1. Chapter 13 Modeling the Credit Spreads Dynamics
FIXED-INCOME SECURITIES
Chapter 13
Modeling the Credit Spreads Dynamics

2. Outline Analyzing Credit Spreads Modeling Credit Spreads Ratings
Probability of Default
Severity of Default
Modeling Credit Spreads
Structural Models
Reduced-Form Models
Historical versus Risk-Adjusted Default Probabilities

3. 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

4. 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?

5. Analyzing Credit Spreads Bond Quality Ratings

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

7. Analyzing Credit Spreads TS of Annual Default Probabilities- Investment Grade

8. Analyzing Credit Spreads TS of Annual Default Probabilities - High Yield

9. 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

10. Analyzing Credit Spreads Size of the Corporate Bond Markets - US and Europe

11. 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 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 basis points and 191 basis points over the 5-year Treasury benchmark bond and the 10-year Treasury benchmark bond respectively

12. 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 BP spread over the maturity nearest US T-Bond
Example

13. 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

14. 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

15. 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.

16. 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.

17. Payoff of Equity

18. 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!

19. Payoff of Debt

20. BSM Equity Value
Under the usual assumptions of log-normality and constant risk-free rate and volatility, we have the stock price

21. 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

22. 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.

23. 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.

24. 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.

25. 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.

26. KMV Case Study: Enron
EDF gives the probability of defaulting within the next year
EDF
S&P Rating
EDF ranges from 0.02% to 20%

27. Pros of Merton Model Uses readily available equity prices.
Equity correlations can indicate default correlations.
EDFs appear to lead credit rating changes!

28. 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.

29. Modeling Credit Spreads Implementation of Merton’s Model – Example
Problem
Value of company equity E0 = $4 M
Volatility sE = 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 sV = 21.08%
D0 = $11.59 –$4 = $7.59 M

30. 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