AN ANALYSIS OF THE DETERMINANTS

Transcription

1 AN ANALYSIS OF THE DETERMINANTS OF CREDIT DEFAULT SWAP SPREADS USING MERTON'S MODEL by Antonio Di Cesare Giovanni Guazzarotti Banca d'italia Servizio Studi Via Nazionale, Roma May 2005 Abstract Empirical work on the determinants of credit risk has traditionally looked at corporate spreads, i.e. spreads between corporate and government bonds. In this paper we analyze what determines the changes in credit default swap spreads using the structural model introduced by Merton (1974) and comparing the results with a classical linear model. The most important finding is that theoretical spreads explain a higher portion of actual CDS spread variations than the traditional linear model, supporting the hypothesis that non-linear relationships implied by Merton's model are relevant in explaining observed credit spreads. Part of the CDS spread changes, however, still remain unexplained.

2 CONTENTS 1. INTRODUCTION A REVIEW OF THE LITERATURE ON CREDIT SPREADS MAIN DETERMINANTS OF CREDIT SPREADS EMPIRICAL STUDIES MERTON'S MODEL METHODOLOGY DATA DESCRIPTION EMPIRICAL MODELS AND TESTING METHODOLOGY RESULTS CONCLUSIONS...17 REFERENCES

3 1. Introduction 1 Empirical work on the determinants of credit risk has traditionally looked at corporate spreads, i.e. spreads between corporate and government bonds. The common finding has been that only a minor part of the spread can be considered as a compensation for the risk of default of the issuers, whereas the greatest part of observed spreads can be traced back to liquidity, tax and other non-identified systematic risk factors (Elton et al., 2001 and Driessen, 2003). Recently, Amato and Remolona (2003) suggested that the credit spread puzzle could be partly explained by the fact that the risk implicit in bond portfolios is much more difficult to diversify than for equities, due to the skewness in returns induced by possible defaults. This difficulty would be compensated with higher spreads. If the bond market has traditionally been regarded as the best place in which the creditworthiness of an issuer may be assessed, in recent years there has been a huge development in instruments that are specifically designed for trading credit risk 2. These new financial instruments are usually known as credit derivatives. According to the International Swap and Derivatives Association (ISDA) the notional value of credit derivatives outstanding at the end of 2004 was USD 8.4 trillion, compared with USD 3.6 trillion a year before and just USD 1.0 trillion in Among the financial derivatives explicitly linked to a company's credit worthiness, credit default swaps (CDSs) have been particularly successful, representing more than a half of the credit derivatives market. Essentially, a CDS looks like an insurance policy: the policy-holder (i.e. the buyer of the CDS) pays a premium to the insurer (i.e. the seller of the CDS) in order to receive a compensation if one or more events occur. The main difference between a CDS and an insurance contract is that, while in the latter the policy-holder get a reimbursement only for damages that are effectively suffered, with a CDS it is possible to buy or sell protection independently from the real exposition to the credit risk of the reference entity. This aspect clearly distinguishes also the CDS market from the bond market. In the latter the short selling of credit risk is limited by the low level of liquidity of the repo market, especially for high-yield bonds, and by the short maturity 1 The views expressed in the articles are those of the authors and do not involve the responsibility of the Bank. 2 See Committee on the Global Financial System (2003). 3

4 of repo contracts. Credit default swaps allow people to short sell credit risk for a long period at a fixed rate, thus making the CDS market potentially more efficient in establishing the right price of credit risk. Some authors investigated the relationships between bond and CDS markets. Blanco et al. (2003) find that, even if in the long run both CDS and bond markets reflect firm-specific variables equally, CDS prices are better integrated with those factors in the short run. Zhu (2004) confirms that credit risk tends to be priced equally in the two markets in the long run, but the derivatives market seems to lead the cash market in anticipating rating events and in price adjustments. Other works on the ability of CDS spreads to anticipate rating changes (Hull et al., 2003, Norden and Weber, 2003, Di Cesare, 2005) show that this market is usually very effective in anticipating changes in the creditworthiness of debtors. Despite of the growing interest in the CDS market, up to now little work has been done as regard the determinants of CDS spreads. As said before, the nature of CDS is such that it is not obvious that the factors that determine corporate spreads are also relevant for CDS spreads. To the best of our knowledge up to now only Aunon-Nerin et al. (2002) have undertook the task of analyzing the determinants of credit default swap spreads. Their work, however, suffers of some caveats. First of all, the sample used is very small and mainly includes financial companies, due also to the fact that the credit derivatives market was still in its infancy when the paper was written. Moreover, the analysis is conducted on the levels of the variables, thus leaving the door open to econometric problems such as the presence of unit roots in the data. Finally, only linear models are used, whereas structural models of credit risk, such as the one introduced by Merton (1974), predict that the relationship between the variables could be highly non-linear. The following section offers a survey of the literature on the determinants of credit spreads and on previous empirical studies. In Section 3 we describe the model introduced by Merton (1974) and the way in which it can be practically used to estimate model-based CDS spreads. In Section 4 we describe the methodology used to compare the ability of Merton's model in explaining CDS spread movements with respect to a classical linear model. The results are reported in Section 5 and Section 6 concludes. 4

5 2. A review of the literature on credit spreads In this section we review some empirical contributions to the literature on the determinants of credit spreads. The first objective of previous work has been to assess how much of observed spreads is explained by structural default factors, that is by factors that theoretical models of default suggest to be linked to credit risk. Then, having observed that predicted spreads are much lower than actual ones, the next objective of the literature has been to investigate the origins of such extra yield, testing for alternative hypotheses. 2.1 Main determinants of credit spreads The first structural default model was presented by Merton in its seminal work in In this model, that will be described in some detail in the next section, default occurs if, at the maturity of the debt, the value of the firm, which is described by a stochastic process, is below the face value of outstanding debt. A debt claim is therefore modeled as the combination of a risk-free debt claim and a European put option sold to equity holders with strike price equal to the face value of the risk-free claim. In this setting, the price of debt is determined through standard option pricing methods. It basically depends on the parameters of the stochastic process driving the firm value and on the level of outstanding debt. The prediction of the model is therefore that credit spread changes should respond to changes in the following variables: i) risk-free interest rate, as a higher rate increases the drift of the firm value process under the risk-neutral measure; ii) outstanding amount of debt, which represents the value at which default is triggered; iii) firm value, as the higher is the distance between firm value and outstanding debt, the lower is the default probability (everything else equal); iv) assets volatility, as it increases the probability of default (everything else equal), i.e. the probability that the value of the firm falls below the value of outstanding debt. 3 For several extensions of Merton's model, see Cossin and Pirotte (2001) and references therein. 5

6 While previous factors should explain the expected default component of credit spreads, these may also respond to other factors not linked to credit risk, which may assume particular relevance in specific markets, such as the bond market: i) taxes, to the extent that credit instruments and government bonds are subject to different tax rates; ii) liquidity, as credit risky instruments have lower volumes and higher transaction costs than government bond markets; iii) systematic risk factors, such as those prevailing in equity or Treasury markets; iv) supply and demand shocks, which could also be specific to the corporate bond market. 2.2 Empirical studies Empirical work on the determinants of credit risk has traditionally looked at corporate spreads. The common finding has been that only a minor part of the spread can be considered as a compensation for the risk of default of the issuers, whereas the greatest part of observed spreads can be traced back to liquidity, tax and other nonidentified systematic risk factors. A first group of these empirical studies (Elton et al., 2001, and Driessen, 2003) tries to estimate the part of the credit spread which is due to expected default losses, using probabilities of default derived from historical data (such as rating transition matrices 4 and data on recovery rates). This approach does not allow, however, to separately assess the impact of each fundamental default factor: after having estimated the expected default component, one can in fact only analyze the influence of non-credit risk factors on the differential between predicted and actual spreads. In addition, as the expected default premium is estimated relying on historical data, rather than on market prices, the approach cannot provide bond-specific and forward looking estimates 5. 4 Rating transition matrices provide historical probabilities of credit transition by rating category and are compiled by a number of different organizations, like Moody s Investors Service and Standard & Poor s. 5 A more recent approach proposes to calibrate a structural default model to observed spreads, accounting data, and equity market information, obtaining estimates for both default probabilities and default expected losses. This approach, has the advantage of explicitly taking 6

7 Elton et al. (2001) provide evidence of the size of the following components of credit spreads: the expected default loss, the tax premium, and the risk premium; the paper presents one of the first attempts to measure the tax and risk components. The authors argue that tax premiums exist in the United States because corporate bonds are subject to state and local taxes, whereas government bonds are not. Risk premiums should instead reflect the systematic, and thus undiversifiable, part of the risk on corporate bonds. They estimate that the expected loss component accounts for from 4 to 35 per cent of the spreads, across various rating and maturity classes, and that the tax premium can explain from 28 to 73 per cent of the spreads. In addition, they show that most of the remaining part is related to a systematic risk premium. They do not however identify the sources of such systematic component. Using different data and methods, Driessen (2003) also provides estimates for credit spreads components. In addition to the tax and risk premiums, he also allows for a liquidity premium, which would account for about 20 per cent of spreads. Amato and Remolona (2003) propose an alternative explanation for the credit spread puzzle. In particular, while most studies estimate only the expected loss component of credit risk, they argue that extra spreads may actually compensate for undiversified credit risk. Unlike equity markets, full diversification in corporate bond markets is prevented by the skewness in returns, which makes portfolio diversification much more difficult to achieve. In other words, the size of portfolios required to reduce the probability of large losses is too large to be attained. They give indirect evidence of the fact that bond portfolios are actually not fully diversified, but they do not provide an estimate of the undiversified credit risk component. A second group of studies tries to explain credit premiums by regressing changes in actual spreads on factors that theory indicates as relevant in determining both default and non-default risks (Collin-Dufresne et al., 2001). This approach has the advantage of estimating, directly and separately, the impact of each fundamental factor into account market information, formulating bond-specific and forward-looking estimates of expected default losses. As in the previous case, however, it is not clear how to obtain estimates on the importance of each default factor (see Cooper and Davydenko, 2003, and Crosbie and Bohn, 2002). 7

8 on credit spreads. In addition, the empirical model can be easily extended to test alternative hypotheses The main results of Collin-Dufresne et al. (2001) are that default factors explain only 25 per cent of the observed changes, while structural models of default predict that they should represent the main explanatory variables. Moreover, they show that the missing component is a common risk factor and that it is independent of equity, swap and Treasury markets. They conclude that credit spread changes may be mostly driven by supply and demand shocks, which are specific to the corporate bond market and independent of both default and liquidity factors. Guazzarotti (2004) investigates the determinants of changes in individual credit spreads of non financial European corporate bonds in the period In particular, he investigates the relevance of structural default factors (like leverage, asset volatility, and the level and slope of the risk-less yield curve) and of some non creditrisk factors (e.g., market liquidity and market risk). The main results are that: i) default risk factors account for less than 20 per cent of total variation of credit spreads; ii) liquidity risk and aggregate market risk factors are significant, but they explain only an additional 10 per cent; iii) the remaining part of the credit spread changes remains unexplained. 3. Merton's model The literature divides the models of credit risk into two classes: models that follow the so-called structural approach and models based on the reduced form approach 6. Under the first approach the liabilities of a firm are seen as contingent claims on the firm's assets. In this case default occurs when the value of the assets, which is explicitly modeled, reaches some bound. The reduced form approach, instead, postulates that default occurs randomly, due to some exogenous factor whose intensity is modeled and calibrated using market data. Merton (1974) introduces the first model of credit risk based on the structural approach. The basic assumptions are that the firm is only financed, besides of equity, by 6 Giesecke (2004) provides a nice survey of both approaches. He also introduces the incomplete information approach that, in many instances, can be just seen as an extension of the structural approach. 8

9 debt represented by a zero coupon bond maturing at time T, with face value D. The value of the assets V follows a geometric Brownian motion whose dynamics, under the risk neutral probability measure, is given by dv t = r V dt + σ V dw (1) t t t where r is the risk-free interest rate and σ is the volatility of the process. If the value of the firm at time T is lower than D then there is a default and the property of the firm is transferred from the stockholders to the creditors. In this framework the creditors of the firm can be seen as the owners of a risk-free zero coupon bond that have granted the stockholders a European put option (that is, a right to sell) on the firm's assets, with strike price equal to D and maturity T. The spread between corporate and government (or risk-free) yields is thus the compensation required by creditors for this put option. Under a similar perspective, the equity value is nothing else than the value of a European call (that is, a right to buy) on the firm's assets, with strike price equal to D and maturity T. The value of equity E is thus given by max(0, V T -D) at time T and, using standard option valuation formulas (Black and Sholes, 1973), by at time t<t, where E t t 1 2 = V N(d ) exp( r(t t)) D N(d ) (2) d 1 ln(v t/d) + (r = σ 2 + σ /2) T - t (T t) d 1 2 = d σ T - t and N(x) is the standard normal cumulative distribution function evaluated at x. The value of the debt at time t, which we denote with D t, is thus equal to V t -E t and the corporate yield, y t, to ln(d/d t )/(T-t). The corporate spread is therefore equal to s = y - r t t t = ln(d/d t )/(T - In order to use Merton's model to estimate the corporate spread one would thus need the current value of the assets of the firm, its volatility, the face value of the debt and the level of the risk-free interest rate. Unfortunately the first two variables are not observable. On the other hand, if the firm is listed, one can both observe the value of the equity and estimate its historical volatility, or even observe its implied volatility (if there t) - r. t 9

10 are options written of the firm's shares). Moreover, using Ito's lemma, it is easy to show that the dynamics of E t is given by a geometric Brownian motion whose drift is given by σ V t de t /dv t. The last term of the drift denotes the derivative of the equity with respect to the value of the assets (also called the delta by option traders), and it is equal to N(d 1 ). Under the assumption that the dynamic of the equity can also be described by a geometric Brownian motion with drift σ E E t one has that E t = σ V N(d1)/σ. (3) t E Using both equations (2) and (3) it is possible to recover the two unknown variable σ and V t, if a solution to the system exists. Figures 1.1 and 1.2 in the appendix show the theoretical spread s t generated by Merton's model for three firms with different leverage (D/E t =0.5, D/E t =1.0 and D/E t =2.0) when the volatility of the equity, σ E, changes. The other assumptions are r=0.04 and T=5. As made apparent by the figures, the main driver of the spread in this model is the level of volatility. When the volatility of the equity falls below about 30 per cent, the spread is almost negligible, no matter what the leverage is. Actually, for σ E =0.3 the spread range from 3 to 9 basis points. As the level of the volatility increases, the spread surge exponentially and the relative difference among firms with different leverages sharply decreases. Merton's model, in its simple form just described, provides a nice tool to understand how fundamental variables, such as leverage and asset volatility of a firm, affect the corporate spread. On the other hand there are several features of the real financial world that are not taken into account by the model and that have led to several extensions. The main drawbacks of the model are of three types: 1. There is only one kind of debt securities. In the model only the case of a single zero coupon bond is considered. Actually most of the corporate bonds pay coupons. Moreover a firm usually issues bonds that can have different level of seniority, covenants, times to maturity, currency, and so on; 2. Default only occurs at the maturity of the debt, if the value of assets is below the face value of the debt. In real world creditors can cause the default of a firm at any time if it fails to meet any kind of obligation (such as the 10

11 payment of coupons) and, in several cases, also if the value of a firm falls below a specified level or if the firm takes on specific corporate actions (such as mergers and acquisitions) without the agreement of the creditors. On the other hand, bankruptcy is a process that usually involves several forms of cost, and this fact can push stockholders and creditors to "play" a non-cooperative game, whose result is a default barrier different from the face value of the debt; 3. Volatility is assumed to be constant through maturities. Empirical work on option pricing shows that implied volatility is not constant, neither for different strikes nor for different maturities. This means that volatility implied by equity options, which have maturity usually up to one year, could be a biased estimator of the volatility of equities over longer horizons. 4. Methodology In this section we describe the data, the model and the statistical methodology used in our analysis. 4.1 Data description We started by selecting all credit default swap contracts available through Bloomberg with maturity equal to 5-years (which is usually the most liquid contract), written on senior debt, denominated in euro, issued by non-financial firms. We did not restrict our sample to firms belonging to a particular region or to a given industrial sector. We then selected those contracts for which CDS data are available from at least January 2003 and for which we could recover quarterly balance sheet data (we used the total of current and non-current liabilities minus cash and equivalents). For the same firms we then recovered the market capitalization and the implied volatility derived from listed options (we use the mean of implied volatilities derived from call and put options). After also dropping a few firms for which we found problems in the data 7 we ended with a sample of 59 companies. As a proxy of the risk-free interest rate we have the 5-year euro swap rate. We use monthly average data for our financial indicators 7 We dropped Energia de Portugal and Scania, due to abnormal behavior of the implied volatility indicator, Pechiney, which was delisted in March 2004, and Telecom Italia, whose 11

12 (credit default swap spread, market capitalization, implied volatility ad swap interest rate) from December 2001 to December Quarterly balance sheet data were linearly interpolated. 4.2 Empirical models and testing methodology Merton's model described in Section 3 posits that credit spreads are a function of the value and volatility of the assets, the face value and maturity of the debt and the risk-free interest rate. Previous literature has mainly tested a linear relationship between credit spreads and the above mentioned variables, concluding that the latter are indeed useful in explaining observed spreads, though a large part of total variations remains unexplained. We first try to replicate previous results by estimating the following linear model (Model 1): CDS_SPREAD(i,t) = (4) = CONST + b1 LEV(i,t) + b2 SWAP_RATE(t) + b3 EQUITY_IMP_VOL(i,t) + e(i,t), where i=1,,n denotes a specific firm, t=1,,m a specific time period. LEV is calculated as total of balance sheet liabilities net of cash divided by the sum of balance sheet liabilities net of cash and market capitalization of the firm, SWAP_RATE is the 5- year euro swap rate and EQUITY_IMP_VOL is equal to the average of the implied volatility calculated from call and put options. In order to test for the relevance of non-linear relationships among the variables, we then compare results from equation (4) with those obtained using as regressor the theoretical corporate spread implied by Merton's model, derived in Section 3 (Model 2): CDS_SPREAD(i,t) = CONST + b1 MERTON_SPREAD(i,t) + e(i,t), (5) where MERTON_SPREAD is calculated using the following assumptions: the face value of the debt, whose maturity is assumed to be equal to 5 years, is equal to the total of balance sheet liabilities net of cash, the value of the equity is equal to the market capitalization of the firm, the volatility of the equity is equal to the average of the implied volatility calculated from call and put options, the risk-free interest rate is given by the 5-year euro swap rate. market capitalization indicator has a discontinuity in August 2003 due to the process of reorganization of the group. 12

13 If Merton's model is correct, we would expect that, once we add the theoretical spread to the traditional model, the portion of total variation explained by the model increases and linear terms loose significance. Thus, we also test other two models. Model 3 is a combination of the two basic models (Model 3): CDS_SPREAD(i,t) = CONST + b1 MERTON_SPREAD(i,t) + (6) + b2 LEV(i,t) + b3 SWAP_RATE(t) + b4 EQUITY_IMP_VOL(i,t) + e(i,t). Model 4 is an extended model in which we include other variables, generally used by the literature on the determinants of credit spreads, such as the slope of the yield curve (the difference between the 10-year and the 2-year euro swap rates), an index of the average level of credit spread (the option-adjusted spread for the Merrill Lynch Euro Corporate Index for non financial companies) and an equity index (the Morgan Stanley World Free stock index) CDS_SPREAD(i,t) = CONST + b1 MERTON_SPREAD(i,t) + (7) + b2 LEV(i,t) + b3 SWAP_RATE(t) + b4 EQUITY_IMP_VOL(i,t) + + b5 SLOPE(t) + b6 OAS(t) + b7 MSWFSI (t) + e(i,t). We abstract from factors not linked to credit risk which the literature has documented to be significant in the corporate bonds market (such as tax, liquidity and supply and demand shocks) on the assumption that for the CDS market they are less relevant. Equations (4) to (7) have been estimated by running a pooled OLS regression for all observations in the sample, with heteroscedasticity robust standard errors 8. Since we found evidence of unit roots in most of the series used, we estimated the equations both in levels and in first-order differences. We also run a set of regressions for subgroups of firms belonging to different sectors and having different leverage, in order to examine possible differences in the behavior of the models across sub-samples. 5. Results In this section, we summarize the main findings of preliminary estimates. We first discuss results for the whole sample, and then those for subgroups. After having shown some descriptive statistics of our data set in Tables 1 to 3, we report in Tables 4 13

14 and 5 the estimates on the whole sample of 1,975 observations of the models presented in Section 4, expressed both in levels and first-order differences. In the traditional linear model (Model 1), estimated coefficients are significant and, in many cases, have the expected sign. In particular, leverage and volatility have a positive effect on spreads, both in levels and differences: a 1 percentage point increase in leverage causes an increase in spreads of 2 to 7 basis points; a similar increase in volatility produces an increase of 4 to 8 basis points. The effect of the interest rate, which the theory would predict to be negative (as it proxies for the drift of the process describing the firm's assets value), is instead ambiguous: negative when the model is expressed in levels and positive when in differences. In Model 2, the coefficient of the theoretical spread is positive and highly significant, ranging from 0.36, when variables are in levels, to 0.15, when variables are in differences. The value is however significantly lower than one, the expected value under the hypothesis that Merton's model provides a good representation of the data. In Model 3, where the theoretical spread is added to the traditional linear factors, its coefficient remains positive and significant, though lower in absolute terms. When variables are expressed in changes, the adjusted R-squared of Model 1 increases from 0.30 to 0.39, a value higher than previous findings of studies on corporate bond spread changes 9. Moreover, the coefficients of equity implied volatility and risk-free interest rate loose their significance at conventional levels. When variables are in levels the adjusted R-squared rises from 0.61 to 0.67, and risk-free interest rate is not significant anymore. After adding the theoretical spreads, however, the linear contribution of leverage does not change in magnitude and remains highly significant, probably due to the fact that Merton's model is not very sensitive to changes in leverage (see Figures 1.1 and 1.2). Model 4 shows that other explanatory variables that the literature has indicated as important determinants of credit spreads (such as, the slope of the curve, the 8 The test by Hausman (1987) implies that the model with random effects cannot be rejected, however a panel data analysis with random effects did not result in appreciable differences. 9 Collin-Dufresne et al. (2001) and Guazzarotti (2004), which use a Mean Group Estimator (running separate time-series regressions for each firm), explain from about 20 to 30 per cent of total variation in corporate bonds' spread changes. 14

15 corporate bond spread and the stock market return) do not seem to improve to the explanatory power of the model, once the theoretical spread is included. In particular, the coefficient of the equity index is positive and not significant, while we would have expected that positive equity returns, by signaling better overall economic conditions and higher recovery rates, had a negative effect on spreads. Estimates for the slope of the curve and the corporate bond spread are not stable across the two specifications, as the signs are negative when the model is in levels and positive when in differences. All in all, results suggest that the theoretical spread does convey specific information on credit risk that cannot be captured by the linear model, supporting the hypothesis that non-linearities implied by Merton's model are important in determining credit spreads. Results of models in levels appear somewhat more problematic. In particular, the sign of the swap rate is unstable, being negative in model 1 and 3, positive in Model 4, while the negative coefficient on bond spreads is highly counter intuitive. Models in differences might be preferred also on the ground that they may isolate unobserved individual factors which do not vary through time and, at the same time, they may account for the possible presence of unit-roots in the processes for CDS spreads 10. Tables 6 to 8 report estimates for models 1 to 3, expressed in first-order differences, by subgroups defined according to sector and leverage. In order to analyze separately the effects of each variable we first neglect nonlinear effects and focus on the traditional linear model (Model 1). As one would expect, the effect of leverage on credit spreads is strongest for the group with the highest leverage (a coefficient of against for the whole sample; Table 6). As this group should be associated with higher credit risk, spreads should in fact be more sensible to changes in leverage. Moreover, the effect of leverage is highest for communications firms (0.134), which could be due to the fact that in the period under analysis such firms tended to be characterized by higher average spreads and equity volatility (Table 2). Coherently, spreads of communications firms are also more sensible than those of other firms to equity volatility (0.044 against 0.035). These results are 10 We actually found that the augmented Dickey-Fuller test, run separately for each CDS time series, does not reject the null hypothesis of the presence of a unit-root in almost all cases. 15

16 generally confirmed by estimates of Model 3, which extend the linear model including theoretical spreads (Table 8), and are coherent with those obtained in previous works on the basis of different models 11. Comparing the performance of different models across sub-samples, the most important result is that Model 2 explains a higher portion of total spread variation than Model 1, the traditional linear model, in almost all subgroups (Table 7). The only exceptions are the firms in the consumer/industry sector (where both models explain 28 per cent) and in the high-leverage sub-sample (where Model 2 explains only 15 per cent, against 21 one of Model 1). Moreover, which extends the linear model by including also the theoretical spread, Model 3, appears to be superior, or at least not worse, than both other models in all subgroups. This improvement is greater for firms belonging to the communication sector, which are characterized by both higher spreads and volatilities (Tables 2 and 3). Focussing on the performance of Model 2 across subgroups, estimates show that theoretical spread changes tend to be most effective in explaining variations of spreads in the communications sector (an adjusted R-squared of 0.46, against 0.34 for the whole sample; Table 7) and least effective in both the energy/utilities sectors and in the highleverage group (an adjusted R-squared of 0.25 and 0.15, respectively). These results can be explained by considering the mechanics and the drawbacks of the Merton indicator, as they have been outlined in Section The poor performance of Model 2 for highleverage firms may be reconducted to the fact that, in our sample, equity volatility is probably much lower than long-term volatility. As a consequence, given that theoretical spreads are less sensible to changes in variables when volatility is low (Figures 1.1 and 1.2), Merton's model assigns extremely low and stable values of asset volatility to highleverage firms, which in turn tend to generate abnormally low and stable theoretical spreads. This is also reflected in the low value of the estimated coefficient. At the same time, the model explain a relatively low proportion of total variation for the energy and 11 See, for example, Collin-Dufresne et al. (2001) and Guazzarotti (2004). 12 First, the theoretical spread derived through the Merton's model tend to be more sensible to both leverage and volatility when the level of volatility is high. Second, short-term equity volatility may be a poorer proxy for long-term volatility in some sectors than in others, depending, for example, on the actual average level of long-term asset volatility. Third, short- 16

17 utilities sectors, as these were characterized by periods in which the short-term equity volatility was probably too much volatile with respect to long-term volatility. On the other hand, as theoretical spread derived through Merton's model tends to be much more sensible to variables when volatility is high and volatile, the model performs better for communications firms, which in the period under analysis have been characterized by high and volatile equity volatility (Table 3). Previous evidence would suggest that short-term equity implied volatility can, in some cases, be a biased estimator of long-term expected volatility implicit in CDS spreads. 6. Conclusions In this paper we analyze the determinants of CDS spreads for a sample of 59 non-financial firms, from different countries and sectors, for the period between December 2001 and December We use the seminal model by Merton (1974) to examine the relevance of non-linear relationships between firms' characteristic factors and CDS spreads. We find that the inclusion of a non linear term in the classical linear model significantly improve the capacity of the changes in the fundamental variables to explain the changes in CDS spreads. The extended model is able to explain about 40 per cent of the variations in CDS spreads, which is higher than previous findings of studies on corporate bond spread changes. Given that Merton's model is very sensible to volatility changes, the improvement is greater in those cases where short-term equity volatility is a better proxy for long-term volatility, as it is probably the case for the communication sector, which is characterized by both higher spreads and volatilities. Moreover, when the theoretical spread calculated using Merton's model is introduced, the equity implied volatility and the risk-free interest rate linear components of CDS spreads decrease their statistical significance, while the linear component of leverage changes maintain their usefulness in explaining CDS spreads changes, probably due to the fact that Merton's model is not very sensitive to variations in leverage. All in all, results suggest that the theoretical spread does convey specific information on credit risk that cannot be captured by the linear model, supporting the term volatility tends to be more volatile than long-term volatility, as confirmed by evidence on swaptions. 17

18 hypothesis that non-linearities implied by Merton's model are important in determining credit spreads. 18

19 Tables and figures 2,000 Figure 1.1: Theoretical Spreads Generated by Merton's Model (1) (basis points) 2,000 1,800 1,600 1,400 1,200 Debt/Equity=0.5 Debt/Equity=1.0 Debt/Equity=2.0 1,800 1,600 1,400 1,200 1,000 1, Figure 1.2: Theoretical Spreads Generated by Merton's Model (1) (basis points) Debt/Equity=0.5 Debt/Equity=1.0 Debt/Equity= (1) Difference between the yield on 5-year zero coupon bonds issued by firms with different leverages and the risk-free interest rate (assumed to be equal to 4.0 per cent), as a function of equity volatility. The risky yield is calculated using Merton's model. 19

20 Table 1.1: Firm by Firm Descriptive Statistics (1) Mean Values Firm Number Nationality Sector CDS Spreads (2) Theoretical spreads (2) Leverage (3) Volatility (4) Market Capitalization (5) R2 (Model1) R2 (Model 2) Germany Spain Spain Netherlands Germany Netherlands France United States France Germany Britain Germany Germany Sweden Germany Italy Britain Spain Germany Spain Basic Materials Utilities Communications Consumer Non-cyclical Industrial Basic Materials Communications Consumer Non-cyclical Communications Utilities Industrial Consumer Cyclical Communications Communications Utilities Utilities Communications Utilities Consumer Cyclical Utilities , ,468 14,564 54,875 12,623 4,951 9,299 12,948 95,215 19,892 18,522 1,488 4,792 57, ,255 34,434 34,954 17,392 4,560 3,250 13, (1) Firms are ordered by R2 (Model 1). - (2) Basis points. - (3) Percentages. - (4) Percentage points. - (5) Millions of euros. 20

21 Table 1.2: Firm by Firm Descriptive Statistics (1) Mean Values Firm Number Nationality Sector CDS Spreads (2) Theoretical spreads (2) Leverage (3) Volatility (4) Market Capitalization (5) R2 (Model1) R2 (Model 2) Britain Germany Germany Netherlands Sweden Spain Germany Germany Switzerland Germany United States Germany Germany Denmark Finland Britain Switzerland Italy Sweden Finland Utilities Consumer Non-cyclical Basic Materials Communications Basic Materials Energy Consumer Cyclical Basic Materials Technology Consumer Cyclical Energy Industrial Consumer Cyclical Communications Basic Materials Consumer Cyclical Industrial Industrial Consumer Cyclical Basic Materials ,928 10,315 23,845 14,583 66,746 18,071 37,232 6,864 17,986 23,161 29,982 49,517 14,634 43,459 8,232 2,144 10,861 5,252 87,782 9, (1) Firms are ordered by R2 (Model 1). - (2) Basis points. - (3) Percentages. - (4) Percentage points. - (5) Millions of euros. 21

22 Table 1.3: Firm by Firm Descriptive Statistics (1) Mean Values Firm Number Nationality Sector CDS Spreads (2) Theoretical spreads (2) Leverage (3) Volatility (4) Market Capitalization (5) R2 (Model1) R2 (Model 2) Netherlands Netherlands Finland Netherlands France Luxembourg Netherlands Sweden Norway Italy Sweden Portugal Germany Sweden Norway Britain Netherlands Britain Switzerland Communications Industrial Communications Basic Materials Energy Basic Materials Consumer Non-cyclical Consumer Non-cyclical Communications Consumer Cyclical Communications Communications Consumer Non-cyclical Consumer Cyclical Energy Consumer Non-cyclical Energy Energy Consumer Non-cyclical ,571 28,450 73,249 3,998 99,624 7,018 31,999 38,254 72,078 5, ,821 9,007 8,614 51, ,763 71,824 92, , , (1) Firms are ordered by R2 (Model 1). - (2) Basis points. - (3) Percentages. - (4) Percentage points. - (5) Millions of euros. 22

23 Table 2: Descriptive Statistics of the Firms: Breakdown by Sector and Leverage (1) Mean Values No. Obs. CDS Spreads (2) Theoretical spreads (2) Leverage (3) Volatility (4) Market Capitalization (5) Whole sample 1, ,229 Sector Consumer/Industry ,597 Communications ,689 Basic Materials ,862 Energy/Utilities ,646 Leverage (3) , , ,164 Over ,079 (1) Classes for leverage have been determined as quartiles. (2) Basis points. - (3) Percentages. - (4) Percentage points. - (5) Millions of euros.

24 Table 3: Descriptive Statistics of Implied Volatility: Breakdown by Sector and Leverage (1) Mean (2) Standard Deviation Min. (2) Max. (2) Whole sample Sector Consumer/Industry Communications Basic Materials Energy/Utilities Leverage (3) Over (1) Classes for leverage have been determined as quartiles. - (2) Percentage points. - (3) Percentages. 24

25 Table 4: The Determinants of Credit Default Swap Spreads The values in the table are obtained by running a pooled OLS regression for all observations in the sample, with heteroscedasticity robust standard errors. The dependent variable is the CDS spread of firm i as of month t, in percentage units. The explanatory variables are: the estimated theoretical spread for firm i, at time t, in percentage units; the leverage ratio, in percentage units, of firm i at time t; the 5-year swap rate at time t; the slope of the Euro swap curve at time t (10-year minus 2-year swap rate); the volatility implied in the option prices written on firm i stocks at time t; the Morgan Stanley World Free stock index at time t; the option-adjusted spread for the Merrill Lynch Euro Corporate Index for non financial bonds at time t (BBB minus AA rated bonds). No. Obs. is the number of observations used in the regression; monthly data from December 2001 to December Significance levels: *** = 1 per cent; ** = 5 per cent; * = 10 per cent. Model 1 Model 2 Model 3 Model 4 Coeff. St. dev. Coeff. St. dev. Coeff. St. dev. Coeff. St. dev. Theoretical spread 0.360*** *** *** Leverage 0.021*** *** *** year swap rate *** *** Volatility 0.075*** *** *** Slope of the swap curve *** Option-adjusted spreads *** MS World Stock Index Intercept *** *** *** *** Adjusted R No. Obs. 1,975 1,975 1,975 1,975

26 Table 5: The Determinants of Changes in Credit Default Swap Spreads The values in the table are obtained by running a pooled OLS regression for all observations in the sample with heteroscedasticity robust standard errors. The dependent variable is the monthly change at time t of the CDS spreads of firm i, in percentage units. The explanatory variables are: the monthly change at time t of the estimated theoretical spread for firm i, in percentage units; the monthly change at time t of the leverage ratio, in percentage units, of firm i; the monthly change at time t of the 5-year swap rate; the monthly change at time t of the slope of the Euro swap curve (10 year minus 2 year swap rate); the monthly change at time t of the volatility implied in the option prices of firm i; the monthly price index at time t of the Morgan Stanley World Free stock index; the monthly change at time t in the option-adjusted spread of the Merrill Lynch Euro Corporate Index for non financial bonds (BBB minus AA rated bonds). No. Obs. is the number of observations used in the regression; monthly data from December 2001 to December Significance levels: *** = 1 per cent; ** = 5 per cent; * = 10 per cent. Model 1 Model 2 Model 3 Model 4 Coeff. St. dev. Coeff. St. dev. Coeff. St. dev. Coeff. St. dev. Theoretical spread 0.145*** *** *** Leverage 0.075*** *** *** year swap rate 0.217*** * Volatility 0.035*** Slope of the swap curve Option-adjusted spreads 0.334*** MS World Stock Index Intercept * Adjusted R No. Obs. 1,916 1,916 1,916 1,916 26

27 Table 6: The Determinants of Changes in CDS Spreads by Sector and Leverage: Model 1 The values in the table are obtained by running a pooled OLS regression for all observations in the selected subgroup, with heteroscedasticity robust standard errors. The dependent variable is the CDS spread of firm i as of month t, in percentage units. The explanatory variables are: the leverage ratio, in percentage units, of firm i at time t; the 5-year swap rate at time t; the volatility implied in the option prices written on firm i stocks at time t; a constant term. No. Obs. is the number of observations used in the regression; monthly data from December 2001 to December Significance levels: *** = 1 per cent; ** = 5 per cent; * = 10 per cent. Leverage Swap rate Volatility Constant Adj. R2 No. Obs. Whole sample 0.075*** 0.217*** 0.035*** ,916 Sector Consumer/Industry 0.088*** 0.314** 0.028*** Communications 0.134*** *** Basic Materials 0.017*** *** * 0.26 Energy/Utilities *** Leverage (percentages) *** 0.228** 0.035** * 0.297** 0.046** *** 0.252** 0.034*** Over *** ***

28 Table 7: The Determinants of Changes in CDS Spreads by Sector and Leverage: Model 2 The values in the table are obtained by running a pooled OLS regression for all observations in the selected subgroup with heteroscedasticity robust standard errors. The dependent variable is the CDS spread of firm i as of month t, in percentage units. The explanatory variables are: the estimated theoretical spread for firm i, at time t, in percentage units, and a constant term. The number of observations is the same as in Table 6; monthly data from December 2001 to December Significance levels: *** = 1 per cent; ** = 5 per cent; * = 10 per cent. Theoretical Spreads Constant Adj. R2 Whole sample 0.145*** * 0.34 Sector Consumer/Industry 0.113*** Communications 0.194*** Basic Materials 0.205*** Energy/Utilities 0.601*** Leverage (percentages) *** *** *** Over *** *

29 Table 8: The Determinants of Changes in CDS Spreads by Sector and Leverage: Model 3 The values in the table are obtained by running a pooled OLS regression for all observations in the selected subgroup, with heteroscedasticity robust standard errors. The dependent variable is the CDS spread of firm i as of month t, in percentage units. The explanatory variables are: the estimated theoretical spread for firm i, at time t, in percentage units; the leverage ratio of firm i, at time t, in percentage units; the 5-year swap rate at time t; the volatility implied in the option prices written on firm i stocks at time t; a constant term. The number of observations is the same as in Table 6; monthly data from December 2001 to December Significance levels: *** = 1 per cent; ** = 5 per cent; * = 10 per cent. Theoretical spreads Leverage Swap rate Volatility Constant Adj. R2 Whole sample 0.115*** 0.071*** 0.103* Sector Consumer/Industry 0.082*** 0.087*** 0.203* Communications 0.183*** 0.115*** Basic Materials 0.146*** 0.015*** * Energy/Utilities 0.505*** Leverage (percentages) *** 0.070*** *** ** 0.067*** Over * 0.095***