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					Workshop on

Credit Rating in View of Basel II
      Prof. Dr. Ralf Korn
      Dept. Mathematics
      University of Kaiserslautern
      Fraunhofer Institute for Industrial Mathematics, ITWM Kaiserslautern
      Dept. Financial Mathematics

      MSc. Fin.Math. Evren Baydar
      Fraunhofer Institute for Industrial Mathematics, ITWM Kaiserslautern
      Dept. Financial Mathematics
                       Credit Products

                                         Prof. Dr. Ralf Korn
                                         Dept. Mathematics
                                         University of Kaiserslautern

                                         Fraunhofer Institute for Industrial Mathematics, ITWM
                                         Dept. Financial Mathematics

                                         MSc. Fin.Math. Evren Baydar
                                         Fraunhofer Institute for Industrial Mathematics, ITWM
                                         Kaiserslautern Dept. Financial Mathematics

                                                                                                 Seite 2
                       Our schedule

                       1. Introduction
                       2. Before we start ...
                       3. Why credit rating and what is it ?
                       4. General steps on our way to a perfect rating
                       5. Statistical problems in credit rating

                       6. The rating in action
                       7. Credit Products
                       8. Further aspects

                                                                         Seite 3
                       A. Definition and Types of Credit Products

                       •rise in trading volumes of risky assets => creation of credit derivatives
                       •credit derivatives market
                                     •exists since early 1990s
                                     •expected to exceed $8.2 trillion by end-2006*
                       •92% of world‘s 500 largest firms use them for hedging their credit risk (ISDA
                       Credit derivatives are: [Schönbucher]
                       • securities which are primarily used to transfer, hedge or manage credit

                       • security whose payoff is materially affected by credit risk

                       narrower definition:
                       •security whose payoff is conditioned on the occurence of a credit event
                       *) According to Celent Comminications; measured by notional outstanding

                                                                                                        Seite 4
                       A. Terminology:
                       • credit event: bankruptcy, failure of a payment obligation, …
                                               standard definitions by ISDA (2003)
                       • reference credit (or entity): issuer(s), whose default triggers the credit event
                       • reference credit asset: set of assets issued by the reference credit, needed for
                                                defining the credit event
                                                calculating recovery rate
                       • default payment: money transfers upon a credit event
                        Default Digital Swap written on a coupon bond issued by Batar Ltd. Firm;
                        protection Seller (S) agrees to pay 1000 YTL to protection Buyer (B);
                        if Batar Ltd misses a coupon payment
                        credit event: a missed coupon payment

                        reference credit: Batar Ltd.
                        reference credit asset: the coupon bond issued by Batar Ltd
                        default payment: 1000 YTL
                                                  B pays a fee to S for protection
                                                                                                        Seite 5
                       A. Types of Credit Derivatives

                       • Single name products:
                        - have only one reference credit asset
                             Credit Default Swaps, Credit Linked Notes, Total Return Swaps, …
                       • Multi-name (or correlation) products:
                        - have more than one reference credit asset
                        - cover a portfolio of defaultable assets
                        - require more complicated pricing techniques due to the need for

                          default correlation modeling
                             Basket Default Swaps ( 1th to default swaps, 2nd to default swaps,…),
                              Collateralized Debt Obligations (CDO), CDO^2, …

                                                                                               Seite 6
                       B. Credit Default Swaps (CDS)
                       What is it?
                       • most popular (covers 73 % of outstanding notional of the credit derivatives
                       • Over-The-Counter (OTC) type contract between the protection Buyer (B) and
                       protection Seller (S)
                       • B makes fixed payment(s) (CDS premium or spread) to S at fixed intervals or
                       lump sum up-front payment (digital default swap or binary swap)
                                                         premium leg
                       • B pays the premiums to S until the credit event or until CDS matures
                       • if credit event does not occur until maturity of CDS, S pays nothing to B
                       • if credit event occurs before maturity, S guarantees to pay the loss of B

                                                         protection leg
                       • default payment can be optional :
                              • cash settlement
                              • physical delivery
                         *) 2003 Risk magazine survey
                                                                                                       Seite 7
                       B. Payoff diagrams

                                                   CDS premium
                         CDS Buyer (B)                                          CDS Seller (S)

                        Physical delivery
                                                   reference asset
                          CDS Buyer                                              CDS Seller

                                            Face value of the reference asset

                        Cash settlement

                           CDS Buyer                                             CDS Seller

                         R: Recovery rate
                                                                                                 Seite 8
                       B. Example of a CDS

                       •Inception date: 1.April 2006

                       •Maturity: 5 years

                       •Reference asset: Batar Ltd. Bond

                       •Notional amount: 100 million YTL

                       •Credit event: default of the firm Batar Ltd.

                       •Premium: 90 bps, starts at 1. April 2006

                       •B makes quarterly payments to S appr. =100 million x 0.009 x 0.25= 225 000 YTL

                       •If Batar Ltd. defaults after a month, asset has a recovery of 45 YTL per 100 YTL of Face
                       value, then (cash settlement)

                        - S pays to B : 100 million x (1-0.45)= 55 million YTL

                        - B pays the accrued premium to S =100 million x 0.009 x 1/12=75 000 YTL

                                                                                                                   Seite 9
                       C. How to price a CDS…

                       1. Choose a credit risk model for default probability; structural (firm value based)
                          or reduced- form (intensity based)

                       2. Input:

                        -   Payment Data

                                  Inception date t0, premium payment dates t0,t1,… maturity date tN, contract details
                                   (existance of accrued payment, up-front fee)

                        -   Market Data

                                  yield curve, for discount factors Z(t,T)

                                  term structure of hazard rates for survival probabilities Q(t,T) ->bootstrapping

                                  Recovery rate R

                       3. Valuate Premium Leg PremiumPV(t,T) and Protection Leg ProtectionPV(t,T)

                       4. Calculate break-even spread S(tV,T)
                                                                                                                      Seite 10
                       C. Intensity based model by Jarrow and Turnbull (1995)

                       •Credit event is modeled directly as a first event of a Poisson counting process, occurs at 


                         - hazard rate process     (t )   is deterministic function of t

                               independent from interest rate and recovery rate process

                         - recovery rate R is an exogenously given fraction (Recovery of Treasury)

                       •Continuous probability of survival from tV until T, given that default did not occur until tV

                       considering the limit   dt  0

                                                                                                                        Seite 11
                       C. Valuing the Premium Leg I
                       1. premium payments until         or maturity of the CDS, whichever occurs first

                       2. accrued payment from the last premium payment

                        1. PART. Notation and calculation of PV of premium payments:

                        -   n=1,..,N : number of premium payments

                        -   t1,t2,…,tN : premium payment dates, tN: maturity of CDS

                        -   S(t0,tN): the contractual swap rate

                        -   (tn 1 , tn , B) daycount fraction between tn-1 to tn with convention B (e.g. Actual/360, 30/360,…)

                        -   Q(tV,tn) : the arbitrage-free survival prob.

                        -   Z(tV,tn) : the LIBOR discount factor (assume we have the full term structure Z(t,T) in YTL)

                        -   Assume there are no accrued payments here:

                                                                                                                                   Seite 12
                       C. Valuing the Premium Leg II
                       2.Part: Accrued payment calculation: if    tn 1 , tn 

                         1. determine the prob. of survival (PS) from tV until each time s between tn-1 ,tn and
                            prob. of default (PD) in next small next interval ds

                         2. Calculate the accrued payment for each time

                         3. Discount them

                         4. Integrate over all times in premium periods

                         5. Sum over all payment dates from first n=1 to final one n=N

                         6. Assume average accrued premium is the half of the full payment paid on the end of accrual
                            period and approximate the integral:

                                                                                                                  Seite 13
                       C. Valuing the Premium Leg III

                       •1PA: indicator for accrued payment

                       •RPV01: risky PV01

                       Full premium leg value=


                                                             Seite 14
                       C.Valuing the Protection Leg

                       1. Calculate the PS to some future time s       Q(tV,s)

                       2. Compute the PD in next small time interval       ( s)ds
                       3. Payoff: (1-R), discount it with risk-free rate Z(tV,s)

                       4. Consider that default can happen on the first day s=tV until maturity s=tN

                          Expected PV of recovery:

                       5. Assume default can happen only at a finite number M of discrete points per year

                          Discretize the integral with m=1,…,tN X M

                                                                                                            Seite 15
                       C.Calculation of Break Even Spread
                       •CDS price on tV=t0:

                                     Value of CDS = ProtectionPV(tV ,tN) -PremiumPV(tV ,tN)

                       •General pricing rule:

                        - Set a CDS spread S(tV,tN) which makes Value of CDS = 0

                        - PremiumPV(tV ,tN)= ProtectionPV(tV ,tN)

                       •we have direct relation to market quoted CDS spreads and PS it implies

                       •Time to construct the term structure for hazard rates!

                                                                                                 Seite 16
                       C.Constructing the Term Structure of Hazard Rate

                       •Assume the hazard rate is a piecewise constant function of maturity time

                                                                      Source: Lehman Brothers

                       •i.e. given market quoted CDS spreads for 1Y, 3Y, 5Y, 7Y, 10Y contracts, calculate

                            0,1, 1,3 , 3,5 , 5,7 , 7,10   iteratively -> Bootstrap

                       i.e. use 1Y CDS market spread-> 0,1 -> Q(tV,T)->3Y CDS market spread-> 1,3         -> …
                                                                                                                   Seite 17
                       C.Some good reasons to make a CDS contract:

                       •hedging the default risk on the (non-traded) reference asset

                       •having a view on the decline (or improvement) of credit quality of the issuer

                       •Buying a CDS is a synthetic short position in reference asset

                        - Buying a CDS might be easier than to short sell an asset

                       •in emerging markets CDS has more liquidity than cash market

                       • institutions with low funding costs are able to earn return with funding a defaultable asset on
                       balance sheet and buying protection

                       •to buy exposure to the credit via a default swap is cheaper than buying the cash bonds and

                       funding them on balance sheets for the banks with high funding costs

                                                                                                                    Seite 18
                       D. Correlation Products: Collateralized Debt Obligations CDO

                       •Market facts:

                        - Most popular correlation product

                        - Estimated to reach $2 trillion by the end of 2006

                        - Since 1998, average annual growth rate is 150%

                       •Basket of credits

                        - Cash CDO(loans-> CLO; bonds-> CBO;

                        - Synthetic CDOs (CDS)

                       •Credit risk is transferred via tranches

                              Equity tranche: %0 – l %, l >0%

                              Mezzanine tranche(s): l%-u%, u<100%

                              Senior tranche: u-100%
                                                                                      Seite 19
                       D. Correlation Products: Default Correlation

                       • Valuation depends on modeling default correlation

                       • based on structure of dependence

                       • can be modeled with

                        - Fixed time horizon (Static) or Continuous time models (Dynamic, or semi-

                             Moody‘s Binomial Expansion Technique „BET“

                             Factor models; one-factor, multi factor

                                - Firm‘s value models „Merton type“

                                - Intensity based models

                                - Copula approach : the default time of several obligors are modeled
                                  --> implemented by MC simulation or (semi-) analytical techniques
                                                                                                       Seite 20
                       D.Copula Functions
                                     :random variables

                                                                       : marginal distribution function

                                                                        : multivariate distribution function

                                                            : Copula function

                       A function C : [0,1]  [0,1] is a copula if

                       (a) There are random variables U1 ,...,U N taking values in [0,1] such that C is
                            their distribution function

                       (b) C has uniform marginal distributions, i.e. for all i  N , ui  [0,1]

                                                 C (1,...,1, ui ,1,...,1)  ui
                        Sklar‘s theorem: there exists an N-dimensional copula C such that for all              x    N

                                   F ( x1, x2 ,..., xn )  C ( F1 ( x1 ), F2 ( x2 ),..., FN ( xN ))  C ( F ( x ))
                                                                                                                         Seite 21
                       D. Examples of Copula Functions
                       •Elliptical Copulas (Gaussian, student-t), Archimedian Copulas (Clayton, Gumbel,)
                       Gaussian Copula:
                       N variate normal distribution with N x N correlation matrix

                        Sampling of Gaussian Copula

                                                                                                           Seite 22
                       D. ITWM-Copula Visualizing Tool

                                                         Seite 23
                       E. Calibration of Default Correlation in Copula Function

                       Using CreditMetrics approach:

                       •Suppose obligors A, B have one year PD of PDA , PDB respectively,

                       •Obtain ZA and ZB such that       PDA  P[ Z  Z A ]                Z~N(0,1)

                                                         PDB  P[ Z  Z B ]

                       •Joint default probability for A and B , with asset correlation :      
                                                       Z A ZB

                           P[ Z  Z A , Z  Z B ]      
                                                        
                                                             2 ( x, y |  )  N 2 ( Z A , Z B )
                       •use bivariate normal copula function with correlation parameter              
                       •Denote default times with A  B and cdf with FA , FB respectively, then joint default probability:


                        P[ A  1, B  1]  CGauss  N 2 ( N 1 ( FA (1)), N 1 ( FB (1)))

                            PDi  P[ i  1]  Fi (1)  Zi  N 1( PDi ) for i  A, B
                       Notice that if               then joint PDs are same!

                                                                                                                             Seite 24
                       E.CDO Pricing

                       1. Tranche structure-> determine the attachment and detachment points

                       2. Choose a default correlation model -> i.e. one factor gaussian copula

                       3. Input

                           •    Payment Data

                                     Inception date t0, premium payment dates t0,t1,… maturity date tm, contract details
                                      (number of Obligors, Notionals & Recovery rates of each obligor…)

                           •    Market Data

                                     yield curve, for discount factors Z(t,T)

                                     term structure of hazard rates for PS Q(t,T) bootstrapping from CDS spreads“

                       •       Determine the PV of premium and the protection leg

                       •       Find expected value of tranche
                                                                                                                       Seite 25
                       E.CDO Pricing

                                       Seite 26
                       E.CDO Pricing with Monte Carlo Method

                       •Determine the correlated default times       i    for n credits

                       • Ri : recovery rate for i th credit

                       • Ni : nominal for i th credit

                       • [l,u] : the tranche, where l : attachment point, u: detachment point

                       • portfolio loss:
                                           Lportfolio (t )   N i (1  Ri )1 i t
                                                            i 1

                       •Tranche notional:

                                                Ntranche  N portfolio (u  l )
                       • tranche loss:

                                 Ltranche (t )  max( Lportfolio (t )  l ,0)  max( Lportfolio (t )  u,0)
                                                                                                              Seite 27
                       E.CDO Pricing with Monte Carlo Method

                       •Probability that tranche will default:
                                                               E0 [ Ltranche (t )]
                                              PDtranche (t ) 
                                                                  N tranche
                       •Premium Leg PV, where tm: maturity of the CDO tranche
                                                S (tV , tm ) N tranche  (t j 1, t j , B)(1  PDtranche (t j )) Z (tV , t j )
                                                                      j 1
                       •Protection Leg PV:

                                          N tranche  Z (tV , s )dPDtranche ( s)

                         - Discretisize the integral:
                                         N tranche  Z (tV , ti )( PDtranche (ti )  PDtranche (ti 1 ))
                                                    i 1

                                                                                                                                  Seite 28
                       E.Break-even spread of the CDO

                       • From the tranche buyer (protection seller) side:

                           Tranche PV = PremiumLegPV-ProtectionLegPV

                       •   perform these steps m times, add up all PV, devide the sum by m

                                  -   expected value of the tranche [l u]

                                                                                             Seite 29
                       E. CDO pricing with semi analytic techniques

                          calculate the Loss distribution directly

                                   P[ Ltranche  L]
                          ->calculate the E[Ltranche]

                          -> calculate the PDtranche

                       Use factor (i.e, one-factor Gaussian) models, usually with firm‘s value credit risk models,

                          determine the tresholds

                                                                                                                     Seite 30
                       E.CDO Pricing literature with semi-analytical techniques
                       Laurent and Gregory
                       •1 factor Gauss Copula model
                       •Calculation with FFT method

                       Hull and White
                       •1 factor Gauss-Copula model
                       • Loss distribution calculation via recursion
                       •Valuation of Nth to Default is possible
                       Andersen and Sidenius
                       •1 factor Gauss Copula model with factor loading

                       • Loss distribution calculation via recursion
                       •Stochastic recovery is possible

                                                                                  Seite 31

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