1. Advanced Risk Management II
Lecture 3
Intensity based models

2. Reduced form models
In the reduced form models, default probability and recovery rate (LGD) are modelled based on statistical assumptions instead of an economic model of the firm.
The modelling techniques that are used are very close to those applied in insurance mathematics, specifying the frequency of occurrence of the event (default probability) and the severity of loss in case the event takes place (loss given default)
These models use then the concept of intensity and for this reason are also called intensity based

3. Credit spread and survival analysis
Denote, in a structural model, Q the probability of survival of the obligor after the maturity of the obligation, (the default probability is then DP = 1 – Q) and LGD the loss given default figure. Then, the credit spread is given by
Credit Spread = – ln[1 – (1 – Q )LGD]/(T – t)
Assume now the most extreme case in which all the exposure is lost (LGD = 1). We have
Credit Spread = – ln[Q]/(T – t)
Models from survival analysis (actuarial science) can help design the credit spread.

4. Hazard rates
Consider the conditional default probability

5. Prob( > T) = exp (–  (T - t))
Poisson model
Assume the default event to be drawn from a Poisson distribution (remember that it describes the probability of a countable set of events in a period of time).
The Poisson distribution is characterized by a single parameter, called intensity. The probability that no event takes place before time T (in our case meaning survival probability beyond that)is given by the formula
Prob( > T) = exp (–  (T - t))

6. Constant intensity model
Applying the survival probability function
Q = Prob( > T) = exp (–  (T - t))
to the credit spread formula (again under the assumption LGD = 1)
Credit Spread = – ln[Q]/(T – t)
we get
Credit Spread = 

7. Intensity vs structural
Intensity  denotes the probability of an event in an infinitesimal interval of time. The expected time before occurrence of the event is 1/.
Differently from structural models, the default event comes as a “surprise”. Technically, it is said that default is an inaccessible time.
The intensity corresponds to the concept of instantaneous forward rate in interest rate models.

8. Double stochastic models
If the intensity parameter is not fixed, but changes stochastically with time, the model is called Cox model (or double stochastic models)
For every maturity we can consider an average intensity (t,T) and the credit spread curve will be
Credit Spread(t,T) = (t,T)
Notice that the relationship between , that is the instantaneous intensity, and the average intensity is the same as that between instantaneous spot rate and yield to maturity in term structure models

9. Survival probability
The survival probability beyond time T is recovered simply using the zero coupon bond formula

10. Affine models
Assume the dynamics of default intensity is described by a diffusive process like
d (t) = k( – (t))dt + dz(t)
where setting  = 0, 0.5 deliver standard affine term structure models for the credit spread
Debt(t,T) = v(t,T)exp(A(T-t) - B(T -t) (t))
with A and B the affine functions in Vasicek ( = 0) or Cox Ingersoll Ross ( = 0.5) models

11. Positive recovery rate
If we assume positive recovery rate (and so LGD < 1) and independence between interest rate in default intensity we can easily extend the analysis. Denote  the recovery rate and compute
Debt(t,T; )=v(t,T)[Prob( > T)+ Prob(  T)]
Debt(t,T; )=  v(t,T) +(1-) Prob( >T)v(t,T)
Debt(t,T; 0)= Prob( >T)v(t,T), from which...
Debt(t,T; )=  v(t,T) +(1-) D(t,T; 0)
The price is obtained as a portfolio of the risk free asset and a defaultable exposure with recovery rate zero.

14. Default probabilities
The spread of a BBB 10 exposure over the risk-free yield curve is 45 basis points.
Assuming zero recovery rate we get
Prob( >T) = exp (– ) =
and the probability of default is
= %
Assuming a 50% recovery rate we have
Prob( >T) = [exp (– ) - ]/(1- ) =
and default probability is = %

15. Simulating default times (1)
Inverse CDF simulation
Exploit Fisher transforms
F(T1) = P( > T1) = exp(– T1) = u which is uniformly distributed
Generate: u = rnd()
Compute T1 = F-1(u) = – ln(u)/ 

16. Simulating default times (2)

17. Recovery rate
Beyond default probability, the other point involved in modelling the expected loss refers to the recovery rate.
This topic is particularly involved, and the research on the subject is not very developed.
In particular, the point is how to recover
the value of the recovery rate
the dependence of recovery rate and default
whether the recovery rate is computed with respect to face value or market value, or other

18. Recovery rate and default
In case the recovery rate is independent of the default probability expected loss can be computed using whatever distribution defined on the support between 0 and 1
Typical example is the beta distribution that is very often used to study the recovery from samples of defaults.
Altman,Resti and Sironi find that the number of defaults and the amount of recovery are negatively correlated across business cycles

19. Recovery of what?
Recovery of face value: it is the legal concept of recovery. Once in default, payments of interest is stopped and a fraction of principal is allocated to creditors
Recovery of treasury: it refers to a fraction of a risk free bond (Treasury!!) with the same financial structure of the defaultable bond (Jarrow and Turnbull, 1995)
Recovery of market value: it refers to the market value of the bond prior to default (Duffie and Singleton, 1999)

20. Recovery of face value
It is based on the idea of legal default. It is useful for corporate bonds.
Moody’s compute the recovery rate as the ratio of the value of the bond on the secondary market one month after default and the face value.
Technically, default should be considered at every time, assuming that the bond is substituted with a percentage of par.
In practice, it can be approximated assuming payment of the recovery at maturity of the bond, as in the previous example.

21. Recovery at face value
Assume a bond payng coupon c on dates {t1,t2,…,tm} and principal 1 at tm. Denote  the recovery rate fraction and  a n time periods partitioning the time span from today to tm.
The value of the defaultable bond is

22. An alternative derivation
Notice that as an alternative we could refer to the case of recovery rate zero:
and write (Duffie and Singleton, ch. 6)

23. Recovery of market value
It is considered as an approximation that makes the model easier to handle.
In fact, it turns out that intensity can be modified by allowing for the intensity to be adjusted by LGD. Namely, in the simple constant intensity model, with intensity , one could adjust the intensity to * =  LGD.
In a discrete time setting, the one-period interest rate (r +  LGD)/(1 –  LGD)

24. Stochastic interest rates
Assume discrete time model, time span , stochastic interest rate and loss given default defined in terms of market value. R risky rate, r riskless rate

25. Recovery of treasury
It is based on the assumption that in case of default there is rescheduling.
It is typical of sovereign risk applications in which case there is no formal bankruptcy procedure leading to recovery of fraction of par
Typical example is the concept of “haircut” that is currently used in the current discussion on the sovereign debt crisis in the Euro area.

26. Recovery at treasury
Assume a bond payng coupon c on dates {t1,t2,…,tm} and principal 1 at tm. Denote  the recovery rate fraction and  a n time periods partitioning the time span from today to tm.
The value of the defaultable bond is

27. Strategic debt service
Anderson and Sundaresan present a model in which after default, there is a restructuring game at the end of which the stockholder manages to take home the bankrupcy cost.
The result of the restructuring game is a Nash equilibrium in which the creditors face the choice of going through formal bankrupcy or to accept the proposal of the stockholder.