Forward CDS and the Time Decomposition of Credit Spreads

Transcription

1 Forward CDS and the Time Decomposition of Credit Spreads Santiago Forte* November, 2017 Abstract In this paper, I adopt a model-free approach to derive simple closed-form expressions for the pricing of forward CDS and the time decomposition of CDS spreads. These expressions can be practically implemented using a reduced-form model, structural model, or mixed method. I propose one particular form of mixed method that combines the structural model framework and the information provided by the term structure of CDS spreads. The robustness of this estimation method to extreme scenarios significant model misspecification, high CDS levels, and long maturities is illustrated by means of simulation experiments and a case study: the Eurozone crisis. JEL classification: G12, G13. Keywords: Credit default swaps; spot and forward contracts; time decomposition. * ESADE Business School, Ramon Llull University, Av. Torreblanca 59, Sant Cugat del Vallès, Barcelona, Spain; Tel.: ; santiago.forte@esade.edu. I am grateful for the financial support of Banco Sabadell.

2 1. Introduction Hull and White (2003) define a forward credit default swap (CDS) contract as the obligation to buy or sell a CDS on a specified reference entity for a specified spread at a specified future time. The obligation ceases to exist if the reference entity defaults during the life of the forward contract. The spread is set so that the contract has zero value. Hull and White (2003) describe the pricing of forward CDS in the context of reduced-form models. Since the time this paper was published, the academic literature has given very little attention to this type of contract, and the few papers that address this issue have consistently relied on the reduced-form model approach. 1 In this paper, I adopt a model-free setting to derive a simple closed-form expression for the pricing of forward CDS. I show that, analyzed in this way, forward CDS spreads lead to a straightforward time decomposition of spot CDS spreads. The practical implementation of these results can be made using a reduced-form model, a structural model, or mixed method. I consider one specific form of mixed method that combines the structural model framework and the information provided by the term structure of CDS spreads (TSCS henceforth). The following example will help explaining the motivations and contributions of this paper. Figure 1 reproduces one particular tranche of the TSCS 1 to 5 years for Greece, April 15, 2010; Italy, October 4, 2011; and Portugal, July 9, The data have been intentionally chosen for two reasons. First, the 5-year CDS spread is roughly the same in the three cases: 420 bp ( for Greece, for Italy, and for Portugal). Second, outside this coincidence, their credit risk profile is totally different: 1 For example, Leccadito et al. (2015). 2

3 decreasing TSCS in the case of Greece, hump-shaped for Italy, and increasing for Portugal. <Figure 1 about here> The first question that I address in this paper pertains to the pricing of forward CDS in a model-free setting and its practical implementation using a mixed (reducedform-structural) approach. Results for the five consecutive 1-year time periods of our example are provided in Figure 2. They show a remarkable variation not only between time periods (same entity) but also between entities (same time period). Hence, while the CDS spread for a contract providing credit protection for a total period of 5 years is basically the same in the three cases, there is a significant variation in the spread of a forward CDS contract providing credit protection for one specific year. <Figure 2 about here> The second question that I address in this paper has a direct relationship with the previous question and can be formulated in the following way: what percentage of the total cost of credit protection for a reference bond up to a given maturity (premium leg of the contract) can be associated to the protection of one specific time interval? I show that the pricing of forward CDS in a model-free setting offers a straightforward answer to this question. I also show that the resulting time decomposition of the total cost of credit protection up to a given maturity leads to an equivalent time decomposition of the CDS spread for that maturity. Taking into account the aforementioned illustrative example, we can decompose the 5-year CDS spread for the three reference entities as reflected in Figure 3. Evidently, the contribution of each year to the 420 bp differs significantly among the three cases. For instance, while 31% of the total cost of credit protection ( bp) can be attributed to the cost of protecting the first year in the case of Greece, 3

4 this number falls to 16% (69.21 bp) in the case of Portugal. Besides, only 12% of the total cost (51.43 bp) can be attributed to the fifth year in the case of Greece, compared to a 20% (85.08 bp) in the case of Portugal. The overall conclusion is that the time decomposition of the spread of a CDS contract, which emerges from incorporating the information content of the TSCS, provides significant additional information on the credit risk profile of the reference entity during the life of the contract. <Figure 3 about here> The rest of the paper is organized as follows. Section 2 describes the pricing of spot and forward CDS contracts in a model-free setting. Section 3 investigates the time decomposition of CDS spreads. Section 4 presents the estimation approach and, by simulation experiments, explores its robustness to extreme scenarios: significant model misspecification, high CDS levels, and long maturities. Section 5 analyzes the Eurozone crisis as a case study. The particularities of this sovereign debt crisis allow for a further robustness test on the estimation approach, in this case, based on real data. Section 6 summarizes the main conclusions and concludes the paper. 2. Pricing CDS and Forward CDS I assume continuous payments and a constant risk-free interest rate. I also take as given that conditions for the risk-neutral pricing of CDS and forward CDS apply. This pricing will rely on the following two main building blocks: : Price (at current time 0) of a contingent claim paying $1 at time 0, conditional on no default prior to. : Price (at current time 0) of a dollar-in-default claim, that is, a claim paying $1 at default if this happens before. 4

5 2.1. CDS Pricing a CDS contract involves determining the CDS spread that equates the premium leg ( ) and the protection leg ( ) of the contract. If we denote the CDS spread, the risk-free interest rate, the maturity of the contract, the nominal of the protected bond, and the recovery rate, then 2 1, (1) 1. (2) Expression (1) reflects that the premium leg can be initially valued as a perpetuity of premium payments (value 1 within brackets), and then account for the cancellation of this perpetuity due to either the maturity of the contract ( ) or default ( ). On the other hand, expression (2) reflects the delivery of the bond in exchange of the bond s face value in case of default. If we finally equate the premium leg and the protection leg, then 1 1. (3) It is worth noting that the same result would emerge if the par-yield spread of a bond with maturity, nominal, and recovery rate were considered. The value of this bond will be 3, (4) 2 See Ericsson et al. (2015). 3 See Leland and Toft (1996). 5

6 where denotes the constant coupon flow. In fact, it is straightforward to show that the bond will be at par value ( 0 if and only if its yield spread ( / fits the aforementioned CDS spread Forward CDS Consider now a forward CDS contract signed at current time 0 for credit protection between and, with 0. The contract will have no effect in case the company defaults at some point prior to. If we use to denote the spread of this contract, then the present value of the premium leg and the protection leg will be as follows:,, (5), 1. (6) The term in expression (5) reflects that the initiation of premium payments at requires the company to survive until that period. The term reflects the discount to the present value of premium payments associated to the finite maturity of the contract. The last term in expression (5) stands for the discount associated with disrupted payments due to default at some point between and. Expression (6) reflects the delivery of the bond in exchange of the bond s face value in case of default, provided this happens between and. It must be noted that 0, and 0,, that is, the value of the premium leg and the protection leg converge to those of a spot contract as tends to zero. If we now equate both legs of the contract, then 6

7 , 1. (7) Expression (7) offers a closed-form solution for the spread of a forward CDS contract under the assumption of continuous payments and constant interest rates. In line with our previous arguments, 0,. It is worth noting that the theoretical equivalence between CDS spreads and paryield spreads seen in the case of spot contracts, also holds in the case of forward contracts. Consider a forward contract to buy at a bond with maturity at. The contract specifies a strike price equal to the face value, and it will cease to exist in case of default at any time prior to. The present value of a long position in this contract will be,. (8) By equating the strike price to the bond s face value, I assume that it is the coupon that needs to be determined for the present value of the contract to be zero. In fact, it can be easily shown that the resulting yield spread will reproduce the aforementioned forward CDS spread Time Decomposition of CDS Spreads The pricing of forward CDS contracts in a model-free setting allows for the time decomposition of CDS spreads. This decomposition follows from the possibility of representing a long (short) position in a CDS contract as a portfolio of long (short) 4 It can be also verified that 0,, that is, the value of a long position in a forward bond contract converges to the difference between the value of a regular bond and its face value as tends to zero. 7

8 positions in consecutive forward CDS contracts. Particularly, if we define 0 and, then,, (9) that is, the present value of the cost of credit protection up to time should be equal to the present value of the cost of credit protection for an arbitrary number of consecutive (not necessarily identical) time intervals between time 0 and time. As a result,, ;,, (10) where, ; 1 0,1 ; (11) with, ; 1. (12) Expression, ; needs to be positive because the present value of a premium leg (either of a spot or a forward contract) will always be positive. On the other hand, it is easy to verify that the values add up to 1, and therefore,, ; cannot be higher than 1 at an individual level. From expressions (10 12), it is concluded that we can divide the maturity of a CDS contract into an arbitrary number of intermediate time intervals, and express the associated CDS spread as a weighted average of the spreads of the forward CDS contracts corresponding to these time intervals. As a corollary, it is possible 8

9 to estimate the relative contribution of any of these time intervals to that CDS spread; that is,, ;, ;, 0,1, (13) with, ; 1. (14) Expression (13) indicates that the contribution of one particular time interval to the spread of a CDS contract is given by the ratio between the following: the present value of $1 paid at default if this happens during that particular time interval, and the present value of $1 paid at default if this happens at any time during the life of the contract. The proximity of these two values would imply that the risk of default is concentrated in that specific time interval, thereby resulting in a significant contribution to the spread of the CDS contract. Evidently, the opposite result will be achieved if there is a significant difference between the two aforementioned values. One simple form of the time decomposition of CDS spreads will be the following. Let us first define, 0, ; ; then,, 1,,, (15) with 0, and, 1 1 0,1. (16) We can finally define, 0, ;, or 9

10 ,, 0,1. (17) Expression (17) provides a simple representation of the percentage of the CDS spread that can be attributed to the first years of the contract, with 0, 0 and, 1. It is also worth noting that expression (15) allows for the representation of forward CDS spreads as a function of CDS spreads:,, 1, (18) 4. Estimation Approach: Description and First Stress Testing Previous sections provide closed-form expressions for the pricing of forward CDS and the time decomposition of CDS spreads based on estimates of and. The derivation of these values from the TSCS is not straightforward, though. The main difficulty lies in differentiating and from a single CDS spread using expression (3). Such indetermination problem, however, can be solved by imposing a sensible structure on the relationship between and. While invoking a reduced-form model represents a clear option, in the following lines I describe an alternative method that relies on the appealing theoretical relationship between and imposed by structural models. Assume, in line with structural credit risk models, that default is triggered by a state variable,, reaching a specific critical point,, naturally referred to as the default barrier. Assume also that evolves according to the continuous diffusion process 10

11 , (19) where and represent the expected growth rate and the volatility of, respectively, and is a standard Brownian motion. Under these conditions, values for and can be derived (among others) from Leland and Toft (1996): 1, (20) where is the risk-neutral default probability from time 0 up to time, ; and, (21) with ; ; ; ; 2 ; ; 2. The conventional approach for estimating and based on the aforementioned structural model would entail the determination of latent variables and parameters in (20) and (21), namely,,, and. Such estimates of and could be finally used to derive a theoretical CDS spread using expression (3). However, our 11

12 objective is not to price a CDS contract, but to solve the indetermination problem in the estimation of and by taking the CDS spread as given. In this case, since the absolute values of and are found to be irrelevant, and only their ratio is considered in expressions (20) and (21), the state variable can be normalized to 1. Subsequently, a specific default barrier within the interval 0,1 can be selected. The values of and can be finally obtained by calibrating the volatility to fit the observed CDS level. The main question of course is how to select a specific value for the default barrier in the interval 0,1, and to what extent this choice will affect the results. As a simulation experiment, Table 1 provides estimates of and for different default barrier specifications (from 0.1 to 0.9), different maturities (from 0.5 to 30 years), and different CDS spread levels (from 10 to 1,000 bp). The values considered for the risk-free rate and the recovery rate are 2.5% and 40%, respectively. 5 The table includes the corresponding values for as well as the result of adding up and. As indicated in expressions (16) and (18) and considering the observed CDS spreads, the total value of is relevant because it holds significance for the pricing of forward CDS. It is also clear from expression (4) that estimates of and from a given CDS spread will allow for the pricing of a bond,, with the same maturity as the CDS contract (and vice versa). While it is not the focus of the present study, the table also reports the associated values for a bond with a nominal of $100 and 5% annual coupon. As a supplementary information to Table 1 and for the convenience of the reader, Figure 4 provides a graphical representation of the results for a subsample of the considered maturities (1, 5, 10, and 30 years). 5 Alternative values for the risk-free interest rate lead to different numerical results but the same conclusions. Results for the specific values of 0.01% and 5.00% are omitted for the sake of brevity, but available upon request. 12

13 <Table 1 about here> <Figure 4 about here> The main conclusion from Table 1 and Figure 4 is that, once the structural approach is adopted, estimates of and for a given CDS spread and maturity are quite robust to the exact definition of,, and. The robustness of the results is more evident for and for. It is worth emphasizing that the reported numbers have been derived after allowing for extreme cases of model misspecification. Let us consider a spread of 250 bp for a 5-year CDS contract. The assumption that the default barrier could be either at 10% or 90% of the current value of the state variable is equivalent to the assumption that its volatility could be either 59.6% or 6.0%. Even so, estimated values for,,, and remain roughly the same. Overall, the precise definition of,, and has a negligible effect on the results for CDS spreads of up to 250 bp and maturities of up to 5 years. Let us now consider an extreme scenario not only in terms of model misspecification but also in terms of a high CDS spread for a contract with a long-term maturity: a spread of 1,000 bp for a 10-year CDS contract. In this case, would range from $0.123 for a default barrier of 0.1 to $0.242 for a default barrier of 0.9; the corresponding numbers for would be $0.763 and $0.659, respectively. It is clear that the potential error would be higher in this case, and that this would translate into a higher potential error also in and. However, these differences would come with a volatility ranging from a minimum of 12.5% to a maximum of 90.4%. While the exact definition of latent parameters is a well-known problem in structural credit risk models, sensible values usually move into much smaller intervals. 13

14 In the next section, I analyze the Eurozone crisis as a case study. It will reflect the extent to which the estimation of forward CDS spreads and the time decomposition of CDS spreads can be affected by a potential model misspecification. It is observed that, even in very extreme scenarios, potential errors are minimal. 5. Case Study: The Eurozone Crisis The Eurozone crisis seems a particularly appropriate research field for the purposes of the present study. The first and main reason is the extreme CDS levels reached by some of the Eurozone countries during the crisis. Using these data I will be able to approximate the maximum effect that a potential model misspecification could have on the estimation of forward CDS spreads and the time decomposition of CDS spreads. As we have seen in the previous section, the misspecification problem will tend to increase with the CDS level. At the same time I expect the liquidity of a CDS contract to be higher for France or Ireland than for an average corporation (provided the same CDS level). The use of sovereign data will finally allow to illustrate that, contrary to the traditional implementation of structural credit risk models, the application of the mixed estimation method described in the previous section is not restricted to non-financial corporations Data For this empirical analysis, I choose to focus on the following six Eurozone countries: France, Spain, Italy, Ireland, Portugal and Greece. The last five countries (the so-called PIIGS) have been selected because they were in the core of the crisis. France is included in the list for comparison, and it can be considered as a control country. For these entities, and for the inclusive period January 2008 December 2016, I collect from Datastream weekly data on CDS spreads with the following maturities (in years): 0.5, 1, 14

15 2, 3, 4, 5, 7, 10, 20, and 30. As a proxy for the risk-free rate I collect, also from Datastream, data on the German 10-year government bond yield. 6 CDS data is not always available since January This is the case of Ireland (start date of available data: October 14, 2008) and Portugal (start date of available data: November 4, 2008). It can also happen that the availability of data does not reach December This is the case of Greece (data finishing on February 28, 2012). Instead of considering a common time interval, I use all the information available during the chosen sample period. Particularly, in the case of Greece I limit the sample period to September 6, This was the first date in the sample with a 0.5-year CDS spread above 5,000 bp. 7 Table 2 provides main descriptive statistics for the CDS spreads in the sample, while Figure 5 represents the 5-year spreads. The mean 5-year CDS spread ranges from a minimum of bp for France to a maximum of bp for Greece. Differences between minimum and maximum values in Table 2 are a clear reflection of the effect of the crisis in each country. <Table 2 about here> <Figure 5 about here> 6 Zero yields and negative yields are sometimes observed in the second-half of 2016 (nineteen cases in total). To avoid problems associated with zero or negative values for the risk-free rate, I impose a minimum value of 0.01%. 7 The initial proposal of a bond exchange with a nominal discount of 50% on notional Greek debt was made in the Euro Summit held on October 26, 2011, and was formally announced on February 21, 2012 (see Zettelmeyer et al. (2013) for details). On February 28, 2012 (the last day with available CDS spreads), the International Swaps and Derivatives Association (ISDA) accepted a question related to a potential Hellenic Republic credit event. The occurrence of a credit event was initially denied by the ISDA on March 1, 2012, but was finally accepted on March 9, 2012, after a second question was formulated. The election of an upper limit of 5,000 bp for the 0.5-year CDS spread, and therefore September 6, 2011, as the last considered date in the case of Greece may seem arbitrary. The intention, however, is to avoid dealing with potential misleading values. CDS spreads around these dates reflected the situation of an actual default. 15

16 5.2. Estimation Approach: Second Stress Testing Table 3 contains the results on estimated forward CDS spreads between adjacent dates with available CDS spreads. For the sake of brevity, I consider only three possible default barrier specifications, including the more extreme values: 0.1, 0.5, and 0.9. The main conclusion we can derive from the table is that forward CDS spreads are particularly robust to the exact definition of the default barrier. Maximum differences in the mean values after assuming a default barrier of either 0.1 or 0.9 are 1.01 bp for France, 7,10 ; 1.54 bp for Spain, 4,5 ; 1.88 bp for Italy, 7,10 ; 5.35 bp for Ireland, 20,30 ; bp for Portugal, 20,30 ; and bp for Greece, 10,20. <Table 3 about here> Results on the time decomposition of CDS spreads are summarized in Table 4. The cases considered are: 1) time decomposition of the 5-year CDS spreads into equally spaced time intervals of one year, 2) time decomposition of the 10-year CDS spreads into equally spaced time intervals of five years, and 3) time decomposition of the 30-year CDS spreads into equally spaced time intervals of ten years. We observe that maximum differences in the mean values by country after assuming a default barrier of either 0.1 or 0.9 are always in the order of 2%. Hence, the robustness of the results to the exact definition of the default barrier also applies in this case. Another interesting result relates to the slope of the time decomposition of CDS spreads (TDCS, hereafter). If we first focus on the time decomposition of the 5-year spreads, we notice that, on average, France had an increasing TDCS along the sample period. The precise numbers for the first, second, third, fourth, and fifth year are 8%, 13%, 19%, 28%, and 31%, respectively. This means that, on average, investors perceived 16

17 a higher risk at the end of the holding period than at the beginning. A positive slope is also observed in the case of Spain, Italy, and Ireland, but the slope appears less pronounced as the average 5-year CDS level increases. The slope becomes almost flat in the case of Portugal, and it finally turns negative in the case of Greece. The time decomposition of the 10-year CDS spreads shows a similar pattern. In the case of France, about one-third of the total cost of credit protection corresponds to the first five years, while the other two-thirds correspond to the last five years. We observe again that the slope of the TDCS becomes less pronounced as the overall credit risk level increases. This slope is close to zero (flat TDCS) in the case of Spain, Italy, and Ireland, turning negative for Portugal and Greece. In the particular case of Greece, the numbers essentially mimic those of France but in inverse order. Finally, if we analyze the time decomposition of the 30-year CDS spreads we observe a downward sloping TDCS in all the cases. This result can be explained by the discount effect that is incorporated into the time decomposition of CDS spreads. To put it simply, in the theoretical case of an identical probability of default in each of the considered time intervals, the TDCS will be downward sloping. This is because the present value of a dollar in default-at-the-last-time-interval claim will be lower in this case than a dollar in default-at-the-first-time-interval claim. While discounting will affect the time decomposition of any CDS spread, its severity will increase with the maturity of the contract. It is worth noting, however, that the higher the overall credit risk level, the more pronounced is the negative slope of the time decomposition of the 30-year CDS spreads. This result is fully consistent with the previous findings. 17

18 5.3. The ECB Intervention be enough. The ECB is ready to do whatever it takes to preserve the Euro, and believe me, it will Mario Dragui, President of the ECB. July 26, We have seen in the previous section that the estimation of forward CDS spreads and the time decomposition of credit spreads is generally robust to the exact definition of the default barrier. In this section, I will briefly analyze the information provided by these two instruments concerning the evolution of the Eurozone crisis, particularly the market s reaction to Mario Dragui s speech on July 26, 2012, in support of the Euro and the decisions taken by the ECB in the subsequent weeks. 8 In what follows, I will focus on the results obtained after assuming a default barrier of 0.5. For the sake of brevity, and given their higher prevalence in the market, I will also focus on the 5-year maturity contracts for the analysis. For the six selected countries, Figure 6 plots the spread of CDS contracts providing credit protection for a total period of five years, 5, along with the spread of forward CDS contracts providing credit protection only for the first year, 0,1 1, and fifth year, 4,5. The figure reflects not only the peak of the crisis in , but also the effect of Dragui s speech. It is well known that the announcement (to do whatever it takes to preserve the Euro) brought a significant reduction in the credit spreads paid by countries like Spain and Italy. If we focus our analysis on the reduction of the 5-year CDS spreads in Figure 6, the evident drop in the cost of protecting the 8 On August 2, 2012, the ECB announced the introduction of the Outright Monetary Transactions (OMT). Technical details were provided on September 6, 2012 (ECB, Press Release, 2012), but the plan was never activated (Álvarez et. al, 2017). 18

19 immediate next year suggests that the announcement had a significant effect mainly on the perception of the short-term risk. Although there was also a decline in the cost of exclusively protecting the fifth year, it did so to a much lesser extent. Therefore, it seems that CDS traders took Dragui s words as a guarantee of no default in the short-run, but concerns on the future creditworthiness of countries like Spain or Italy remained for a long time. It is also worth noting that, by the end of 2016, 5-year CDS spreads were back around at the same level they were in However, the gap between the cost of protecting the first year and the cost of protecting the fifth year was still significantly higher at the end of 2016 when compared to the period before the crisis, even for France. This result indicates that concerns on the future of the Euro remained. <Figure 6 about here> Figure 7 is a natural extension of Figure 6. It contains the first and last elements in the time decomposition of the 5-year CDS spreads; that is, 0,1; 5 and 4,5; 5. Owing to Dragui s announcement, the change in the composition of the 5-year CDS spreads is evident, and, apparently, has been consolidated. <Figure 7 about here> 6. Conclusions A forward CDS contract implies the obligation to enter into a CDS contract on a specified reference entity for a specified spread at a specified future time. The obligation ceases to exist in case of default during the life of the forward contract. A forward CDS represents an appealing hedging tool. It allows to hedge today a future credit risk exposure without the need to initiate today the payment of a protection fee. Another interesting aspect of forward CDS is their informational content: forward CDS spreads can be 19

20 naturally interpreted as the market s expectation on the creditworthiness of a reference entity in a future time period, conditional on no previous default. In this paper, I contribute to the relatively scarce academic literature on forward CDS in several ways. First, I derive a simple closed-form expression for the pricing of forward CDS in a model-free setting. Second, I demonstrate that this model-free approach leads to a straightforward time decomposition of CDS spreads. To the best of my knowledge, this is the first paper that provides a theoretical framework for the time decomposition of credit spreads. The model-free setting implies that the estimation of forward CDS spreads and the time decomposition of CDS spreads can be implemented using a reduced-form model, a structural model, or a mixed method. As a third contribution of the paper, I propose one particular form of mixed method that combines the structural model framework and the information provided by the term structure of CDS spreads. Both the simulation analysis and the empirical results from the Eurozone crisis indicate that the proposed estimation method is robust to extreme scenarios: significant model misspecification, high CDS levels, and long maturities. Empirical results from the Eurozone crisis represents the fourth and last contribution of the paper. I find that the estimation of forward CDS spreads and the time decomposition of CDS spreads provide new insights on the effects of the ECB intervention in the sovereign debt markets since July The main conclusion is that a clear segmentation between the perception of the short-term risk and the perception of the long-term risk, which was not present before the ECB intervention, has consolidated since then in the Eurozone. 20

21 References Álvarez, I., Casavecchia, F., De Luca, M., Duering, A., Eser, F., Helmus, C., Hemous, C., Herrala, N., Jakovicka, J., Lo Russo, M., Pasqualone, F., Rubens, M., Soares, R., and Zennaro, F., 2017, The Use of the Eurosystem s Monetary Policy Instruments and the Operational Framework since 2012, ECB Occasional Paper No 188, May. Ericsson, J., Reneby, J., and Wang, H., 2015, Can Structural Models Price Default Risk? Evidence from Bond and Credit Derivative Markets, Quarterly Journal of Finance, 5 (3), European Central Bank, 2012, Technical Features of Outright Money Transactions, Press Release, 6 September. Hull, J., and White, A., 2003, The Valuation of Credit Default Swap Options, Journal of Derivatives, 10 (3), Leccadito, A., Tunaru, R.S., and Urga, G., 2015, Trading Strategies with Implied Forward Credit Default Swaps, Journal of Banking and Finance, 58, Leland, H.E., and Toft, K.B., 1996, Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads, Journal of Finance, 51 (3), Zettelmeyer, J., Trebesch, C., and Gulati, M., 2013, The Greek Debt Restructuring: An Autopsy, Economic Policy, 28 (3),

22 Tables and Figures Table 1. This table provides estimates of,,,,, and for different maturities and CDS spreads. A risk-free interest rate of 2.5% and a recovery rate of 40% are assumed. The state variable is normalized to 1. The volatility is calibrated to fit the CDS spread, given different assumptions on the default barrier. values reflect the estimated price for a bond with a nominal of $100 and 5% annual coupon. t Vb cds = 10 cds = 50 cds = 100 cds = 250 cds = 500 cds = 1,000 A B A+B d A B A+B d A B A+B d A B A+B d A B A+B d A B A+B d

23 Table 1 (cont). t Vb cds = 10 cds = 50 cds = 100 cds = 250 cds = 500 cds = 1,000 A B A+B d A B A+B d A B A+B d A B A+B d A B A+B d A B A+B d

24 Table 1 (cont). t Vb cds = 10 cds = 50 cds = 100 cds = 250 cds = 500 cds = 1,000 A B A+B d A B A+B d A B A+B d A B A+B d A B A+B d A B A+B d

25 Table 2. This table provides main descriptive statistics for each country and contract maturity. The number of weekly observations is shown in parenthesis. Country Statistic cds(0.5) cds(1) cds(2) cds(3) cds(4) cds(5) cds(7) cds(10) cds(20) cds(30) Mean Median France Min (N=470) Max SD Mean Median Spain Min (N=470) Max SD Mean Median Italy Min (N=470) Max SD Mean Median Ireland Min (N=429) Max 1, , , , , , , , SD Mean Median Portugal Min (N=426) Max 1, , , , , , , , SD Mean Median Greece Min (N=193) Max 5, , , , , , , , , , SD

26 Table 3. This table contains the results on the estimated forward CDS spreads between adjacent dates with available CDS spreads (mean values). Three possible values for the default barrier are considered: 0.1, 0.5, and 0.9. Country France Spain Italy Ireland Portugal Greece Vb fcds(0.5,1) fcds(1,2) fcds(2,3) fcds(3,4) fcds(4,5) fcds(5,7) fcds(7,10) fcds(10,20) fcds(20,30)

27 Table 4. This table provides results on the time decomposition of CDS spreads (mean values). The analyzed cases are: 1) time decomposition of the 5-year CDS spreads into equally spaced time intervals of one year, 2) time decomposition of the 10-year CDS spreads into equally spaced time intervals of five years, and 3) time decomposition of the 30-year CDS spreads into equally spaced time intervals of ten years. Three possible values for the default barrier are considered: 0.1, 0.5 and 0.9. Country France Spain Italy Ireland Portugal Greece Vb Q(0,1;5) Q(1,2;5) Q(2,3;5) Q(3,4;5) Q(4,5;5) Q(0,5;10) Q(5,10;10) Q(0,10;30) Q(10,20;30) Q(20,30;30)

28 Figure 1. This figure reproduces the term structure of CDS spreads (1 to 5 years) for Greece, April 15, 2010; Italy, October 4, 2011; and Portugal, July 9, Term Structure of CDS Spreads 500 bp Maturity Greece: Italy: Portugal:

29 Figure 2. This figure represents a particular set of forward CDS spreads for Greece, April 15, 2010; Italy, October 4, 2011; and Portugal, July 9, The considered time intervals are (in years): (0,1), (1,2), (2,3), (3,4) and (4,5). 29