How Safe are European Safe Bonds? An Analysis from the Perspective of Modern Portfolio Credit Risk Models
HHow Safe are European Safe Bonds? An Analysis from thePerspective of Modern Credit Risk Models R¨udiger Frey, Kevin Kurt, Camilla Damian, Institute for Statistics and Mathematics, Vienna University of Economics and Business (WU) July 14, 2020
Abstract
Several proposals for the reform of the euro area advocate the creation of a marketin synthetic securities backed by portfolios of sovereign bonds. Most debated are theso-called European Safe Bonds or ESBies proposed by Brunnermeier, Langfield, Pagano,Reis, Van Nieuwerburgh and Vayanos (2017). The potential benefits of ESBies and otherbond-backed securities hinge on the assertion that these products are really safe. In thispaper we provide a comprehensive quantitative study of the risks associated with ESBiesand related products, using an affine credit risk model with regime switching as vehiclefor our analysis. We discuss a recent proposal of Standard and Poors for the rating ofESBies, we analyse the impact of model parameters and attachment points on the size andthe volatility of the credit spread of ESBies and we consider several approaches to assessthe market risk of ESBies. Moreover, we compare ESBies to synthetic securities createdby pooling the senior tranche of national bonds as suggested by Leandro and Zettelmeyer(2019). The paper concludes with a brief discussion of the policy implications from ouranalysis.
Keywords.
European Safe Bonds, European monetary union, Securitization of credit risk,Markov modulated affine models
Synthetic securities backed by portfolios of sovereign bonds from the euro area have recentlybeen proposed as a tool to improve the stability of the European monetary union and toincrease the amount of safe assets in the euro area, see for instance Dombrovskis and Moscovici(2017) or B´enassy-Qu´er´e et al. (2018). The most debated proposal is due to Brunnermeier et al.(2017), who advocate the creation of a market in so-called European Safe Bonds or ESBies.In credit risk terminology, ESBies form the senior tranche of a CDO backed by a diversifiedportfolio of sovereign bonds from all members of the euro area. According to Brunnermeieret al. (2017), ESBies would be standardized and issued by tightly regulated private institutionsor by a public agency. The junior tranche of the underlying bond portfolio would be sold inthe form of European Junior Bonds (EJBies) to investors traditionally bearing default risk,such as hedge funds or insurance companies.Brunnermeier et al. (2017) argue that a liquid market in ESBies would enhance the stabilityof the euro area in a number of ways: first, it would increase the supply of safe assets in We are grateful to Sam Langfield and to two anonymous referees for very useful comments and suggestions. Email: [email protected] , [email protected] , [email protected] . Postal address: Welthandelsplatz 1, A-1020 Vienna a r X i v : . [ q -f i n . P R ] J u l he euro area; second, it would help to break the vicious circle between bank solvency andthe credit quality of sovereigns created by the fact that most euro area banks hold largeamounts of risky sovereign bonds of the nation state in which they reside; third, it mightreduce the distortions on bond markets caused by the flight-to-safety behavior of investors incrisis times. Moreover, ESBies respect the no-bailout clause and their introduction would notdistort market discipline , as the agency issuing ESBies would buy these bonds at market pricesand as sovereigns would remain responsible for their own bonds, which exerts discipline onborrowing decisions. Another important approach for creating a safe asset for the euro areaconsistent with the no-bailout clause is to issue national sovereign bonds in several senioritylevels and to pool the bonds from the senior tranche, see for instance Leandro and Zettelmeyer(2019). These products and ESBies are therefore different from eurobonds that are currentlydiscussed in the context of the Corona crisis. Eurobonds are jointly issued and guaranteedby all euro area member states so that every member state is liable for the entire issuance.Loosely speaking, ESBies are designed for improving the functioning of the euro area “innormal times”, whereas eurobonds are crisis-intervention instruments.The potential benefits of ESBies hinge on the assertion that these products are really safe.To address this issue, Brunnermeier et al. (2017) carry out a simulation study in an one-period mixture model where defaults are independent given the aggregate state of the euroarea economy. They find that, with reasonably high levels of subordination, the expected lossof ESBies is comparable to that of triple-A rated bonds. However, their model is calibrated ina fairly ad hoc manner. More importantly, Brunnermeier et al. (2017) do not study the marketrisk of ESBies (the risk of a change in the market value of these products due to changes inthe credit quality of the underlying bonds or in the state of the euro area economy). Now, thebad performance of many highly rated rated senior CDO tranches during the financial crisishas shown that the market risk of such products can be substantial. Clearly, a high amount ofmarket risk is inappropriate for a safe asset intended to serve as collateral in security markettransactions, as an investment vehicle for money market funds or as a crisis-resilient store ofvalue on the balance sheet of banks. A thorough quantitative analysis of the risks associatedwith ESBies is thus needed to assess if these securities can in fact perform the function of asafe asset for the euro area. This is the aim of the present paper.We propose to work in a novel dynamic credit risk model that captures salient featuresof the credit spread dynamics of euro area member states and that is at the same time fairlytractable. Such a model is a prerequisite for the analysis of the market risk associated withESBies. In mathematical terms, we consider a reduced-form model with conditionally indepen-dent default times; the hazard rate or default intensity of the different obligors is modelled byCIR-type processes whose mean-reversion level is a function of a common finite state Markovchain. Considering a Markov modulated mean-reversion level permits us to model differentregimes, such as a crisis regime where the default intensity of all sovereigns is high and anexpansionary regime where all default intensities are low. This generates default dependencein a natural way. We successfully calibrate the model to a time series of euro area CDS spreadsover the period January 2009 until September 2018. The main part of the paper is devotedto the risk analysis of ESBies and EJBies. We begin by discussing a recent proposal of S&Pfor the rating of ESBies, see Kraemer (2017). Using novel results on model-independent pricebounds for ESBies, we show that the S&P proposal is ultra-conservative in the sense that itattributes to an ESB the worst rating that is logically consistent with the ratings attributedto the euro area sovereigns. As a next step, we study the robustness of the credit spread (or2quivalently the risk-neutral expected loss) of ESBies and EJBies with respect to subordina-tion levels and model parameters. In particular, we consider several parameterizations for thetransition intensities of the common Markov chain, as these largely drive the default depen-dence in our model. It turns out that, from this perspective, ESBies are very safe already forlow subordination levels (around 15%), in line with the findings of Brunnermeier et al. (2017).We use several approaches to gauge the market risk of ESBies. First, we compute spread-trajectories for ESBies via historical simulation, using as input the calibrated trajectories ofthe default intensities and of the common Markov chain, and we analyse the relation betweenthe attachment point of an ESB and the volatility of the ESB-spreads. Second, we carry outa scenario analysis and study how the risk-neutral default probability of these products isaffected by changes in the underlying risk factors. To robustify our conclusions, we consideralso various contagion scenarios. The results of this analysis are more nuanced. For lowsubordination levels and adverse scenarios (such as the case where the default of a major euroarea sovereign leads to a recession in the euro area), the loss probability of ESBies can be fairlylarge and spread trajectories can be quite volatile. For high subordination levels exceeding30–35%, on the other hand, ESBies remain ‘safe’ even in these adverse scenarios. Third, wecompare the risk characteristics of ESBies to those of a safe asset created by pooling the seniortranche of national bonds. Finally, we use simulations to compute value at risk and expectedshortfall for the return distribution of ESBies. For this we need to estimate the historicaldynamics of the default intensities and the common Markov chain which is done via a suitablevariant of the EM algorithm. From this perspective, the market risk of ESBies is fairly low.Summarizing, we find that while in normal times ESBies are indeed very safe (in fact saferthan assets created by pooling the senior tranche of national bonds), they may become riskyunder extreme circumstances and in contagion scenarios, in particular if the attachment pointis not sufficiently high.We continue with a discussion of the relevant literature. The report of the EuropeanSystemic Risk Board (2018) extends the quantitative analysis of Brunnermeier et al. (2017)and considers risk and return characteristics of ESBies and EJBies in various stress scenariosfor default correlation and loss given default; similar issues are studied in Barucci, Brigo,Francischello and Marazzina (2019) in the context of standard copula models for defaults.The relevance of market risk for ESBies is discussed verbally in de Grauwe and Ji (2019).An interesting quantitative analysis of the market associated with ESBies is de Sola Perea,Dunne and Reininger (2019). They compute hypothetical spread trajectories for tranchesof sovereign bond-backed securities in a copula framework, using observed bond spreads asinput. Techniques from time series analysis (a VAR for VaR analysis and multivariate GARCHmodelling) are used to compute value at risk and marginal expected shortfall for the dailyspread change of these tranches. Further interesting contributions on sovereign bond-backedsecurities for the euro area are Langfield (2020) or Cronin and Dunne (2019).Our work is also related to other strands of the literature on sovereign credit risk, securitiza-tion and financial innovation. Ang and Longstaff (2013) and A¨ıt-Sahalia, Laeven and Pelizzon(2014) carry out interesting empirical work on euro area credit spreads. Brigo, Pallaviciniand Torresetti (2010) give an extensive discussion of CDO pricing models and their empiricalproperties before and during the financial crisis, see also McNeil, Frey and Embrechts (2015).We also use insights from Gennaioli, Shleifer and Vishny (2012) or Golec and Perotti (2015)regarding safe assets and financial innovation. Mathematical results on affine processes withMarkov modulated mean reversion level can be found in Elliott and Siu (2009) and in van3eek, Mandjes, Spreij and Winands (2020).The remainder of the paper is structured as follows. In Section 2 we formally introducethe model and the relevant credit products. Section 3 outlines the calibration of our model tomarket data. The main part of the paper is Section 4 where we carry out a thorough analysis ofthe risks associated with ESBies: in Section 4.1 we discuss the S & P proposal for the rating ofESBies and we relate this to model-independent price bounds, Section 4.2 focuses on expectedloss, while Sections 4.3 to 4.6 deal with the market risk of ESBies. In Section 5 we summarizethe findings from the risk analysis and discuss policy implications. Default model.
Throughout we consider a portfolio of J sovereigns with default times τ j and default indicators { τ j ≤ t } , 1 ≤ j ≤ J , defined on a probability space (Ω , F , Q ) withfiltration G = ( G t ) t ≥ . G is the global filtration, that is all processes introduced are G adapted. In financial terms the σ -field G t describes the information available to investorsat time t . We assume that (Ω , F , Q ) supports a J -dimensional standard Brownian motion W = ( W t , . . . , W Jt ) t ≥ and a finite-state Markov chain X , independent of W , with statespace S X = { , , . . . , K } and generator matrix Q = ( q kl ) ≤ k,l ≤ K . The chain X will be usedto model transitions between K different states or regimes of the euro area economy, and for k (cid:54) = l , q kl gives the intensity of a jump from state k to state l . The measure Q is the risk-neutralmeasure used for the valuation of ESBies; price dynamics under the historical measure P areconsidered in Section 4.6. In the analysis of the model we also use the filtration F = ( F t ) t ≥ that is generated by the Brownian motion W and the Markov chain X .Our default model under the pricing measure Q is outlined in the following two assumptions. A1)
The default times τ , . . . , τ J are conditionally independent doubly stochastic defaulttimes with F adapted hazard rate processes γ , . . . , γ J , see for instance McNeil et al.(2015, Chapter 17). In mathematical terms, for all t , . . . , t J > Q (cid:0) τ ≥ t , . . . , τ J > t j | F ∞ ) = J (cid:89) j =1 exp (cid:16) − (cid:90) t j γ js ds (cid:17) . A2)
The processes γ , . . . , γ J follow CIR-type dynamics with Markov modulated and time-dependent mean-reversion level, that is dγ jt = κ j ( µ j ( X t ) e ω j t − γ jt ) dt + σ j (cid:113) γ jt dW jt , ≤ j ≤ J, (2.1)for constants κ j , σ j > , ω j ≥ µ j : S X → (0 , ∞ ). For notational conve-nience, we introduce the vector process γ = ( γ t , . . . , γ Jt ) t ≥ . Discussion.
For small ∆ t the quantity { τ j >t } γ jt ∆ t gives the probability that firm j defaultsin the period ( t, t + ∆ t ], that is γ j is the default intensity of firm j . Assumption A1 impliesthat given the path ( γ s ( ω )) ∞ s =0 of the hazard rate process, τ , . . . , τ J are independent defaulttimes. Dependence of default events is caused by the special form of the hazard rate dynamicsin A2. More precisely, the assumption that the mean-reversion levels µ , . . . , µ J of the hazardrate processes depend on the common finite-state Markov chain X creates co-movement in the4azard rate of different sovereigns, so that, unconditionally, default times are dependent. Oursetup permits also country-specific fluctuations in hazard rates; these are generated by theindependent Brownian motions W , . . . , W J driving the hazard rate dynamics. Adding thefactor e ω j t implies that the mean-reversion level of the hazard rates is upward-sloping betweentransitions of X . This helps to calibrate the model to the observed term structures of sovereignCDS spreads which are typically upward-sloping as well; see Section 3 for details.Following Brunnermeier et al. (2017), we usually consider K = 3 states of the euro areaeconomy. In the model calibration in Section 3 we find that, for the vast majority of euro areamembers, µ j (1) < µ j (2) < µ j (3); that is, the mean reversion level of the hazard rates of euroarea members is lowest in state one and highest in state three. This allows us to interpret thesestates as expansion (state one), mild recession (state two) and strong recession (state three).A statistical analysis in Section 4.6 shows that a model of the form (2.1) with K = 3 statescan also be used to describe the evolution of the calibrated hazard rates under the historicalmeasure P .The default model outlined in Assumptions A1) and A2) is well-suited for a risk analysisof ESBies. In contrast to the copula models used for instance in Barucci et al. (2019) or inthe work of the ESRB (2018), we model the dynamic evolution of hazard rates and creditspreads. This allows us to generate future spread trajectories, which is important in theanalysis of market risk. By assuming that the hazard rates depend on the common state ofthe Euro area economy we generate default dependence in a natural way. This gives a lotof flexibility for the valuation of ESBies. In fact, the whole range of arbitrage-free prices ofESBies and EJBies consistent with observed CDS spreads can be obtained within our model ifparameters are chosen appropriately; see Section 4.1 for details. At the same time the modelis fairly tractable: due to the conditional independence assumption it is possible to calibratethe model simultaneously to CDS spreads of all euro area sovereigns and the form of hazardrate dynamics allows for a fairly efficient computation of credit derivative prices.On the other hand, our pricing model with conditionally independent defaults does notallow for contagion effects (upward jumps in the credit spreads of non-defaulted sovereignsin reaction to a default event in the euro area), which might arise if insufficient measures aretaken to mitigate the economic fallout from the default of a major euro area member, seeB´enassy-Qu´er´e et al. (2018). This is, however, not an issue for studying the risks associatedwith ESBies. In fact, with appropriately chosen hazard rate dynamics our pricing model is ableto generate arbitrarily conservative (low) valuations for ESBies. Moreover, contagion mattersmost in the analysis of short term price fluctuations and market risk in Section 4.4, and we doconsider contagion scenarios in that context. Loss process and credit default swaps.
The payoff of credit default swaps (CDSs),ESBies and EJBies depends on the exact form of the losses generated by defaults in theunderlying sovereign-debt portfolio. Next we therefore describe the mathematical model forthe loss processes that we use in our analysis. We fix a horizon
T > T of paymentdates 0 = t < t < · · · < t N = T which, in practical applications, usually correspond to Without conditional independence, the price of single-name credit derivatives depends on the default stateand the hazard rate of other sovereigns in the portfolio, and the calibration of the model to single-name CDSspreads is practically possible only for very small portfolios. For instance, in the Hawkes process model ofA¨ıt-Sahalia et al. (2014) spillover effects are only studied for the bivariate case. ≤ j ≤ J the cumulative loss process L j of sovereign j by L jt = N (cid:88) n =1 { t n − <τ j ≤ t n } { t ≥ t n } δ jt n , t ∈ [0 , T ] , (2.2)where the random variable δ jt n gives the loss given default (LGD) of sovereign j at time t n . We assume that, given F t n , the LGD δ jt n is beta distributed with E (cid:0) δ jt n | F t n (cid:1) = δ j ( X t n ) for afunction δ j : S X → (0 , X t n , δ jt n is independent of all othermodel quantities. Working with a random LGD is realistic and, at the same time, helps torobustify our analysis with respect to the exact values chosen for δ j . Given portfolio weights w j > (cid:80) Jj =1 w j = 1, we define the portfolio loss by L t = J (cid:88) j =1 w j L jt , t ≤ T . (2.3)The cash flow stream of the protection-buyer position in a CDS on sovereign j with spread x and premium payment dates T can be described in terms of the process L j ; it is given by L jt − (cid:88) t n ≤ t x ( t n − t n − ) { τ j >t n } , ≤ t ≤ T. (2.4) ESBies and EJBies.
ESBies have not been issued so far, so there is no description of thepayment structure of an actual product and no term sheet. Therefore, we consider stylizedversions of these products. These stylized ESBies and EJBies do capture the essential featuresof every CDO structure, namely pooling and tranching of default risk, so they suffice to analyzethe qualitative properties of ESBies. Denote by V T = 1 − L T the normalized value of the assetpool and note that V T = 1 if there are no defaults in the portfolio. The constant κ ∈ (0 , T is defined to beESB T = min( V T , − κ ) = V T − ( V T − (1 − κ )) + = (1 − L T ) − ( κ − L T ) + , (2.5)EJB T = ( V T − (1 − κ )) + = ( κ − L T ) + . (2.6)In this way, the EJB bears the first 100 κ percent of the loss in the portfolio, if the loss exceeds κ , the ESB is affected as well. While stylized ESBies and EJBies are path independent, inthe sense that their payoff is a function of the portfolio loss at the maturity date T only, ouranalysis is easily extended to path dependent payoffs.Note that, by definition, we have the following put-call-parity-type relation for the payoffof a stylized ESB and a stylized EJB with identical attachment point κ ESB T + EJB T = V T and hence ESB T = (1 − L T ) − EJB T . (2.7) Pricing.
For simplicity, we assume that the risk-free short rate is constant and equal to r ≥
0. We introduce the money market account B t,s = exp( r ( s − t )), s > t , so that B − t,s is thediscount factor at time t for a payoff due at time s . We use standard risk-neutral valuation We prefer to work with (2.2) instead of the more standard definition L jt = { τ j ≤ t } δ j ( X τ j ) as (2.2) is moreconvenient for CDS pricing. In any case, for ( t n − t n − ) small the two definitions of L j are close to each other. t of any integrable G s measurablecontingent claim H is equal to H t = E (cid:0) B ( t, s ) − H | G t (cid:1) , where the expectation is taken underthe risk-neutral measure Q .For further use we introduce some notation related to the pricing of ESBies. Let L t =( L t , . . . , L Jt ). The price of an ESB at time t ∈ { t , t , . . . , t N } is given by E (cid:16) B − t,T ((1 − L T ) − ( κ − L T ) + ) | G t (cid:17) =: h ESB ,κ ( t, X t , γ t , L t ) (2.8)for a suitable function h ESB ,κ . This follows from the fact that the processes (cid:0) X t n , γ t n , L t n ) Nn =0 are jointly Markov; we omit the details. Similarly, the price of an EJB is given by h EJB ,κ ( t, X t , γ t , L t ) := E (cid:16) B − t,T ( κ − L T ) + ) | G t (cid:17) . (2.9)The key tool for the numerical computation of derivative prices is the extended Laplace trans-form of the hazard rates. For Markov modulated CIR processes this transform is available inalmost closed form; see Appendix A for details. Data and calibration design.
The available data consist of weekly CDS spread quotes fromICE data services for ten euro area sovereigns and times-to-maturity equal to 1, 2, 3, 4 and 5years over the period January 7, 2009 until September 3, 2018, giving rise to 510 observationdates. The sovereigns used in our analysis are Austria (AUT), Belgium (BEL), Germany(DEU), Spain (ESP), Finland (FIN), France (FRA), the Republic of Ireland (IRL), Italy(ITA), the Netherlands (NLD) and Portugal (PRT), making up more than 90% of the euro areaGDP in 2018. Table 4 reports summary statistics (sample mean, sample standard deviation,minimum and maximum) of the CDS spreads, together with the most recent Standard & Poor’scredit-rating of the ten sovereigns. Average spreads vary considerably across countries and,with the exception of Ireland, the term structures of the average spreads is upward sloping.We calibrate the model by minimizing the sum of squared differences between the CDSspreads observed on the market and the spreads generated by the model. In order to reduce thedimension of the parameter space, we fix the mean function and the concentration parameterof the beta distribution of the loss given default δ jt n at the outset. The distinct values for themean function δ j ( · ) can be found in Table 5 in the appendix. Following Brunnermeier et al.(2017) we assume that the mean LGD is highest in state 3 (the recession state) and lowestin state 1. Moreover, we work with a concentration parameter ν = 1 .
5; this is a conservativechoice as it leads to a fairly widespread LGD distribution (and we will see in Section 4 thata widespread LGD distribution makes ESBies riskier). While the order of magnitude of themean LGD is in line with the sovereign-debt literature, the exact numerical values for the meanLGD and the concentration parameter ν were handpicked by the authors. In Section 4.5 we The parametrization in terms of mean and concentration parameter is a useful alternative to the standardrepresentation of the beta distribution. Denote by g ( x ; a, b ) = β ( a, b ) x a − (1 − x ) b − { x ∈ [0 , } the beta densityfor given parameters a, b >
0. Then the mean is given by a/ ( a + b ) and the concentration parameter is ν := a + b .A high value of ν implies that the LGD is very concentrated around its conditional mean. In fact, ν cannot be calibrated from CDS spreads, since model CDS spreads depend only on the conditionalmean of the LGD. K = 3 states of X and we use the EONIA at date t as aproxy for r t . We have to determine the trajectories of X and γ and the parameters (Θ j , σ j , Q )with Θ j = ( µ j (1) , µ j (2) , µ j (3) , κ j , ω j ). We impose the restriction that all parameters arenonnegative, and, to preserve the interpretation of µ j ( · ) as mean-reversion level, we imposethe uniform lower bound κ j > . j . We use s < s < · · · < s M to denote theobservation dates and we write { γ s m } = { γ s , . . . , γ s M } and { X s m } = { X s , . . . , X s M } fortrajectories of γ and X . Denote by cds js m ,u the market CDS spread with time to maturity u at time s m and by (cid:99) cds( u, γ js m , Θ j , σ j , Q, X s m ) the corresponding model spread as function of γ js m , X s m and of the model parameters. We determine the model parameters and the realizedtrajectories { γ s m } and { X s m } by minimizing the global calibration error M (cid:88) m =1 J (cid:88) j =1 (cid:16) cds js m ,u − (cid:99) cds( u, γ js m , Θ j , σ j , Q, X s m ) (cid:17) , using a set of modern optimization algorithms. For this we use an iterative approach which isdescribed in detail in Appendix C. Results.
We implement the calibration methodology on the full time series of available CDSdata. We use maturities of one and five years since one-year CDS spreads are particularlyinformative regarding the current value of the hazard rates whereas five-year CDS marketsare most liquid. To assess the quality of the calibration, we report in Table 1 the root meansquared error (RMSE) for all countries and both maturities. As RMSE is scale-dependent, wealso report a relative measure for the calibration error, namely the mean absolute percentageerror (MAPE). The quality of the calibration is illustrated further in Figure 9 in Appendix C,where we plot the time series of CDS spreads together with the model prices and the absolutepricing errors for the Germany and Italy. Given the complexity of the calibration task, weconclude that the calibrated model fits the observed CDS spreads reasonably well.
Mat. AUT BEL DEU ESP FIN FRA IRL ITA NLD PRTRMSE (bp)1 6 .
36 8 .
70 4 .
73 39 .
72 3 .
58 6 .
75 4 .
58 0 .
45 2 .
90 1 .
495 15 .
58 15 .
30 9 .
26 45 .
41 6 .
73 13 .
07 40 .
10 34 .
76 10 .
60 66 . .
25 27 .
48 20 .
55 34 .
47 29 .
60 23 .
94 5 .
27 0 .
45 6 .
17 1 .
865 26 .
29 24 .
36 20 .
85 20 .
11 15 .
79 21 .
62 28 .
54 15 .
38 20 .
74 27 . Table 1:
Calibration error in basis points for maturities of one and five years.
Tables 2 and 3 report the parameter values resulting from the calibration. First, note that µ j (1) < µ j (2) < µ j (3) for all sovereigns except Germany, where µ (1) ≥ µ (2). The uniformordering of the mean-reversion levels allows us to interpret the states of X as expansion, mildand strong recession, and it provides clear evidence that there is strong co-movement in themarket’s perception of the credit quality of euro area members. The resulting ordering of the This reverse ordering is easily explained by Germany’s prominent role as the euro area’s safe haven in timesof financial distress. δ j (1) ≤ δ j (2) ≤ δ j (3) for all sovereigns). The mean reversion speed κ j is quite low for all countries,and for four of them (Austria, Belgium, Finland and France) it is equal to the exogenouslyimposed lower bound of 0 .
1. Consequently, market participants expect idiosyncratic creditshocks to have a long-lasting effect across the term structure of CDS spreads. The motivationfor including the parameter ω j is to better capture the upward sloping term structure of mostof the CDS series. In fact, for ω j = 0 and unrestricted κ j , the calibration frequently leadsto negative values for κ j — a common phenomenon also reported e.g. in Ang and Longstaff(2013). Table 3 reports the estimate of the generator matrix Q . Note that, for the estimated Q , transitions to non-neighbouring states have zero probability.Figure 1 plots the calibrated hazard rates together with the calibrated trajectory of theMarkov chain X . The process X remains in state one for most of the sample period, the onlyexceptions occur at the height of the European sovereign debt crisis from mid-2010 until late2013, when the chain visits states two and three before settling in state one again. In general,the paths of the hazard rates are in line with the movement of the Markov chain; exceptionalindividual events such as the rise of the Portuguese hazard rates at the beginning of 2016 orthe sudden upward movement of Italian rates during mid-2018 are of idiosyncratic nature. Param. AUT BEL DEU ESP FIN FRA IRL ITA NLD PRT µ (1) 0 . . . . . . . . . . µ (2) 0 . . . . . . . . . . µ (3) 0 . . . . . . . . . . κ . . . . . . . . . . ω . . . . . . . . . . σ . . . . . . . . . . Table 2:
Calibration results: parameters of hazard rate dynamics .
State 1 State 2 State 3State 1 (expansion) − . . . . − . . . . − . Calibration results: generator matrix Q of X . After the successful calibration of our model, we may now analyze the risks associated withESBies. We begin with a short overview. In Section 4.1 we discuss a recent proposal ofthe rating agency Standard and Poors (S&P) for the rating of ESBies (Kraemer (2017)) andwe relate the S&P proposal to a worst-case default scenario where the arbitrage-free price ofESBies attains its lower bound. In Section 4.2 we compute the risk-neutral expected loss (orequivalently the credit spread) of ESBies as a function of the attachment point κ for differentparameter sets. We consider a base parameter set corresponding largely to the parametersobtained in the model calibration of Section 3, two crisis sets with higher default correlationand an extremal distribution that corresponds to the worst-case default scenario.9
010 2012 2014 2016 2018 . . . . . . . Estimated Hazard Rates as.Date(dates) g t AUTBELDEUESPFINFRAIRLITANLDPRT
Estimated Markov chain X t Figure 1:
Time series plots of the estimated hazard rates and the calibrated Markov chain. Note thatwe graph √ γ t as this is the natural scale for a CIR process. The subprime credit crisis has shown that the expected loss at maturity gives only limitedinformation regarding the riskiness of tranched credit products such as ESBies. In fact, themarket value of AAA-rated senior tranches of mortgage backed securities (MBS) fell sharplyduring the crisis (some were even downgraded), creating huge losses for many MBS investors.To analyze if ESBies can perform all functions of a safe asset, we thus need to take a closer lookat the associated market risk. We do this in several ways. First, we use a historical simulationapproach and compute credit spread trajectories of ESBies for different attachment points,using as input the calibrated trajectories { X s m } and { γ s m } from Section 3. This analysis givesuseful information on the relation between κ and the volatility of ESB credit spreads. Second,many potential ESB investors, such as managers of money market funds, are extremely riskaverse so that “behavior in (quasi) safe asset markets may be subject to sudden runs whennew information suggests even a minimal chance of a loss” (Golec and Perotti 2015). InSection 4.4 we therefore study how the risk-neutral loss probability Q ( L T > κ ) of ESBies isaffected by changes in the underlying risk factors. To guard against model misspecification andto incorporate stylized facts regarding investor behavior on markets for safe assets, we includevarious contagion scenarios into this analysis. Third, we compare the risk profile of ESBies tothat of a safe asset created by pooling the senior tranche of national bonds and we study therobustness of both product classes with respect to the LGD distribution, see Section 4.5. InSection 4.6 we finally use simulations to study Value at Risk and Expected Shortfall for themark-to-market loss of ESBies. For this we resort to the model dynamics under the real-worldmeasure. 10 .1 The weak-link approach of S&P and worst-case default scenarios In a recent technical document, Kraemer (2017) discusses how the rating agency Standard andPoors (S&P) would determine a rating for ESBies and EJBies. The proposed methodology istermed weak-link approach . The S&P proposal has led to a lot of discussion since it associatesa BBB- rating to an ESB with attachment point κ = 30% (given sovereign-bond ratings of2017), which is at odds with the idea that ESBies are safe assets meriting top ratings.To facilitate the description of the approach, we assume that the sovereigns are orderedaccording to their rating, so that sovereign one has the best rating and sovereign J has theworst rating. Given an ESB with attachment point κ , define the index j ∗ by j ∗ = max { ≤ j ≤ J : (cid:80) Ji = j w i ≥ κ } . Then, under the weak-link approach, the ESB is assigned the rating ofsovereign j ∗ . The assumption underlying this approach is that “sovereigns will default in theorder of their ratings, with lowest rated sovereigns defaulting first” (Kraemer 2017, Page 4)and that the LGD of all sovereigns is equal to one, so that the ESB incurs a loss as soon asthe sovereign j ∗ defaults.In this section we show that the weak-link approach is extremely conservative in variousrespects. We begin by a concise mathematical description. We drop the time index andconsider sovereign debt portfolios with generic loss variables L j = δ j { τ j ≤ T } , 1 ≤ j ≤ J , withvalues in the interval [0 , E (cid:0) L j (cid:1) = ¯ (cid:96) j for a constant ¯ (cid:96) j ∈ [0 , (cid:96) ≤ ¯ (cid:96) · · · ≤ ¯ (cid:96) J . Next, we define loss variables that represent the default scenarioof the weak link approach. Fix some standard uniform random variable U and define the lossvector L ∗ = ( L ∗ , . . . , L ∗ J ) by L ∗ j = { U> − ¯ (cid:96) j } , ≤ j ≤ J (4.1)Clearly, E (cid:0) L ∗ j (cid:1) = Q ( L ∗ j = 1) Q ( U > − ¯ (cid:96) j ) = ¯ (cid:96) j , so that L ∗ respects the expected-lossconstraint. Moreover, under (4.1) sovereigns default exactly in the order of their credit qualitywith sovereign J defaulting first and δ j = 1 for all j , that is L ∗ is indeed a mathematical modelfor the weak link approach. Note that the loss vector L ∗ is comonotonic since its componentsare given by increasing functions of the same one-dimensional random variable U , see McNeilet al. (2015, Chapter 7). Hence, under the weak-link approach, diversification effects betweeneuro area members are ignored completely.The next result shows that the loss variables in (4.1) can be interpreted as worst-casedefault scenario in the sense that they minimize the value of ESBies over all loss variables thatrespect the expected loss constraints. Hence, the price of an ESB under the worst-case scenariois a lower bound for the arbitrage-free price of that bond in any model consistent with theseconstraints. In the rating context this means that the weak link approach associates with anESB the worst rating logically consistent with the ratings of the individual euro area sovereigns. Proposition 4.1.
Define for generic loss variables L = ( L , . . . , L J ) such that L j takes valuesin the interval [0 , and E (cid:0) L j (cid:1) = ¯ (cid:96) j and fixed weights w , . . . , w J summing to one the portfolioloss by L = (cid:80) Jj =1 w j L j . Then it holds for κ ∈ [0 , that E (cid:16) − L ∗ − (cid:0) κ − L ∗ (cid:1) + (cid:17) ≤ E (cid:16) − L − (cid:0) κ − L (cid:1) + (cid:17) . (4.2)The proof can be found in Appendix B. 11e now discuss several economic implications. First, under the worst-case default scenariothe LGD of all sovereigns is almost surely equal to one. Note that under constraints on theexpected loss a high LGD implies a low value for the default probability of a given sovereign,so that L ∗ corresponds to a default scenario with ‘few but large losses’. Second, the worst-casedefault scenario maximizes the probability of large default “clusters” given the expected lossconstraints. This is explained in detail in Appendix B where we discuss properties of thedistribution π ∗ of L ∗ . Third, note that it is possible to approximate the worst-case defaultscenario by properly parameterized versions of the model introduced in Section 2; a preciseconstruction is given in Appendix B. This shows that it is possible to generate arbitrarilyconservative valuations for ESBies in our setup.The qualitative properties of L ∗ suggest that, in the dynamic default model from Section 2,an ESB is more risky for a given expected-loss level of the sovereigns if one chooses highvalues for the mean reversion level of the default intensities in the recession state K , so thatmany defaults are quite likely in that state; at the same time the generator matrix has to beparameterized in such a way that state K is visited relatively infrequently in order to meet theexpected loss constraints. This intuition underlies the construction of the crisis scenarios inthe numerical experiments reported in the next sections. More generally, Proposition 4.1 givesa theoretical justification for the qualitative properties of ESBie prices observed in Section 4.2and in the work of Barucci et al. (2019), Brunnermeier et al. (2017) or ESRB (2018): for agiven expected-loss level of the sovereigns higher default correlations and a higher LGD leadsto a higher expected loss for ESBies. From now on we consider ESBies with a time to maturity of five years and, for simplicity, arisk free interest rate r = 0. In order to make the prices of ESBies with different attachmentpoints κ comparable, we consider normalized ESBies with payoff − κ min( V T , − κ ), so thatthe payoff of a normalized ESB is equal to one if there is no default, i.e. for L T ≤ κ . Moreover,we introduce the risk-neutral expected tranche loss (cid:96) ESB ,κ (0 , X , γ , L ; Q, µ ) = 1 − − κ h ESB ,κ (0 , X , γ , L ; Q, µ ) . (4.3)Here µ = { µ j ( k )) , ≤ k ≤ , ≤ j ≤ J } , Q is the generator matrix of X and the function h ESB ,κ ( t, X t , γ t , L t ; Q, µ ) gives the price of an ESB with attachment point κ at time t , seeequation (2.8). We have made the parameters Q and µ explicit in (4.3) since we want to studyhow variations in their values affect the expected loss of ESBies. Note that we may interpretthe annualized expected loss T (cid:96) ESB ,κ as credit spread c ESB ,κ (0 , T ) of a normalized ESB withattachment point κ . In fact, since r = 0 and since for x close to one ln x ≈ x −
1, it holds that c ESB ,κ (0 , T ) = − T ln (cid:16) − κ h ESB ,κ (cid:17) ≈ T (cid:96)
ESB ,κ . Parameters.
As before, we work with K = 3 states of X . We choose the portfolio weights w j according to the GDP proportions within the euro area; numerical values are given inTable 6 in Appendix C. We use the mean LGD from Table 5, the volatility parameters σ j and the calibrated trajectories { γ s m } and { X s m } obtained in Section 3. In our numericalexperiments we consider three different parameter sets and the worst-case distribution π ∗ (the12istribution of the worst case default scenario from Proposition 4.1). In the base parameterset we use the generator matrix from Section 3. We take ω j = 0 and calibrate µ and κ j tothe full CDS term structure at the valuation date, so that the parameterized model accuratelyreflects the market’s expectation at that date. The generator matrix Q is hard to calibrate from historical data, essentially since productsdepending on the default correlation of euro area countries are not traded. To deal withthe ensuing model risk, we introduce two crisis parameter sets . In these parameterizations therecession state (state three) occurs less frequently than under the base parametrization, but if itoccurs default intensities are (on average) substantially larger than for the base parameter set.To achieve this, we consider two generator matrices (cid:101) Q and (cid:101) Q chosen such that, on average, X spends less time in state three than under the base parametrization. The correspondingmean reversion levels (cid:101) µ and (cid:101) µ and are determined from the constraint that the expectedloss E ( L jT ) is identical for all parameter sets; this typically leads to µ j (3) < (cid:101) µ j (3) < (cid:101) µ j (3).The entries of (cid:101) Q , (cid:101) Q are provided in Table C.4. Results.
In the top panel of Figure 2, we graph the average expected loss of ESBies overthe period from 2014 to September 2018 as a function of the threshold κ . We do this for thebase parametrization, the two crisis parameterizations and the worst-case distribution fromProposition 4.1. The scale for the y -axis is logarithmic and values are given in percentagepoints. In addition, we consider AAA- and A- rated sovereigns (DEU, NLD and IRL, ESP,respectively) and compute the 1%- and 99%-quantile of the risk-neutral expected loss over theperiod from 2014 to September 2018. Those quantiles form the boundaries of the colored areasin Figure 2; they are supposed to give an indication of the credit quality for the ESBies on arating scale. From Figure 2 we draw the following conclusions. First, the average risk-neutral expectedloss of ESBies is indeed small. For example, the average expected loss corresponding to theproposed attachment point of 0.3 is below 0.1%. Most strikingly, except for the worst-casedistribution, the average expected loss of ESBies with thresholds of 0.15 or higher is well belowthe lower bound of the AAA-region. Second, the expected loss is lowest for the base parameters,followed by crisis parameterizations 1 and 2; this is fully in line with the economic intuitionunderlying the construction of these parameter sets. Third, the expected loss for the worst-casedistribution (which is highest by construction) is substantially higher than the expected lossin the crisis parameterizations, underlining the fact that the worst-case distribution, and theassociated weak-link approach of Kraemer (2017), are extremely conservative. Nonetheless, for κ > .
25 the average expected loss for the worst-case distribution is still comparable in size tothat of AAA-rated sovereigns. Fourth, the expected loss of an ESB is decreasing approximatelyat an exponential rate in κ in all four parameter sets (recall that we use a logarithmic scalefor the y -axis). Summarizing, these findings show that an investor willing to hold ESBies with Using a different value for ω j has only a very minor impact on the spread and the loss probability of ESBies. The calibration in Section 3, on the other hand, yields a fixed set of parameters giving a reasonable fitthroughout the entire observation period. This provides evidence for the good performance of our model inexplaining market data, but is of course subject to small pricing errors at any given date. Here the term “average” refers to the average over observation dates, but with a fixed time to maturity offive years, that is we plot the function κ (cid:55)→ M (cid:80) Mm =1 (cid:96) ESB ,κ (0 , X s m , γ s m , ; Q, µ ). We stress that these indicative ratings should not be taken as actual ratings of ESBies, since they arecomputed from risk-neutral expected losses and not from historical ones, and since a rating is more than amechanical mapping of expected loss to some rating scale.
13n attachment point of 0.15 or higher until maturity faces little risk of default-induced losses,which is in agreement with the analysis of Brunnermeier et al. (2017) or Barucci et al. (2019).The bottom panel of Figure 2 shows the average expected loss of EJBies for varying attach-ment points. With five-year expected loss levels ranging from 6% to around 15% (and henceannualized credit spreads between 1.2% and 3%) the risk of EJBies is comparable to that oflower-quality euro area sovereigns. Comparing the expected loss of ESBies and EJBies, wesee that, in line with the proposal of Brunnermeier et al. (2017), EJBies bear the bulk of thecredit risk associated to the eurozone sovereigns. Note that the reverse ordering of the lines inthe two panels of Figure 2 is an immediate consequence of the put-call parity relation (2.7). . . . . . . . threshold % Static Risk AnalysisAvg. Expected Loss of ESBies (2014 − 2018)
AAAA worst case distributioncrisis generator Matrix 2crisis generator Matrix 1base threshold % Avg. Expected Loss of EJBies (2014 − 2018) crisis generator Matrix 2crisis generator Matrix 1base
Figure 2:
Average expected loss of ESBies (top) and of EJBs (bottom) for different thresholds andparameterizations (in %). Note that both graphs use a logarithmic scale on the y -axis. In Figure 3 we plot trajectories of the annualized credit spread c ESB ,κ of ESBies over the wholesample period for different levels of κ . These spreads were computed from our model by a14istorical simulation approach using the calibrated trajectories { γ s m } and { X s m } as input.The solid line gives the spread of an ESB with attachment point κ = 0 . κ ∈ [0 . , . κ closeto 0.2 spreads are very volatile; for κ > . . . . . . % Time Series Analysis of ESBiesCredit Spreads for varying Thresholds t h r e s ho l d − . . . % Difference in Spreads in Comparison with Reference Threshold (0.3) t h r e s ho l d Figure 3:
Spread trajectories of ESBies with varying threshold levels. The solid black line representsthe reference threshold of 0.3.
In this section, we analyze how the risk-neutral loss probability Q ( L T > κ ) of ESBies is affectedby changes in the underlying risk factors X , γ and L . In mathematical terms, we considerthe function κ (cid:55)→ p κ ( X , γ , L ; Q, µ ) := Q ( L T > κ | X , γ , L ; Q, µ ) . We consider different sets of risk factor changes or scenarios . First we study scenarios whichare included in the support of the default model from Section 2, such as a change in the stateof X . Moreover, we consider several contagion scenarios where, in reaction to a default ofItaly, the market becomes more risk averse and changes its perception of the state of X andthe parameter set used to value ESBies. In fact, investors on markets for (quasi) safe assetsfrequently change their expectations in reaction to adverse events, putting more mass on badoutcomes; see for instance Gennaioli et al. (2012). We consider a default of Italy since on the day we used for this analysis (September 3, 2018, the lastobservation date in our sample) Italy had the highest CDS spread of all major euro area economies. A defaultof another major ‘risky’ euro area sovereign would yield similar results. on-contagion scenarios. In the left panel of Figure 4, we graph the function p κ ona log-scale using the parameters of the base scenarios and the calibrated values of γ and X for September 3, 2018 (the last observation date in our sample). The full circles givethe loss probability for varying κ for the base scenario , where the chain is in state one (thegood economic state) and no euro area member is in default (in mathematical terms this isthe function κ (cid:55)→ p κ (1 , γ , L ; Q, µ )). Moreover, we consider four types of changes in theunderlying risk factors:(i) the scenario where all hazard rates experience an upward jump of 10%, that is we plotthe function κ (cid:55)→ p κ (1 , γ × . , L ; Q, µ );(ii) the scenario where the economy moves to a light recession, corresponding to the function κ (cid:55)→ p κ (2 , γ , L ; Q, µ );(iii) the scenario where the economy moves to a severe recession ( κ (cid:55)→ p κ (3 , γ , L ; Q, µ ));(iv) the scenario where Italy defaults with random LGD δ ITA . We assume that δ ITA isbeta distributed with mean 0 .
5, i.e. the loss vector at t = 0 takes the form L =(0 , . . . , , δ ITA , , . . . , κ > .
25, the loss probability remains small evenafter a major default. The most important risk factor changes are clearly changes in the stateof the economy. For instance, for κ = 0 . κ the threshold probabilities are decreasingin κ roughly at an exponential rate, similarly as the expected loss does. In fact, the lossprobability is quite sensitive with respect to the choice of the attachment point (to see this,one may compare the values of p κ for κ = 0 .
35 and κ = 0 . Contagion scenarios.
In the right panel of Figure 4 we graph the function p κ (again on alog-scale) for the base parametrization and for three different contagion scenarios, namely(i) the case where Italy defaults and where, as a reaction, X jumps to state two (mildrecession);(ii) the case where Italy defaults and where, as a reaction, X jumps to state three (strongrecession);(iii) the case where Italy defaults and where, as a reaction, X jumps to state three and themarket uses the crisis parametrization two (instead of the base parameter set). Thisscenario is motivated by the observation that, in the subprime crisis, investors usedmuch more conservative assumptions for default dependence than before the crisis, seefor instance Brigo et al. (2010) for details. We found this exponential decay for a wide range of parameter values, but we do not have a fully convincingtheoretical justification for this effect.
16e see that, for an attachment point κ ≤ .
3, the change in the loss probability caused byone of the contagion scenarios is quite substantial. For instance, in the extreme scenario (iii),the risk-neutral loss probability is of the order of 5%. For attachment points κ > .
35 theimpact is less drastic. However, under scenario (iii), even for κ = 0 .
35 we get a risk-neutralthreshold probability of around 2%, which is definitely non-negligible for a safe asset. This isin stark contrast to the analysis of the expected loss in Section 4.2, where ESBies appeared‘safe’ already for κ > . l l l l l l l l l l l l l l l l l l l . . . . . . . . threshold p r obab ili t y l l l l l l l l l l l l l l l l l l l D E U BE L I R L ll switch to state 3switch to state 2default of Italyrise of hazard ratesbase l l l l l l l l l l l l l l l l l l l . . . . . . . . threshold p r obab ili t y l l l l l l l l l l l l l l l l l l l D E U BE L I R L ll default of Italy + switch to state 3 (cris. gen. mat. 2)default of Italy + switch to state 3default of Italy + switch to state 2base Probability of Default Events (2018−09−03)
Figure 4:
Loss probability of ESBies for different κ and various scenarios. Note that the plot uses alogarithmic scale on the y -axis. Leandro and Zettelmeyer (2019) and a few other recent contributions suggest an alternativeapproach for constructing a safe asset for the euro area. In these proposals, the euro areasovereigns issue national bonds in (at least) two tranches, a senior and a junior tranche. Asafe asset is then formed by pooling the senior tranche of the national debt, so that we usethe acronym PSNT (pooled senior national tranche) for these products. In this section wecompare the risk-neutral expected loss and the risk-neutral loss probability of PSNTs to thoseof ESBies. In particular, we focus on the impact of the random LGD since we cannot fullycalibrate the distribution of the LGD from available market data. We begin with a formaldescription of the payoff of PSNTs. Given some attachment level κ that marks the splitbetween the junior and the senior national bond tranches, we model the payoff of the seniortranche issued by sovereign j as L j,κT := (1 − κ ) − ( L jT − κ ) + . Note that the senior nationaltranche suffers a loss if L jT exceeds the threshold κ . For fixed weights w , . . . , w J , the payoff From a financial engineering viewpoint also the E-Bonds of Monti (2010) fall in the category of nationaltranching followed by pooling; ESBies on the other hand correspond to pooling followed by tranching.
17f the PSNT is then given by J (cid:88) j =1 w j L j,κT = (1 − κ ) − J (cid:88) j =1 w j ( L jT − κ ) + . Hence the PSNT consists of a safe payment of size 1 − κ and a short position in a weightedportfolio of options on the national losses. The normalized risk-neutral expected loss or equiv-alently the non-annualized credit spread of a PSNT is given by11 − κ J (cid:88) j =1 w j E (cid:0) ( L jT − κ ) + (cid:1) . It follows that the credit spread of PSNTs is independent of the dependence structure of thenational losses L T , . . . , L JT . Assumptions on the distribution of the loss given default, on theother hand, have a huge impact on the spread of PSNTs. We begin with a few qualitativeobservations: first, it is easily seen that for fixed expected loss E ( L jT ), the option price E (cid:0) ( L jT − κ ) + (cid:1) is maximal if L jT ∈ { , } . Hence, for fixed expected loss level of the sovereigns, theexpected loss of a PSNT is maximal if the LGD of all sovereigns is equal to one. In fact,in that case the tranching on the national level offers no additional protection for the PSNTcompared to simply pooling the national bonds. Second, if the LGD of all sovereigns is almostsurely smaller than κ , the PSNT is entirely riskless. Finally, due to the convexity of thefunction (cid:96) (cid:55)→ ( (cid:96) − κ ) + the expected loss of the PNST increases with increasing variance of theLGD distribution (keeping E ( L jT ) fixed).Next, we provide quantitative results comparing the behavior of PSNT credit spreads tothat of ESBies. Throughout this section, whenever we vary the mean of the random LGD,we also recalibrate the remaining model parameters such that any considered LGD setup isstill in line with market data. Figure 5 shows the average spread for ESBies (grey) and forPSNTs (black) over the period 2014-2018 for three different LGD distributions with identicalmean function given in Table 5 and different variance/concentration parameter. We makethe following observations. First, the spread of PNSTs is very sensitive to assumptions onthe variance of the LGD distribution whereas the spread of ESBies is comparatively stable.Second, for the given mean function the spread of PNSTs is substantially higher than forESBies. In fact, even with deterministic LGD the expected loss of an ESBie with κ = 0 . κ ≈ .
6; with higher LGD variance the two expectedlosses are equal only if the attachment point of the PSNT is close to one. The differencebetween the spreads of ESBies and of PSNTs is due to the fact that the default of a singleeuro area sovereign is sufficient to cause a loss for a PSNT, whereas ESBies are only affectedin a severe default scenario with multiple defaults. Moreover, for the payoff of a PSNT itmakes a substantial difference if a sovereign defaults only on its junior bond tranche or onboth tranches which explains the sensitivity with respect to the LGD variance.Finally, we consider the risk-neutral loss probability. Note first that the risk-neutral lossprobability of PSNTs is affected by the dependence structure of L T , . . . , L JT (other than thespread). In fact, for fixed marginal distributions of the sovereign losses, the risk-neutral lossprobability of PSNTs decreases with increasing default correlation. This is akin to the be-haviour of the equity tranche in standard CDO structures. In Figure 6 we graph the risk-neutralloss probability of ESBies and PSNTs for various values of the mean and the concentrationparameter of the LGD distribution. We observe the following: first, for the parameter values18 .0 0.2 0.4 0.6 0.8 1.0 . . . . . . threshold % Avg. Expected Loss of ESBies and PSNTs (2014 − 2018) E L ( ESB i e s ) f o r k = . baselower varianceno varianceGray (black) lines correspond to ESBies (PSNTs) Figure 5:
Spread of ESBies (grey) and of PSNTs (black) for varying κ and different value for theconcentration parameter of the LGD distribution. The graph labelled base case corresponds to ν = 1 . ν = 3 . T = 5. Note that the plot uses a logarithmic scale on the y -axis. considered the loss probability of PSNTs is substantially higher than that of ESBies. Second,changes in the mean and in the concentration parameter have a profound impact on the lossprobability of PSNTs; for ESBies these effects are less pronounced. For a κ = 0 .
3, the lossprobability of an ESB increases from 0 . . κ = 0 .
9, the risk-neutral loss probability of an PSNT increases from0 .
089 to 0 . So far we were concerned with the value of ESBies and EJBies in different scenarios; valueswere computed using the risk-neutral measure Q , so that model parameters were derived viacalibration. On the other hand, for computing measures of market risk for the return ofESBies, we have to simulate their loss distribution under the historical measure P , so that weneed to estimate the P dynamics of X and γ using statistical methods. This issue is addressednext. EM estimation of hazard rate dynamics.
In this section, we report the results of anempirical study where a model of the form (2.1) is estimated from the calibrated hazard ratesof the euro area countries (the trajectories { γ s m } generated in the calibration procedure ofSection 3). Here we assume that the trajectory of the Markov chain is not directly observable;rather, the available information is carried by the filtration F γ = ( F γ t ) t ≥ generated by thehazard rates process γ . This assumption is motivated by the fact that the calibration of the19 llllllllllllllllllllllllllllllllllllll . . . . . r obab ili t y lllllllllllllllllllllllllllllllllllll l basehigher mean lllllllllllllllllllllllllllllllllllllll . . . . . r obab ili t y lllllllllllllllllllllllllllllllllllll l baselower variance PSNTsProbability of Default Events for varying LGD assumptions (2018−09−03)
Figure 6:
Loss probability of PSNTs (black) and ESBies (grey) for varying κ and different value forthe mean of the LGD distribution (left) and for the concentration parameter ν of the LGD distribution(right). The graph labelled base case corresponds to ν = 1 . ν = 3 .
3. We fix T = 5. The plot uses a logarithmic scaleon the y -axis and different scales on the x axis (grey for ESBies and black for PSNTs). trajectory { X s m } in Section 3 is quite sensitive with respect to the chosen model parameters,whereas the calibration of { γ s m } is very robust (essentially due to the close connection betweenhazard rates and one-year CDS spreads).Using stochastic filtering and a version of the EM algorithm adapted to our setting, weobtain the filtered and smoothed estimate for the trajectory of X , an estimate of the generatormatrix of X and of country-specific parameters such as mean reversion levels and speed, allunder the real-world measure P . In the EM algorithm we use robust filtering techniques, whichperform well in a situation where observations are only approximately of the form (2.1). Forfurther details on the methodology see Elliott (1993) or Damian, Eksi-Altay and Frey (2018).We consider K = 3 possible states of X , corresponding to a expansionary regime, a lightrecession and a strong recession, respectively. The EM estimates for the generator matrix Q of X are given in Tables 8 and 9 in Appendix C, together with country-specific parameters suchas mean reversion speed and levels. Note that we do not estimate the volatility, but we workwith quadratic variation instead. Overall the estimates appear reasonable. In particular, theestimated mean reversions levels for most countries respect the ordering µ j (1) < µ j (2) < µ j (3),supporting the interpretation of the states of X . As expected, for any given state of theeconomy the estimated levels are lowest for the stronger euro area countries. In Figure 7, wegive a trajectory of the filtered and the smoothed state of X , that is we plot the trajectories t (cid:55)→ E P ( X t | F γ t ) and t (cid:55)→ E P ( X t | F γ T ). These results show that the proposed model describes In order to robustify the EM estimation procedure, we scale the quadratic variation of the strong euro areacountries slightly upward. For Spain and Portugal, the highest mean reversion level is estimated for state 2, which is probably due tothe idiosyncratic behavior these two countries, particularly Portugal, exhibit in the first months of 2012.
010 2012 2014 2016 2018 E s t i m a t ed t r a j e c t o r y Figure 7:
Filtered ( E P ( X t |F γt )) and smoothed ( E P ( X t |F γT )) estimates of the Markov chain trajectory. the qualitative properties of euro area credit spreads and, in particular, the co-movement ofspread levels of the weaker euro-area members reasonably well. The frequent transitions inand out of the middle state are not surprising, given that this state reflects a situation whereonly a few countries experience a rise in default intensities. Measures of market risk.
We use two popular risk measures, Value at Risk (VaR α ) andExpected Shortfall (ES α ) at confidence level α , to study the tail of the loss distribution ofESBies over a horizon of three months. Denote by γ and X the calibrated hazard rates andthe state of X for September 3, 2018. We generate N = 100 000 realizations of the hazardrates and the Markov chain with initial values γ and X over a three-month horizon, usingthe P -parameters estimated in the previous paragraph, and we index the simulation outcomeby i ∈ { , . . . , N } . We then compute the corresponding relative loss R κ,i := 1 − h ESB ,κ (0 . , X ( i )0 . , γ ( i )0 . , L ) h ESB ,κ (0 , X , γ , L ) . VaR and expected shortfall are then computed from the empirical distribution of the sampledrelative losses { R κ,i , i = 1 , . . . , N } , see McNeil et al. (2015, Section 9.2.6) for details.Figure 8 summarizes our analysis. We plot estimates of VaR α (left) and of ES α (right)for the three-month distribution of negative ESB-returns for different κ and confidence levels α = 0 .
95 (points) and α = 0 .
99 (crosses). We see that both risk measure estimates decreaseapproximately at an exponential rate in κ . The horizontal lines in each plot represent the 95%and 99% level of the corresponding risk measure estimate for a German zero coupon bond.We observe that both the VaR α and the ES α of ESBies with κ ≥ . l l l l l l l l l l l l l l l l l l . . . . . threshold % D E U ( % ) D E U ( % ) l l l l l l l l l l l l l l l l l l l l . . . . . threshold % D E U ( % ) D E U ( % ) l Dynamic Risk Analysis(2018−09−03)
Figure 8:
Risk measure estimates VaR α (left) and ES α (right) for the three-month distribution ofnegative ESB-returns for different κ and confidence levels α ∈ { . , . } . Note that risk measures aregiven in percent and that the plot uses a logarithmic scale on the y -axis. We draw the following key conclusions from the risk analysis of ESBies and PSNTs. Both thestatic risk analysis of Section 4.2 and the investigation of the loss distribution in Section 4.6suggest that, in normal circumstances, diversification works and ESBies with κ > .
25 areindeed very safe products. In line with this finding, we showed that the weak link approachfor the rating of ESBies proposed by S&P is extremely conservative as it corresponds to aworst-case default scenario. We have also seen that for typical parameter values, the spreadand the loss probability of ESBies are substantially lower than those of PSNTs (securitiescreated by pooling the senior tranche of national debt). Moreover, for PSNTs, spread and lossprobability are more sensitive to changes in the LGD distribution than for ESBies. This showsthat from a risk perspective ESBies might be preferable to PSNTs.However, considering solely the results of Section 4.2 and Section 4.6 could lead to an overlyoptimistic picture. The analysis of credit spread trajectories in Section 4.3 and the scenariobased analysis of Section 4.4 show that the attachment point κ needs to be chosen moreconservatively in order to make ESBies robust with respect to fluctuations in the underlyingrisk factors or to changes in the market perception of default dependence. In fact, one hasto take attachment points κ > .
35 for ESBies to be safe even in very adverse scenarios.Moreover, ESBies are most likely to generate large market losses in the aftermath of severeeconomic shocks and in contagion scenarios.From a policy perspective, it is therefore important that a large-scale introduction of ESBiesis accompanied by appropriate policy measures to limit the economic implications of external In fact, from the perspective of an expected loss analysis, already an attachment point κ = 0 .
15 mightsuffice to make ESBies safe.
A Pricing methodology
Our main tool for computing prices of credit derivatives is the following extended Laplacetransform for Markov modulated CIR processes. A related result was derived in Elliott andSiu (2009) for the case a single CIR-type process, see also van Beek et al. (2020)
Proposition A.1.
Denote by F = ( F t ) t ≥ the filtration generated by the Brownian motion W and the Markov chain X . Consider vectors a , u ∈ R J + and a function ξ : S X → R . Fix somehorizon date s ≤ T . Then it holds that for ≤ t < sE (cid:18) ξ ( X s ) exp (cid:18) − (cid:90) st a (cid:48) γ θ dθ (cid:19) e − u (cid:48) γ s | F t (cid:19) = v ( t, X t ) exp (cid:0) β ( s − t, u ) (cid:48) γ t (cid:1) . (A.1) Here β ( · , u ) = ( β ( · , u ) , . . . , β J ( · , u )) (cid:48) and the functions β j ( · , u ) , ≤ j ≤ J , solve the Riccatiequation ∂ t β j ( t, u ) = − κ j β j ( t, u ) + 12 ( σ j ) β j ( t, u ) − a j , < t ≤ s , (A.2) with initial condition β (0 , u ) = − u . Moreover, with v ( t ) = (cid:0) v ( t, , . . . , v ( t, K ) (cid:1) (cid:48) , the function v : [0 , s ] × S X → R satisfies the linear ODE system − ddt v ( t ) − diag (¯ µ ( t ) , . . . , ¯ µ K ( t )) v ( t ) = Q v ( t ) , on [0 , s ] , (A.3) with terminal condition v ( s ) = ξ and with ¯ µ k ( t ) = (cid:80) Jj =1 e ω j t κ j µ j ( k ) β j ( s − t, u ) . The functions β j ( t, u ) are known explicitly, see for instance Filipovic (2009) for details.Essentially, Proposition A.1 shows that computing the extended Laplace transform of γ isnot much more complicated than in the classical case of independent CIR processes; the onlyadditional step is to solve the K -dimensional linear ODE system (A.3) for the function v ( t ),which is straightforward to do numerically. Proof.
We start by conditioning in (A.1) on F t ∨F X ∞ . Due to the independence of the Brownianmotions W , . . . , W J , we have conditional independence of γ , . . . , γ J given F X ∞ , which in turnleads to E (cid:18) ξ ( X s ) exp (cid:18) − (cid:90) st a (cid:48) γ θ dθ (cid:19) e − u (cid:48) γ s | F t ∨ F X ∞ (cid:19) = ξ ( X s ) J (cid:89) j =1 E (cid:18) exp (cid:18) − (cid:90) st a j γ jθ dθ (cid:19) e − u j γ js | F t ∨ F X ∞ (cid:19) . (A.4)23onditional on F X ∞ , the hazard rates γ j are time-inhomogeneous affine diffusions. Standardreferences on affine models, such as Duffie, Pan and Singleton (2000), consequently give that E (cid:18) exp (cid:18) − (cid:90) st a j γ jθ dθ (cid:19) e − u j γ js | F t ∨ F X ∞ (cid:19) = exp (cid:16) α j ( t, s ; X ) + β j ( s − t, u ) γ jt (cid:17) , (A.5)where β j solves (A.2) and where ddt α j ( t, s ; X ) = − e ω j t κ j µ j ( X t ) β j ( s − t ) and α j ( s, s ; X ) = 0;see for instance Duffie et al. (2000) or Section 10.6 of McNeil et al. (2015) for a proof. Integra-tion thus gives α j ( t, s ; X ) = (cid:82) st e ω j θ κ j µ j ( X θ ) β j ( s − θ ) dθ . By iterated conditional expectation,we hence get E (cid:18) ξ ( X s ) exp (cid:18) − (cid:90) st a (cid:48) γ θ dθ (cid:19) e − u (cid:48) γ s | F t (cid:19) = exp (cid:16) J (cid:88) j =1 β j ( s − t ) γ jt (cid:17) E (cid:18) ξ ( X s ) exp (cid:18)(cid:90) st ¯ µ X θ ( θ ) dθ (cid:19) | F t (cid:19) The Feynman Kac formula for functions of the Markov chain X finally gives that E (cid:18) ξ ( X s ) exp (cid:18)(cid:90) st ¯ µ X θ ( θ ) dθ (cid:19) | F t (cid:19) = v ( t, X t ) , and hence the result.Next we consider the pricing of a survival claim and of a CDS on sovereign j . Survival claim.
The payoff of a survival claim on sovereign j with maturity date s andpayoff function f : S X → R is of the form { τ j >s } f ( X s ). Using standard results on doublystochastic default times, the price of this claim at time t ≤ s is E (cid:0) B − t,s { τ j >s } f ( X s ) | G t (cid:1) = { τ j >t } B − t,s E (cid:16) e − (cid:82) st γ js ds f ( X s ) | F t (cid:17) , and the expectation on the right can be computed from Proposition A.1 with a = e j , u = and ξ = f . Credit default swap.
We briefly discuss CDS pricing in our setup, since this is crucialfor model calibration. From the payoff description (2.4), pricing a CDS contract amounts tocomputing the conditional expectation E (cid:32) N (cid:88) n =1 B − t,t n { τ j ∈ ( t n − ,t n ] } δ jt n − N (cid:88) n =1 x ( t n − t n − ) B − t,t n { τ j >t n } | G t (cid:33) . (A.6)Denote by V prem t and V def t the present value of the premium and the default leg, that is V prem t ( x ) = N (cid:88) n =1 B − t,t n x ( t n − t n − ) E (cid:16) { τ j >t n } | G t (cid:17) ,V def t = N (cid:88) n =1 B − t,t n E (cid:16) { τ j ∈ ( t n − ,t n ] } δ jt n ) | G t (cid:17) = N (cid:88) n =1 B − t,t n E (cid:16) { τ j ∈ ( t n − ,t n ] } δ j ( X t n ) | G t (cid:17) . (A.7)24o obtain (A.7), we have used the fact that the default leg of the CDS is linear in the lossgiven default, so that we can replace δ jt n with its conditional expectation. The premium legis simply the sum of survival claims. The evaluation of (A.7) is more involved, and we nowshow how this can be achieved via Proposition A.1. Fix any two consecutive payment dates t n − , t n of T and assume w.l.o.g. that t ≤ t n − . Since { t n − <τ j ≤ t n } = { τ j >t n − } − { τ j >t n } ,we can write the term E (cid:16) { τ j ∈ ( t n − ,t n ] } δ j ( X t n ) | G t (cid:17) in the form E (cid:0) { τ j >t n − } δ j ( X t n ) | G t (cid:1) − E (cid:0) { τ j >t n } δ j ( X t n ) | G t (cid:1) . (A.8)The second term in (A.8) is a survival claim. By iterated conditional expectations, we get thatthe first term is equal to E (cid:0) { τ j >t n − } E (cid:0) δ j ( X t n ) | G t n − (cid:1) | G t (cid:1) . (A.9)Since X is Markov, it holds that E (cid:0) δ j ( X t n ) | G t n − (cid:1) = v δ ( t n − , X t n − ) for a suitable function v δ : [0 , t n ] × S X → R (given by the solution of an ODE system), so (A.9) reduces to computing E (cid:0) { τ j >t n − } v δ ( t n − , X t n − ) | G t (cid:1) , which is a standard pricing problem for a survival claim.Finally, we turn to the pricing of ESBies. In order to evaluate the function h EJB ,κ we use Monte Carlo simulation. For the computation of the function h ESB ,κ we use that h ESB ,κ = E (cid:16) B − t,T (1 − L T ) | G t (cid:17) − h EJB ,κ and we compute the expected discounted portfolioloss analytically. B Worst-case default scenario and price bounds
In this section we provide some additional results underpinning our discussion of the worst-casedefault scenario and lower price bounds for ESBies in Section 4.1.
Proof of Proposition 4.1.
By the put call parity (2.7) for ESBies and EJBies, the claim ofthe proposition is equivalent to showing that L ∗ maximizes the value of EJBies. More precisely,we show that for any random vector ( L , . . . , L J ) (cid:48) ∈ [0 , J with E ( L j ) = ¯ (cid:96) j , 1 ≤ j ≤ J , andany κ >
0, it holds that E (cid:16)(cid:0) J (cid:88) j =1 w j L j − κ (cid:1) + (cid:17) ≤ E (cid:16)(cid:0) J (cid:88) j =1 w j L ∗ j − κ (cid:1) + (cid:17) . (B.1)We may use call options instead of put options in (B.1) since E ( (cid:80) Jj =1 w j L j ) is fixed. Toestablish the inequality (B.1) we use a result on stochastic orders from B¨auerle and M¨uller(2006). According to the equivalence ((iii) ⇔ (iv)) in Theorem 2.2 of that paper, (B.1) isequivalent to the inequalityES α (cid:16) J (cid:88) i =1 w j L j (cid:17) ≤ ES α (cid:16) J (cid:88) i =1 w j L ∗ j (cid:17) for all α ∈ [0 , , (B.2)where for a generic random variable Z , ES α ( Z ) = − α (cid:82) α q u ( Z ) du gives the expected shortfallof Z at confidence level α and where q u ( Z ) denotes the quantile of Z at level u .To establish (B.2) we show first that L ∗ j maximizes the quantity ES α ( L j ) over all rvs L j with value in the interval [0 ,
1] and expectation E ( L j ) = ¯ (cid:96) j , simultaneously for all α ∈ [0 , L j has to satisfy the constraints q u ( L j ) ≤ L j ∈ [0 , (cid:82) q u ( L j ) du = ¯ (cid:96) j (since E ( L j ) = ¯ (cid:96) j , so thatES α ( L j ) ≤ − α min { − α, ¯ (cid:96) j } = ES α ( L ∗ j ) . Moreover, we get from the coherence of expected shortfall thatES α (cid:16) J (cid:88) j =1 w j L j (cid:17) ≤ J (cid:88) j =1 w j ES α (cid:0) L j (cid:1) ≤ J (cid:88) j =1 w j ES α (cid:0) L ∗ j (cid:1) = ES α (cid:16) J (cid:88) j =1 w j L ∗ j (cid:17) , where the last equality follows since L ∗ , . . . , L ∗ m are comonotonic. This gives inequality (B.2)and hence the result. Distribution of L ∗ . Next we discuss properties of the distribution π ∗ of the worst-casedefault scenario. This distribution is a discrete probability measure on [0 , m which charges J + 1 points; it is given by π ∗ (cid:0) (1 , . . . , (cid:1) = ¯ (cid:96) , π ∗ (cid:0) (0 , , . . . , (cid:1) = ¯ (cid:96) − ¯ (cid:96) , · · · , π ∗ (cid:0) (0 , . . . , , (cid:1) = ¯ (cid:96) J − ¯ (cid:96) J − ,π ∗ (cid:0) (0 , . . . , (cid:1) = 1 − ¯ (cid:96) J . We call π ∗ the worst-case distribution. Note that, under π ∗ , the probability of large default“clusters” is maximal given the expected loss constraints. First, under π ∗ the event where allsovereigns default has probability ¯ (cid:96) . Since Q ( L = · · · = L J = 1) ≤ Q ( L = 1) ≤ E ( L ) = ¯ (cid:96) , this is the maximum value possible. Next, under π ∗ the default scenario where all sovereignsexcept the first default has probability ¯ (cid:96) − ¯ (cid:96) . It is easily seen that this is the maximumpossible value given the expected-loss constraints and the probability attributed to the firstcluster (the cluster where all sovereigns default). Similarly, the probability of the ( n + 1)-thcluster, where all but the first n sovereigns default, is maximal given the probability attributedto the first n clusters.Finally we sketch an approach for the approximation of the worst-case distribution π ∗ within our model. Note first that, for κ j large and σ j small, the hazard-rate trajectory ( γ jt ) isessentially determined by the trajectory of X and by the choice of the mean reversion level µ j ( · ),so that we concentrate on these quantities. We consider a model with K = J + 1 states of X that correspond to the different default “clusters” under π ∗ . Choose some large n and define themean reversion level µ j ( · ) by µ (1) = · · · = µ J (1) = n ; µ (2) = · · · = µ J − (2) = n , µ J (2) = n ;. . . ; µ ( J + 1) = · · · = µ J ( J + 1) = n . Note that in state k the default probability of obligor1 to obligor J − k + 1 is small, the default probability of obligor J + 2 − k up to obligor J islarge; that is, the state corresponds to the ( J + 2 − k )-th default cluster.Next we define the generator matrix of X . We assume that states 2 to J + 1 are absorbing,so that q ik = 0 for 2 ≤ i ≤ J + 1 and all k . Define probabilities p , . . . , p J +1 by p = 1 − ¯ l J , p k = ¯ (cid:96) J +2 − k − ¯ (cid:96) J +1 − k for 2 ≤ k ≤ J , and finally p J +1 = ¯ (cid:96) , that is p k corresponds to theprobability of the ( J +2 − k )-th default cluster under π ∗ . Since states 2 , . . . , J +1 are absorbing,we get for any valid choice for the first row of Q that Q ( X T = 1) = e q T and Q ( X T = k ) = (1 − e q T ) q k − q , k = 2 , . . . , J + 1 , q = − (cid:80) J +1 k =2 q k ). We want to choose q , . . . , q J +1 so that Q ( X T = k ) = p k for all k . This gives q = 1 T ln p and q k = − p k q − p , k = 2 , . . . , J + 1 . (B.3)Since (cid:80) J +1 k =1 p k = 1, we get that q = − (cid:80) K +1 k =2 q k so that (B.3) defines indeed a valid generatormatrix. Moreover, for n → ∞ , κ j → ∞ and σ j → Q (cid:0) { τ ≤ T } = · · · = { τ J − k +1 ≤ T } = 0 , { τ J − k +2 ≤ T } = · · · = { τ J ≤ T } = 1 (cid:1) converges to Q ( X T = k ) = p k which gives the result by definition of the p k . C Details on Calibration
C.1 Data
In Table 4 below we present summary statistics of the data we use in the model calibration.
Yrs. AUT BEL DEU ESP FIN FRA IRL ITA NLD PRTAA AA AAA A AA AA A BBB AAA BBBPanel A: Mean1 31 .
071 44 .
063 12 .
819 113 .
637 13 .
273 26 .
590 204 .
380 115 .
934 20 .
269 307 . .
341 54 .
593 16 .
700 138 .
474 17 .
780 35 .
106 220 .
752 143 .
971 25 .
232 346 . .
016 66 .
843 21 .
918 153 .
769 22 .
148 44 .
961 224 .
791 165 .
881 30 .
413 352 . .
657 77 .
339 29 .
151 165 .
480 28 .
219 56 .
684 223 .
055 181 .
786 38 .
267 354 . .
675 85 .
437 34 .
562 174 .
373 33 .
048 65 .
906 222 .
409 193 .
090 43 .
902 359 . .
589 59 .
330 12 .
418 113 .
398 12 .
932 30 .
303 309 .
597 110 .
190 21 .
720 436 . .
347 67 .
134 14 .
520 129 .
415 14 .
257 34 .
529 308 .
774 113 .
689 23 .
782 449 . .
800 74 .
987 17 .
157 131 .
520 15 .
065 39 .
920 293 .
590 115 .
705 24 .
563 399 . .
974 76 .
066 21 .
330 130 .
483 16 .
769 45 .
769 264 .
432 113 .
967 27 .
595 352 . .
023 76 .
099 24 .
114 129 .
470 17 .
407 49 .
149 244 .
132 112 .
625 29 .
439 324 . .
080 3 .
840 2 .
920 10 .
450 2 .
250 3 .
550 7 .
830 21 .
620 3 .
120 12 . .
190 7 .
020 3 .
980 18 .
900 3 .
800 6 .
570 12 .
330 33 .
880 4 .
840 28 . .
820 9 .
430 6 .
230 25 .
050 6 .
040 9 .
620 15 .
860 48 .
990 7 .
230 40 . .
890 11 .
910 8 .
330 30 .
730 10 .
290 12 .
630 19 .
670 56 .
490 9 .
140 47 . .
270 16 .
480 9 .
510 37 .
230 13 .
020 17 .
400 23 .
970 59 .
830 11 .
240 47 . .
960 301 .
620 74 .
840 489 .
430 66 .
530 160 .
660 1629 .
340 619 .
540 110 .
870 2598 . .
440 337 .
600 81 .
080 608 .
330 74 .
600 177 .
440 1614 .
480 591 .
030 125 .
040 2494 . .
490 375 .
700 90 .
350 619 .
920 82 .
550 198 .
200 1572 .
800 581 .
050 130 .
650 2102 . .
430 379 .
090 108 .
250 622 .
220 90 .
460 222 .
510 1419 .
750 575 .
930 132 .
710 1846 . .
180 380 .
940 119 .
060 624 .
290 95 .
000 237 .
300 1318 .
590 573 .
030 136 .
960 1802 . Table 4:
Summary statistics of CDS spreads (in bp).
C.2 Methodology
In order to determine the parameters (Θ j , σ j ), 1 ≤ j ≤ J , the generator matrix Q andthe realised trajectories { γ s m } and { X s m } , we use an iterative approach which is compactlysummarized in Algorithm 1 below. We set Θ = (Θ , . . . , Θ J ) and we use { γ t m } ( i ) , { X t m } ( i ) and (Θ j ) ( i ) to denote the i -th estimate of the distinct variables within the iteration.27 lgorithm 1: Detailed description of calibration step
Data:
Market CDS spreads for maturities u ∈ T for each sovereign 1 ≤ j ≤ J Result:
Estimates for { γ s m } , { X s m } and Θ Initialization for { γ s m } (0) , { X s m } (0) , ( Θ ) (0) and Q (0) i = 0 while (cid:80) Jj =1 (cid:80) Mm =0 l j (cid:16) ( γ js m ) ( i ) , (Θ j ) ( i ) , ( σ j ) ( i ) , Q ( i ) , X ( i ) s m (cid:17) ≥ (cid:15) do for j ← to J do for m ← to M do ( γ js m ) ( i +1) = arg min γ l js m ( γ, (Θ j ) ( i ) , ( σ j ) ( i ) , Q ( i ) , X ( i ) s m ) end Estimate ( σ j ) ( i +1) based on the quadratic variation of ( γ j ) ( i +1) end for m ← to M do X ( i +1) s m = arg min x (cid:80) Jj =1 l js m (cid:16) ( γ js m ) ( i +1) , (Θ j ) ( i ) , ( σ j ) ( i +1) , Q ( i ) , x ) (cid:17) end Estimate Q ( i +1) via MLE based on X ( i +1) for j ← to J do (Θ j ) ( i +1) = arg min Θ (cid:80) Mm =0 l js m (cid:16) ( γ js m ) ( i +1) , Θ , ( σ j ) ( i +1) , Q ( i +1) X ( i +1) s m ) (cid:17) end Set i ← i + 1 end The assumption of conditionally independent defaults substantially facilitates the calibra-tion procedure: given an estimate for Q and { X s m } , estimation of { γ jt m } and of the parametervector Θ j can be done independently for each sovereign j . We initiate the calibration byapplying k -means clustering on the relevant CDS spreads to get an estimate for X (0) . Forsmall maturities T , it holds that (cid:99) cds jT ≈ δ j ( X ) γ j . We use this approximation along with theinitial estimate X (0) to get an estimate for ( γ j ) (0) and we consequently solve the optimizationproblem of line 15 in Algorithm 1 to obtain the initial value Θ (0) . To compute the estimatesfor σ j , we use that the quadratic variation of γ j satisfies[ γ j , γ j ] t = ( σ j ) (cid:90) t γ js ds, and we approximate the integral with Riemann sums. For a given (estimated) realisation ofthe Markov chain, we use the standard MLE estimator for continuous-time Markov chains toget an estimate of Q .The main numerical challenge in the application of Algorithm 1 is to solve the optimizationproblem min Θ M (cid:88) m =0 l j (cid:16) ( γ js m ) ( i +1) , Θ , ( σ j ) ( i +1) , Q, X ( i +1) s m ) (cid:17) . (C.1)We impose the restriction that all parameters are non-negative and, for regularization purposes,we set the lower bound of the mean-reversion speed κ j to 0 . j . During the firstiteration of Algorithm 1, we employ an algorithm for constrained optimization as presentedin Runarsson and Yao (2005). The algorithm uses heuristics to escape local optima. In order28o refine the estimation, in the subsequent calibration steps (i.e. for steps i >
1) we use thelocal optimizer of Powell (1994), which provides a derivative-free optimization method basedon linear approximations of the target function. After successful convergence of Algorithm 1,we perform a final refinement step in which we keep all input variables except Θ j , 1 ≤ j ≤ J ,fixed. C.3 Results
State AUT BEL DEU ESP FIN FRA IRL ITA NLD PRT1 0 .
55 0 .
55 0 .
50 0 .
55 0 .
50 0 .
50 0 .
55 0 .
50 0 .
50 0 .
552 0 .
55 0 .
55 0 .
50 0 .
55 0 .
50 0 .
50 0 .
55 0 .
50 0 .
50 0 .
553 0 .
65 0 .
65 0 .
60 0 .
65 0 .
60 0 .
60 0 .
65 0 .
60 0 .
60 0 . Table 5:
Fixed conditional means of LGDs for different sovereigns and varying states.
The following figure illustrates the quality of the model fit for two different sovereigns.
C.4 Parameters used in Risk Analysis
AUT BEL DEU ESP FIN FRA IRL ITA NLD PRT0 .
04 0 .
04 0 .
29 0 .
12 0 .
02 0 .
20 0 .
03 0 .
18 0 .
07 0 . Table 6:
Portfolio weights of ESBies and EJBies, based on proportion of sovereigns on euro area GDPas of 2018. (cid:101) Q (cid:101) Q State 1 State 2 State 3 State 1 State 2 State 3State 1 (expansion) − . . . − . . . . − . . . − . . . . − . . . − . Table 7:
Generator matrices (cid:101) Q and (cid:101) Q for crisis scenarios. C.5 Results of EM Estimation
Param. AUT BEL DEU ESP FIN FRA IRL ITA NLD PRT µ (1) 0 . . . . . . . . . . µ (2) 0 . . . . . . . . . . µ (3) 0 . . . . . . . . . . κ . . . . . . . . . . Table 8:
Estimation results: parameters of hazard rate dynamics.
010 2012 2014 2016 2018
CDS Spreads (1 yr) ba s i s po i n t s DEU datamodel2010 2012 2014 2016 2018
Absolute Pricing Errors ba s i s po i n t s CDS Spreads (5 yrs) ba s i s po i n t s DEU datamodel2010 2012 2014 2016 2018
Absolute Pricing Errors ba s i s po i n t s CDS Spreads (1 yr) ba s i s po i n t s ITA datamodel2010 2012 2014 2016 2018
Absolute Pricing Errors ba s i s po i n t s CDS Spreads (5 yrs) ba s i s po i n t s ITA datamodel2010 2012 2014 2016 2018
Absolute Pricing Errors ba s i s po i n t s Figure 9:
Time series plots of market CDS spreads against model values. The solid (dashed) linescorrespond to the market (model) values of the distinct CDS spreads.
State 1 State 2 State 3State 1 (expansion) − . . . . − . . . . − . Estimation results: generator matrix Q of X . References
A¨ıt-Sahalia, Y., Laeven, J. and Pelizzon, L.: 2014, Mutual excitation in Eurozone sovereignCDS,
Journal of Econometrics (2), 151–167.30ltman, E., Brady, B., Resti, A. and Sironi, A.: 2005, The Link between Default and RecoveryRates: Theory, Empirical Evidence, and Implications,
Journal of Business (6), 2203–2228.Ang, A. and Longstaff, F. A.: 2013, Systemic sovereign credit risk: Lessons from the U.S. andEurope, Journal of Monetary Economics (5), 493–510. URL: http://dx.doi.org/10.1016/j.jmoneco.2013.04.009
Barucci, E., Brigo, D., Francischello, M. and Marazzina, D.: 2019, On the design of sovereignbond backed securities, working paper, available via SSRN.B¨auerle, N. and M¨uller, A.: 2006, Stochastic orders and risk measures: Consistency andbounds,
Insurance: Mathematics and Economics , 132–148.B´enassy-Qu´er´e, A., Brunnermeier, M., Enderlein, H., Farhi, E., Marcel Fratzscher, M., Fuest,C., Gourinchas, P., Martin, P., Pisani-Ferry, J., Rey, H., Schnabel, I., Vron, N., Weder diMauro, B. and Zettelmeyer, J.: 2018, Reconciling risk sharing with market discipline: Aconstructive approach to euro area reform, CEPR Policy Insight No 91.Brigo, D., Pallavicini, A. and Torresetti, R.: 2010, Credit Models and the Crisis: A Journeyinto CDOs, Copulas, Correlations and Dynamic Models , Wiley Finance Series, Wiley.Brunnermeier, M., Langfield, S., Pagano, M., Reis, R., Van Nieuwerburgh, S. and Vayanos,D.: 2017, ESBies: Safety in the tranches,
Economic Policy .Cronin, D. and Dunne, P.: 2019, How effective are sovereign bond-backed securities as aspillover prevention device?,
Journal of International Money and Finance , 49–66.Damian, C., Eksi-Altay, Z. and Frey, R.: 2018, EM algorithm for Markov chains observed viaGaussian noise and point process information: Theory and case studies, Statistics andRisk Modelling , 51–72.de Grauwe, P. and Ji, Y.: 2019, Making the eurozone sustainable by financial engineering orpolitical union?, Journal of Common Market Studies , 40–48.de Sola Perea, M., Dunne, P. and Puhl, M. and Reininger, T.: 2019, Sovereign bond-backedsecurities: A VAR for VaR and marginal expected shortfall assessment, Journal of Em-pirical Finance , 33–52.Dombrovskis, V. and Moscovici, P.: 2017, Reflecting Paper on the Deepening of the Economicand Monetary Union, Technical report , European Commission.Duffie, D., Pan, J. and Singleton, K.: 2000, Transform analysis and asset pricing for affinejump diffusions,
Econometrica (6), 1343–1376.Elliott, R. J.: 1993, New finite-dimensional filters and smoothers for noisily observed Markovchains, IEEE Trans. Info. theory (1), 265–271.Elliott, R. and Siu, T. K.: 2009, On Markov-modulated exponential-affine bond price formulae, Applied Mathematical Finance (1), 1–15.31SRB High-Level Task Force on Safe Assets: 2018, Sovereign bond backed securities: Afeasibility study, Technical report , European Systemic Risk Board.Filipovic, D.: 2009,
Term-Structure Models. A Graduate Course. , Springer.Gennaioli, N., Shleifer, A. and Vishny, R.: 2012, Neglected risks, financial innovation, andfinancial fragility,
Journal of Financial Economics (3), 452468.Golec, P. and Perotti, E.: 2015, Safe assets: a review, ECB working paper No 2035, EuropeanCentral Bank.Kraemer, M.: 2017, How S&P Global Ratings Would Assess European ”Safe” Bonds (ESBies),
Technical report , Standard & Poors.Langfield, S.: 2020, Bridge over troubled monetary union: A reply to de Grauwe and Li.Leandro, A. and Zettelmeyer, J.: 2019, Creating a Euro area safe asset without mutualizingrisk (much),
Capital Markets Law Journal (4), 488–517.McNeil, A. J., Frey, R. and Embrechts, P.: 2015, Quantitative Risk Management: Concepts,Techniques and Tools , 2nd edn, Princeton University Press, Princeton.Monti, M.: 2010, A new strategy for the single market at the service of Europe’s economy andsociety, Report to the President of the European Commission.Powell, M. J.: 1994, A direct search optimization method that models the objective and con-straint functions by linear interpolation, in S. Gomez and J.-P. Hennart (eds),
Advancesin optimization and numerical analysis , pp. 51–67.Runarsson, T. P. and Yao, X.: 2005, Search biases in constrained evolutionary optimiza-tion,
IEEE Trans. on Systems, Man, and Cybernetics Part C: Applications and Reviews (2), 233–243.van Beek, M., Mandjes, M., Spreij, P. and Winands, E.: 2020, Regime switching affine pro-cesses with applications to finance, Finance and Stochastics24