MEAN REVERSION IN SOVEREIGN CDS AND BOND MARKETS OF ADVANCED ECONOMIES: MEASURING LONG-TERM IMPACT OF 2019-2022 CRISES ON VOLATILITY USING GARCH AND HALF-LIFE MODELS

​ABSTRACT

The purpose of this study is to test whether there is a long-run impact of the external shocks in 2019 2022 on the volatility dynamics of sovereign bonds and CDS, namely the onset of the COVID-19 pandemic in 2020, the abrupt change in the monetary policy by the Fed and ECB in 2022, and the global energy crisis in 2021 2022. An important task in this case is to measure how fast the volatility returns to its mean values, or the speed of mean reversion. To address it, the bond and CDS markets of seven developed countries were selected for the period from January 6, 2019, to January 1, 2023. The research methodology included univariate ARMA-(E)GARCH models with and without endogenous structural breaks and measures of mean reversion speed using a half-life metric. The results indicate that sovereign bonds and CDS of developed countries exhibit characteristics of a mean reversion process in the level of volatility, with an observed acceleration in the dynamics of the process after accounting for external shocks, which points to the absence of long-run effects of crises. This conclusion indicates that sovereign bond and CDS pricing does not obey the efficient market hypothesis. In other words, yields and volatility of such financial instruments inevitably come to their long-run historical values, successfully absorbing external shocks that do not have a long-run impact on pricing process. The reaction of volatility to external shocks among countries seems to be homogeneous, pointing at close integration of sovereign CDS and bond markets.

Keywords: sovereign CDS and bond markets, external shocks, efficient market hypothesis, GARCH-modelling, mean reversion process.

JEL G01, G15, F30

1. Introduction

Exploring volatility of sovereign bonds and credit derivatives (CDS) in advanced economies in the presence of external shocks to financial system from 2019 to 2022 is an important and challenging task. External shocks can significantly impact the pricing of these assets, which in turn can have long-lasting effects on their volatility. Crises can alter fundamental characteristics of markets, including a speed of mean reversion (mean reversion phenomenon). This phenomenon describes how quickly the volatility, return, or price of an asset returns to its long-term average, measured in the number of trading periods.

From a practical point of view, exploring mean reversion properties in the presence of external shocks is of major importance to the investors as it opens up opportunities for better arbitrage strategies and more efficient risk management. Firstly, by using certain metrics it is possible to calculate the average time with which the return of an asset goes back to its historical mean, which makes it possible to determine the optimal time to exit a position. Secondly, if there are long-term effects of the shocks on returns distribution, VaR estimates will be biased because such effects imply changes in the fundamental characteristics of the distribution of yields, therefore VaR calculation on a unite sample will not be correct. Thirdly, understanding the asymmetric reaction of assets of different countries to external shocks within the mean reversion process creates opportunities for better diversification of portfolios and optimization of portfolio VaR by giving some insights into changes of correlation dynamics.

The aim of the study is to confirm or refute the long-term impact of external shocks on volatility dynamics of sovereign CDS and bond markets by analysing mean reversion phenomenon via measuring mean reversion speed.

The first hypothesis ( 1) is that both sovereign CDS and bond markets of advanced economies exhibit the presence of mean reversion process in volatility dynamics.

The second hypothesis ( 2) is that there is no long-term impact of 2019 2022 crises on the dynamics of volatility of sovereign bonds and CDS of advanced markets.

The third hypothesis ( 3) suggests that there are no significant heterogeneous effects on volatility dynamics of sovereign bonds and CDS of advanced markets.

The paper is organised as follows: in the first section the background of the study as well as its aim and hypotheses are described; the second section includes literature review with respect to the research gap and current views on the topic of volatility dynamics in the context of mean reversion phenomenon; the third section describes the process of data collection, explains estimation techniques and provides detailed methodology of the paper; the fourth section presents estimation procedure with subsequent empirical and substantive analysis of the obtained results; the final section includes discussion of the results with delimitations of the research and conclusion with respect to the opportunities for future research.

2. Literature Review

The existing literature on the topic of volatility dynamics in the context of mean reversion phenomenon has devoted considerable attention to the stock market. Some work focus on currency and commodity markets, while research on mean reversion in the bond and CDS markets is insufficient, which is the issue this paper tries to solve.

De Bondt & Thaler [1] carried out key research, which shows that there is a tendency of the returns to come back to their mean values after a certain period. This suggests that future stock prices can be predicted based on their past values.

Rizwan et al. [2] conducted extensive research on mean reversion in the stock indices returns of emerging and advanced markets and concluded that all markets follow mean reversion process, with the former reverting faster to its mean historical values than the latter.

Caporale & Bezares [3] investigated European stock market in the 2018-2023 period with respect to various timeframes. They found presence of mean reversion for both daily and weekly data but no signs of mean reversion for monthly data. This conclusion may advocate efficiency of the markets on a long-time horizon.

Nguyen et al. [4] made use of US, UK and Japan stock indices in 1929-2016 period to run a series of unit root tests on them. Researchers concluded that all three indices are trend-stationary, which is a feature of mean reversion hypothesis rather than efficient market hypothesis (EMH) which implies trend-stationarity of time-series.

Some researchers tried to explain possible key drivers for the predictability of market prices in the mean reversion paradigm by analysing psychology of investors. Kim et al. [5] argue that there is a strong obtainability prejudice of market participants, which is mainly based on investors` tendency to make investments on the basis of available information. In practice, this tendency leads to overreaction of market players to certain news and events, which often accelerates mean reversion.

Trypsteen [6] points to the phenomenon of the attraction of investors to low equity prices. In his view, investors are searching for an opportunity to make excessive gains in the stock market, thereby being more inclined to buy stocks at lower prices.

Schmeck & Schmerin [7] support this view by analysing supply/demand dynamic when stocks are perceived as expensive by investors. Authors stress that buyers become more reluctant to buy stocks on higher prices while suppliers seek to capitalize their gains, which explains mean reversion during periods of elevated stock valuations.

Tabot [8] demonstrates irrational decision-making of the investors by analysing weekend anomaly phenomenon referring to a tendency of market assets to open lower on Monday compared to Friday close price. Tabot [9] claims that some investors still try to exploit such phenomenon, implementing strategies found to be unprofitable and bear the risk of unpredicted losses. To sum up, risk-aversion, obtainability prejudice and attraction to low equity prices are claimed to be the main drivers of mean reversion.

Mean reversion process has also been reported in a limited number of papers on the topic of bond and CDS volatility. Ho et al. [10] identify significant long-memory effects in the volatility of sovereign CDS of four developing countries using a TV-FIGARCH model that accounts for potential structural breaks in data.

Ngene et al. [11] observed high volatility persistence in the sovereign CDS and emerging bond markets, indicating the presence of mean reversion. They argue that the rate of mean reversion is higher in the CDS market compared to the bond market. Authors state that this feature of the bond market can be associated with either informed trading or the presence of asymmetric information.

A common classification divides a rate of return to the mean into a short-term and a long-term. Chi & Dong [12] state that the short-term mean reversion is present for the stocks which returns are reverting to the mean values within one-month period. Chaves & Viswanathan [13] demonstrate that unlike the long-term mean reversion, it allows to generate better investment performance with less severe drawdowns and superior Sharpe ratios.

Mean reversion phenomenon attracted much attention from a view of investment opportunities it offers to market participants. Zhang [14] analysed the investment performance of different bond classes applying traditional mean-reversion strategy. The author came to the conclusion that sovereign bonds are the most volatile compared to corporate and municipal bonds, while both three showing negative returns and Sharpe ratios if traditional mean-reversion strategy is used. Buzzacchi & Ghezzi [15] analyses mean blur phenomenon, which refers to impossibility to accurately measure mean rates of return of an asset by using statistical methods.

Despite plenty of empirical evidence in favour of mean reversion hypothesis including statistical properties of time-series, half-life analysis and psychological drivers behind market inefficiencies, some researchers tried to incorporate these facts into broader and nuanced version of EMH. Schmidhuber [16] came to the conclusion that trends present in various asset classes (equities, interest rates, currencies and commodities) tend to revert before they become statistically significant in a period between 1990 and 2019. Besides, a key observation of the paper is that tomorrow`s expected return follows a cubic polynomial of today`s trend strength. However, the most intriguing content of the work is the analogy author draws between financial markets and critical phenomenon in physics by comparing trend strength and buy-sell orders to the parameters of statistical-mechanical system near second-order phase transitions .

Fama [17] discussed a form of EMH, which exhibits the same properties. In other words, inefficiencies resulting in price deviations are eliminated before they become strongly statistically significant. In author`s view, mean reversion is present in short/medium term periods and represent temporal inefficiencies of the market.

Most of the studies regarding price formation of the markets tend to describe it either as mean reversion or efficient market hypothesis. Mili [18] argues that the returns of sovereign CDS in Europe follow random walk hypothesis (consistent with EMH) in times of crises while being consistent with mean reversion hypothesis in pre-crises periods. This view suggests unpredictable behaviour of CDS returns during highly volatile periods, prohibiting investors from exploiting extreme deviation of CDS spreads from its historical mean returns.

Danielson et al. [19] justify this finding. Authors analyse the phenomenon of endogenous risk and apply its concept to VaR-amplifying mechanism , which exacerbates volatility during extreme volatility periods in the market by the means of simultaneous sell-off of assets by market participants.

Vera-Valdes [20] shows that persistence of volatility increased significantly for the European stock markets during COVID-19 pandemic announcement and during subsequent news of beta-variant. As a matter of fact, increase in the volatility persistence parameter in classic univariate GARCH models decreases mean reversion, making them more efficient and less predictable.

Some research has been conducted to capture spillover effects of external shocks on volatility dynamics in CDS and bond markets, which may be useful to perform cross-country and cross-market analysis in this paper. Papers in which cointegration between CDS and bond markets is analysed in terms of volatility are mainly focused on old crises. Delatte et al. [21] investigated the interaction between bonds and CDS using a small sample of 2009/10s and found that the interconnectedness and cointegration of the markets increases during stressful periods, with CDS market volatility potentially spilling over into the bond market. Most researchers agree that the CDS market is leading in the pricing process, i.e. its volatility can be used to predict volatility of the bond market.

Tiwari et al. [22] argue that in the context of global asset classes, namely equities, CDS, sovereign bonds and currency markets, the first two instruments are transmitters of volatility, while the remaining instruments are recipients of volatility.

Frinjs & Zwinkels [23] rationalize this view by decomposing sovereign bond and CDS prices during 2011 European sovereign debt crisis. They conclude that bond and CDS prices are divided into three components in the following proportions: 80% for bonds vs. 49% for CDS is liquid trading, 13% vs. 45% for credit news, and 5.4% vs. 5.5% for speculative trading. Thus, it can be considered that the CDS market has a greater informational function than the bond market, as its pricing is not dominated by the liquidity factor, which is highly volatile in the context of external shocks.

Overall, the issue of mean reversion of sovereign CDS and bond markets under presence of external shocks is fairly unexplored in literature. Most of the papers have their focus either on finding evidence for mean reversion on a specific period, excluding crisis analysis or trying to quantify the impact of shocks by analysing spillover effects using more sophisticated multivariate models with less attention to mean reversion phenomenon. Besides, most of attention in mean reversion analysis is devoted to the stock market with rare research done on the topic of sovereign CDS and bond markets.

3. Data and Methodology

3.1. Country selection

The paper analyzed weekly series of yields of seven developed countries. These are the markets of 5-year sovereign bonds and CDS of Germany, Italy, Spain, France, Great Britain, the USA and Japan. The choice of these countries is explained by their special importance to the global financial system: the USA is the first economy of the world, and its debt yield is considered risk-free and serves as a benchmark for pricing of many financial products in the world; Germany is an analog of the American debt in Europe; Great Britain, Italy, Spain and France are the largest economies in Europe, possessing sufficiently liquid debt markets and a significant volume of the repo market in comparison with other European countries; Japan and its currency and debt markets are of paramount importance for the global financial system in terms of carry-trade.

3.2. Data collection

Five-year maturity bonds and CDS were chosen for analysis because of their high liquidity. Five-year CDS on sovereign debt are actively traded in both the US and Europe and are close in pricing to the arbitrage-free condition, as manifested by the CDS basis being equal to zero. These characteristics make five-year instruments the most popular instruments to study in literature, for example in the work of Palladini & Portes [24].

Discrete time series of bond and CDS yields for the period from January 6, 2019, to January 1, 2023, were used for the analysis. The total number of time series analyzed was 14. Each sample includes 208 weekly observations for the logarithmic series of bond and CDS returns. Each observation represents the weekly return of the asset, calculated as the ratio of the closing price of the current Friday to the closing price of the previous Friday. The formula is following:

                                    (1)

where ln(price_it) is the natural log of price in period t; i 1, 2; 1 CDS, 2 bond.

3.3. Estimation techniques

The empirical analysis in this paper was performed using various statistical methods, including descriptive statistics, ARMA-(E)GARCH models, the Bai-Perron test to identify endogenous structural breaks, and the rolling window procedure for their validation. The mean reversion rate and the size of volatility shocks were estimated using the half-life metric in Chaudhuri & Wu [26] and Kuttu [27]. These methods provide a deeper understanding of volatility dynamics and the response of financial assets to exogenous shocks.

The use of ARMA-(E)GARCH models helps to account for time dependence and variability in the data, while the Bai-Perron test helps to identify structural breaks in the series. The rolling window procedure provides an additional check on the stability of the analysis results over time. The half-life metric is used to assess the rate of return to the mean, which is a key aspect in understanding the behavior of volatility in financial markets.

3.4. Preliminary statistical tests verifying the applicability of ARMA-(E)GARCH models

To fulfil the necessary conditions for the application of GARCH class models, various diagnostic tests were conducted. These include: (1) ARCH-LM test of Engle [28] to test for conditional heteroscedasticity, or the presence of ARCH effects; (2) BG test of Breush & Godfrey [29] to assess serial autocorrelation; (3) Bartlett and Portmone test by Box & Pierce [30] to check residuals against the white noise process.

The ARCH-LM test has the null hypothesis of constant variance (homoscedasticity) or no serial correlation of heteroscedasticity as a function of the number of lags. The ARCH-effect implies that there is a statistically significant effect of squared past return shocks on the conditional variance. The basic model for testing ARCH effects is as follows:

,                                                (2)

,                                           (3)

where _{, ,} regression coefficients, return shocks.

Conditional variance refers to the conditional variance of the error with k-th lag and is formalized as:

                     (4)

where conditional variance, _-i return shocks.

The ARCH effect with lag p is modelled as follows:

,                                                    (5)

where _, regression coefficients, conditional variance, return shocks.

The ARCH-LM test tests the significance of the coefficient α₁, which characterizes the long memory volatility effect. The first lag of returns was used as regressor in the equation.

3.5. Univariate GARCH models

The paper uses ARMA-GARCH and ARMA-EGARCH models depending on the visual estimation of asymmetry in return shocks and tests for determining distribution of errors. The specification of the models is as follows:

           (6)

where returns, return shocks, a set of endogenous dummy variables, which equal 1 after break date, 0 otherwise, _i , regression coefficients; i 1, 2; 1 CDS, 2 bond.

,                                               (7)

where white noise process with 0 mean and conditional variance

,                                              (8)

where standardized return shocks, .

_i,           (9)

where unconditional variance; module of standardized past return shock, standardized past return shock; _i a set of endogenous dummy variables, which equal 1 after break date, 0 otherwise; natural log of unconditional variance, , , regression coefficients.

,             (10)

where unconditional variance; past return shock; _i a set of endogenous dummy variables, which equal 1 after break date, 0 otherwise; unconditional variance; , regression coefficients.

The first equation specifies the ARMA (r,s) process for the conditional mean return, r is the number of return lags, s is the number of error lags in the model, ω is the unconditional (long-run) variance parameter; α measures the effect of shock size on conditional volatility; shows the leverage effect (if, conditional volatility reacts more strongly to negative shocks than to positive shocks and vice versa if); β volatility persistence parameter: the closer β is to 1, the longer it takes (trading weeks) for conditional volatility to return to its long-term average value after a shock).

3.6. Identifying structural breaks in returns and volatility time-series

External shocks can have a significant impact on both bond and CDS yields and volatility, which can be manifested in changes in the average yield level or its trend after crisis events. To identify structural breaks, an algorithm that includes statistical tests to detect endogenous changes and their subsequent validation through graphical analysis is used. Since the introduction of exogenous breaks does not always guarantee their significance in the sample, searching for endogenous breaks, which usually correspond to periods of crises, is a more robust method. In this paper, such breaks were detected using the BP test, which aims to identify multiple structural breaks. It allows a more accurate assessment of the impact of crisis events on the volatility and returns of financial assets. Its specification has the following form (the algorithm is used in the work of Göktaş & Disbudak [31]):

_i,                                    (11)

where , 0 otherwise; n is the number of breaks, is their location in the variance equation, w is a constant return shock; a set of dummy variables.

_, (12)

where , 0 otherwise; for 0 otherwise; is the location in the mean equation; ,_mI and regression coefficients; returns of either i 1(bond) or i 2(CDS); _, a set of dummy variables.

First, the BP test is used to find structural breaks in the mean return equation, assuming there is a shift in both the slope and the value of the constant θ (equation 12). Next, a structural break in the mean variance is tested (equation 11). According to Göktaş & Disbudak [31], changing the specification of ε_t² in Equation 11 does not affect the estimation results due to the robustness of the BP test.

3.7. Measuring mean reversion speed in volatility using half-life metric

The mean reversion rate and its corresponding half-life metric in classic symmetric GARCH model (equation 6) is given by the following equation of Rizwan et al. [2]:

                                            (13)

                   (14)

where μ_t is the vector of average returns (defined in the paper by the sum of lagged returns), ε_t is the return shock, h_t is the conditional variance.

The half-life metric indicates the number of periods (weeks) it takes for the conditional variance shock to halve. In symmetric GARCH, α is responsible for trailing variance and β for conditional variance, which is why the sum of the coefficients is important for the half-life estimation. On the contrary, in the EGARCH model the coefficients α and γ are responsible for the shocks of returns, so that they do not affect the rate of return of volatility to the mean. In other words, half-life in EGARCH model is estimated differently:

(15)

In terms of pricing hypotheses:

1) or the process is nonstationary and the EMH hypothesis is true. There are no long memory effects in the market, and there is no return to the mean. The value of the half-life metric is undefined, given .

2)or the process is stationary, long memory effects are present, mean reversion hypothesis is true, mean reversion is inherent in asset volatility, the same is true for prices/yields (trend reversion).

Confirmation of second inequality proves the existence of mean reversion process in the markets of sovereign bonds and CDS. If the mean reversion hypothesis is relevant for pricing in these markets, the rate of mean reversion should increase in case of a significant deviation from the trend caused by the crises of 2020 2022 and thus, refute the existence of a long-term impact of crises on the volatility of the analysed assets.

The presence of long-run impact of crises in this paper is understood as a significant decrease in the speed of mean reversion. In such case, it can be argued that key asset characteristics, such as returns and volatility, have deviated substantially from their mean values of past periods. Assuming that CDS and bond pricing was subject to a mean reversion process prior to the 2020 2022 crisis events, such dynamics would indicate that market players are less willing to speculate on mean reversion. In such a case, external shocks will have a long-term impact on the pricing process.

3.8. Comparing model parameters with and without external shocks for analyzing long-term impact of crises

In the context of the presented study, the successful estimation of models with external shocks allows to identify volatility persistence, size and leverage effects (Nelson [32]), unconditional variance, and the half-life metric indicating the mean reversion rate more precisely. Comparison of these parameters with similar values obtained from the models that do not take into account structural breaks tests the second hypothesis of the study (H2).

If the hypothesis is confirmed, it would imply that the effects of long memory (expressed through reduced half-life) are amplified after the shocks, which provides investors with additional opportunities for arbitrage trades by detecting deviations from long-run trends in returns and volatility. In case of obtaining robust results showing the growth of half-life values after taking external shocks into consideration, it will be possible to state statistically significant influence of external shocks on long term memory of returns and volatility of analyzed assets.

If the volatility persistence parameter is increased after accounting for structural breaks, this would indicate that past shocks of returns (periods t-1, t-2, ..., t-n) have a long-run effect on the volatility and returns of the current period t. Generally, this tendency indicates greater market efficiency, which implies a difficulty in forecasting future prices based on past their values, since current prices already incorporate most of the past information. In case of or , the half-life metric will be uncertain: . This will indicate the impossibility of estimating mean reversion and confirm the efficient market hypothesis (EMH).

4. Estimation and results

4.1. Descriptive analysis

The descriptive statistics presented in Table 1 show several key features: (1) CDS premiums are, on average, significantly higher than bond yields; (2) The standard deviation of CDS premiums is also significantly higher than that of bonds; (3) The average bond yields of all countries are negative. The first and the second facts correlate with stylized facts about the CDS market, which is traditionally considered more liquid and speculative than the bond market, resulting in higher yields and volatility. The third fact is explained by the onset of the global cycle of interest rate hikes of central banks worldwide, which decreased the market value of the 5-year zero-coupon bonds considered in the paper.

Table 1.

Descriptive statistics

Source: author`s calculations

Various diagnostic tests are summarized in Table 2. These include: ARCH-LM test for conditional heteroscedasticity (so-called ARCH effects) by Engle [28], BG test for serial autocorrelation, Bartlett and Portmone tests for residuals following a white noise process.

Table 2.

Diagnostics tests for CDS and bonds time-series

Note: Figures correspond to p-values. (***) - 1% significance level; (**) - 5%-significance level; (*) - 10% significance level.

Source: author`s calculations

For the CDS series, the null hypothesis is rejected at the 1% significance level for 10 lags, indicating the presence of statistically significant ARCH effects. This means that past shocks of CDS premiums have a significant effect on current period volatility. Thus, all CDS series exhibit long memory effects and can be efficiently estimated using the A.R.C.H family of models.

The Breusch-Godfrey test for autocorrelation of residuals showed that the 10 lag autocorrelation is significant for CDS of Germany (at the 5% level) and CDS of Japan (at 1% level). For the remaining countries, the null hypothesis of no autocorrelation of the residuals at 10 lags was confirmed.

White noise tests showed that the errors in the main equation of CDS premiums follow a white noise process; the null hypothesis was not rejected for any CDS series. Combined with the results of the ARCH-LM and BG tests, this allows us to use A.R.C.H. family models with a normal error distribution for any CDS premium series.

Significant ARCH effects were found only for Spanish and Japanese bonds. Significant autocorrelation was found only for US bonds. The white noise tests showed that the errors of the US, German and UK bond series do not obey the white noise process at the levels of significance of 1%, 10% and 10%, respectively. These results indicate that for a number of countries the errors will not be normal, therefore requiring special error distribution to adequately estimate the GARCH models.

4.2. Exploring features of CDS/bonds returns and volatility time-series on a uniform sample using GARCH model

The estimation results allow to make cross-country comparison of the coefficients as well as an analysis of volatility metrics in bond and CDS markets:

1) In general, CDS market have significantly lower volatility persistence (β) than the bond market. Accordingly, the half-life of CDS is, on average, significantly lower than that of bonds. The only exception is the UK CDS, which half-life is the highest among all analyzed series at 692. This result correlates with the findings of Ngene et al. [11] who found greater volatility persistence of bonds compared to CDS both with and without structural breaks in the sample of developing countries at 2001 2012 period. Presumably, high market liquidity and the popularity of CDS as a financial instrument providing opportunities for trading credit risk independently of market risk contributes to high speculative activity. CDS market does not allow prices (CDS premiums) to stay at extreme values for a long time, as there are many players on the market willing to use arbitrage opportunities. In contrast, the bond market is less liquid, which means that yield shocks will be absorbed slower due to various phenomena that limit arbitrage activity (such as flight-to-liquidity and short-selling frictions , Fontana & Scheicher [33]).

2) Among CDS series, Germany has the lowest parameter β = 0.46 (volatility persistence), accordingly, German CDS have the lowest half-life of only 0.89 weeks. Thus, the mean reversion of Germany's CDS in the unite sample (2019-2022 excluding structural breaks) is the highest among CDSs of all developed countries.

3) In general, all series of 5-year bonds have high volatility persistence above 0.9. The minimum half-life indicator is in the UK bond series, the highest the US. The rate of mean reversion for all bond series can be considered long-term (more than a month).

Table 3.

Estimation results of E(GARCH) models without external shocks for bonds

Note: Coefficients and their significance are obtained from the equation of conditional variance.

Source: authors calculations.

Table 4.

Estimation results of the (E)GARCH models without external shocks for CDS

Note: Coefficients and their significance are obtained from the equation of conditional variance.

Source: author`s calculations

4.3. Properties of yield/volatility series on the sample with the account of external shocks in the model

In order to test the hypothesis of the study about the absence of long-term impact of crises, it is necessary to compare the coefficients of two models for each of the 14 analysed series of sovereign bond yields and CDS of developed countries. The first version of the model does not take into account external shocks modelled through structural break variables.

A similar procedure to find the optimal ARMA-(E)GARCH specification was done in the samples that include structural breaks. Adding them is necessary to obtain a valid model of each of the bond/CDS series. The second type of the model allows for a better interpretation of the impact of past crises, directly incorporating dummy variables tied to the detected dates of structural breaks into yield and volatility series. Adding dummy variables provides more accurate estimates of the ARMA-(E)GARCH model parameters and to improves the quality of the model in terms of information criteria.

4.4. Dates of structural breaks

In the context of the presented study, structural breaks are needed to obtain statistically significant parameter estimates in the variance (conditional volatility) equation of the ARMA-(E)GARCH model for comparison with the estimates of the model that does not account for external shocks. The aim is to estimate the impact of crises on the speed of mean reversion.

Structural breaks in volatility were found in each of the 14 analysed time series. The same is true for structural breaks in the returns, except for Italy and US bonds.

Based on the dates of the structural breaks and corresponding observation numbers, some patterns can be identified:

1) 31-61 observations, from 04.08.2019 to 01.03.2020. This period can be named pre-covid . It was characterized by a slowdown in global economic growth and corresponding decline in capacity utilization in developed countries. Important events in the financial sector during this period include the inversion of the US yield curve (the spread of ten-year and two-year Treasury bonds went into the negative zone on 29.08.2019, which corresponds to 34-th observation in the sample), as well as the short-term crisis of the US banking system related to the repo market (there was a sharp increase in rates in the repo market on 17-19 September, which required an immediate response from the Fed to stabilize the situation), corresponding to 37th observation in the sample;

2) 62-65 observations the epicenter of coronacrisis in March 2020, accompanied by large-scale liquidations in global capital markets;

3) 66-78 observations post-coronacrisis shock;

4) 146-165 observations the beginning of the energy crisis in Europe (October 2021), which turned into a global energy crisis due to the beginning of escalation and armed conflict between Russia and Ukraine and the accompanying sanctions confrontation between Russia and Europe, USA;

5) 166-177 observations a period of increased volatility in the global financial system, expressed in the rapid growth of the dollar index (DXY), the beginning of the Fed interest rate hike cycle (167 observation), rapid decline of the SPX and NASDAQ stock indices, the acceleration of inflation indicators in developed countries and tensions in the geopolitical situation against the background of sanctions wars and the global energy crisis.

Thus, the majority of endogenous breaks detected by the BP test corresponded to the expected range of dates of external shocks analyzed in this paper. In most cases, such breaks have a significant impact on the parameter estimates of the models, improving the goodness of fit of the regression.

Table 5.

Dates of structural breaks in the return and variance equations for bonds

Source: author`s calculations

Table 6.

Dates of structural breaks in the return and variance equations for CDS

Source: author`s calculations

4.5. Validation of structural breaks and statistical conclusions regarding GARCH estimates with external shocks

In order to validate structural breaks, a rolling window procedure was performed. Its idea is to construct a one-week forecast of returns and compare them with the realization of returns over a given time horizon (analogous to the back testing procedure). Rolling window can be better understood by a certain dynamic range of observations with a fixed sample size. In this paper, the window size is equal to 100 observations. 100 observations, on the one hand is a large enough number to minimize model variance, but on the other hand contains outdated observations that potentially distort model estimates. However, there are advantages and disadvantages of choosing almost any window size in the goal of forecasting.

The models within each of the windows have already taken into account the dates of structural breaks in their specification. The described procedure was also necessary for post-estimation validation of breaks in model diagnostics. The results of the structural break validation are indicated in Tables 8 and 9 with an appropriate label (e.g., D145v). D62, D81, etc. correspond to a dummy variable equal to 1 starting with number 62, 81, etc., and 0 otherwise. The notation D62v indicates that the break was confirmed by the rolling window procedure.

The obtained results of model estimation allow us to draw certain statistical conclusions regarding the impact of external shocks on the series of sovereign bonds and CDS of the countries under consideration:

1) The volatility persistence parameter fell significantly for most of the studied assets. The exceptions are Spanish bonds (0.971 vs. 0.96), Spanish CDS (0.959 vs. 0.9) and German CDS (0.521 vs. 0.46). This result is in line with the findings of Ngene et al. [11] who investigated sovereign bonds and CDS on a sample of developing countries in the 2001 - 2012 sample. The variation in the half-life of the CDS market was: 0.5 weeks (France) - 16.55 weeks (Spain); for the bond market: 1.44 weeks (UK) - 23.55 weeks (Spain).

2) The largest drop in the volatility persistence parameter and the associated half-life metric is observed for UK CDS (2.19 weeks vs. 692 weeks). The addition of external shocks to the model has significantly reduced the impact of past volatility shocks on its current level.

3) The largest drop in volatility persistence and half-life in the bond market is found for the US (2.37 weeks vs. 68.96 weeks). Similar to UK CDS, US bonds had a volatility persistence indicator of almost 1 in the model without external shocks. The addition of external shocks to the model significantly accelerated the mean reversion process, which most likely indicates to serious distortions in the estimation of volatility persistence in the unite sample due to anomalous volatility spikes corresponding to the dates of the structural breaks found.

4) All coefficients are significant at least at 10% level except for constant in the regression for SpainCDS, γ and α in ItalyBond regression and γ in UScds regression.

5) γ parameter remained positive after accounting for structural breaks in the model. The only exception is the regression for Italy bond, where γ is negative and insignificant. It can be concluded that leverage effect associated with negative values of γ is absent in both CDS and bond markets. Therefore, the conditional volatility of analysed financial instruments tends to react more to positive past return shocks rather than negative ones.

Table 7.

Estimation results of (E)GARCH models for bonds with structural breaks.

Note: The coefficients and their significance correspond to the conditional variance equation from (9) and (10). The values of the half-life period are compared with the values of this metric from tables 3 and 4.

Source: author`s calculations

Table 8.

Estimation results of (E)GARCH models for CDS with structural breaks.

Note: The coefficients and their significance correspond to the conditional variance equation from (9) and (10). The values of the half-life period are compared with the values of this metric in the model without breaks.

Source: author`s calculations

4.6. Substantive analysis of the long-run impact of crises on the volatility of sovereign bonds and CDS of developed countries

The results of model estimation allow us to draw a number of meaningful conclusions regarding pricing in bond and CDS markets, as well as to conduct a cross-country analysis of the impact of external shocks on the studied assets:

1) All the studied assets are subject to a mean reversion process in the level of volatility. This indicates the existence of a long memory in asset pricing associated with the existence of some long term average/long term trend in returns. Thus, unlike the EMH hypothesis, under mean reversion the current price depends not only on unexpected news of the current period but also on past shocks or, in other words, price has a long memory. In addition, current prices do not reflect all available information, which is due to the numerous phenomena such as investor risk phobia, insider trading, fear and greed in the decision making, amplifying VaR mechanism in capital markets by Danielson et al. [19] (past negative/positive events increase/decrease VaR, amplifying return fluctuations, realized in volatility clustering) and volatility spillovers resulting in ever-changing asset correlations. The results of this study, combined with empirically proven phenomena listed above, are very likely to refute the efficient market hypothesis. The most obvious conclusion from the existence of mean reversion is that prices of future periods can be predicted on the basis of past values, but the creation of effective predictive models is difficult due to constantly changing market conditions (expressed in the destruction of previously observed correlations under the influence of some deterministic factors).

2) CDSs have higher mean reversion rate compared to bonds both with and without external shocks. In addition to the greater liquidity of the CDS market and the resulting lesser restrictions on arbitrage activity, another explanation for the strength of the mean reversion process in the market of default insurance can be found. It is related to the much more stable long-run characteristics of the market, since the credit characteristics of a sovereign borrower are generally more stable than the cost of money in the economy (which can be estimated by the 5-year sovereign bonds serving as a proxy variable). According to Frinjs & Zwinkels [24], bond prices are more dependent on liquidity trading than on credit news. This means that while bond prices are more dependent on changes in the conjuncture of financial conditions in the market, they also provide more information regarding financial conditions and the time value of money, thus possessing a greater information function than the CDS market in a common sense (e.g., yield curve inversion in the US bond market has successfully predicted all ten recessions since 1955).

3) The external shocks considered in this paper have no statistically confirmed long-run impact on sovereign bond and CDS markets. The second hypothesis (H2) is confirmed by the econometric analysis. Some empirical results, such as the increase in the half-life of Spanish bonds or German and Spanish CDSs, point to a slowdown in mean reversion and are not in line with overall results of the study. However, they can be explained by either mediocre quality of the model compared to its version in the unite sample (as in the case of Spanish CDSs). Since the statistical significance of the coefficients was not tested in the study, it is not possible to draw unambiguous conclusions regarding the long-term impact of crises in these assets. The most adequate interpretation for these assets is that there is no tangible impact of crises on pricing.

4) Differences in responses of volatility and yields of sovereign bonds and CDS to external shocks are mostly insignificant: the general tendency of increased speed of mean reversion prevails for all studied assets. This result is consistent with the conclusions reached by Nasir et al [34] and Brunnermeier & Pedersen [35]. According to the latter, there are implicit costs of holding positions in securities on the balance sheet of a financial institution (e.g. capital, margin and liquidity regulatory requirements, costs of fair value measurement of the instruments used). Researchers prove that in periods of stable market condition and predictable volatility regime such costs are always positive and can rise substantially during crises. In the context of CDS and bonds, such costs can explain the risk-off regime in periods of stress, which significantly distort the fair value of assets, solely due to the factor of insufficient liquidity provided by financial intermediaries.

5) The findings of this paper demonstrate that bond yields of developed countries behave quite similarly during the periods of recovery from crises, because the reaction of volatility to external shocks is similar. This is probably explained by the fact that the Central Banks of developed countries conducted similar monetary policies in response to the 2020-2022 crises (except for the Central Bank of Japan). Global cycles of rate hikes and rate cuts were fairly synchronized, which was due to relatively similar dynamics of economic data (inflation, unemployment, business activity indices, consumer sentiment, etc.). For the most part, 2019 and 2021 years can be considered periods of soft financial conditions, while 2020 and 2022 can be considered periods of tight financial conditions (as indicated by the Financial Conditions Index from the US FRB Chicago)^[1], which gives a liquidity explanation both for monetary policy response to crises as well as similar reaction of CDS/bonds to the external shocks across countries in this paper.

5. Discussions

The main rationale for the research was that the issue of mean reversion in sovereign CDS and bond markets under presence of external shocks is fairly unexplored in literature. Most of the papers have their focus either on finding evidence for mean reversion on a specific period, excluding crisis analysis (for example, in Rizwan et al. [2] or Chaves & Vizwanathan [13]) or trying to quantify the impact of shocks by analysing spillover effects by using more sophisticated multivariate models with less attention to mean reversion phenomenon (e.g. Tiwari et al. [23]). Presented research tries to capture the main features of these approaches while remaining specific.

The overall findings of this paper suggest that mean reversion is present in sovereign and CDS markets of advanced economies, while there is no long-term impact of external shocks of 2019-2022 on volatility dynamics. These results are consistent with Ngene et al. [11], who showed that sovereign CDS and bond markets of emerging markets exhibit the presence of mean reversion process, which accelerates after accounting for structural breaks in data, usually associated with crises. This means that volatility and returns of studied assets tend to return to their historical mean values after a certain period, while crisis events accelerate this process. In other words, market behaves in a somewhat predictable manner as past shocks of crises periods affect volatility of a current period less, which is shown by decreased half-life and increased volatility persistence parameter in GARCH models after accounting for external shocks. Therefore, investors seem to effectively exploit the opportunities created by external shocks, believing that the true fundamentals such as volatility and returns of CDS and bond markets remained stable after highly volatile 2019-2022 period. As a consequence, findings of the paper can be useful in risk-management purposes since they allow to leave risk measures (such as Value-at-risk, VaR or Expected Shortfall, ES) unconstrained a decision, market participants cannot make if using unite sample without accounting for external shocks in modelling.

Another result of presented research is that differences in the response of volatility and yields of sovereign bonds and CDS to external shocks are mostly insignificant: the general tendency of increased speed of mean reversion prevails for all studied assets. This result is consistent with the conclusions reached earlier by Nasir et al. [34]. Cross-country analysis implies that France CDS market can be considered the best for arbitrage transactions, as the speed of volatility returning to its mean value is the highest with half-life equal to half a week. The best bond market in this regard is the UK market, where the speed of mean reversion exceeds other markets with the half-life being equal to 1.44 weeks. The worst markets in terms of arbitrage opportunities are the Spanish bond and CDS markets, as they require the longest time to close a position at the moment when past volatility shocks are fully absorbed and the window for arbitrage closes (with half-life of 23,55 and 16,55 weeks respectively).

Despite the results of this paper in favour of mean reversion phenomenon presence in CDS and bond markets, there is some mixed empirical evidence for the existence of efficient market hypothesis. Spierdijk & Bikker [36] point that one should analyse big enough dataset to confirm the presence of mean reversion as a clear mechanism for pricing in the market. Caporale & Bezzares [3] find that mean reversion may be present only for some timeframes in the analysis. Thus, it is imperative that future research focusing on the nature of price formation of the market includes sufficient size of a dataset to capture potential temporal deviations from EMH and market inefficiencies. The work of Schmidhuber [16] appears to present powerful evidence for the validity of EMH hypothesis for most of the financial assets with the caveat of market inefficiencies being present until they become statistically significant. Therefore, mean reversion may be a valid hypothesis for price mechanism only on medium-size datasets, which may be the case for presented research.

The limitations of presented research are concentrated within its dataset size and the choice of weekly data for GARCH modelling. As discussed above, it is difficult to provide enough evidence to refute EMH using medium-size dataset. Moreover, conclusions of the paper regarding the absence of long-run impact of shocks are based on changes of half-life metric in different models, which significance was not tested, since there is little research done on such methods. Therefore, only sizable deviations in half-life values before and after incorporating external shocks were considered valid to refute long-run impact of crises. This led to some blur in conclusions about German and Spain CDS markets, which could have been avoided by estimating GARCH-model coefficients significance by constructing specific statistical test for that matter. The choice of weekly data for GARCH modelling may be justified by providing smoothness to the data by excluding potential outliers, which may be responsible for autocorrelation and biased estimations. However, it is highly likely that some endogenous structural breaks were missed due to the lack of observations, which could be corrected by using daily data in future research.

6. Conclusion

The results of the study indicate that there is no long-run impact of crises on volatility dynamics. The models with and without external shocks have mean reversion properties in volatility, with the speed of mean reversion increasing after accounting for external shocks. This finding implies that all three hypotheses of the study are accepted as mean reversion is evident both for CDS and bond markets, while there are no signs of long-run effects of external shocks on volatility. The speed of mean reversion in the unite sample differed significantly across countries and markets, whereas in sample separated by structural breaks the speed of mean reversion shows a much smaller variation in absolute values, indicating that there is no significant heterogeneous effect of external shocks on volatility dynamics.

Results provide evidence that the pricing of sovereign bonds and CDS does not obey the efficient market hypothesis. From a theoretical point of view, it can be concluded that volatility of future periods can be predicted on the basis of past values. From a practical standpoint, the findings of the paper may allow investors to calculate optimal timing to open and close positions in CDS and bond markets based on country-specific mean reversion speed. Market participants can use arbitrage strategies, exploiting mean reversion phenomenon and enhance their risk management, implementing unconstrained VaR estimates based on the absence of long-run impact of 2019-2022 crises on volatility dynamics. In addition, obtained results may be useful for regulatory institutions engaged in market pricing research to implement effective policies to counter liquidity crises.

The area of future research may capture impact of mean reversion on market stability as well as investigate the microstructure of CDS and bond markets with respect to the roles of different market participants in driving volatility persistence. It is also important to include shocks of the 2023 and 2024, the response to which is likely to be more divergent in bonds and CDS due to differences in monetary policy/business cycle phase and economic data in the studied advanced economies. In addition, the results of the presented work suggest a unidirectional response of sovereign bond and CDS market volatility to external shocks, which may indicate that the markets are substantially integrated. This potential relationship can best be investigated by cross-country analysis of volatility spillover effects within the framework of multivariate GARCH models.

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