Credit Risk. June 2014

Transcription

1 Credit Risk Dr. Sudheer Chava Professor of Finance Director, Quantitative and Computational Finance Georgia Tech, Ernest Scheller Jr. College of Business June 2014

2 The views expressed in the following material are the author s and do not necessarily represent the views of the Global Association of Risk Professionals (GARP), its Membership or its Management. 2

3 Information for Credit Risk Evaluation Multiple Sources of information Credit Rating Agencies! Accounting Information Stock Prices Credit Default Swap prices Bond Markets Sudheer Chava GARP Atlanta June / 28

4 Methods for Credit Risk Evaluation Multiple methods to evaluate credit risk. If available directly use Credit Rating Agencies ratings If not rated, compute synthetic credit ratings based on Credit Rating Agencies ratings Independent Internal credit score models Implied default probabilities from market prices (bond market, stock market, credit default swap market) Statistical Models Risk-Neutral vs Physical default probabilities Sudheer Chava GARP Atlanta June / 28

5 Credit Ratings If credit ratings are not available can try to replicate credit ratings many methods Comparables (similar to matrix pricing in bond market) Statistical Models Steps involved identify plausible factors used by credit rating agencies (reverse engineer) project the firm s characteristics onto the rated universe calculate a synthetic credit rating Sudheer Chava GARP Atlanta June / 28

6 Statistical Models for Predicting Defaults Static Models Linear Discriminant Analysis (DA) (eg: Altman s Z-score) Logistic Regression and Probit Models (eg: Static models) Hazard Models (eg: Chava and Jarrow (2004), Chava, Stefanescu and Turnbull (2012)) Sudheer Chava GARP Atlanta June / 28

7 Statistical Models: Steps Steps involved in implementing statistical models of default 1 Define default 2 Decide on the sample selection criteria 3 Decide on a set of explanatory variables (eg. accounting data) that may have an impact on the credit risk of the firm 4 Identify the default status of all the firms in the sample 5 Gather data on the explanatory variables for all sample firms 6 Run the statistical model (eg. DA, logistic model or hazard model) 7 Evaluate the in-sample performance of the model 8 Evaluate the out-of-sample performance of the model Sudheer Chava GARP Atlanta June / 28

8 Equity Based Estimation: Intuition Structural Model of Default: Debt in Merton model can be decomposed into A risk-free security with the same face value, F and the same maturity T as the risky debt in the firm s capital structure A put option on the firm s assets struck at the face value of debt. The lender or purchaser of the firm s debt implicitly writes this put option to the firm s shareholders, who can put the firm s assets back to the debt holder in case V < F. This option is similar to a credit default swap (CDS) D = e rt F CDS E + e rt F = CDS + V Sudheer Chava GARP Atlanta June / 28

9 Distance to Default model Based on the Black-Scholes formula, value of the equity is E = V N (d 1 ) e rt F N (d 2 ) where E is the market value of the firm s equity, F is the face value of the firm s debt, r is the instantaneous risk-free rate, N (.) is the cumulative standard normal distribution function, d 1 = log(v /F ) + (r + σ2 V /2)T σ V T d 2 = d 1 σ V T Sudheer Chava GARP Atlanta June / 28

10 Distance to Default model In this model, the second equation, using an application of Ito s lemma and the fact that E V = N (d 1), links the volatility of the firm value and the volatility of the equity. σ E = V E N (d 1)σ V Sudheer Chava GARP Atlanta June / 28

11 Distance to Default model The unknowns in these two equations are the firm value V and the asset volatility σ V. The known quantities are equity value E, face value of debt or the default boundary F, risk-free interest rate r, time to maturity T. Sudheer Chava GARP Atlanta June / 28

12 Distance to Default Computation Once we compute V, σ V, the probability of first passage time to the default boundary is given by EDF EDF = N ( DD) where DD is the distance to default and is defined as DD log(v /F ) + (µ σ2 V /2)T σ V T V is the total value of the firm; F is a face value of firm s debt; µ is the expected rate of return on the firm s assets; σ V is the volatility of the firm value, and T is the time horizon that is set to one year. Sudheer Chava GARP Atlanta June / 28

13 CDS Based Estimation: Intuition 1-period CDS contract, Notional Amount N = 1, probability of default p, recovery rate R Expected payout of protection seller: L = p(1 R) CDS spread: S = p(1 R) 1+r Implied Probability of Default: p = S(1+r) (1 R) Sudheer Chava GARP Atlanta June / 28

14 CDS Based Estimation: Simple Model 1 CDS pricing equation: Prem T = Prot T Simple Model Assumptions: Constant default intensity λ, Constant known recovery rate R, Ignoring accrued premium between 2 payments f.t Prot T = (1 R) exp( r i t i )(Q(τ t i 1 ) Q(τ t i )) i=1 f.t Prem T = exp( r i t i ) s m Q(τ t i) i=1 Survival probability Q(τ t n ) = exp( λt n ) Solve for λ based on (liquid) CDS spreads observed in the markets Sudheer Chava GARP Atlanta June / 28

15 CDS Based Estimation: Simple Model 2 CDS pricing equation: Prem T = Prot T Simple Model Assumptions: Polynomial form for default intensity λ, Constant known recovery rate R, Take into account accrued premium f.t Prem T = exp( r i t i ) s m Q(τ t i) i=1 f.t Prot T = L(i) exp( r i t i )(Q(τ t i 1 ) Q(τ t i )) i=1 Default occurs at t i then: L(i) = (1 R) s m (t i t last coupon ) t coupon Where: λ(t) = a + bt + ct 2 ; Q(τ t n ) = exp( n i=1 λ(t i) t i ) Solve for a, b, and c based on (liquid) CDS spreads (1-yr, 3-yr, 5-yr CDS spreads) observed in the markets Sudheer Chava GARP Atlanta June / 28

16 Default probability CDS Based Estimation: Simple Model % 1.60% Default Probability: Polynomial Method 1.40% 1.20% 1.00% 0.80% 0.60% 0.40% 0.20% 0.00% Time in years Sudheer Chava GARP Atlanta June / 28

17 CDS Based Estimation: Bootstrapping General form of protection leg: Prot T t = (1 R) t+t E Q t t [ ( λ u exp u t )] (r s + λ s ) ds du Assume: Deterministic interest rates and default intensities. Let t = 0 then: Prot T = (1 R) T 0 e t 0 (λs +rs ) ds λ t dt For Bootstrapping procedure: Let λ and r be piecewise continuous functions and constant between two CDS tenors that trade in the market. N ( ) Then: Prot T = (1 R) λj S(T j 1 )D(T j 1 ) [1 ] e (λ j +r j )h λ j + r j j=1 Where: S(T j ) = e j k=1 λ j h, D(T j ) = e j k=1 r j h, and T j ɛ{t 1, T 3, T 5, T 7, T 10 } λ j, r j are default intensities and forward rates between T j 1 and T j, h = T j T j 1 and N is the N th CDS of tenor T Sudheer Chava GARP Atlanta June / 28

18 CDS Based Estimation: Bootstrapping General form of premium leg: Prem T t = S T t RPV T t Where: RPV T t = N n=1 N δ(t n 1, t n )E Q t n=1 tn t n 1 δ(t n 1, u)e Q t [ ( exp tn [ ( λ u exp t )] (r s + λ s ) ds + u t )] (r s + λ s ) ds du let δ(t n 1, t n ) is the day count fraction between two consecutive premium payment dates, Frequency of premium payment: m per year, t = 0, N = m T then: RPV T = N n=1 1 tn m e 0 (λs +rs ) ds + N n=1 tn t n 1 (t t n 1 )e t 0 (λs +rs ) ds λ t dt Sudheer Chava GARP Atlanta June / 28

19 CDS Based Estimation: Bootstrapping Solving we get: N 1 Payment Term = m S(t n)d(t n ) Accrued Term = n=1 N λ n S(t n 1 )D(t n 1 ) n=1 SoRPV T = N n=1 N n=1 1 m 1 m S(t n)d(t n )+ tn λ n (λ n + r n ) S(t n 1)D(t n 1 ) t n 1 e (λn+rn)(t t n 1) (t t n 1 ) dt ( ) 1 e (λn+rn). 1 m Where: CDS T = ProtT RPV T Sudheer Chava GARP Atlanta June / 28

20 Bootstrapping Forward Curve of Interest Rates Bootstrapping Interest Rates data: interest rate swaps with maturities of 1,2,3,4,5,7 and 10 years and US Libor money market deposits with maturities 1,3,6,9 months. Let swap rate s, frequency of swap payments m (usually quarterly), maturity of swap T, Number of payments N = m T Swaps are priced such that s is the coupon payment on a bond trading at par with coupon payment frequency m and maturity T Pricing equation: 1 = N n=1 s tn m e r 0 t dt + 1.e t N 0 r t dt Bootstrap forward curve using above equation and assuming forward rate r t to be a piecewise linear continuous function and constant between any 2 swap payments Sudheer Chava GARP Atlanta June / 28

21 Default probability CDS Based Estimation: Bootstrapping 5.00% 4.50% Default Probability: Bootstrapping Method 4.00% 3.50% 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% Time in years Sudheer Chava GARP Atlanta June / 28

22 Bond Based Estimation: Intuition 1-period Bond contract, Principal Amount N = 1, probability of default p, recovery rate R Bond price: B = (1 p)+p(r) 1+r Implied Probability of Default: p = 1 (1+r)B (1 R) All methods above used to estimate default probabilites from CDS can be used for Bonds as well. Sudheer Chava GARP Atlanta June / 28

23 Bond Based Estimation: Exponential Spline Applied widely in the industry Bond Price: N B = exp( r i t i )CF tot (t i )Q(τ t i ) + i=1 Let Q(τ t) = 3 β 3 e kαt k=1 Survival probability at t=0 is 1 implies N exp( r i t i )(1.R principal + CF i=1 3 β i = 1 k=1 Decay parameter α interpreted as long-maturity asymptotic limit of hazard rate Sudheer Chava GARP Atlanta June / 28

24 Equity, CDS Based Estimation: Fair-value CDS spreads Risky Debt = Default-free Debt - Expected Loss Value B = Fe rt ELV Put option = Expected Loss Value(ELV): ELV = Be rt N ( d 2 ) V N ( d 1 ) Fair Value Credit Spread: log(f /D) S = y r = r = 1 T T log(1 ELV Be rt ) Government subsidy = Equity implied FVCDS - market CDS Sudheer Chava GARP Atlanta June / 28

25 CDS Implied Ratings: Intuition Estimate CDS boundaries separating two adjacent (closest) rating groups in a non-parametric manner Misclassifications: CDS spreads of bonds with higher rating is higher than CDS spreads of bonds with lower rating Estimation of CDS boundary: minimize a penalty function with the objective of reducing the number of such misclassifications Example: Minimize penalty function F to estimate boundary between AA and A rating categories is defined by: F (b AA A ) = 1 m m i=1 [max(s i,aa b AA A, 0)] n n j=1 [max(b AA A s j,a, 0)] 2 Sudheer Chava GARP Atlanta June / 28

26 CDS Implied Ratings Construction F (b AA A ) = 1 m m i=1 [max(s i,aa b AA A, 0)] n n j=1 [max(b AA A s j,a, 0)] 2 Where: s i,aa is the CDS spread of AA-rated firm i s j,a is the CDS spread of A-rated firm j m is number of firms in the AA rating class n is number of firms in the A rating class The penalty function for estimating boundaries between other adjacent rating classes are defined similarly Sudheer Chava GARP Atlanta June / 28

27 CDS Implied Rating Model F = r n r i (s i b + r ) 2 n r (b r s i ) 2 n r else 0 if s i > b + r if s i < b r Minimize F where: i iterates over all spread observations r iterates over all rating categories s i is the i th spread, which will be in some r for all cases b + r b r is the upper spread boundary for rating category r is the lower spread boundary for rating category r n r number of spreads in rating category r Note: b r + = b r+1 The upper bound of a category is the lower bound for the next higher category Sudheer Chava GARP Atlanta June / 28

28 CDS Implied Rating Model The Fitch CDS-IR model first penalizes spreads that are above the rating boundary for its rating category (i.e s i > b + r ) A symmetrical penalty is assessed for spreads that are below the applicable rating boundary (i.e s i < b r ) The penalty rises with the square of the distance making this function differentiable in all cases It is summed across all rating categories and across all CDS spreads within each category With such a specification, the boundaries would cannot cross which can sometimes be a problem when the penalty function is minimized individually Sudheer Chava GARP Atlanta June / 28

29 CDS Implied Rating Scale vs CRA Rating Scale Are Credit Ratings Stil Relevant? Chava, Ganduri and Ornthanlai (2013) 4.5 Downgrade from BBB 5.5 Downgrade from BB 4 5 Credit ratings Upgrade from BB 4 Upgrade from B 4.8 Credit ratings Event day Event day Rating agencies CDS implied ratings Sudheer Chava GARP Atlanta June / 28