Credit Value Adjustment for Counterparties with Illiquid CDS
aa r X i v : . [ q -f i n . M F ] J un Credit Value Adjustment for Counterpartieswith Illiquid CDS
Ola HammarlidMarta LeniecJune 21, 2018
Abstract
Credit Value Adjustment (CVA) is the difference between the valueof the default-free and credit-risky derivative portfolio, which can be re-garded as the cost of the credit hedge. Default probabilities are there-fore needed, as input parameters to the valuation. When liquid CDS areavailable, then implied probabilities of default can be derived and used.However, in small markets, like the Nordic region of Europe, there arepractically no CDS to use. We study the following problem: given thatno liquid contracts written on the default event are available, choose amodel for the default time and estimate the model parameters. We usethe minimum variance hedge to show that we should use the real-worldprobabilities, first in a discrete time setting and later in the continuoustime setting. We also argue that this approach should fulfil the require-ments of IFRS 13, which means it could be used in accounting as well. Wealso present a method that can be used to estimate the real-world prob-abilities of default, making maximal use of market information (IFRSrequirement).
Credit Value Adjustment (CVA) is defined as the difference in value ofa portfolio, without credit risk and when exposed to default risk, see forexample (3.1) in Greeen [11] or (12.1) in Gregory [12]. This difference canbe regarded as a derivative, and as such its value is equal to the cost ofthe hedge.The Probability of Default (PD) and Loss Given Default (LGD) aretwo of the most important parameters that impact CVA. If derivativesof the default event exist, like Credit Default Swaps (CDS), then impliedprobability of default can be derived and used in pricing. Our aim is toanswer the following question: How should we model and price the creditrisk of derivatives (CVA) for counterparties for which there does not existany derivatives that can be used for hedging default risk.We argue that in that case it is not the extended implied methodology(e.g. Nomura model in [5]) that should be used. The basic reasoning is ased on minimum variance hedging, which leads to the use of real-worldprobabilities. To illustrate our line of argument, when a derivatives onthe default is lacking, we start with a discrete time example. In this introductory example of variance minimization, the time of default τ can only occur in discrete time T = { t , t , ... } . Let p t denote theprobability that the default is equal t , for any t ∈ T , i.e. p t = P ( τ = t ) , for all t ∈ T . Let V t denote the value at time t of a portfolio of derivatives with maturitytimes less or equal than T >
0. The positive part of the portfolio isdenoted by V + t = max { V t , } , and is called the positive exposure at time t . The positive exposure atdefault is therefore given by P E τ = V + τ = X t ∈T ,t P E, V + t i (cid:1) = Cov X t ∈T ,t CV A = T X t = t E Q ∗ (cid:2) e − rt V ∗ t (cid:3) · p t , because the value of the contract with payoff V ∗ t is given by E Q ∗ (cid:2) e − rt V ∗ t (cid:3) in the default-free market, where r is a deterministic and constant interestrate. .2 Minimum variance hedging There are several models of default, where the most popular approachesare either in the category the structural approach or the reduced-formapproach (see for example [18]). We consider the latter and assume thatthe default time is an exponentially distributed random variable with apiecewise-constant intensity.A commonly used method to estmimate the implied probability ofdefault, when CDS does not exist, is proposed by Nomura in [5]. Inthis model CDS spreads of liquid names are used to construct proxy CDSspreads for illiquid names, by a mapping process (for a presentation of themodel see Appendix 4). However, this method is based on the assumptionthat a CDS spread exists and results in implied default probabilities. Inour opinion we should build the model from a hedging assumption, usingonly the available contracts and not hypothetical ones.As it was pointed out in Green, the number of credit default swapsin the US market is much larger than in other regions and therefore onecould question how appropriate the Nomura method is. Green mentionsthat the XVA trading desks should understand that if the proxy CDS isused for hedging, it will not be effective at the actual default time andhence it does not hedge the default risk. Moreover, Green claims in [11]that ”risk warehousing is inevitable and this leads directly to incompletemarkets and the physical measure”. This article supports Green’s claim.Furthermore, in markets like the Northern Europe, the number of CDSis negligible and therefore it is sub-optimal to model the default proba-bilities of the majority of counterparties based on such a small sample.Especially, since CVA is a portfolio effect that is different from a standalone derivative.Since perfect hedging is not possible in incomplete markets, the com-mon approach in the literature is to determine a hedging policy accordingto some criterion. Starting with the Markowitz optimal portfolio selection(see [22]), the variance-minimizing criterion has widely been employed inthe literature in various economic contexts. The main references for thegeneral case of hedging in the incomplete markets are Hull [14], McDonald[23] and Stulz [24].Moreover, F¨ollmer and Sondermannn [10] and then F¨ollmer and Schweizerin [8] presented the connection of variance hedging and the minimal mar-tingale measure, where the minimal martingale measure preserves thestructure of the real-world measure as far as possible, under the con-straint that the discounted underlying stock price is a martingale. It wasdiscussed in [9] that the decomposition of any contingent claim under theminimal martingale measure provides the so-called F¨ollmer-Schweizer (seefor example [8]) decomposition of the contingent claims under the real-world measure, and this in turn immediately gives the variance-minimizinghedging strategy for the claim. In practise it means that if one aims tofind a hedging strategy that minimizes the variance of the hedging error,then one should use the minimal martingale measure.Jeanblanc and Rutkowski [18] give an overview on default modelling,which is consistent with our approach, and shows that when no defaultablehedging claims exist (i.e. no liquid CDS), the market is incomplete and eplication of defaultable claims is not possible. They also suggest theminimum variance hedging of credit derivatives(see Bielecki, Jeanblancand Rutkowski [2] for more details). They also verify that in a completemarket, their mean-variance price is equal to the unique arbitrage-freeprice.In incomplete markets, there are infinitely many martingale measureswhich are consistent with the no-arbitrage condition. El Karoui et al [1]showed that in case of default risk and incomplete markets, the varianceminimization leads to the minimal martingale measure that removes thedrift of the underlying stock but leaves the probability of the defaultunchanged, i.e. the default probability is under the real-world measure.They study the minimal martingale measure approach (mentioned above)as well as the minimal entropy martingale measureHence, following this approach, we employ the variance minimizationhedging strategy for the problem of pricing and hedging CVA. The inter-pretation of our result is that an investor who wants to hedge a derivativein the presence of default risk in the incomplete market, prices the con-tracts as she was risk-neutral with respect to the default risk. To ourknowledge, this line of argument, has so far not been applied before toCVA to support the use of real-world probabilities of default. This paper studies an applied problem that is present for various banksand financial institutions around the world that fall under the Interna-tional Financial Reporting Standards (IFRS) regulations. We argue thatthe requirements presented in IFRS 13 are satisfied by the proposed frame-work.Firstly, IFRS 13, § § 22 states that: ”An entity shall measure the fair value of an asset or aliability using the assumptions that market participants would use whenpricing the asset or liability, assuming that market participants act intheir economic best interest.” The cost of the hedge is equal to the priceof a derivative, hence a price which all participants in the market couldagree upon.Lastly, when estimating the input parameters that the model con-sumes IFRS 13, § 67 states: Valuation ”techniques used to measure fairvalue shall maximise the use of relevant observable inputs and minimisethe use of unobservable inputs.” This apply to the second step of themodelling, the estimation of the real-world probabilities of default. Wepresent how to make maximum use of market observable information toestimate probability of default. Pricing and hedging The minimal martingale measure and the minimal entropy martingalemeasure lead to a particular choice of a measure from the set of equivalentmartingale measures (see El Karoui et al. [1]). We connect this conceptsto the variance-minimizing hedging strategy and consequently CVA.Let (Ω , F , F = ( F t ) t ≥ , P ) be a filtered probability space, where T > F is the filtration (satisfying the usual condi-tions) generated by, for example a geometric Brownian motion W t , thatis, dS t = µ ( t, S t ) dt + σ ( t, S t ) dW t , S = s (7)where µ is the drift and σ > F T ⊂ F . Moreover, leta default time τ be an exponential random variable with intensity λ P > , F ) and denote by G = ( G ) t ∈ [0 ,T ] a filtration (satisfying theusual conditions) such that G t = F t ∨ σ ( τ ∧ t ) for all t ∈ [0 , T ] . The stock price uncertainty is market risk and the uncertainty comingfrom the default is default risk. We still assume independence between τ and W t for all t ≥ . The filtration G = ( G t ) t ≥ represents the information S and the infor-mation about the default time τ, i.e. at any time t ≥ t ≥ τ has already occurredor not. This is a standard way of modelling information flow in the areaof financial mathematics and more details can be found for example in[20] and [4]. Moreover, the practitioners use this way of modelling theinformation level in case of default risk. The standard reference is [11]which is the handbook for CVA calculations. Let us begin with introducing a default-free market consisting of a price S defined by (7) and a bank account B = ( B t ) t ∈ [0 ,T ] , where dB t = rB t dt, B = 1and r is a constant interest-rate. Since the coefficients of the geomet-ric Brownian motion S are constant and σ > , the Assumption 1 andAssumption 2 of Blanchet-Scalliet et al. [1] are satisfied and the default-free market is complete and arbitrage-free. For some more details see forexample Karatzas [21].The information flow in this case is the filtration F generated by theprice S and the unique equivalent martingale measure Q ∗ on F is givenby d Q ∗ | F t = Z ∗ t d P | F t for all t ∈ [0 , T ] , where Z ∗ = ( Z ∗ t ) t ∈ [0 ,T ] is the Radon-Nikodym derivative of Q ∗ with re-spect to P given by Z ∗ t = exp (cid:26) − θ t + θW t (cid:27) , Z ∗ = 1 (8) nd θ = − ( µ − r ) σ − . We see that the only source of randomness in thismarket is the market risk coming from the Brownian motion W, i.e. fromthe fluctuations of the price S. Let us denote by M ( F ) the set of equivalent martingale measures on F . Then we have that M ( F ) = { Q ∗ } . Consequently, any F T -measurable contingent claim X T has a unique pricegiven by E Q ∗ h e − rT X T i = E P h e − rT X T Z ∗ T i . Now we extend the default-free market defined in Subsection 2.1 by in-troducing the random default time τ which is exponentially distributedwith an intensity λ P > . As discussed above, the information level in thiscase is given by the filtration G and thus the set of equivalent measuresmaking the discounted price process a martingale has to be defined on thefiltration G (see for example [20]).Since we assume that the default time and the price are independent,the Assumption 3 of [1] is satisfied. Moreover, the exponential distributionassumption of τ makes the Assumption 4 and Assumption 5 of [1] satisfied.Let M ( G ) denote the set of equivalent martingale measures for thefiltration G . Since we assumed that τ is independent of the Brownianmotion W, the Jacod’s hypothesis is satisfied (see for example [20] for theJacod’s hypothesis) and consequently there exists at least one equivalentmartingale measure in M ( G ) . As shown in [1], [16], [19] and other articles in the area of credit risk,if Q H ∈ M ( G ) , then the Radon-Nikodym derivative L H = ( L Ht ) t ∈ [0 ,T ] forthe change of probability measure from Q H to P on G is given by L Ht = L ( t ) · L H ( t ) , (9)where L ( t ) = exp (cid:26) − θ t + θW t (cid:27) and L H ( t ) = exp (cid:26) H τ I τ ≤ t − λ P Z t ∧ t (cid:16) e H s − (cid:17) ds (cid:27) , where H = ( H t ) t ∈ [0 ,T ] is a G -adapted process satisfying some technicalconditions (see for example [19]).Moreover, it was shown for example in [1] that the intensity λ Q H t ofthe default time τ under measure Q H satisfies λ Q H t = e H t · λ P for any t ∈ [0 , T ] . We see that L ( t ) is equal to the Radon-Nikodym derivative for theunique equivalent change of measure from Q ∗ to P on F and since τ isindependent of W t then L ( t ) is independent of τ. Since H is a G -adapted process, then in general L H ( t ) is not indepen-dent of W t . Also, in general L H ( t ) is not independent of τ. .2.1 Incompleteness of the defaultable market It was shown for example in [16] and [19] that the set M ( G ) has in-finitely many elements, which means that introducing the default riskto the default-free market brings some form of incompleteness. Specifi-cally, in [19] the authors show that the measure Q (i.e. a measure from M ( G ) such that H t = 0 for any t ∈ [0 , T ]) belongs to M ( G ) and theauthors in [16] show that there exists a measure Q H in M ( G ) for which H t = 0 and hence due to the fact that any convex combination of mea-sures from M ( G ) also belongs to M ( G ) we get that M ( G ) has infinitelymany elements. As a result, one faces the problem of narrowing the setof equivalent martingale measures.One method for narrowing down the set M ( G ) would be completingthe defaultable market by introducing so-called generalized risk-free as-sets. A generalised risk-free asset would be for example an asset paying1 at the default time τ, i.e. a CDS can be one of them. Then, if suchcontracts were dynamically traded, then one would be able to extract therisk-neutral intensity λ Q H for a particular counterparty from the pricesof these contracts and use it to calculate CVA. This intensity may be re-garded as an implied risk-neutral intensity. The authors of [1] argue thatthe presence of dynamically traded generalized risk-free assets implies aunique specification of the equivalent martingale measure in M ( G ) andhence a complete market. Consequently, if it possible to extract λ Q H fromthe prices of the dynamically traded CDS, then λ Q H should be used forpricing purposes. Hence, if our aim is to calculate CVA of a counterpartywith liquid CDS contracts, then we should extract the risk-neutral inten-sity from the CDS prices (for example by the bootstrap technique) anduse it in the CVA calculations. However, if we consider a counterpartywithout liquid CDS or without any CDS at all, then we should use othertechniques for choosing the equivalent martingale measure from the set M ( G ) . We present some of these methods in the following subsections. In this subsection we consider the case of a counterparty without liquidCDS, and as a result, we deal with the problem of narrowing the set ofequivalent martingale measures M ( G ).The connection between the minimal martingale measure and thevariance-minimization hedging between the payoff h ( S τ ) and the terminalwealth generated from a self-financing strategy, was introduced by F¨ollmerand Sondermann in [10]. In economical terms; an approximation of thecontingent claim in terms of a self-financing strategy with the replicationerror (”the tracking error”) as small as possible.As in El Karoui et al. [1], the imperfect hedging is connected with aminimal martingale measure, which is defined by the following two con-ditions: an equivalent martingale measure Q H ∈ M ( G ) is called minimalmartingale measure if Q H = P on G and if every ( P , G )-square martingaleorthogonal to W under P is a ( Q H , G )-martingale. By Proposition 6 in[1] we have that the minimal martingale measure is equal to Q , i.e. the quivalent martingale measure defined by H t = 0 for every t ∈ [0 , T ] . Thiscorresponds to a zero risk premium associated with the default risk. Inother words we get that the pricing measure can be chosen to be the min-imal martingale measure Q defined by the Radon-Nikodym derivative L = ( L t ) t ∈ [0 ,T ] , where L t = exp (cid:26) − θ t + θW t (cid:27) , for any t ∈ [0 , T ] . (10) CVA is given by the following formula (see for example (3 . 11) in Green[11]) CV A = E Q H (cid:2) e − rτ (1 − R ) V + τ (cid:3) , (11)where Q H ∈ M ( G ) , the constant R is the recovery rate ( LGD = 1 − R )and V + is the positive exposure. It is also assumed that V = ( V t ) t ∈ [0 ,T ] is an F -adapted process, i.e. the only uncertainty in V is the randomnesscoming from the price S given by (7).Since there do not exist any dynamically traded hedging instrumentsfor the counterparty, we decide on narrowing the set M ( G ) by the well-studied method summarized in Subsection 2.2.2. As a result, we use theequivalent martingale measure Q defined by (10) and we get that CV A = E Q (cid:2) e − rτ (1 − R ) V + τ (cid:3) = (1 − R ) E P (cid:2) e − rτ V + τ L τ (cid:3) = (1 − R ) Z ∞ E P (cid:2) e − rt V + t L t (cid:3) f P ( t ) dt = (1 − R ) Z ∞ E Q ∗ (cid:2) e − rt V + t (cid:3) f P ( t ) dt = (1 − R ) Z ∞ E Q ∗ (cid:2) e − rt V + t (cid:3) λ P e − λ P t dt, where the second equality is by the change of equivalent martingale mea-sure and the third equality is by the fact that L t is independent of τ andthat τ is independent of the Brownian motion W . The equivalent mar-tingale measure Q ∗ is given by the Radon-Nikodym derivative Z ∗ definedin (8).We note that in the case of a counterparty with liquid CDS we wouldfind the implied risk-neutral default intensity λ Q H from the CDS spreadsand use it to calculate the CVA, i.e. we would have CV A = E Q H (cid:2) e − rτ (1 − R ) V + τ (cid:3) = (1 − R ) Z ∞ E Q ∗ (cid:2) e − rt V + t (cid:3) λ Q H e − λ Q H t dt. We choose a set J of firms that have both: liquid 5-year CDSs and ex-pected default frequencies (EDF) published on Moody’s CreditEdge portal n a given time interval I given in days. EDF is a firm specific forward-looking measure of real-world probability of default that is calculated byMoody’s based on the Kealhofer-McQuown-Vasicek (KMV) model. Themain idea is that firms equity can be seen as a call option on the under-lying asset with the strike price equal to the face value of the firms debt.Then a mapping is used to transfer the distance-to-default to historicaldefaults.As it was discussed in [3], the CDS spreads are not pure measuresof credit risk and a methodology is needed to disentangle the liquiditypremium from them. Hence, the cleaned CDS spreads will be used toderive the implied probabilities of default, where a cleaned CDS spreadmeans a CDS spread without the liquidity premium. Then, for every i ∈ I for every firm j ∈ J we bootstrap the default intensity from thecleaned CDS spread and denote it by λ CDSij . Moreover, we can calculatethe EDF-implied default intensity λ EDFij by using the following formula λ EDFij = − ln(1 − p EDFif ) , where p EDFif is the EDF -implied 1-year default probability.Hence, we have a sequence of pairs ( λ EDFij , λ CDSij ) ( i ∈ I,j ∈ J ) . Similarly to [6] we assume a simple linear model between the naturallogarithm of the EDF-implied default intensity λ EDFij and the naturallogarithm of the CDS-implied default intensities λ CDSij , i.e.ln( λ EDFij ) = γ i + γ i ln( λ CDSij ) + ǫ i , where ǫ i is a standard normal random variable.Then, we use the obtained parameters γ i and γ i to calculate the de-fault intensity for a counterparty that does not have a liquid CDS in thefollowing way: Let C denote the set of Swedbank’s counterparties withoutliquid CDS. Firstly, we calculate a cleaned CDS proxy as discussed in Ap-pendix 4 and then we assume that the above mentioned linear relationshipholds also for the pairs ( λ CDS proxyic , λ P ic ) i ∈ I,c ∈ C , i.e. that we haveln( λ P ic ) = γ i + γ i ln( λ CDS proxy ic ) + ǫ i , where for every i ∈ I and c ∈ C we have that λ P ic is the real-world intensity.Hence, for every day i ∈ I and for every counterparty c ∈ C, thecumulative distribution function p ic ( t ) of the default time τ is calculatedby the following formula p ic ( t ) = P ( τ ≤ t ) = 1 − e − λ P ic t , where λ P ic = e γ i + γ i ln( λ CDSproxyic ) . References [1] C. Blanchet-Scalliet, N. El Karoui and L. Martellini Dynamic AssetPricing Theory with Uncertain Time-Horizon, Journal of EconomicDynamics and Control (2005) 2] T. Bielecki, M. Jeanblanc and M. Rutkowski Pricing and hedging ofcredit risk: replication and mean-variance approaches. (2004) [3] D. Brigo, M. Predescu and A. Capponi Credit Default Swaps LiquidityModelling: A survey. (2010) [4] G. Callegaro, M. Jeanblanc and B. Zargari Carthaginian enlargementof Filtrations. ESAIM: Probability and Statistics 17, 550-566 (2013) [5] K. Chourdakis, E. Epperlein, M. Jeannin and J. McEwen A cross-section across CVA.Nomura (2013) [6] D. Duffie, A. Berndt and R. Douglas Measuring Default Risk Premiafrom Default Swap Rates and EDFs. (2004) [7] D. Duffie and H. Richardson Mean-variance hedging in ContinuousTime, The Annals of Applied Probability (1991) [8] H. F¨ollmer and M. Schweizer Hedging contingent claims under incom-plete information, Allied Stochastic Analysis, Stochastic Monographs,5, 389-414 (1990) [9] H. F¨ollmer and M. Schweizer The Minimal Martingale Measure. En-cyclopaedia of Quantitative Finance. (2010) [10] H. F¨ollmer and D. Sondermann Hedging of non-redundant contingentclaims. (1985) [11] A. Green XVA. Credit, Funding and Capital Valuation Adjustments.Wiley Finance Series. (2016) [12] J. Gregory Counterparty credit risk and credit value adjustment. Wi-ley Finance. Second Edition. (2012) [13] W. Heynderickx, J. Cariboni, W. Schoutens and B. Smits The rela-tionship between risk-neutral and actual default probabilities: the creditrisk premium. Journal of Applied Economics. (2016) [14] J. Hull Options, Futures and Other Derivatives, Seventh Edition,Prentice Hall. (2008) [15] H. Hult, F. Lindskog, O. Hammarlid and C. Rehn Risk and PortfolioAnalysis. Principles and Methods. Springer (2012) [16] B. Iftimie, M. Jeanblanc, T. Lim and H. Nguyen Optimization prob-lem under change of regime of interest rate arXiv:1305.7309 (2013) [17] R. Jarrow, D. Lando, S. Turnbull A Markov Model for the TermStructure of Credit Risk Spreads. (1997) [18] M. Jeanblanc and M. Rutkowski Modelling and Hedging of DefaultRisk. (2003) [19] M. Jeanblanc and M. Leniec Role of Information in Pricing Default-Sensitive Contingent Claims. International Journal of Theoretical andApplied Finance (2015) [20] M. Jeanblanc, M. Yor and M. Chesney, Mathematics Methods forFinancial Markets New York: Springer-Verlag (2009) [21] I. Karatzas Lectures on the Mathematics in Finance, CRM Mono-graph Series, Montreal, vol.8. (1997) 22] H. Markovitz Portfolio Selection: Efficient Diversification of Invest-ments. (1967) [23] R. McDonald Derivatives Markets, Second Edition, Addison Wesley.(2006) [24] R. Stulz Risk Management and Derivatives, First Edition, SouthWestern College Publishing. (2003) The cross-sectional methodology for calculating proxy CDS spreads isbased on [5] and can be summarized as follows. We fix a set of ratings,regions and sectors and choose a universe of counterparties with liquidCDS spreads. We label every counterparty with a rating, region andsector. The proxy CDS spread for a given counterparty is S proxyi = M global M rating ( i ) M region ( i ) M sector ( i ) , where: • M global is a global factor • M rating ( i ) is the factor for the rating of counterparty i, • M region ( i ) is the factor for the region of counterparty i, • M sector ( i ) is the factor for the sector of counterparty i, For example, for a counterparty from North America with sector F IN and rating AA, we would have S proxyi = M global M AA M NorthAmerica M F IN . As it was discussed in [3], the CDS spreads are not pure measures ofcredit risk and a methodology is needed to disentangle the liquiditypremium from them. Hence, the cleaned CDS proxy spreads shouldbe used to derive the implied probabilities of default, where a cleanedCDS proxy spread means a CDS proxy spread without the liquiditypremium. To find the factors of the cross-sectional method we can follow thefollowing steps. • Enumerate all the factors starting with M global , • Let n be the total number of all factors (for example for 7 regions,7 ratings and 11 sectors we have n = 26) • Denote y i = ln( S proxyi ) and x j = ln( M j ) • Let A = [ A ij ] be a matrix of 0s and 1s hen we can write the model as y i = n X j =1 A ij x j . We want to find the optimal x that makes the proxy spreads as closeas possible to the market spreads. We can define ”as close as possible”to mean ”minimizing total squared difference in log spreads.” Thenfinding the optimal x simply means performing a linear regression. The idea can be summarized in the following steps.Step 1 Fix a set of ratings, regions and sectorsStep 2 Choose a universe of firms with liquid CDSStep 3 Label every firm with a rating, region and sector chosen in Step1Step 4 Label every counterparty with a rating, region and sector chosenin Step 1Step 5 Calculate a CDS proxy for every counterparty with a cross-sectional method (see Appendix 4)Step 6 Bootstrap default intensity from the CDS proxy for every coun-terpartyStep 7 Calculate the real-world default intensity λ P Step 8 Use the default intensity λ P to calculate default probabilitiesIn Appendix 4 we present a method for calculating CDS proxy spreadswhat are needed in the Step 5 above and in Subsection 3.1 we presenta method that can be used for calculating real-world default probabil-ities based on the CDS proxy spreads and Expected Default Frequen-cies (EDF).As an example we present in the following section a methodology thatcan be used for estimating real-world default probabilities from theCDS market with the use of Expected Default Frequencies (EDF).to calculate default probabilitiesIn Appendix 4 we present a method for calculating CDS proxy spreadswhat are needed in the Step 5 above and in Subsection 3.1 we presenta method that can be used for calculating real-world default probabil-ities based on the CDS proxy spreads and Expected Default Frequen-cies (EDF).As an example we present in the following section a methodology thatcan be used for estimating real-world default probabilities from theCDS market with the use of Expected Default Frequencies (EDF).