index is included as explanatory variable. Moreover, it is coherent with the introduction of the extreme volatility regime in the Markov switching model. The sign of its coefficient is predicted negative, because increases in the shipping costs means higher demand for raw materials, implying an increase in economic activity which translates into increased sales for companies and therefore a decrease in the probability of default. The predicted signs of each regressor are summarized in Table 1.
2.2
Preliminary data analysis
Since the time series have different trading days, I standardized them considering the business days between the 21th October 2011 and the 16th April 2020. Each time series is daily, composed by 2134 observations and data is downloaded from Bloomberg, investing.com and stoxx.com. Before the estimation results, some preliminary statistics are conducted and collected in Table 2. For all the considered time series I do not accept the null hypothesis of the Augmented Dickey-Fuller test at 1% significance level, meaning that the time series do not present a unit root and so are all stationary. The same happen for the Jarque-Bera test, in which the null hypothesis of Normal distribution is always rejected. This result is supported by the kurtosis and skewness measures, which significantly differ from the Normal distribution’s ones. In conclusion, it is noticeable the presence of multicollinearity in the explanatory variables matrix. In particular, the correlation between equity index returns and changes in implied volatility is about 78%, representing a possible issue for my estimates.
2.3
Preliminary analysis of the variance
I estimated a GARCH(1,1) model on the iTraxx spreads time series with the error term assumed to be distributed as a standard Normal, in order to have an idea of how the conditional variance behaves. The estimation results are collected in Table 3, while in Figure 1 the fitted model is shown.
The aim is to have a benchmark about the presence of volatility clustering, which signals both the presence of either an extreme or high volatility regime and its persistence. As possible to see in Figure 1, the vertical lines represent four events that had a significant impact on the iTraxx index. The time series covers three main financial events. The first one is the 2011 sovereign debt crisis in which is evident a long period of volatility clustering. After the Mario Draghi’s speech, in which he stated that the ECB was ready to preserve both the Euro and the European economic and financial stability, the volatility started to decline until the beginning of 2014. The second important financial event is represented by two
T able 2: Preliminary stati stics on the explanator y v ariable s. In the second column are com puted the sample means, in the third one the standard deviati ons, in the fourth and fifth sk ewness and kurtosis . In the last tw o columns are rep orted the p-v alues of b oth Augmen ted Dic k ey-F u ller and Jarque-Bera test s (at significan t lev el of 1%) Explanatory v ariable Mea n Std dev Sk ewness Kurtosis p-v alue ADF test p-v alue JB test iT raxx index spreads -0.0429 3.0312 0,2393 14.1489 <0.001 <0.001 VSto xx index spreads 0.0012 1.6959 1,054 17.6043 <0.001 <0.001 Euro Sto xx 50 index returns 0.0165 1.2449 -0,6515 12.585 <0.001 <0.00 1 MSCI Europ e index returns 0,0182 1.0396 -0,9376 16.0024 <0.001 <0.00 1 WTI crude oil returns -0,0394 1.0396 0.0567 23.7965 <0.001 <0.001 Bond mark et liquidit y changes -0.0001 0.0224 0.4045 17.8575 <0.001 <0.001 BDI changes -0,6687 29,8933 0.2057 10.1419 <0.001 <0.001 PC1 -0,0066 0.1777 0.2205 6.1945 <0.001 <0.001 PC2 -0,0009 0.0365 -0.0013 8.9156 <0 .001 <0.001
Table 3: GARCH(1,1) model estimation results, where *** means significant at 1% level
const GARCH(1) ARCH(1)
Estimate 0.1526 0.8411 0.1533
t-value 10.1860*** 100.8500*** 16.7120***
close facts, which are the Japanese negative interest rate policy and the Brexit referendum. The period ranging from the beginning up to the mid of 2016 is characterized by the presence of volatility clustering. The last event is the COVID-19 pandemic. The financial markets started to react after the mid of February 2020, and at the same time the volatility has suffered a dizzying increase. In proximity of these events either high or extreme volatility regimes are expected. The time series is also characterized by some, even though less relevant from a financial point of view, peaks, such as the Donbass war and decrease in the Dow Jones index in 2014; the British government announcement of article 50 triggering, which started the process of withdrawn from the EU in 2017; and the Italian political uncertainty which raised the spread in 2018. The Markov switching regression model will treat these events as extreme volatility regime spikes, even though they have lasted less and are less important than the previous ones. Clustering of large and small deviations of the process from the mean (i.e. volatility clustering effect) is coherent with the hidden Markov chain process assumed for modelling changes in the short-term relationships.
01/2012 01/2014 01/2016 01/2018 01/2020 Time -20 -15 -10 -5 0 5 10 15 20 25 iTraxx spreads iTraxx spreads Conditional volatility Whatever it takes Japan negative i.r. policy Brexit
COVID-19 market impact
Figure 1: Fitted GARCH(1,1) model on the differenced iTraxx time series. From left to right, the vertical lines represent the Mario Draghi "whatever it takes" speech, Japan’s negative interest rate policy, Brexit, and impact of COVID-19 pandemic on the financial markets
3
Multiple regression models
In this section, I estimate two regression models considering as many different stock indexes. Then, I make a comparison with an update version of the regression model estimated in Alexander and Kaeck (2008). All the multiple regression models are estimated via OLS method and are the following:
∆iT raxxt= β0+ β1∆iT raxxt−1+ β2∆Vt+ β3ES50rett+
+ β4∆liqt+ β5W T Irett+ β6∆BDIt+
+ β7P C1t+ β8P C2t+ t
(1)
∆iT raxxt= β0+ β1∆iT raxxt−1+ β2∆Vt+ β3M SCIrett+
+ β4∆liqt+ β5W T Irett+ β6∆BDIt+
+ β7P C1t+ β8P C2t+ t
where, in both models t is the error term and is assumed to be independent and
identically distributed as a Normal, with mean 0 and variance σ2. Moreover, a
Durbin-Watson test is employed in order to check if the residuals are correlated. The result (Model (1) p − value = 0.0731 and Model (2) p − value = 0.4704) is that the null hypothesis of no-autocorrelation is not rejected for both models. The estimation results are collected in Table 4 and in Table 5 are summarized the models’ diagnostics.
Now, I estimate the update version of the Alexander and Kaeck (2008) multiple regression model:
∆iT raxxt= β0+ β1∆iT raxxt−1+ β2∆Vt+ β3ES50rett+
+ β7P C1t+ β8P C2t+ t
(3)
∆iT raxxt= β0+ β1∆iT raxxt−1+ β2∆Vt+ β3M SCIrett+
+ β7P C1t+ β8P C2t+ t
(4)
As in the previous models, both tare assumed to be i.i.d. as a Normal with mean 0
and variance σ2. Model (3)’s residual present significant autocorrelation, in fact the
Durbin-Watson test falls to not reject the null hypothesis (p−value = 0.0350), and so the variance-covariance matrix is estimated with the Newey-West methodology. Model (4) on the other hand, does not present significant autocorrelation in the
residulas (p − value = 0.3612). Considering Euro Stoxx 50 returns, the ∆V
parameter continues to not be significant due to the presence of multicollinearity. Nevertheless, the parameters have the predicted sign, except for the second principal component and changes in BDI which are not significant. The results in the updated version are similar to the ones previous found, in particular for Model (4).
T able 4: Multiple regression mo del s es timation results. In paren thesis are rep orted the t-v alues, where. *** indicates parameter significance at 1%, ** at 5% and * at 10% Mo del (1 ) const ∆ iT r axx t− 1 ∆ Vt E S 50 r ett ∆ liq t W T I r ett ∆ B D It P C 1t P C 2t -0.0263 -0.0169 0.026 0 -1.6896 4.3046 -0.0952 0.0023 -1.3 857 0.7611 (-0.6269) (-1.193 4) (0.6444) (-29 .4 926 *** ) (2.2317 ** ) (-5.2522 *** ) (1.6475 * ) (-5.5279 *** ) (0.6415) Mo del (2 ) const ∆ iT r axx t− 1 ∆ Vt M S C I r ett ∆ liq t W T I r ett ∆ B D It P C 1t P C 2t -0.0281 -0.0266 0.161 5 -1.7130 8.0878 -0.0838 0.0023 -1.9 901 -0.3933 (-0.6327) (-1.7796 * ) (3.7 928 ** ) (-23.6984 *** ) (3.9936 *** ) (-4.3550 *** ) (1.5357) (-7.60 43 *** ) (-0.3122) Mo del (3 ) const ∆ iT r axx t− 1 ∆ Vt E S 50 r ett P C 1t P C 2t -0.0242 -0.2390 0.038 5 -1.7381 -1.4756 1.1787 (-0.5743) (-0.780 3) (0.5503) (-18 .5 289 *** ) (-3.528 *** ) (0.6610) Mo del (4 ) const ∆ iT r axx t− 1 ∆ Vt M S C I r ett P C 1t P C 2t -0.0259 -0.0316 0.173 7 -1.7810 -2.0655 0.1090 (-0.5782) (-2.1130 ** ) (4 .0562 *** ) (-24.7844 *** ) (-7.8548 *** ) (0.0861)
T able 5: Multiple regressi on mo de ls diagnostics. I first re-e stimated a mo del without the last 45 observ ations. T hen, b o th MSE and MAE in sample are referred to T-45 observ ation s, while the out-of -sample are referred to the last 45 observ ati ons Mo del R 2 adj AIC BIC MSE in-sample MA E in-sample MSE out-of-sample MAE out-of-sample Mo del (1 ) 0.589 8898.15 8949.15 3.34 1.25 24.21 3.47 Mo del (2 ) 0.543 9129.17 9180.16 3.72 1.31 26.49 3.67 Mo del (3 ) 0.584 8925.97 8959.97 3.37 1.25 25.20 3.51 Mo del (4 ) 0.536 9158.65 9192.64 3.77 1.31 27.04 3.70
3.1
Regression models results discussion
The autoregressive parameter significance depends upon the equity index considered. In fact, if MSCI index returns are considered it is significant, while if I consider the Euro Stoxx 50 returns it turns to be non-significant. This problem could be associated to the presence of multicollinearity. Nevertheless, regressing ∆iT raxxton ∆iT raxxt−1, the parameter is strongly insignificant. It is not evident
if the autocorrelation is still persistent in the post subprime crisis in the iTraxx Europe index.
Similarly, multicollinearity affects also the ∆V parameter. However, regressing ∆iT raxxt on ∆Vt, the implied volatility parameter is equal to 1.0822 and turns
to be highly significant (t − value = 36.0733). So, positive changes in implied volatility are associated with positive changes in corporate CDS spreads.
All the equity indexes returns are strongly significant with the predicted sign. Therefore, upside movements in stock returns cause a downside movement in CDS spreads. Since the iTraxx index is composed by blue chip companies, the significance is in line with the Collin-Dufresne et al. (2001) results, in which the companies with the best rating are more sensible to changes in the equity benchmark. Moreover, the result is similar to the one find using the Alexander and Kaeck (2008) model.
Oil shocks have been studied only considering sovereign CDS spreads, revealing their significance in explaining sovereign CDS changes during turmoil periods (Sabhka et al. (2019)). Similarly to Baltic Dry Index, they can be seen as a world economic health indicator. Considering the high significance of changes in WTI crude oil price in both models, they also have a relevant explanatory power for corporate CDS spreads, affecting them negatively.
In contrast, changes in Baltic Dry Index are estimated with positive sign, contrasting with theory. Nevertheless, it is only significant in Model (1) and at 10% significance level. It is noticeable that the index started to drop from mid December 2019, as the Chinese network CCTV reported the news of a new outbreak in Wuhan. The BDI continued to decline for the first two months of
2020, following the worsening news about COVID-19 pandemic, anticipating the imminent economic recession.
Changes in corporate bond liquidity are always significant. It is the regressor with the higher impact on CDS spreads. During the range of time considered, several monetary policies have been undertaken by the ECB to tackle both the impact of the sovereign debt crisis (i.e. APP) and the Coronavirus impact. Such policies have pushed the interest rates below 0% and this may be influenced the investors’ risk-aversion, re-allocating their investments toward stock markets.
The first principal components impacts, with the expected sign, significantly the corporate CDS spreads, meaning that upward shifts of the term structure imply a decrease in CDS spreads The same result had been found both in Alexander and Kaeck (2008) and in their updated model. The second principal components are not significant in all the estimated models coherently with the original model of Alexander and Kaeck. Therefore, changes in the yield curve’s slope do not affect CDS spreads. The sensitivity of CDS spreads to changes in interest rates can also be seen empirically as in Figure 2 on the following page. In fact, after the announcement of each TLTRO followed a decline in the corporate iTraxx index. An interesting point is the comparison of the adjusted R2 in order to see if the added explanatory variables could be useful to explain more variation of iTraxx spreads. Model (1) and (2) achieved an adjusted R2 of 58.9% and 54.3%
respectively. The adjusted R2 achieved with the update regression models of
Alexander and Kaeck (2008) are slightly lower, meaning that considering both macro and liquidity variables are useful to explain variation in corporate CDS
spreads. Moreover, considering Euro Stoxx 50 returns instead of MSCI ones,
allows to increase the adjusted R2 of about 4% other than achieving better in-sample and out-of-in-sample diagnostics. A possible reason is given by the fact that the former equity index reflects more the composition of the iTraxx index than the latter equity benchmark. Considering the previous results found in literature, even though with different specifications, frequency and time frame, Collin-Dufresne et al. (2001) estimated a regression model in which was able to explain approximately 34% of credit spreads (i.e. adjusted R2=0.34). A similar result was achieved by
01/2012 01/2014 01/2016 01/2018 01/2020 Time 40 60 80 100 120 140 160 180 200 220 iTraxx index
TLTRO 1 TLTRO 2 TLTRO 3
ECB announces easing on TLTRO repayments
Figure 2: This figure represents the corporate iTraxx index reaction to ECB
announcements. In particular, from left to right, TLTRO 1, TLTRO 2, TLTRO 3 and the more recent announcement of policies to tackle the COVID-19 pandemic recession (ecb.europa.eu)
Ericsson et al. (2009). Abid e Naifar (2006), including credit rating increased the adjusted R2 to 66%. Zhang et al. (2009) considering jump-risk, they explained 73% of the variation on single-name CDSs. Alexander and Kaeck (2008) explained 28%. Generally, in light of the previous findings, I can say that I have achieved satisfactory results.
To end this section, following the procedure of Alexander and Kaeck a rolling Chow test is applied to the models in order to check the parameters constancy and stability. The results are collected in Table 6. The test is applied leaving out ∆V in order to reduce multicollinearity. The results evidence strong instability in the estimated parameters similarly as found in Alexander and Kaeck (2008). This is a possible signal of a non-linear relationship. Therefore, in the next section the regression model will be extended, considering an estimation method which accounts for non-linearity and regime dependent behaviour of CDS spreads.
Table 6: Rolling Chow test results. The test is conducted leaving out the first and last 150 observations at 5% significance level
Model % of rejection
Model (1) 75.90%
Model (2) 82.61%
Model (3) 79.99%
Model (4) 80.32%
4
Markov switching technique
Before the estimation results, this section has the aim to give a detailed theoretical explanation about the functioning of Markov switching models.
State space models allow for analysing dynamic phenomena which evolve over time. A time varying system can be represented through a dynamic model, which is constituted by an observable component and an unobservable internal state. The main advantage in using the general Bayesian state space representation of a dynamic model, is that it accounts also for non-linear and non-Gaussian models. Assume that state space, observation space and parameters space respectively are X ⊂ Rnx, Y ⊂ Rny and Θ ⊂ Rnθ respectively. The Bayesian state space
representation is given by an initial distribution p(x0|θ), a measurement density
p(yt|xt, y1:t−1, θ) and a transition density p(xt|x0;t−1, y1:t−1, θ). Therefore, the
dynamic model is:
yt∼ p(yt|xt, y1:t−1, θ) (5)
xt∼ p(xt|x0;t−1, y1:t−1, θ) (6)
x0 ∼ p(x0|θ), t = 1, . . . , T (7)
The distribution p(x0|θ) can be interpreted as the prior distribution of the initial
state of the system. When the transition density depends on the past only through the last value of the hidden state vector, the dynamic model is defined first-order
Markovian, and system becomes:
yt|xt∼ p(yt|xt, y1:t−1, θ) (8)
xt|xt−1 ∼ p(xt|xt−1, y1:t−1, θ) (9)
x0 ∼ p(x0|θ), t = 1, . . . , T (10)
The system in (8),(9) and (10) is the one that I consider in order to estimate the Markov switching regression model.
4.1
Markov chain process
The state variable, st, is modelled as a Markov chain of order N , meaning that it
can only assume the integer numbers {1,2,. . . ,N}. A Markov chain is a stochastic process characterized by:
P {st= j|st−1= i, st−1= k, . . . } = P {st= j|st−1 = i} = pij (11)
Equation (11) implies that the probability to be in state j at time t will only depends on the state at time t − 1. The probabilities, pij, are called transition
probabilities and are collected in the transition matrix P :
P = p11 p21 . . . pN 1 p12 p22 . . . pN 2 .. . ... . .. ... p1N p2N . . . pN N (12)
A Markov chain can be represented as a vector autoregressive process. Let ξ represent a random (N × 1) vector, whose j − th element is equal to unity if st= j
of IN, the N × N identity matrix. ξt = (10 · · ·0)0 if st = 1 (01 · · ·0)0 if st = 2 .. . ... ... (00 · · ·1)0 if st= N (13)
If st= i, then the j-th element of ξt+1 is a random variable that takes value 1 with
probability pij and value 0 otherwise. Consequently, the conditional expectation
of ξt+1 given st= i is equal to:
E(ξt+1|st= i) = pi1 pi2 .. . piN (14)
Representing the product between the i-th raw of P and a (N × 1) vector, in which at the j-th raw is present 1 and 0 elsewhere, that is the i-th column of transition matrix P . Therefore:
E(ξt+1|st) = P ξt (15)
Implying that:
E(ξt+1|ξt) = P ξt (16)
From the Markov chain property described at equation (15):
E(ξt+1|ξt, ξt−1, . . . ) = P ξt (17)
From this result, we express the Markov chain in the following form:
ξt+1 = P ξt+ vt+1 (18)
This expression has the form of a first-order vector autoregression, where the innovation vt+1:
is a martingale difference sequence. Although the vector v can take on only a finite set of values, on average it is zero. The m-period step ahead forecasts for a Markov chain is:
E(ξt+m|ξt, ξt−1, . . . ) = Pmξt (20)
As stated before, since the j-th element of ξt+m will be unity if st+m= j and zero
otherwise, the jth element of the (N × 1) vector E(ξt+m|ξt, ξt−1, . . . ) indicates the
probability that st+m takes on the value j, conditional on the state of the system
at time t. So, assuming that the process is in state i at date t and denoting with ei the ith column of the N × N identity matrix, then (20) claims that:
Pmei = P {st+m = 1|st = i} P {st+m = 2|st = i} .. . P {st+m= N |st= i} (21)
For the ergodicity and reducibility properties see Time series analysis, p. 680-685.
4.2
Hamilton filter
Markov switching models are useful since they allow a given set of independent variables to follow a different time series over different subsamples. In order to estimate the regimes at which the time series is at each time step, the Hamilton filter is employed. The advantage of modelling the regimes as Markov chains is their flexibility. The model investigated is:
yt = Xt0β + t with t = 1, 2, . . . , T (22)
where yt represents the dependent variable, Xt is the (k × 1) vector of independent
variables, β is the (k × 1) vector of regression parameters and t is the (N × 1)
vector of the error terms, which are assumed to be i.i.d as a Normal with mean zero and variance σ2. Let Ψ
t = (yt, X) be a matrix containing all the observed
follows that the conditional density of yt is assumed to be given by:
f (yt|st = j, Xt, Ψt−1; α) (23)
Where α contains the parameters which characterize the conditional density of yt.
These densities are collected in the vector ηt.
For example, considering 3 regimes and so N = 3, which corresponds to my case, α is composed by:
α =hβ10 σ1 β20 σ2 β30 σ3
i
(24) Where each βj and σ2j are regime-specific parameters. Consequently, ηt is a vector
containing 3 conditional density functions:
ηt = 1 √ 2πσ2 1 exp{−(yt−Xt0β1)2 2σ2 1 } 1 √ 2πσ2 2 exp{−(yt−Xt0β2)2 2σ2 2 } 1 √ 2πσ2 3 exp{−(yt−Xt0β3)2 2σ2 3 } (25)
The parameters that describe a time series modelled by (23) α and the transition probabilities pij. These parameters are collected in the vector θ. The aim is to
estimate θ based on observation Ψt. Even if θ is supposed known at time t, the
regime st is not. But, is possible to make a guess about it, which correspond to
inference st based on the observed data:
P (st= j|Ψt; θ) (26)
This is simply a conditional probability assigned to the possibility that tth observation is generated by regime j. These conditional probabilities are collected in the vector bξt|t. Moreover, it is also possible to forecasts how likely the process is
to be in regime j at date t + 1, denoting it by bξt+1|t. Finally, the optimal inference
and forecasts for each time step in the sample is found iterating the following pair of equations: b ξt|t = (bξt|t−1 ηt) 10(bξt|t−1 ηt) (27)
b
ξt+1|t = P bξt|t (28)
In (27), ηt represents the (N × 1) vector containing the conditional densities,
is the Hadamaard element-by-element product, 1 is a (2 × 1) vector of ones and P is the transition matrix. In the two equations it is noticeable the application of the Bayes theorem. In fact, in (27), the numerator corresponds to the joint distribution:
f (∆iT raxxt, st = j|Xt, Ψt; α) = f (yt|st = j, Xt, Ψt; α)p(st = j|Xt, Ψt; α)
= bξt|t−1 ηt
(29)
The denominator corresponds to the marginal distribution:
f (yt|Xt, Ψt; α) = N
X
j=1
f (yt, st= j|Xt, Ψt; α) = 10( ˆξt|t−1 ηt) (30)
In any case the algorithm needs to be initiated, and the initial values of both b
ξ1|0 and the transition probabilities pij are estimated either via ML method or
chosen arbitrary. Usually, bξ1|0 is either equal to the ergodic probability vector or
the discrete uniform distribution. In order to get the posterior estimates of the regimes, the smoothing backward recursion is applied (see Kim (1993)):
b
ξt|T = bξt|t (P0(bξt+1|T bξt+1|t)) (31)
4.3
Gibbs sampler
In order to estimate the parameters, a data augmentation framework is followed: f (y1, . . . , yT, s1, . . . , sT|θ) = T Y t=1 1 p2πσ2(s t) exp{−(yt− µ(st)) 2 2σ2(s t) } M Y l=1 M Y k=1 pI(st=k)I(st−1=l) lk (32) where θ = (µj, σ2j, plk) and: µ(st) = E(yt|st) = Xt0βst and: σ21 < σ22 < · · · < σN2
is the identification constraint. I assume the following prior distribution for my parameters:
βj ∼ N (m, s2)
σj−2 ∼ Ga(a, b) plk∼ Dir(vl1, . . . , vlM)
The joint parameter and latent variable posterior distribution is not tractable thus a multi-move Gibbs sampler is applied. The hidden states are sampled jointly thanks to the forward filtering backward sampling (FFBS) procedure. The details about the full conditionals are in the following:
Full conditional of βj: p(µj|θ−µ, ¯S, ¯X) ∝ ∝ T Y t=1 exp{−(yt− X 0 tβj)2 2σ2(s t) } exp{−(βj− m) 2 2S2 } ∝ Y t∈τj exp{− 1 2σ2 j (yt− Xt0βj)2} exp{− 1 2s2(βj − m) 2} ∝ exp{− 1 2σ2 j (βj2(Xt0Xt) − 2βj T X t=1 ytXt) − 1 2s2(β 2 j − 2β 0 jm)} ∝ N ( ¯mj, ¯s2j) with: ¯ mj = ¯s2j( P t∈τjX 0 tyt σ2 j +mj s2 j ) ¯ s2j = ( P t∈τjX 0 tXt σ2 j + 1 s2 j )−1 Where τj = {t|st = j}, Tj = Card(τj), j = 1, . . . , N . Full conditional of σj2: p(σ−2j |θ−σ2 j, ¯S, ¯X) ∝ ∝ T Y t=1 1 pσ2(s t) exp{−(yt− µ(st)) 2 2σ2(s t) }( 1 σ2 j )a−1exp{− 1 σ2 j b} ∝ ( 1 σ2 j )Tj2+a−1exp{− 1 σ2 j (b + 1 2 X t∈τj (yt− µj)2)} ∝ Ga(¯a, ¯b) with: ¯ a = a + Tj 2 ¯b = b + 1 2 X t∈τj (yt− µj)2
Full conditional of (pl1, . . . , plM): p((pl1, . . . , plM)|θ−pl, ¯S, ¯X) ∝ ∝ T Y t=1 M Y l=1 M Y k pI(st=k)I(st=l) lk ∝ M Y l=1 ( M Y k=1 p PT t=1I(st=k)I(st−1=l) lk ) ∝ M Y l=1 Dir(Nl1, . . . , NlM) with: Nlk= T X t=1 I(st= k)I(st−1= l) Full conditional of ¯S:
I obtain the vector of smoothed probabilities ξt|T = p(st|θ, ¯X) from the Hamilton
(HMM) filter and draws a state vector St from the smoothed, by using a
5
Markov switching regression models
The posterior distributions for the 2-regime MS model are in Appendix A, while the ones for the 3-regime MS are in Appendix B.
5.1
2-regime Markov switching regression
In this section, I make a comparison between Model (3) and Model (1), both estimated by applying the methodology described in the previous section. The reason about considering only the Euro Stoxx 50 returns for the further models, is that they allow to achieve an higher level of explained variation than MSCI returns.
The estimated Markov switching regression models are the following: ∆iT raxxt= β0,st+ β1,st∆iT raxxt−1+ β2,st∆Vt+ β3,stES50rett+
+ β4,st∆liqt+ β5,stW T Irett+ β6,st∆BDIt+
+ β7,stP C1t+ β8,stP C2t+ t,st with st= {1, 2}
(33)
∆iT raxxt= β0,st+ β1,st∆iT raxxt−1+ β2,st∆Vt+ β3,stES50rett+
+ β4,stP C1t+ β5,stP C2t+ t,st with st= {1, 2}
(34)
I start estimating the 2-regime version, the results are collected in Table 7, the models diagnostics are in Table 8 and in Figure 3 and Figure 4 are displayed the two fitted time series. The parameter significance is verified estimating the quantiles of the parameter’s posterior distribution. If there is a change in the quantiles sign, the parameter is not significant at the considered level.
T able 7: 2-regime Mark o v switc hin g m ult iple regression mo dels estimati on results. * means signi fican t a t 10% lev el, ** at 5% and *** at 1% const ∆ iT r axx t− 1 ∆ Vt E S 50 r ett ∆ liq t W T I r et t ∆ B D It P C 1t P C 2t pii σ 2 Mo del (33) Regime 1 -0.0484 * 0.0260 0.1983 *** -0.9405 *** 2.2598 -0.0312 ** 0.0003 -0.3891 ** 0.6265 0.9747 0.9783 Regime 2 0.0155 -0.0260 0.034 6 -2.2881 *** -0.0631 -0.1259 *** 0.0080 * -1.3767 ** 0.5548 0.9387 8.3758 Mo del (34) Regime 1 -0.0492 * 0.0269 0.2027 *** -0.9495 *** -0.4800 *** 0.5766 0.9745 0.9806 Regime 2 0.0317 -0.0402 * 0.0460 -2.3606 *** -1.3013 ** 0.6703 0.9385 8.5386 T able 8: 2-regimes Mark o v switc hing m ultiple regress ion mo del s diagnostics. T is the sample size, and 45 is the n um b er of observ ati ons whic h falls within to the CO VID-19 p erio d Mo del MSE in-samp le T-45 MSE in-sample last 45 obs MAE in-sample T-45 MAE in-sample last 45 obs Mo del (33) 17.08 92 .72 3.63 7.33 Mo del (34) 14.74 89 .76 3.61 7.30
01/2012 01/2013 01/2014 01/2015 01/2016 01/2017 01/2018 01/2019 01/2020 Time -30 -20 -10 0 10 20 30
Changes in iTraxx index
0 2 4 6 8 10 12 Regimes y posterior mean of y regimes
Figure 3: Model (33) Markov switching regression fitted to the ∆iT raxx time series
01/2012 01/2013 01/2014 01/2015 01/2016 01/2017 01/2018 01/2019 01/2020 Time -30 -20 -10 0 10 20 30
Changes in iTraxx index
0 2 4 6 8 10 12 Regimes y posterior mean of y regimes
5.2
2-regimes
Markov
switching
regression
results
discussion
From a graphical point of view (Figures 3 and 4), the two models find similar regimes. In particular, is possible to see a persistent regime of high volatility at the beginning of the time series representing the sovereign debt crisis. In the central part, which corresponds to the Brexit and the negative interest rate policy undertaken by Japan, the high volatility regime appears to be less persistent and in the last part is evident the impact of COVID-19 pandemic, in which the high volatility regime becomes persistent. The results are coherent with the GARCH(1,1) model previously estimated.
In Model (34), the autoregressive component is weakly significant only in the high volatility regime and with negative sign.
The implied volatility parameter is significant in regime 1, while it is not in the other one. The same behaviour is seen in Model (33). This finding is opposite to the one in Alexander and Kaeck (2008) and a possible reason is that CDS spreads are more sensitive to implied volatility variations during tranquil periods, while during turmoil periods they become sensitive to other explanatory variables. Such variable could be represented by WTI crude oil returns. In fact, its parameter is significant and has a greater effect in the volatile regime, as Sabhka et al. (2019) found. Moreover, it is significant also in the low volatility regime but with a lower impact. Compared to the OLS regressions, the liquidity variable does not influence changes in iTraxx index in any regime. Likewise, the second principal component, confirming the OLS regressions and the Alexander and Kaeck (2008) result. Changes in the Baltic Dry Index is significant at 10% level in the high volatile regime (coherently with Lin et al. (2019)), and is still estimated with the opposite sign respect to the one expected.
As in Alexander and Kaeck(2008), the first principal component, which represents shifts in the level of interest rates and the equity index returns are the two most significant and with the higher effect on iTraxx spreads in both models and both regimes. Moreover, their power increases as the process enters in the high volatility
Table 9: Wald test for each regression coefficient estimated in regime 2 of models (33) and (34)
Model (33) Model (34)
Coefficient Wald statistic Coefficient Wald statistic
const 0.5338 const 0.6739
∆iT raxxt−1 -2.1599 ∆iT raxxt−1 -2.7545
∆Vt -1.8194 ∆Vt -1.7532 ES50rett -10.7854 ES50rett -11.4170 ∆liqt -1.2183 W T Irett -2.2944 ∆BDIt 1.6793 P C1t -1.6262 P C1t -1.3649 P C2t -0.0424 P C2t 0.0557 regime.
Looking at the models diagnostics (Table 8), the updated 2-regime MS regression model fits the time series slightly better.
Following the Alexander and Kaeck (2008) methodology, in order to check if the second regime is relevant, a Wald test is applied. The aim is to test if the estimated βs in the second regime are significantly different from the ones estimated in regime one, suggesting the need of two different regimes, against the hypothesis that different regimes are not necessary, i.e.
H0 : βj,st=1 = βj,st=2 H1 : βj,st=1 6= βj,st=2 ∀j
The test statistic, under the null and taking the square root, is distributed as a Standard Normal. The test statistics are collected in table 9. For both models, for at least one coefficient, the test statistic rejects the null hypothesis at 1% significance level and therefore there is evidence of a regime-specific behaviour.
5.3
3-regimes Markov switching regression
In this section, I add the new regime representing the extreme volatility, which characterizes crisis periods. I still consider the specification of models (33) and (34), therefore:
∆iT raxxt= β0,st+ β1,st∆iT raxxt−1+ β2,st∆Vt+ β3,stES50rett+
+ β4,st∆liqt+ β5,stW T Irett+ β6,st∆BDIt+
+ β7,stP C1t+ β8,stP C2t+ t,st with st= {1, 2, 3}
(35)
∆iT raxxt= β0,st+ β1,st∆iT raxxt−1+ β2,st∆Vt+ β3,stES50rett+
+ β4,stP C1t+ β5,stP C2t+ t,st with st= {1, 2, 3}
(36)
The estimation results are collected in Table 10, the models diagnostics are in Table 11 and in Figure 5 and Figure 6 are displayed the two fitted time series. Parameter’s significance is checked in the same method described in the previous section.
5.4
3-regimes
Markov
switching
regression
results
discussion
As before, the models fit the ∆iT raxx time series in an analogous way. Indeed, Model (36) still has slightly better diagnostics than Model (35). There is also small differences in the estimated regimes.
Neither the lagged value of iTraxx spreads nor the second principal component are significant in both models.
For the 3-regimes MS models, the changes in implied volatility coefficients cease to be significant in the extreme volatility regime, while it is significant in both low and high volatility regimes. The result is coherent with the 2-regimes Markov switching model, and in fact the estimated variances in regime 3 of Model (35)
T able 10 : 3-regime Mark o v switc hing m ultiple regression mo dels estimation results. *** mean significan t at 1%, ** at 5% and * at 10% const ∆ iT r axx t− 1 ∆ Vt E S 50 r ett ∆ liq t W T I r ett ∆ B D It P C 1t P C 2t pii σ 2 Mo del (35) Regime 1 -0.0917 *** 0.0010 0.2404 *** -0.5379 *** 0.2091 -0.0164 -0.0004 -0.1914 0.34 51 0.9586 0.4517 Regime 2 -0.0473 0.0253 0.2286 *** -1.1582 *** 2.8943 * -0.0329 * 0.0006 -0.7091 *** -0.2335 0.9607 1.2674 Regime 3 0.0170 -0.0278 -0.0262 -2.3055 *** 0.1529 -0.1434 *** 0.0107 * -1.6799 ** 0.8091 0 .9282 9.5335 Mo del (36) Regime 1 -0.0891 ** 0.0019 0.2466 *** -0.5447 *** -0.1695 0.1997 0.9576 0.4538 Regime 2 -0.0489 0.0239 0.2158 *** -1.1945 *** -0.8180 *** -0.1202 0.9585 1.2879 Regime 3 0.0436 0.0239 -0.0031 -2.3937 *** -1.5675 ** 0.8709 0 .9258 9.8078 T able 11 : 3-regimes Mark o v switc hing m ultiple regression mo dels diagnostics. T is the sample size, and 4 5 is the n um b er of observ ati ons whic h falls within to the CO VID-19 p erio d Mo del MSE in-samp le T-45 MSE in-sample last 45 obs MAE in-sample T-45 MAE in-sample last 45 obs Mo del (33) 17.03 94 .17 3.44 7.38 Mo del (34) 14.89 89 .47 3.13 7.13
01/2012 01/2013 01/2014 01/2015 01/2016 01/2017 01/2018 01/2019 01/2020 Time -30 -20 -10 0 10 20 30
Changes in iTraxx index
0 2 4 6 8 10 12 Regimes y posterior mean of y regimes
Figure 5: Model (35) Markov switching regression fitted to the ∆iT raxx time series
01/2012 01/2013 01/2014 01/2015 01/2016 01/2017 01/2018 01/2019 01/2020 Time -30 -20 -10 0 10 20 30
Changes in iTraxx index
0 2 4 6 8 10 12 Regimes y posterior mean of y regimes
and (36) are similar to the ones in regime 2 of Model (33) and (34). The first principal component does the opposite. Indeed, during the high volatile regimes its coefficient is significant, while, as the iTraxx index enters in the low volatile regime, it is no longer significant.
Changes in corporate sector liquidity has low significance and only in regime 2. The same happen for ∆BDI parameter, which is significant in the extreme volatility regime, shifting its significance from the high volatility regime to the one of extreme volatility. However, the sign continues to be positive.
The most significant macroeconomic variable is W T Iret. In fact, during regimes 2 and 3, representing the higher volatility regimes, the estimated coefficient is significant in explaining variations of corporate CDS spreads. Therefore, as the WTI crude oil price increases during turmoil periods, a decrease in iTraxx spreads is expected.
Also in these models, the main corporate CDS spreads determinant is represented by the equity index returns. Its explanatory power increases with increasing levels of volatility.
Furthermore, there are different peaks of extreme volatility captured by the model. Except for the initial, central and final parts of the time series, representing the sovereign debt crisis, Brexit and COVID-19 pandemic respectively, the other peaks should be classified as high volatility. Therefore, I repeat the Wald test in order to check if each estimated parameter in the third regime is significant. The test statistics are in Table 15. Also in this case, for at least one coefficient I do not accept the null hypothesis at 1%, therefore the third regime is significant.
Comparing diagnostics of 2-regime and 3-regime MS regression models, the inclusion of a new regime does not improve markedly the in-sample diagnostics. Nevertheless, for the most significant explanatory variables (i.e. ES50rett, ∆Vt,
W T Irett and P C1t) the third regime is significant, looking both at their Wald
statistic and explanatory power.
The addition of macroeconomic variables to explain CDS spreads, as suggested by Collin-Dufresne et al. (2001) and Annaert et al. (2010), are of benefit in the linear
Table 12: Wald test for each regression coefficient estimated in regime 3 of models (35) and (36)
Model (35) Model (36)
Coefficient Wald statistic Coefficient Wald statistic
const 0.7796 const 0.9389
∆iT raxxt−1 -1.0764 ∆iT raxxt−1 -1.6686
∆Vt -2.5003 ∆Vt -2.3431 ES50rett -12.2188 ES50rett -13.1541 ∆liqt -0.0287 W T Irett -2.6771 ∆BDIt 1.9282 P C1t -2.1285 P C1t -1.9944 P C2t 0.2610 P C2t 0.3758
regression, allowing to increase the explained variations. In the Markov switching framework, 2 out of 3 turn to be not very significant.
The updated version of Alexander and Kaeck (2008) Markov switching regression model, differs from the one previously estimated. In particular, the lagged value of the dependent variable coefficient is significant just in regime 2 of Model (34) at 10% level. Moreover, the authors find that during the low volatile regimes iTraxx spreads are more sensitive to changes in the level of interest rate than changes in implied volatility, and vice versa in high volatility regimes. My empirical results suggest the opposite.
6
Conclusions
In my thesis I contribute to extend the line of research about the daily
iTraxx spreads determinants. To do so, I update the multiple regression
model of Alexander and Kaeck (2008) and I also estimate a regression model including macroeconomic variables, as Collin-Dufresne (2001), among others, found necessary to explain credit spreads. Two out of three of the macroeconomic variables considered are novel to literature of CDS spreads determinants and are Baltic Dry Index changes and WTI crude oil returns. Moreover, I extend the work of Collin-Dufresne (2001) in a Markov switching framework, introducing a third regime, which is novel in literature, in order to take into account the extreme levels of volatility.
I estimated two multiple regression models, one which updates the Alexander and Kaeck (2008) version, and the second in which macroeconomics variables are included. Most of them (except the BDI variations) are highly significant. The latter model achieve an higher adjusted R2, meaning that macroeconomic
factors allow to increase the explained variations of CDS spreads as Collin-Dufresne (2001), among others, previously found.
However, there is evidence of a regime dependent behaviour of iTraxx spreads, highlighted by the high percentage of rejection of the Chow test. According to Alexander and Kaeck (2008), this behaviour is also associated to the unknown
systematic factor found in Collin-Dufresne (2001). Moreover, the estimated
parameters in the second regime of the 2-regime Markov switching regression are significant different from the ones in the first regime. In a similar way, the estimated coefficients in the third regime of the 3-regime Markov switching regression are significant different from the ones estimated in the second regime. This is further confirmation of the regime-dependent behaviour of the iTraxx index spreads. In addition, the third regime is significant, even though there is a limited benefit from an estimation point of view and there are several peaks, which can be seen as a "model error". In general, the models consistently estimate the correct regimes (i.e. high and extreme volatility) in correspondence of the volatility clustering periods determined by the GARCH(1,1) model.
From my analysis emerge that the iTraxx index variations does not present the autocorrelated structure found in the past by Byström (2005,2006) and Avino and Nneji (2014).
Among the novel explanatory variables introduced to study corporate CDS spreads
(i.e. W T Iret, ∆liq and ∆BDI), the one which is the most significant is
represented by WTI crude oil returns. In particular, during periods of high volatility the coefficient is highly significant and impacts the iTraxx spreads negatively. Therefore, WTI crude oil returns should be considered in analysing the corporate CDS spreads determinants. Changes in corporate sector liquidity parameter is not significant in the 2-regime framework, while it is in the 3-regime model at 10% level, and only during the high volatility regime. The coefficient has one of the largest magnitude, and could lead the iTraxx spreads either to the low or the extreme volatility regime. Baltic Dry Index variations are significant during both the high volatility of the 2-regimes MS and the extreme volatility of the 3-regimes MS model, and always at 10% level.
Coherently with the previous findings, the second principal component,
representing changes in the yield curve slope, has no effect on corporate CDS spreads. While, the first principal component is significant and in particular during the higher volatility regimes. This imply, that corporate CDS spreads are highly sensitive to interest rate and monetary policies, as empirically demonstrated by the ECB’s APP and the negative interest rate policy undertaken in 2016 by Japan. Therefore, an increase in interest rates induce a decrease in the iTraxx level. Vice versa, changes in European implied volatility are significant during the lower volatile regimes. Therefore, positive changes in the VStoxx index are associated with an upward movement in the iTraxx index leading it in both high and extreme volatility regimes. The most significant explanatory variable is the Euro Stoxx 50 index returns. The estimated coefficient is always significant at 1% level, meaning that is the main driver of European iTraxx corporate index. In conclusion, I summarize my findings: i) the updated models of Alexander and Kaeck (2008) show that the iTraxx index does not present autocorrelation; ii) there is evidence that macroeconomics factors are significant in explaining corporate
CDS spreads; iii) even if the model’s diagnostics does not improve markedly, the third regime of extreme volatility is significant to understand the explanatory variables’ magnitude during different volatility regimes.
