Chapter13-Modelling Spread Dynamics by Salim Kasap

FIXED-INCOME SECURITIES

Chapter 13

Modeling the Credit Spreads Dynamics

Outline • Analyzing Credit Spreads – Ratings – Probability of Default – Severity of Default

• Modeling Credit Spreads – Structural Models – Reduced-Form Models – Historical versus Risk-Adjusted Default Probabilities

Analyzing Credit Spreads Corporate Bonds • Bonds issued by a corporation • Typically pay semi-annual coupons • 3 Sources of Risk – Interest Rate Risk – Default Risk – Liquidity Risk

• Bond indenture contracts stipulate collateral and specify terms • Different “seniority” classes

Analyzing Credit Spreads Bond Quality •

•

Standard & Poor, Moody’s and other firms score ‘the probability of continued & uninterrupted streams of interest & principal payments to investors’ Classes of grades – Moody’s Investment Grades: Aaa,Aa,A,Baa – Moody’s Speculative Grades: Ba, B, Caa, Ca, C – Moody’s Default Class: D

•

Are ratings agencies better able to discern default risk or simply react to events?

Analyzing Credit Spreads Bond Quality Ratings Investment Grade (High Creditworthiness) Moody's S&P Definition Aaa AAA Gilt-edged, best quality, extremely strong creditworthiness

Aa1 Aa2 Aa3 A1 A2 A3 Baa1 Baa2 Baa3

AA+ AA Very high grade, high quality, very strong creditworthiness AAA+ A Upper medium grade, strong creditworthiness ABBB+ BBB Lower medium grade, adequate creditworthiness BBB-

Moody's Ba1 Ba2 Ba3 B1 B2 B3

S&P BB+ BB BBB+ B BCCC+ CCC CCCCC C D

Caa Ca C

Speculative Grade (Low Creditworthiness) Definition Low grade, speculative, vulnerable to non-payment

Highly speculative, more vulnerable to non-payment

Substantial risk, in poor standing, currently vulnerable to non-payment May be in default, extremely speculative, currently highly vulnerable to non-payment Even more speculative Default

The modifiers 1, 2, 3 or +, - account for relative standing within the major rating categories.

Analyzing Credit Spreads Moody’s and S&P Rating Transition (US, end of 1999) TO

F R O M

Moody's Aaa Aa A Baa Ba B Caa D

Aaa 88.66% 1.08% 0.06% 0.05% 0.03% 0.01% 0.00% 0.00%

Aa 10.29% 88.70% 2.88% 0.34% 0.08% 0.04% 0.00% 0.00%

A 1.02% 9.55% 90.21% 7.07% 0.56% 0.17% 0.66% 0.00%

Baa 0.00% 0.34% 5.92% 85.24% 5.68% 0.65% 1.05% 0.00%

Ba 0.03% 0.15% 0.74% 6.05% 83.57% 6.59% 3.05% 0.00%

B 0.00% 0.15% 0.18% 1.01% 8.08% 82.70% 6.11% 0.00%

Caa 0.00% 0.00% 0.01% 0.08% 0.54% 2.76% 62.97% 0.00%

D 0.00% 0.03% 0.01% 0.16% 1.46% 7.06% 26.16% 100.00%

F R O M

S&P AAA AA A BBB BB B CCC D

AAA 91.94% 0.64% 0.07% 0.04% 0.04% 0.00% 0.19% 0.00%

AA 7.46% 91.80% 2.27% 0.27% 0.10% 0.10% 0.00% 0.00%

A 0.48% 6.75% 91.69% 5.56% 0.61% 0.28% 0.37% 0.00%

BBB 0.08% 0.60% 5.11% 87.87% 7.75% 0.46% 0.75% 0.00%

BB 0.04% 0.06% 0.56% 4.83% 81.49% 6.95% 2.43% 0.00%

B 0.00% 0.12% 0.25% 1.02% 7.89% 82.80% 12.13% 0.00%

CCC 0.00% 0.03% 0.01% 0.17% 1.11% 3.96% 60.44% 0.00%

D 0.00% 0.00% 0.04% 0.24% 1.01% 5.45% 23.69% 100.00%

Analyzing Credit Spreads TS of Annual Default Probabilities- Investment Grade 1.65 1.50

Annual Default Probability in %

1.35 1.20 1.05

AAA AA A BBB

0.90 0.75 0.60 0.45 0.30 0.15 0.00 1

Maturity in Years

Analyzing Credit Spreads TS of Annual Default Probabilities - High Yield 40

Annual Default Probability in %

BB B CCC

0 1

Maturity in Years

Analyzing Credit Spreads Recovery Rates • Not only likelihood of default matters but also severity of default • The higher the seniority of a bond the lower the loss incurred, that is the higher its recovery rate • Recovery statistics in table below Seniority class Recovery rate Standard deviation Senior Secured 54% 27% Senior Unsecured 51% 25% Senior Subordinated 39% 24% Subordinated 33% 20% Junior Subordinated 17% 11%

Analyzing Credit Spreads Size of the Corporate Bond Markets - US and Europe Description USD Broad Investment Grade bond market USD Govt./Govt. Sponsored USD Collateralized USD Corporate USD Corporate (Large Capitalizations) USD Corporate AAA AA A BBB Description EUR Broad Investment Grade bond market EUR Govt./Govt. Sponsored EUR Collateralized EUR Corporate EUR Corporate (Large Capitalizations) EUR Corporate AAA AA A BBB

Par Amount (in billion USD) 6,110.51 2,498.23 2,216.73 1,395.56 869.96 1,395.56 35.33 200.81 653.76 505.65 Par Amount (in billion USD) 3,740.77 2,455.24 685.78 599.75 416.96 599.75 143.10 151.55 220.36 84.74

Weight (in %) 100.00 40.88 36.28 22.84 14.24 100.00 2.53 14.39 46.90 36.23 Weight (in %) 100.00 65.63 18.33 16.03 11.15 100.00 23.86 25.27 36.74 14.13

Analyzing Credit Spreads Bond Quoted Spread • Corporate bonds are usually quoted in price and in spread over a given benchmark bond rather than in yield • So as to recover the corresponding yield, you simply have to add this spread to the yield of the underlying benchmark bond • The table hereafter gives an example of a bond yield and spread analysis as can be seen on a Bloomberg screen – The bond bears a spread of 156.4 basis points over the interpolated US swap yield, whereas it bears a spread of 234 basis points over the interpolated US Treasury benchmark bond yield – Furthermore, its spread over the maturity nearest US Treasury benchmark bonds amounts to 259.3 basis points and 191 basis points over the 5-year Treasury benchmark bond and the 10-year Treasury benchmark bond respectively

• The bond bears a 156.4 BP spread over interpolated US swap yield • The bond bears a 234 BP spread over interpolated US T-Bond yield • The bond bears a 259.3 BP spread over the maturity nearest US T-Bond

Example

Analyzing Credit Spreads Term Structure of Credit Spreads • In what follows, we are going to discuss models of the TS of credit spreads, needed for pricing and hedging credit derivatives • On the other hand, to price and hedge risky bonds, it is sufficient to estimate the TS of credit spreads • TS of credit spreads for a given rating class and a given economic sector can be derived from market data through two different methods – Disjoint method: separately deriving the term structure of non-default zerocoupon yields and the term structure of risky zero-coupon yields so as to obtain by differentiation the term structure of zero-coupon credit spreads – Joint method: generating both term structures of zero-coupon yields through a one-step procedure

Analyzing Credit Spreads Disjoint Method • 3 steps procedure – Step 1: derive the benchmark risk-free zero-coupon yield curve (see above) – Step 2: derive the corresponding risky zero-coupon yield curve from a basket of homogenous corporate bonds – Step 3: obtain the TS of credit spreads by substraction

• Drawback of this method – Estimated credit spreads are sensitive to model assumptions like the choice of the discount function, the number of splines and the localisation of pasting points. – Estimated credit spreads may be unsmooth functions of time to maturity which is not realistic and contradictory to the smooth functions (monotonically increasing, humped-shaped or downward sloping) obtained in the theoretical models of credit bond prices like the models by Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995) and others

Analyzing Credit Spreads Joint Method • Define – J i the number of bonds of the ith risk class j

i – Pt∧ market price at date t of the jth bond of the ith risk class

– Pt i theoretical price at date t of the jth bond of the ith risk class –

the coupon and/or principal payment of the jth bond of the ji Fithsrisk class

–

the discount factor associated to the ith risk class, Bi (t ,the s ) price at date t of a zero-coupon bond of the ith risk i.e., class paying $1 at date s (the 0th risk class is the benchmark riskfree class)

Analyzing Credit Spreads Additive and Multiplicative Spreads •

There are two ways of modeling the relationship between discount factors

Bi (t , s ) = B0 (t , s ) + S i (t , s ) – Multiplicative spread: Bi (t , s ) = B0 (t , s ) × Ti (t , s ) – Note that S 0 (t , s ) = 0 and T0 (t , s ) = 1 – Additive spread:

•

Additive model allows us to keep the linear character of the problem for the minimization program (write the discount function as a linear function of parameters to be estimated

•

Multiplicative model leads to a non-linear minimization program, allows us to write the continuously compounded risky zero-coupon rate as the sum of the benchmark zero-coupon rate plus a spread

RiC (t , s ) = R0C (t , s ) + tiC (t , s )

Analyzing Credit Spreads Comparison - Example •

We derive the zero coupon spread curve for the bank sector in the Eurozone as of May 31th 2000, using the interbank zero-coupon curve as benchmark curve

•

For that purpose, we use two different methods – Disjoint method: we consider the standard cubic B-splines to model the two discount functions associated respectively to the risky zero-coupon yield curve and the benchmark curve; we consider the following splines [0;1], [1;5], [5;10] for the benchmark curve and [0,3], [3,10] for the risky class – Joint method: we consider the joint method using an additive spread and use again the standard cubic B-splines to model the discount function associated to the benchmark curve and the spread function associated to the risky spread curve

Analyzing Credit Spreads Disjoint Method - Results Disjoint Method - Cubic B-Splines DATE

Sum of squared spreads Average spread

Rate or Instrument

31/05/2000

0.595 0.146

Maturity

Market Price

Theoretical Price

Spread

07/06/00 30/06/00 31/07/00 31/08/00 30/11/00 28/02/01 31/05/01 31/05/02 30/05/03 31/05/04 31/05/05 31/05/06 31/05/07 30/05/08 29/05/09 31/05/10 07/06/01 02/10/01 14/05/03 21/09/04 22/11/04 03/12/04 06/08/07 01/10/07 10/03/08 21/04/09 06/07/09 27/07/09

99.918 99.646 99.267 98.874 97.648 96.418 95.189 100.000 100.000 100.000 100.000 100.000 100.000 100.000 100.000 100.000 106.666 111.033 104.643 99.411 103.733 106.184 103.457 103.467 96.582 89.518 95.564 97.591

99.917 99.639 99.256 98.865 97.674 96.455 95.189 99.994 100.015 99.999 100.011 99.993 99.990 100.001 100.012 99.994 106.763 111.115 104.384 99.338 103.851 105.857 103.368 103.929 96.813 89.347 95.315 97.720

0.002 0.007 0.011 0.009 -0.026 -0.036 0.000 0.006 -0.015 0.001 -0.011 0.007 0.010 -0.001 -0.012 0.006 -0.097 -0.082 0.259 0.074 -0.118 0.327 0.089 -0.462 -0.230 0.171 0.249 -0.129

Procedure of minimization 1-week Euribor 1-month Euribor 2-month Euribor 3-month Euribor 6-month derived from Euribor futures contract 9-month derived from Euribor futures contract 1-year derived from Euribor futures contract 2-year swap 3-year swap 4-year swap 5-year swap 6-year swap 7-year swap 8-year swap 9-year swap 10-year swap BNP PARIBAS 6 07/06/01 CREDIT NATIONAL 9.25 10/02/01 CREDIT NATIONAL 7.25 05/14/03 SNS BANK 4.75 09/21/04 CREDIT NATIONAL 6 11/22/04 BNP PARIBAS 6.5 12/03/04 BNP PARIBAS 5.75 08/06/07 ING BANK NV 6 10/01/07 ING BANK NV 5.375 03/10/08 COMMERZBANK AG 4.75 04/21/09 BSCH ISSUANCES 5.125 07/06/09 BANK OF SCOTLAND 5.5 07/27/09

Analyzing Credit Spreads Joint (Additive) Method â&#x20AC;&#x201C; Lower Quality of Fit Joint Method - Cubic B-Splines DATE

Sum of squared spreads Average spread

Rate or Instrument

31/05/2000

1.818 0.255

Maturity

Market Price

Theoretical Price

Spread

99.918 99.645 99.266 98.876 97.680 96.445 95.163 99.934 99.961 100.021 100.090 100.111 100.115 100.083 99.976 99.784 106.643 111.004 104.770 99.626 104.088 106.085 102.605 103.168 96.141 89.532 95.784 98.272

0.000 0.001 0.001 -0.003 -0.032 -0.027 0.026 0.066 0.039 -0.021 -0.090 -0.111 -0.115 -0.083 0.024 0.216 0.024 0.029 -0.127 -0.214 -0.355 0.099 0.852 0.298 0.441 -0.015 -0.220 -0.682

Analyzing Credit Spreads Comparison â&#x20AC;&#x201C; Smoother Fit with Joint Method Euro Bank Sector A-Swap 0 Coupon Spread 80 70

Spread (in bps)

60 50 40 30 20 Disjoint Estimation Method

Joint Estimation Method 0 0

Maturity

Modeling Credit Spreads Structural Models - Merton’s Model • Merton’s approach: structural approach • Merton’s model regards the equity as an option on the assets of the firm • In a simple situation the equity value is ET =max(VT -F, 0) where VT is the value of the firm and F is the debt repayment required • Then value of the risky debt: D0 = V0 - E0

Modeling Credit Spreads Option Pricing Model • An option pricing model enables the value of the firm’s equity today, E0, to be related to the value of its assets today, V0, and the volatility of its assets, σV • Popular example is Black-Scholes-Merton option pricing model E0 =V0 Φ( d1 ) − Fe −rT Φ( d 2 ) ( 1 )

where ln V0 / F + ( r +σV2 2)T d1 = ; d 2 = d1 −σV σV T Φ( d ) =

1 2π

∫e

−∞

−

x2 2

Modeling Credit Spreads Implementation of Merton’s Model • This approach is intuitively appealing and seems to be of easy use in practice • One problem, however, is that asset value and the volatility of its dynamics are not directly observable – At least, market value of equity and equity volatility are easily observable if the equity and options on that equity are traded – Good news: one can show that the volatility of equity and the volatility of assets are related through the following equation

∂E σ E E0 = σ V V0 = Φ( d1 )σ V V0 ∂V

( 2)

• From equations (1) and (2) and estimates for E and σE (from market prices of the stock and option on the stock), one may obtain V and σV

Modeling Credit Spreads Implementation of Merton’s Model – Example • Problem – – – –

Value of company equity E0 = $4 M Volatility σE = 60% Face value of debt (1 year maturity) F = $8 M Risk-free rate: r = 5%

• Solve (1) and (2) • Need to impose that the value of the Black-Scholes formula is equal to the value of equity E=4, subject to the constraint that N(d1)σVV0 = 4x60% = 2.4$ • We obtain – V0 = $11.59 M and σV = 21.08% – D0 = $11.59 –$4 = $7.59 M

Modeling Credit Spreads Subsequent Structural Models • Merton’s interpretation of default is very narrow – Corporate bond is a zero-coupon – Default cannot occur before the debt matures – Focus on credit risk: interest rate is assumed to be constant

• Extensions – Geske (1977, 1979) extends the analysis to coupon bonds – Black and Cox (1976) extended Merton's model to cases where creditors can force the firm into bankruptcy at any time (when asset value falls below an exogenous threshold defined in the indenture) – Ramaswamy and Sundaresan (1993) and Briys and De Varenne (1997) introduce interest rate risk – See also Longstaff and Schwartz (1995), Collin-Dufresne and Goldstein (2001) for more recent models

• Empirical shortcoming of (most) structural models – Predicted spreads are too low – In particular, short-term spread is zero

Modeling Credit Spreads Reduced-Form Models • They don’t try to explain why and how default occurs • Just assume default occurs at a random time with instantaneous probability

1 pt = lim Pr (τ < t + δt τ > t ) δt →0 δt

• Pricing formula

– Between t and t+δt, the value of $1 received at date t unless default occurs is expected discounted value of the cash-flow

Pt ,t + dt = (1 − pt dt ) × e − rt dt + pt dt × 0 ≈ e − ( rt + pt ) dt

– The value of a defaultable payoff is equal to the value of an otherwise default-free payoff, discounted at a discount rate augmented by the instantaneous probability of default

Modeling Credit Spreads Reduced-Form Models – Con’t • General formula in the case of a percentage recovery Rt = 1 – Lt (Duffie and Singleton (1999))

P0,T

T   = E exp− ∫ ( rt + pt Lt ) dt  0  

• Prediction: spread over Treasuries is equal to probability of default times percentage loss in case of default • Calibration – Calibrate the models so as to fit corporate bond prices – Use the model to price derivatives, e.g., credit derivatives

Risk-Adjusted Default Probabilities Empirical Puzzle • Historical probabilities of default are (much) higher than implicit probabilities of default • Assume a 50 BP spread for a grade A bond and use s = (1-R)p to get estimate of implicit default proba Recovery Rate

10%

20%

30%

40%

50%

60%

Implicit Default Proability 0.56% 0.63% 0.71% 0.83% 1.00% 1.25%

• Historical probability of default for a grade A bond is typically around 0.04% • Explanations – Peso problem – Liquidity and tax effects – Risk-adjustment

Risk-Adjusted Default Probabilities Default Risk is not Diversifiable • Agents do not simply take an expectation; because they are risk-averse, they adjust for risk • The default probabilities estimated from bond prices are risk-adjusted default probabilities • The default probabilities estimated from historical data are real-world default probabilities • Agents overweight the probability of default when pricing bonds because default risk is not diversifiable • Default hurts more as it is more likely to occur when return on your portfolio tends to be low