1. Based on Risk Management, Crouhy, Galai, Mark, McGraw-Hill, 2000 Credit Risk Modeling Economic Models of Credit Risk Lectures 10 & 11

2. The Contingent Claim Approach - Structural Approach: KMV (Kealhofer / McQuown / Vasicek)

3. 3 KMV challenges CreditMetrics on several fronts: 1. Firms within the same rating class have the same default rate 2. The actual default rate (migration probabilities) are equal to the historical default rate (migration frequencies) Default rates change continuously while ratings are adjusted in a discrete fashion.Default rates change continuously while ratings are adjusted in a discrete fashion. Default rates vary with current economic and financial conditions of the firm.Default rates vary with current economic and financial conditions of the firm. The Option Pricing Approach: KMV

4. 4 KMV challenges CreditMetrics on several fronts: 3. Default is only defined in a statistical sense without explicit reference to the process which leads to default. KMV proposes a structural model which relates default to balance sheet dynamicsKMV proposes a structural model which relates default to balance sheet dynamics Microeconomic approach to default: a firm is in default when it cannot meet its financial obligationsMicroeconomic approach to default: a firm is in default when it cannot meet its financial obligations This happens when the value of the firm’s assets falls below some critical levelThis happens when the value of the firm’s assets falls below some critical level The Option Pricing Approach: KMV

5. 5 KMV’s model is based on the option pricing approach to credit risk as originated by Merton (1974) 1. The firm’s asset value follows a standard geometric Brownian motion, i.e.:       tt ttVV    ) 2 (exp 2 0 t t t dZdt V dV  The Option Pricing Approach: KMV

6. 6 Equity: S t AssetsLiabilities / Equity Risky Assets: V t Debt: B t (F) Total: V t V t 2. Balance sheet of Merton’s firm

7. 7 Equity value at maturity of debt obligation:  0,maxFVS TT  Firm defaults if FV T  with probability of default (“real world” probability measure)  2 2 0 0 2 dN T T F V Ln PFVP TT                             The Option Pricing Approach: KMV

8. 8 Assets Value V T V 0 Probability of default Time T F E V Ve T O T ()    VVTT TOT              exp   2 2  3. Probability of default (“real world” probability measure) The Option Pricing Approach: KMV Distribution of asset values at maturity of the debt obligation

9. 9 Time 0T Value of Assets V 0 V T  F V T > F Bank’s Position: · make a loan -B 0 V T F buy a put -P 0 F - V T O Total -B 0 -P 0 FF B 0 + P 0 = Fe -rT Bank’s pay-off matrix at times 0 and T for making a loan to Firm ABC and buying a put on the value of ABC Corporate loan = Treasury bond + short a put The Option Pricing Approach: KMV ·

10. 10 Firm ABC is structured as follows: V t = Value of Assets (at time t) S t = Value of Equity B t = Value of Debt (zero-coupon) F = Face Value of Debt 1P o = f ( V o, F,  v, r, T )(Black-Scholes option price) 2B o = Fe -rT - P o 3S o = V o - B o (assuming markets are frictionless) 4B o = Fe -Y T T where Y T is yield to maturity 5 Probability of Default = g (V o, F,  v, r, T) = N ( - d 2 ) (“risk neutral” probability measure) 6Conditional recovery when default = V T KMV: Merton’s Model

11. 11 Problem: V o ( say =100 ), F ( say = 77 ),  v ( say = 40% ), r ( say =10% ) and T ( say = 1 year) Solve for B o,S o,Y T and Probability of Default KMV: Merton’s Model

12. 12 ` P 0 ( = 3.37)  B o ( = 66.63)  S o ( = 33.37)  Y T ( =15.6%)   T ( = 5.6%) Note: In solving for P 0 we get Probability of Default ( = 24.4% ) BFeP o rT o   SVB ooo YL F B Yr TN o TT         Solution: PfVT oo  (,)  KMV: Merton’s Model

13. 13 Default spread ( ) for corporate debt ( For V 0 = 100, T = 1, and r = 10% ) rY TT  0 V Fe LR rT   Leverage ratio: The Option Pricing Approach: KMV  LR 0.050.100.200.40 0.50001.0 0.6000.1%2.5% 0.7000.4%5.6% 0.800.1%1.5%8.4% 0.90.1%0.8%4.1%12.5% 1.02.1%3.1%8.3%17.3%

14. 14 KMV: EDFs (Expected Default Frequencies) 4. Default point and distance to default Observation: Firms more likely to default when their asset values reach a certain level of total liabilities and value of short-term debt. Default point is defined as DPT=STD+0.5LTD STD - short-term debt LTD - long-term debt

15. 15 Asset Value Time 1 year DPT = STD + ½ LTD Expected growth of assets, net E(V) 1 DD Probability distribution of V 0 V0V0 KMV: EDFs (Expected Default Frequencies) Default point (DPT)

16. 16 KMV: EDFs (Expected Default Frequencies) Distance-to-default (DD) DD - is the distance between the expected asset value in T years, E(V T ), and the default point, DPT, expressed in standard deviation of future asset returns:

17. 17 5. Derivation of the probabilities of default from the distance to default 5 6 4 3 DD 21 EDF 40 bp KMV also uses historical data to compute EDFs KMV: EDFs (Expected Default Frequencies)

18. 18 KMV: EDFs (Expected Default Frequencies) Example: V0 = 1,000 20% V 1 = V0(1.20) = 1,200 Current market value of assets: Net expected growth of assets per annum: Expected asset value in one year: Annualized asset volatility, Default point  : 100 800 DD    1200800 100 4, Assume that among the population of all the firms with DD of 4 at one point in time, e.g. 5,000, 20 defaulted one year later, then: EDF year1 20 5000 000404% ,.. or 40 bp The implied rating for this probability of default is BB +

19. 19 KMV: EDFs (Expected Default Frequencies) Example: Federal Express ($ figures are in billions of US$) November 1997 February 1998 Market capitalization (S 0 ) (price* shares outstanding) Book liabilities Market value of assets (V 0 ) Asset volatility Default point Distance to default (DD) EDF $ 7.8 $ 4.8 $ 12.6 15% $ 3.4 12.6-3.40.15·12.60.06%(6bp) $ 7.3 $4.9 $ 12.2 17% $ 3.5 12.2-3.50.17·12.2 0.11%(11bp) 0.11%(11bp)  = 4.9 = 4.2 

20. 20 KMV: EDFs (Expected Default Frequencies) 4.EDF as a predictor of default EDF of a firm which actually defaulted versus EDFs of firms in various quartiles and the lower decile. The quartiles and decile represent a range of EDFs for a specific credit class.

21. 21 KMV: EDFs (Expected Default Frequencies) 4.EDF as a predictor of default EDF of a firm which actually defaulted versus Standard & Poor’s rating.

22. 22 KMV: EDFs (Expected Default Frequencies) 4.EDF as a predictor of default Assets value, equity value, short term debt and long term debt of a firm which actually defaulted.

23. IV The Actuarial Approach: CreditRisk+ IV The Actuarial Approach: CreditRisk+ Credit Suisse Financial Products

24. 24 In CreditRisk+ no assumption is made about the causes of default: an obligor A is either in default with probability P A, or it is not in default with probability 1-P A. It is assumed that: for a loan, the probability of default in a given period, say one month, is the same for any other month for a loan, the probability of default in a given period, say one month, is the same for any other month for a large number of obligors, the probability of default by any particular obligor is small and the number of defaults that occur in any given period is independent of the number, of defaults that occur in any other period for a large number of obligors, the probability of default by any particular obligor is small and the number of defaults that occur in any given period is independent of the number, of defaults that occur in any other period The Actuarial Approach: CreditRisk+

25. 25 Under those circumstances, the probability distribution for the number of defaults, during a given period of time (say one year) is well represented by a Poisson distribution:  = average number of defaults per year         A A P  where It is shown that can be approximated as  The Actuarial Approach: CreditRisk+

26. 26 CreditRisk+: Frequency of default events Credit Rating Aaa Aa A Baa Ba B Average (%) 0.00 0.03 0.01 0.13 1.42 7.62 Standard deviation (%) 0.0 0.1 0.0 0.3 1.3 5.1 Note, that standard deviation of a Poisson distribution is. For instance, for rating B:. CreditRisk+ assumes that default rate is random and has Gamma distribution with given mean and standard deviation. One year default rate Source: Carty and Lieberman (1996) 1.5 76.262.7versus  

27. 27 Distribution of default events CreditRisk+: Frequency of default events Probability Excluding default rate volatility Including default rate volatility Number of defaults Source: CreditRisk+

28. 28 CreditRisk+: Loss distribution In CreditRisk+, the exposure for each obligor is adjusted by the anticipated recovery rate in order to produce a loss given default (exogenous to the model)

29. 29 1.Losses (exposures, net of recovery) are divided into bands, with the level of exposure in each band being approximated by a single number. Notation Obligor A Exposure (net of recovery) Probability of default Expected loss A =L A xP A LALA PAPA CreditRisk+: Loss distribution

30. 30 Obligor A Exposure ($) (loss given default) L A Exposure (in $100,000) j Round-off exposure (in $100,000) j Band j 1150,0001.522 2460,0004.655 3435,0004.3555 4370,0003.744 5190,0001.922 6480,0004.855 The unit of exposure is assumed to be L=$100,000. Each band j, j=1, …, m, with m=10, has an average common exposure: v j =$100,000j Example: 500 obligors with exposures between $50,000 and $1M (6 obligors are shown in the table) CreditRisk+: Loss distribution

31. 31 Notation Common exposure in band j in units of L j Expected loss in band j in units of L  j (for all obligors in band j) Expected number of defaults in band j  j In Credit Risk+ each band is viewed as an independent portfolio of loans/bonds, for which we introduce the following notation: j = $100,000, $200,000, …, $1M  j can be expressed in terms of the individual loan characteristics  j = j x  j CreditRisk+: Loss distribution

32. 32 Band: j Number of obligors e j m j 1301.5 (1.5x1)1.5 2408 (4x2)4 3506 (2x3)2 47025.26.3 5100357 66014.42.4 75038.55.5 84019.22.4 94025.22.8 10204 (0.4x10) 0.4 CreditRisk+: Loss distribution

33. 33 To derive the distribution of losses for the entire portfolio we proceed as follows: Step 1: Probability generating function for each band. Each band is viewed as a portfolio of exposures by itself. The probability generating function for any band, say band j, is by definition: j n n n n j zdefaultsnPznLloss j PzG       00 )()()( where the losses are expressed in the unit L of exposure. Since we have assumed that the number of defaults follows a Poisson distribution (see expression 30) then: j jjj j zn n j n j ez n e zG           ! )( 0 CreditRisk+: Loss distribution

34. 34 Step 2: Probability generating function for the entire portfolio. Since we have assumed that each band is a portfolio of exposures, independent from the other bands, the probability generating function for the entire portfolio is just the product of the probability generating functions for all bands.         m j j m j j j jj jz z m j eezG 11 1 )(   where    m j j 1   denotes the expected number of defaults for the entire portfolio. CreditRisk+: Loss distribution

35. 35 CreditRisk+: Loss distribution Given the probability generating function (33) it is straightforward to derive the loss distribution, since these probabilities can be expressed in closed form, and depend only on 2 sets of parameters:  j and j. (See Credit Suisse 1997 p.26),...2,1| )( ! 1 )( 0   nfor dz zGd n nLoflossP z n n Step 3: loss distribution for the entire portfolio ()  L vnP n nLP j nvj j j  =   of loss of loss :    ()()  ==  j j j v ee GP  0 0

36. V Reduced Form Approach Duffie-Singleton - Jarrow-Turnbull

37. 37 Reduced Form Approach Reduced form approach uses a Poisson process like environment to describe default. Contrary to the structural approach the timing of default takes the bond-holders by surprise. Default is treated as a stopping time with a hazard rate process. Reduced form approach is less intuitive than the structural model from an economic standpoint, but its calibration is based on credit spreads that are “observable”.

38. 38 Example: a two-year defaultable zero-coupon bond that pays 100 if no default, probability of default, LGD=L=60%. The annual (risk- neutral) risk-free rate process is : Reduced Form Approach 5.0 1  p 08.86 12.1 1004.006.010094.0 11    V 64.87 1.1 1004.006.010094.0 12    V  52.77 08.1 4.006.094.05.04.006.094.05.0 12 11 0    VVVV V %8  r %10  r 5.0 2  p %12  r

39. 39 Reduced Form Approach “Default-adjusted” interest at the tree nodes is: %2.161 08.86 100 11  R %1.141 64.87 100 12  R % 1 52.77 64.875.008.865.0 0    R )1(1Ltt tLtr tR    In all three cases R is solution of the equation ( ):  )1()1( 1 1 1 1 Ltt trtR     1  t If, then, where is the risk-neutral expected loss rate, which can be interpreted as the spread over the risk-free rate to compensate the investor for the risk of default. 0  t LrR  L

40. 40 Reduced Form Approach General case: is hazard rate, so that if denotes the time to default, the survival probability at horizon t is  t          t dssEtProb 0 ))(exp()(  E is expectation under risk-neutral measure. For the constant we have:   t )exp()(tEtProb  The probability of default over the interval provided no default has happened until time t is:  ttt , tttttProb  )(  (similar to the example above).            

41. 41 Reduced Form Approach Term structure of interest rates t  tR Treasury curve Corporate curve Yield spread = L r R rR, Maturity

42. 42 Reduced Form Approach By modelling the default adjusted rate we can incorporate other factors which affect spreads such as liquidity: lLrR  where l denotes the “liquidity” adjustment premium. if there is a shortage of bonds and one can benefit from holding the bond in inventory, if it becomes difficult to sell the bond. 0l  0l  Identification problem : how to separate and in. Usually is assumed to be given. Implementations differ with respect to assumptions made regarding default intensity. L L L

43. 43 Reduced Form Approach How to compute default probabilities and Example. Derive the term structure of implied default probabilities from the term structure of credit spreads (assume L=50%). Maturity t (years) Treasury curve (%) Company X one-year forward rates (%) One-year forward credit spreads FS t (%) 15.525.76 0.24 26.306.74 0.44 36.407.05 0.65 46.567.64 1.08 56.567.71 1.15 66.818.21 1.40 76.818,47 1.65

44. 44 Reduced Form Approach For example, for year 4:, then %08.1 44  LFS 16.2 4  Cumulative probability:  74.41 4334  PPP Conditional probability:  10.21 434  Pp Maturity t (years) Forward probabilities of default t (%) Cumulative defauilt probabilities t P (%) Conditional default probabilities t p (%) 10.48 20.881.360.88 31.302.641.28 42.164.742.10 52.306.932.19 62.809.542.61 7 3.3012.522.99

45. 45 Reduced Form Approach Generalizations: Intensity of the default is modeled as a Cox process (CIR model), conditional on vector of state variables, such as default free interest rates, stock market indices, etc. where is a standard Brownian motion, is the long-run mean of is mean rate of reversion to the long-run mean, is a volatility coefficient. Properties:, Conditional survival probability, where and are known time-dependent functions of time, The volatility of is  t  tX , tdBtdttktd      tB k   0  t   tstst estp  ,    stp,  ttsstp ,

46. 46 Reduced Form Approach Generalizations: Intensity of the default can be modeled as a jump process: where, - cumulative jumps by at Poisson arrival times, is mean arrival rate, is mean jump size.  t , tdZdttktd   JttNtZ  t  J  tN (b.p.) Years 0 Take jumps sizes to be, say, independent and exponentially distributed.

47. 47 Reduced Form Approach Generalizations: Risk free spot rate is modeled as one factor extended Vasicek process. where,, are similar to parameters in CIR model, is a function defined from current term structure of interest rates. is correlated with Brownian motion of the default intensity process. Closed form solutions for the bond prices.  t 1   tr , 1111 tdBdttrtktdr   tB 1 1 k 1   tB 1

48. 48 Reduced Form Approach Inputs : the term structure of default-free rates the term structure of credit spreads for each credit category the loss rate for each credit category Model assumptions : zero correlations between credit events and interest rates deterministic credit spreads as long as there are no credit events constant recovery rates