JEAN-DAVID FERMANIAN AND MOHAMMED SBAI 137(Cox, 1972), where the (conditionally on the covariates) default intensities
are multiplied by some unobservable random effects. Thus, in the basic
version of frailty models, we set for every timetand every firmi:
λi(t,Xi,Z)=Zλ 0 (t) exp (βTXi), (7.3)whereβis a vector-valued parameter of interest.Xiis the vector of observ-
able covariates of the firmi. They may be firm specific and/or systemic
(macro-economic indices).λ 0 is the deterministic baseline hazard function.Z
is a frailty, an unobservable gamma distributed random variable. We assume
it is the same for every obligor.
The random variableZcan be interpreted as a synthetic macro-economic
factor that has not been included into the observable covariatesXi. For the
sake of simplicity, we assume thatλ 0 is a constant function and thatβis
equal to 0 (no observable covariates). Thus, the dependence is driven byZ
only. Moreover, theZrealizations are assumed constant. This constancy is
clearly a strong assumption, but it is realistic when we restrict ourselves to
a one or two year horizon. This is indeed the case in this section. Then:
λi(t)=λi=Zλ0,i whereZis following a gamma lawG(α,θ) (7.4)
This implies that the expectation ofZisα/θand that its variance isα/θ^2.
The default probabilities are taken from the same source as in the Merton
model. We consider one year as the time unit, sayTis expressed in years.
Thus,λicanbeidentifiedwiththeyearlydefaultintensity. Wegettherandom
default probability at timeTas:
pi(T|λi)=P(τ≤T|λi)= 1 −exp(−λiT) (7.5)When we take the expectation with respects toZ, we have:
E(1−exp(−Tλi))= 1 −(
θ
θ+Tλ0,i)α
=pi(T) (7.6)This provides a first condition on the parameters (α,θ) andλ0,isince we
know the mean historical probabilitiespi(T). In order to make the baseline
hazard functionλ0,iidentifiable, we normalize the frailty variable :E(Z)=1,
i.eα=θ. In this case, Var(Z)=1/α. Now, the key parameter isα.
We consider the same portfolio as in the Merton model and we follow the
following steps to get the loss distribution: for every timeT
1 we invokepi(T), the mean default probability (see(7.6)) to deduceλ0,i;
2 we simulateZand deduceλ0,ifor each obligori(see (7.4));3 we draw a uniform random variable and we compare it topi(T|λi)tosee
if a default is triggered or not; see (7.5); and finally,4 we cumulate the losses and we repeat the same procedure many times.