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Restructuring Risk in Credit Default Swaps:
An Empirical Analysis∗
Antje Berndt†^ Robert A. Jarrow‡^ ChoongOh Kang§
Current Version: November 18, 2005
Preliminary and Incomplete
Abstract This paper estimates the price for bearing exposure to restructuring risk in the U.S. corporate bond market during 2000-2005, based on the relationship between quotes for default swap (CDS) contracts that include restructuring as a covered default event and contracts that do not. We find that on average the premium for exposure to restructuring risk amounts to 6% to 8% of the value of protection against non-restructuring default events. The increase in the restructuring premium in response to an increase in rates on default swaps that do not include restructuring as a covered event is higher for high-yield CDS and lower for investment-grade firms, and depends on firm-specific balance-sheet and macroeconomic variables. We observe that firms that offer a distressed exchange often experience a steep decline in their distance to default prior to the completion of the exchange. As an application, we propose a reduced-form arbitrage-free pricing model for default swaps, allowing for a potential jump in the risk-neutral non-restructuring default intensity if debt restructuring occurs.
∗We thank Lombard Risk for Default Swap data. We are grateful to Jean Helwege, Yongmiao Hong, Philip Protter and Roberto Perli for useful comments.
†Tepper School of Business, Carnegie Mellon University. ‡Johnson Graduate School of Management, Cornell University. §Department of Economics, Cornell University.
1 Introduction
This paper estimates the price for bearing exposure to restructuring risk in the U.S. corporate bond market during 2000-2005, based on the relationship between quotes for default swap (CDS) contracts that include restructuring as a covered default event and contracts that do not. We find that on average the premium for expo- sure to restructuring risk amounts to 6% to 8% of the value of protection against non-restructuring default events. The increase in the restructuring premium in re- sponse to an increases in rates on default swaps that do not include restructuring as a covered event is higher for high-yield CDS and lower for investment-grade firms, and it depends on firm-specific balance-sheet and macroeconomic variables. We find restructuring premia to be the highest in the Telephone, Service & Leisure and Rail- road sectors, and much lower for firms in the Oil and Gas industry and for Gas utility firms. We also extend the empirical work to explore the determinants of the default swap rates by controlling for the restructuring clause and the time to maturity of the contract, together with the time period with regard to regulations by the International Swaps and Derivatives Association (ISDA). To proxy for firm-specific default risk, we use distance to default, Merton default probabilities, and leverage ratios. Market variables include the level and slope of the risk-free interest rate, a volatility (VIX) index, Moody’s Baa corporate yield, and the spread between Moody’s Aaa yield and 20-year Treasury yield. After controlling for firm-specific and macroeconomic variables, we obtain a regression coefficient of determination of 60%, and even over 71% after taking logarithms. We observe that firms which complete a distressed exchange often experience a steep decline in their distance to default prior to the completion of the exchange. As an application, we develop a reduced-form arbitrage-free pricing model for default swaps that explicitly takes into account the distinct restructuring clause of the CDS contract. We incorporate the effect of the restructuring event on the default risk by allowing for a jump in the default intensity should restructuring occur. The jump is allowed to be both positive or negative depending on investors expectation of the firm’s financial health after the debt restructuring. (The model is similar to the primary-secondary framework in Jarrow and Yu (2001), where the primary firm’s default causes the default intensity of the secondary firm to jump upward.) The restructuring event, if it happens prior to default, may directly affect the firm’s
be the most contentious credit events. Restructuring has been at the center of debate because it may constitute a soft credit event that would not necessarily result in losses to the owner of the reference obligation.^2 In relation to this soft credit event, restructuring retains a various coupon and maturity structures, so that bonds with lower coupons and longer maturities trade less favorably than others. Thus, the protection buyer’s cheapest-to-deliver option has greater value under restructuring than other non-restructuring default events. As a response, the International Swaps and Derivatives Association (ISDA) pro- vides four choices under restructuring as a credit event:
- Full restructuring (FR), based on the ISDA 1999 Definition
- Modified restructuring (MR), based on the ISDA 2001 Supplement Definition
- Modified-modified restructuring (MMR), based on the ISDA 2003 Definition, and
- No restructuring (NR).
Each restructuring rule has different clauses regarding the maturity and transfer- ability of deliverable obligations as shown in Table 1. We can see that the value of the cheapest to deliver option is more limited under MR or MMR than FR. Also, MR and MMR are more restrictive on the confirmation of restructuring event. Thus, the soft restructuring problem is alleviated under these rules. Detailed discussions on the contractual terms regarding restructuring can be found in FitchRatings (2003) and Packer and Zhu (2005).
Table 1: Limits on Deliverable Obligation. T is the maturity of CDS contract, and T¯ denotes the maturity of the deliverable obligation.
Restructuring Deliverable Obligation Clause FR Any bond of maturity up to 30 years MR T ≤ T <¯ (T + 30 months ) MMR Allow additional 30 months for the restructured bond. For other obligations, same as MR.
(^2) For a description of the Conseco debt restructuring case, refer to Bomfim (2005), p.294.
3 Data
The main data source is the ValuSpread Credit Data (VSCD) provided by Lombard Risk Systems. For each reference name on a particular date, the data reports the “average” mid-market CDS rate derived from the available quotes information con- tributed by about 25 market makers. Additional information includes the seniority (senior/subordinated) and the currency of the underlying debt, the maturity of the CDS contract (1, 3, 5, 7, or 10 years), the standard deviation of the mid-market quotes, and most importantly the restructuring clause applied in the contract. Also reported is the average expected recovery rate under each restructuring clause. The frequency of the data has increased over time: monthly (month-end quote) from 1999 to 2001, biweekly from January 2002 to June 2002, weekly from July 2002 to May 2003, and daily from 15 May 2003. The version of the database we use covers the period from 31 July 1999 through 30 June 2005. The standard deviation of the mid-market quotes can be a measure of reliability of each observation. If the standard deviation is too small, it is highly likely that there is only one or two contributors. In this case, the data can be biased due to small sample size. On the other hand, if the standard deviation is too large, it indicates that there are outliers in the sample. So we filter out observations with standard deviation of greater than 20% or less than 1% of the mean CDS rate. The industry information for each reference name is obtained from the Fixed In- vestment Securities Database (FISD) from LJS Global Information Systems. Among the 2,781 tickers listed in VSCD, we could identify the industry information and CUSIP numbers for 1,521 tickers, of which 929 are U.S. names, 532 are non-U.S. names, and 60 are CDS indices such as TRAC-X and iBoxx. The number of iden- tified tickers in each industry for U.S. and non-U.S. tickers are shown in Table 9 in Appendix D.^3 Table 2 presents the number of CDS quotes by restructuring clause and adjust- ments to the ISDA definitions, and Table 3 provides the number of quotes per industry for U.S. firms. From Table 2 we see that the U.S. market has selected to transact according to the modified restructuring clause. Contrary to to the European market, the modified-modified restructuring clause (MMR) is the least popular in the U.S. market. In Table 4, it is of interest to note that under current market convention, (^3) Note that the number of reference name is about 2,100 which is less than the number of tickers because tickers may change according to the event such as mergers and acquisitions even though the company name does not change.
Table 3: Number of quotes by industry. FISD Industry code Number of observations† Total NR FR MR MMR Industrial 10 Manufacturing 258,355 40,648 46,008 170,240 1, 11 Media/Communications 48,693 12,192 7,682 28,819 0 12 Oil & Gas 38,429 5,972 6,269 26,188 0 13 Railroad 1,961 395 247 1,319 0 14 Retail 58,842 10,551 10,855 37,424 12 15 Service/Leisure 21,358 12,184 7,855 1,319 0 16 Transportation 20,301 3,613 3,950 12,738 0 32 Telephone 14,738 3,457 1,474 9,807 0 Finance 20 Banking 30,990 3,391 7,528 20,071 0 21 Credit/Financing 28,256 5,130 6,193 16,933 0 22 Financial Services 38,567 4,323 6,283 27,961 0 23 Insurance 41,358 5,957 4,474 30,927 0 24 Real Estate 26,256 2,397 4,478 18,416 965 25 Savings & Loan 137 0 0 137 0 26 Leasing 1,629 273 108 1,248 0 Utility 30 Electric 39,685 6,019 4,731 28,935 0 31 Gas 7,148 1,356 1,069 4,723 0 33 Water 0 0 0 0 0 Government 40 Foreign Agencies 0 0 0 0 0 41 Foreign 0 0 0 0 0 42 Supranational 835 0 593 242 0 43 U.S. Treasuries 0 0 0 0 0 44 U.S. Agencies 2,151 429 248 1,474 0 45 Taxable Municipal 0 0 0 0 0 Miscellaneous 60 Miscellaneous 0 0 0 0 0 99 Unassigned 0 0 0 0 0 Total 679,689 118,287 120,045 438,921 2,
† (^) These are the number of observations for U.S. names used in the analysis whose industry information is verified by the author using FISD data.
Table 4: Number of 5-year CDS rate quotes for U.S. firms by rating. For both investment-grade (IG) and speculative-grade (SG) firms, the entries under each re- structuring clause are for number of quotes, percentage of total number of quotes, row percentage, and column percentage.
Restructuring clause Total FR MM MR NR IG 42,228 733 152,212 61,869 257, 13.7 0.2 49.4 20.1 83. 16.4 0.3 59.2 24. 87.9 98.0 85.0 76. SG 5,818 15 26,921 18,581 51, 1.9 0.0 8.7 6.0 16. 11.3 0.0 52.4 36. 12.1 2.0 15.0 23. Total 48,046 748 179,133 80,450 308, 15.6 0.2 58.1 26.1 100.
FR has positive mean and median premia over both MR and MMR. While the mean and median of restructuring premia of FR, MR and MMR over NR are all positive, negative premia are observed for all restructuring rules for certain firms and dates. Although, in theory, the existence of negative premia could be explained if investors were to believe that a restructuring credit event will cause a non-restructuring default event such as bankruptcy or failure to pay soon after, and subject to recovery rates under restructurings being sufficiently higher than under bankruptcy or failure to pay, we believe that these occurrences are more likely due to different default-swap brokers and investment banks contributing to the composite quote for default swap contracts under different restructuring rules for a given firm on a given date. In what follows, therefore, we remove quotes with negative outcomes for the restructuring premium from the sample.
4 Regression Analysis
In order to obtain a simple and robust model of the relationship between the re- structuring risk premia and the CDS rate under no restructuring, we undertake a regression analysis of 10,020 paired 5-year cN R^ and cM R^ observations from May 2002 through December 2004, for all U.S. firms in the Industrial and Utilities sector listed
Table 5: Results of OLS regression of the modified restructuring premium, cM R^ −cN R, on the CDS rate under no restructuring, cN R, as well as credit-quality and sector fixed effects. The reference sector is Manufacturing. Results for the full restructuring risk premia are available upon request.
estimate SD estimate SD estimate SD Intercept 0.537 0.045 0.748 0.059 0.683 0. cN R^ 0.051 0.000 0.050 0.001 0.049 0. SG -2.010 0.162 -2.093 0. SG×cN R^ 0.006 0.001 0.007 0. Media & Comm 0.397 0. Oil & Gas -0.167 0. Railroad 0.678 0. Retail -0.040 0. Service & Leisure 0.641 0. Transportation 0.197 0. Telephone 0.668 0. Electric 0.099 0. Gas -0.210 0. R^2 0.518^ 0.521^ 0. no obs 25814 25814 25814
that the base CDS rate alone explains 51.8% of the restructuring premium of MR over NR. We can also consider the market leverage^5 or the distance to default based on Merton (1974), but the base CDS rate shows the highest performance. In addition, we combine ratings information with the base CDS rate by adding speculative grade dummy variable for both intercept and slope with respect to the base CDS rate. It is defined to be 1 if the firm is rated to be speculative grade (rated BB or lower by Standard and Poor’s) and 0 otherwise. Note that the likelihood of default and the recovery rate also depend on the state of the economy. Thus, we also consider the 5-year constant maturity Treasury rate to control the state of the economy, and the Moody’s seasoned Baa corporate bond yield to control the state of the overall credit market. Since the restructuring is a method for a firm to overcome financially distressed situation, it is reasonable to assume that the likelihood of restructuring is propor- tional to the likelihood of default. More relevant information is the likelihood of restructuring relative to the likelihood of default, and this can be interpreted as the likelihood that the decision makers (debtors and creditors) come to agree to choose a pre-default debt restructuring as the “first attempt” to handle a financial distress. The restructuring event relevant to CDS contracts can be considered as a soft version of private workouts in the sense that we only consider the debt restructuring prior to any violation of the contract: if the firm violates the terms of contract, the event is classified as default such as failure to pay, and this cannot constitute a restructuring event. Noting this, we still rely on existing literature on the choice between private workouts and formal bankruptcy under the subject of financial distress^6 to obtain rea- sonable variables that might affect the relative likelihood of restructuring. In theory, it is known that private workouts are more likely than bankruptcy for firms with the following characteristics: higher economic viability, less severe coordination problem, and relatively high leverage^7. For our research, we consider market and balance-sheet (^5) Here we define the market leverage as D/(D + E), where D is the book value of total debt, E is the market value of equity following Collin-Dufresne, Goldstein, and Martin (2001). 6 See John (1993) for an overview on the methods to deal with financial distress. Chatterjee, Dhillon, and Ramirez (1996) provides an empirical examination on the determinants of the choice of debt restructuring methods: Chapter 11 reorganizations, prepackaged bankruptcies (prepacks), and workouts. 7 Firms with relatively higher leverage (more debt) have an incentive to prefer the private debt restructuring to the formal bankruptcy since these firms are expected to suffer less erosion in eco- nomic value before default is triggered as Jensen (1989) suggests. Also, Ross, Westerfield, and Jaffe (1996) argue that firms with more debt will experience financial distress earlier than firms with less debt, and will have more time for private workouts to handle the distressed situation appropriately.
base CDS rate is around 50%, and the coefficient of the volatility is statistically insignificant in the regression analysis. For recovery rate at restructuring, it is reasonable to assume that the determinants are almost the same as those of the recovery rate at default. However, we should also consider the cheapest to deliver option inherent in the CDS with restructuring because the protection buyer would deliver the debt which is the cheapest among the deliverable debts in the market. If the value of this option is higher, the CDS with restructuring is more valuable. The cheapest debt will often be the debt with the longer maturity and lower coupon rate. Under modified restructuring clause, the maturity of the delivered debt should not be earlier than the original maturity of the CDS contract and must not exceed thirty months after the original maturity date of the CDS contract. Thus, the deliverable obligations under MR should mature in 5 to 7.5 years for 5-year CDS contracts. Under full restructuring clause, the maturity limitation is much more generous to allow all the obligations maturing in 30-years. We try the ratio of the debt maturing after five years to long-term debt as a proxy for the value of the cheapest to deliver option. However, since the value of the deliverable debt also depends on its coupon structure, it is still arguable whether this variable can proxy the value of the delivery option or not. The expected change in the likelihood of default after the restructuring event depends on the market’s expectation on the probability that the firm’s pre-default debt restructuring will be successful or unsuccessful. This is closely related to the likelihood of restructuring: the debtors and creditors would agree to take private pre- default debt restructuring if they expect it to be successful. Thus, we expect that the variables for this are practically the same as those of the likelihood of restructuring. Table 6 shows the result of the regressions of the restructuring premium, cM R^ − cN R, on the covariates discussed so far. In summary, the regression results show that the coefficients of the covariates suggested by the literatures on financial distress are statistically significant, but have very little impact for explaining the restructuring premium compared to the base CDS rate itself: the R^2 increases from 51.8% (with the base CDS rate cN R^ alone) to 54.6% (with all the covariates). *** To be developed further. ***
4.1 The Effect of Restructuring Rules on CDS Rates
In this section, we examine to what extent different restructuring clauses impact CDS rate quotes, along with other possible determinants, in a panel-regression setting.
Table 6: Results of OLS regression of the modified restructuring premium, cM R^ − cN R, on the CDS rate under no restructuring, cN R, as well as credit-quality, firm- specific accounting data and macro-economic variables. The covariates are described in Table 11 in the appendix. Results for the full restructuring risk premia are available upon request.
estimate SD estimate SD estimate SD Intercept -4.980 0.791 -10.495 1.113 -9.005 1. cN R^ 0.046 0.001 0.045 0.001 0.045 0. SG -1.779 0.162 -4.062 0.227 -4.391 0. SG×cN R^ 0.009 0.001 0.019 0.001 0.019 0. Gov5yr -1.865 0.119 -2.274 0.155 -2.281 0. Baa 1.899 0.135 2.399 0.175 2.383 0. EBITDA/TtlDebt 6.783 1.439 7.778 1. StockRet20days 128.519 11.677 124.144 11. log(sales) 0.428 0.065 0.390 0. log(no employee) -0.311 0.056 -0.335 0. SubDebt/TtlDebt 11.338 1.191 12.822 1. SecDebt/TtlDebt 3.340 0.640 4.011 0. AuditorOp 0.354 0.104 0.259 0. Intangible/TtlAsset 2.542 0. Collateral/TtlAsset -1.079 0. Deliverable 0.198 0.037 0.165 0. R^2 0.527 0.546 0. no obs 25,814 14,539 14,
- Maturity Dummy (Tyr). The CDS rate depends on the time to maturity. Though the distance to default measure already considers the maturity of the CDS, it may not be sufficient to capture the whole effect of the time to maturity. Also, the maturity dummies can capture the different liquidity premium on the CDS with different maturities.
Tables 12, 13, and 14, and Table 15 show the result of the regression of CDS and logarithm of CDS, using leverage, Merton default probabilities, and distance to default, respectively. We also report the results with the CDS rate divided by the reported loss given default in Tables 16 and 17. Finally, Table 18 regresses the reported loss given default on distance to default in order to gain intuition about the relationship between recovery estimates and expected default frequencies. We use CDS rate with 5 years maturity (T5) under no restructuring (NR) in finance industry (IND4) observed in ISDA03 period as the base. The proxies of default probability used in the tables are the 1 year distance to default (DD1), the Merton 1 year default probability (NDD1) and leverage (Lev). The regressions using T-years DD and NDD where T is the corresponding CDS maturity are not reported here. Although not all estimates are significant, the signs and magnitudes of coefficients of restructuring dummies and maturity dummies are all coincides with our expectation. In each time period between changes to ISDA regulations, CDS rates are, on average, highest under full restructuring and lowest under no restructuring. *** To be developed further. ***
4.2 The Effect of Restructuring Events on Default Probabil-
ities
The restructuring event, if it happens prior to default, may directly affect the firm’s default risk in two ways. On the one hand, it is possible that restructuring can successfully reduce the firm’s financial burden to improve overall financial health of the firm, and its default probability can be lowered (successful restructuring). On the other hand, the restructuring event can serve as a signal that the firm is in a financially weak condition, in which case, investors will raise their estimates of the firm’s default risk, and eventually the firm will become financially distressed (unsuccessful restructuring). Examples for both successful and unsuccessful restructuring are provided in Fig- ure 1 which plots the distance to default for eight firms that did experience a distressed
exchange^9 , starting six months prior to the completion of the distressed exchange to six months afterward. The distance to default is a proxy of survival probability of a firm based on Merton (1974). Note that the exchange offer is usually announced several weeks to months before the completion date. From the plots, we can find that six out of eight firms experienced notable drop in the distance to default upon the distressed exchange event. This implies that the probability of default jumps upward at the time of the restructuring event.
5 A Reduced-Form Pricing Model for CDS Con-
tracts Under Different Restructuring Clauses
In this section, we develop a reduced-form arbitrage-free pricing model for default swaps that explicitly takes into account the restructuring clause in the contract. To keep notation simple, we will distinguish between two categories of credit events: re- structuring and non-restructuring default events, where the latter includes bankruptcy and a material failure by the obligor to make payments on its debt issue. We suppose that the restructuring of a given firm occurs at the first event time of a (non-explosive) counting process NR, relative to a probability space with measure P (actual or data-generating measure) and an increasing family {Ft}t≥ 0 of information sets defining the resolution of information over time, that satisfy the usual conditions (see, for example, Protter (2004)). Assuming the arbitrage-free and frictionless mar- ket, Harrison and Kreps (1979) and Delbaen and Schachermayer (1999) show that, under mild technical conditions, there exists a “risk-neutral”(or “equivalent martin- gale”) measure P˜, under which the price Pt at time t of a security paying a single, possibly random, amount Z at some stopping time τ > t is
Pt = E˜
e−^
R (^) τ t rs^ dsZ|Ft
where r is the short-term interest rate process,^10 and E˜ denotes expectation under (^9) Distressed exchange, as a category of default by Moody’s, is defined to occur when “(i) the issuer offers bondholders a new security or package of securities that amount to a diminished financial obligation (such as preferred or common stock, or debt with a lower coupon or par amount), or (ii) the exchange had the apparent purpose of helping the borrower avoid default,”see Keenan, Hamilton, Shtogrin, Zarin, and Stumpp (2000) for details. Since the debt restructuring is processed through exchange offer, we consider the distressed exchange to be almost equivalent to the restructuring event. (^10) The short-rate process r is progressively measurable with respect to {Ft}t≥ 0 with ∫^0 t |rs| ds
the risk-neutral measure P˜. Note that the market is not required to be complete, so the martingale measure P˜ is not assumed to be unique. However, we suppose that the measure is determined uniquely by the market in equilibrium. Restructur- ing of the firm occurs at time τ R, the first jump time of the counting process NR, with a risk-neutral restructuring intensity process λR, for which we will assume the doubly-stochastic property under P˜. The doubly-stochastic, or Cox-process, assump- tion implies that the risk-neutral conditional probability at time t that the obligor will not restructure on or before time T is
sR(t, T ) = P˜
τ R^ > T |Ft
= E˜
e−^
R (^) T t λRs ds|Ft
Similarly, we assume that the non-restructuring default occurs at the first event time τ D^ of a (non-explosive) counting process ND, with a risk-neutral non-restructuring default intensity process hD^. We will extend the doubly stochastic setting of arrival of credit events under the risk-neutral measure under ˜P to include hD. Motivated by the observed negative jumps in the distance-to-default around the time a distressed exchange offer was made for several of the companies, as shown in Figure 1 in Sec- tion 4, we opted to specify a model under which hD^ allows for a, possibly random, jump in the risk-neutral non-restructuring default intensity. Specifically, we assume
hDt = λDt + k (^1 1) {t≥τ R} + k 2 λDt 1 {t≥τ R}, (5)
where k 1 and k 2 , k 2 > −1, are random variables. Note that, in our model, a debt restructuring event, if it occurs, always precedes non-restructuring default events such as bankruptcy and failure to pay. In fact, the debt issues are also restructured under bankruptcy or other default processes. However, at the time of these restructuring events, all the default swap contracts are already terminated by the other default events. So it is meaningless to consider any restructuring after a non-restructuring default event. Model specification (5) allows for both upward and downward jumps in the risk- neutral non-restructuring default intensity, capturing the possibility for both unsuc- cessful and successful debt restructurings.^11 It may be compared to the primary-
P^ ˜-almost surely and E˜(e−^ R^0 t^ rs^ ds) < ∞, for all t. See Protter (2004) for details. (^11) Recall that Figure 1 exhibits a negative jump to the distance to default, and therefore a possible positive jump in the default intensities, for six out of eight firms. For the other two firms, Cellstar and Focal Communications, the impact was somewhat in the opposite direction as the distance to default did increase or, in the later case, stayed relatively flat.
secondary framework of Jarrow and Yu (2001), where a primary firm’s default causes the default intensity of the secondary firm to jump upward by a constant amount. The primary-secondary structure violates the standard Cox process framework in Lando (1998). However, as discussed in Collin-Dufresne, Goldstein, and Hugonnier (2004), the no-jump condition in Duffie and Singleton (1999) is still satisfied. This enables us to utilize the standard pricing machinery, in the sense that the fundamen- tal relationship between the conditional survival probability and the default intensity holds
sD(t, T ) = ˜P(τ D^ > T |Ft) = E˜
e−^
R (^) T t hDs ds|Ft
The conditional risk-neutral probability of survival until time T , given that a credit event (including both restructuring and non-restructuring default) did not occur by time t, is
s(t, T ) = P˜(τ D^ ∧ τ R^ > T |Ft) = E˜
e−^
R (^) T t λDs^ +λRs^ ds|Ft
where τ D^ ∧ τ R^ = min
τ D^ , τ R
which will be denoted by τ hereafter. In our doubly- stochastic setting, conditional on the paths of the intensities, the probability that both restructuring and non-restructuring credit events happen at the same time is zero. Equation (6) can be rewritten as
sD(t, T ) = P˜(τ > T |Ft) + P˜(τ D^ > T, τ R^ ≤ T |Ft)
= E˜
e−^
R (^) T t λDs^ +λRs^ ds|Ft
+ E˜
e−^
R (^) T t λDs^ ds
∫ T
t
e−(k^1 (T^ −v)+k^2
R (^) T v λDs^ ds)λRv e−^
R (^) v t λRs^ dsdv|Ft
= E˜
⎣e
− R^ tT λDs ds
e−^
R (^) T t λRs^ ds^ +
∫ T
t
e−(k^1 (T^ −v)+k^2
R (^) T v λDs^ ds)^ λRv e−^
R (^) v t λRs^ dsdv
RF
∣∣Ft
where RF can be interpreted as an adjustment factor due to the restructuring risk. It equals 1 if a restructuring event has no direct impact on the non-restructuring default intensity, that is, when k 1 = k 2 = 0, and it is different from 1 otherwise. If the jump size is positive (k 1 ≥ 0 and k 2 ≥ 0), the restructuring adjustment factor (RF ) falls between 0 and 1, implying a decrease in the risk-neutral survival probability sD(t, T ).
