Chapter 13

Modeling the Credit Spreads Dynamics

FIXED-INCOME SECURITIES

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 SpreadsCorporate 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

Moody's S&P DefinitionAaa AAA Gilt-edged, best quality, extremely strong creditworthiness

Aa1 AA+Aa2 AA Very high grade, high quality, very strong creditworthinessAa3 AA-A1 A+A2 A Upper medium grade, strong creditworthinessA3 A-Baa1 BBB+Baa2 BBB Lower medium grade, adequate creditworthinessBaa3 BBB-

Moody's S&P DefinitionBa1 BB+Ba2 BB Low grade, speculative, vulnerable to non-paymentBa3 BB-B1 B+B2 B Highly speculative, more vulnerable to non-paymentB3 B-

CCC+Caa CCC Substantial risk, in poor standing, currently vulnerable to non-payment

CCC-Ca CC May be in default, extremely speculative, currently highly vulnerable to non-paymentC C Even more speculative

D Default

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

Investment Grade (High Creditworthiness)

Speculative Grade (Low Creditworthiness)

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

S&P AAA AA A BBB BB B CCC DAAA 91.94% 7.46% 0.48% 0.08% 0.04% 0.00% 0.00% 0.00%AA 0.64% 91.80% 6.75% 0.60% 0.06% 0.12% 0.03% 0.00%

F A 0.07% 2.27% 91.69% 5.11% 0.56% 0.25% 0.01% 0.04%R BBB 0.04% 0.27% 5.56% 87.87% 4.83% 1.02% 0.17% 0.24%O BB 0.04% 0.10% 0.61% 7.75% 81.49% 7.89% 1.11% 1.01%M B 0.00% 0.10% 0.28% 0.46% 6.95% 82.80% 3.96% 5.45%

CCC 0.19% 0.00% 0.37% 0.75% 2.43% 12.13% 60.44% 23.69%D 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00%

TO

Moody's Aaa Aa A Baa Ba B Caa DAaa 88.66% 10.29% 1.02% 0.00% 0.03% 0.00% 0.00% 0.00%Aa 1.08% 88.70% 9.55% 0.34% 0.15% 0.15% 0.00% 0.03%

F A 0.06% 2.88% 90.21% 5.92% 0.74% 0.18% 0.01% 0.01%R Baa 0.05% 0.34% 7.07% 85.24% 6.05% 1.01% 0.08% 0.16%O Ba 0.03% 0.08% 0.56% 5.68% 83.57% 8.08% 0.54% 1.46%M B 0.01% 0.04% 0.17% 0.65% 6.59% 82.70% 2.76% 7.06%

Caa 0.00% 0.00% 0.66% 1.05% 3.05% 6.11% 62.97% 26.16%D 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00%

TO

Analyzing Credit Spreads TS of Annual Default Probabilities- Investment Grade

0.00

0.15

0.30

0.45

0.60

0.75

0.90

1.05

1.20

1.35

1.50

1.65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Maturity in Years

An

nu

al D

efau

lt P

rob

abil

ity

in %

AAA

AA

A

BBB

Analyzing Credit Spreads TS of Annual Default Probabilities - High Yield

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Maturity in Years

An

nu

al

Defa

ult

Pro

bab

ilit

y i

n %

BBBCCC

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 deviationSenior 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 Par Amount Weight(in billion USD) (in %)

USD Broad Investment Grade bond market 6,110.51 100.00 USD Govt./Govt. Sponsored 2,498.23 40.88 USD Collateralized 2,216.73 36.28 USD Corporate 1,395.56 22.84 USD Corporate (Large Capitalizations) 869.96 14.24 USD Corporate 1,395.56 100.00 AAA 35.33 2.53 AA 200.81 14.39 A 653.76 46.90 BBB 505.65 36.23 Description Par Amount Weight

(in billion USD) (in %) EUR Broad Investment Grade bond market 3,740.77 100.00 EUR Govt./Govt. Sponsored 2,455.24 65.63 EUR Collateralized 685.78 18.33 EUR Corporate 599.75 16.03 EUR Corporate (Large Capitalizations) 416.96 11.15 EUR Corporate 599.75 100.00 AAA 143.10 23.86 AA 151.55 25.27 A 220.36 36.74 BBB 84.74 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

Example

• 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

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 zero-coupon 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 subtraction

• 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 localization 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

Using Equity Price Information

• Corporate bond market yield data is often not available.

• Bonds may not be publicly traded or may have option features.

• Trading may be thin.• Stock prices, on the other hand, are available for any

listed company, and are usually updated frequently.

The Merton Model (I)

• Consider a firm with market value, V, and debt with face value K on date T.

• The value of the equity in the firm, S, at maturity of the debt, T, is the price of a hypothetical call option on the firm value

• Limited liability is an option.

0,KVMaxS TT

Payoff of Equity

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

V(T)

S(T)

The Merton Model (II)

• The value of a corporate bond, B, on its maturity date, T, is

• The risky bond equals the value of a risk-free bond with face value K and a short put on the value of the firm with strike price K.

• A corporate bond is a credit derivative!

0,or

,0,

TT

TTTTTT

VKMaxKB

KVMinKVMaxVSVB

Payoff of Debt

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

V(T)

B(T)

BSM Equity Value

• Under the usual assumptions of log-normality and constant risk-free rate and volatility, we have the stock price

)(

2

)(

)(

/ln

where

)1

12

)(

1

2)(

1

tTdd

tT

tT

KeVd

dNKedVNS

V

V

V

tTr

tTr

Calculating Value and Volatility

• Notice we need firm asset value V, and the volatility of firm asset value, and not just equity volatility from the stock price.

• It can be shown that the value of the firm and the value of the stock are related by

• And we can relate the volatilities by

dVdNdVdVV

SdS )( 1

)()2 1dVNVS VVS

Iterate to Solution

• We observe the market price of equity and its volatility. We can back out firm asset value and its volatility from the call option price and the preceding equation, which links the two volatilities to each other.

• Two nonlinear equations in two unknowns. Solve iteratively.

BSM Bond Value

• Given the basic balance sheet identity which states that B=V-S, we have

• The risk neutral probability of default is equal to N(-d2)=1-N(d2). This is the probability of V falling below K at T.

1)(

2)(

12)(

//

or

1

dNKeVdNKeB

dNVdNKeB

tTrtTr

tTr

Expected Credit Loss (ECL)

• The current value of the risk-free less the risky bond is the ECL:

• Can be thought of as the probability of default times the loss when defaulting.

21)(

2

12)(

12)()(

/

1

dNdVNKedN

dVNdNKe

dNVdNKeKeBB

tTr

tTr

tTrtTrF

Applying Merton: KMV

• The KMV, now acquired by Moody’s Analytics provides Estimated Default Frequencies (EDF) on a large number of firms across the World.

• EDF

• They use a proprietary version of the Merton model allowing for early exercise and time-varying strike price capturing dynamics in the firm’s liabilities.

KMV Case Study: Enron

EDF

S&P Rating

EDF ranges from 0.02% to 20%

EDF gives the probability of defaulting within the next year

Pros of Merton Model

• Uses readily available equity prices.• Equity correlations can indicate default correlations.• EDFs appear to lead credit rating changes!

Cons of Merton Model

• Cannot be used to price sovereign credit risk.• Static model of balance sheet with constant horizon• Time-varying volatility. Management can chose to

increase volatility.• It predicts credit spreads systematically different

from those observed.

Modeling Credit Spreads Implementation of Merton’s Model – Example

• Problem– Value of company equity E0 = $4 M– Volatility E = 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)VV0 = 4x60% = 2.4$

• We obtain – V0 = $11.59 M and V = 21.08% – D0 = $11.59 –$4 = $7.59 M

Modeling Credit Spreads Subsequent Structural Models

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