Model Framework

In a default intensity model, default occurs at the first jump time of a conditional Poisson process. Loosely speaking, for a small time period ∆t, conditional on loan obligor i having not defaulted by time t, the probability that default will occur before time t+ ∆t is approximately given by λi_t∆t. Here, λi_t is called the default intensity of obligor i. The Poisson process is said to be doubly-stochastic if λi_t follows a stochastic process. The term doubly-stochastic comes from the fact that both the default time and future values of λi_t are stochastic. Duffie (2005) provides a detailed description of the intensity approach to modeling default.

In the doubly-stochastic Poisson process for default that we will employ, λi_t depends on a vector-valued Markov state process, Xt. The state vector, Xt, consists of both

observable macroeconomic variables and unobserved frailty variables. This makes the framework used here similar in principle to Duffie, Eckner, Horel, and Saita (2009). Frailties induce additional correlation between individual defaults and hence introduce additional variability in aggregate default rates. Alternative ways of introducing default correlation and hence aggregate default rate volatility include contagion, whereby the default of one obligor triggers an increase in the default probability of other borrowers (Davis and Lo (2001)). In our context, contagion does not seem a sensible modeling approach. We apply our model to bank charge-off data for such exposures as credit 1_{In some related research, Pesaran, Schuermann, Treutler, and Weiner (2006) and Pesaran, Schuer-}

cards and consumer loans for which direct default contagion is implausible.

We assume that the portfolio under consideration is homogeneous in that obligors have identical default intensities. That is, λi_t = Λ(Xt, t) for all i. Of course this does not

imply that obligors have the same default times. It implies that any two obligors who have not yet defaulted have the same probability of defaulting by any given time in the future. The homogeneity assumption is not unusual. For instance, the well known and commonly used static loan loss distribution model of Vasicek (1987) makes this assumption. We discuss the case of heterogeneous portfolios in Section 4.4, and show that the homogeneous model derived here can be reinterpreted to account for heterogeneities across obligors.

In doubly-stochastic intensity based default models such as Duffie, Saita, and Wang (2007) it is assumed that, conditional on the path of a fully observable state vector, the random default times of the different obligors are independent of each other. We also make this assumption.

Given sufficient data it is possible in this framework to estimate a rich model with mul- tiple frailty factors and more complex stochastic processes for the frailties to follow. For simplicity and due to limitations on the amount of available data, we restrict attention to the case in which the state process Xt consists of a single frailty (the first element of

Xt, denoted X1,t) and multiple observable macroeconomic risk factors (X2,t, . . . , Xn,t).

The next step is to specify (i) the functional form Λ(Xt, t) and (ii) the stochastic process

followed by the state variables. Fully specifying the stochastic processes for the state variables is necessary if one is to make forward-looking statements such as calculating the term structure of the loan loss distribution or forecasting future default probabil- ities. The chosen specifications also have implications for the econometric estimation of the model. This will become clear in the next subsection. We adopt the following assumption.

AssumptionThe default intensity of obligoriat timetdepends linearly on an n-dimensional vectorXt and a time trend:

λi_t= Λ(Xt)≡µ0Xt+γt, (4.1)

whereXt follows the multivariate Ornstein-Uhlenbeck process

Here, Wt = (Wt,1, . . . , Wt,n)T is an n-dimensional vector of independent

Brownian motions. µ, γ, K, Σ and θ are parameters to be estimated. µ is an n-vector, γ is a scalar, K is a diagonal n×n matrix, Σ is an n×n matrix andθ is ann-vector. Furthermore, suppose that the frailty variable, X1,t, is orthogonal to the remaining state variables (X2,t, . . . , Xn,t). That

is: Σ1,j = 0 for j = 2, . . . , n. Given that X1,t is latent, for econometric

identification µ1 is set to 1.

An immediate disadvantage of the above specification where the state variables follow an Ornstein-Uhlenbeck process is that there is a non-zero probability that the intensity λi_t becomes negative over a discrete time horizon. This could result in negative probabilities of default. However, once the model has been implemented, it will become clear that, for the parameter estimates found, the probability that λi

t becomes negative is extremely

low.

One might also question the Ornstein-Uhlenbeck distributional restrictions imposed on the macroeconomic variables. In principle, the autoregressive dynamics of the processes could be generalized without difficulty and one may employ transformations of the macro variables so that innovations have the properties well approximated by Gaussian distri- butions.2

More generally, non-Gaussian processes may also be modeled statistically or more com- plex functional forms for Λ(·) can be used. However, implementing such approaches requires computationally intensive non-linear filtering techniques. In the current study, we follow a simpler approach so that the model may be implemented using a linear Kalman filter, and defer an investigation into non-linear models to future research. The Ornstein-Uhlenbeck specification for the frailty variable adopted here is common to Duffie, Eckner, Horel, and Saita (2009), although these authors use a proportional hazards form for Λ(·). They also provide some justification for Ornstein-Uhlenbeck specification in the context of corporate default. In the context of credit cards and consumer loans this variable could capture, for example, shocks to the costs associated with default. Such shocks may decay over time leading to mean reversion. The mean- reversion parameter K11captures the expected rate of decay of such shocks.

2

Note that it is necessary to specify an explicit process for the macroeconomic variables even though we will estimate the model conditional on these variables. The reason is that since the model is formulated in continuous time yet the data is time-aggregated and discrete time, estimation involves a filtering problem and this requires a full specification of the processes.

The model also allows for a time trend in default intensities. This feature captures the possibility that a gradual loosening of lending criteria has made the underlying obligor pool more risky. The time trend may also capture the possibility that the social costs associated with default may have decreased over time.