Model

In the previous section, the necessity for industry-specific analysis was established by explor- ing default cluster patterns. Here we expand on the frailty concept described in Duffie, Eckner, Horel, and Saita (2009) to industry-specific terms. This section provides a detailed specification for a model on default intensity. The model is first introduced, followed by a discussion on the observed and hidden common factors adopted in this essay.

2.3.1 Mixed Effect Model with Correlated Industry-Specific Default Intensity

As for the typical intensity model, we suppose default intensities of all firms at timetdepend on a state vectorXt and that, given the path of the state vector process X, the default times for different firms are conditionally independent. By assuming that all firms’ default intensity share the same Xt at each time t, we can incorporate a correlation structure for entire companies into the model. We assume that, conditional on the path of X determining the default intensities, the default times of firms are the first event times of an independent Poisson process with time-varying intensities determined by the path ofX.

The hidden factor idea proposed in Duffie, Eckner, Horel, and Saita (2009), however, departs in an important way from the traditional setting by assuming that X is not fully observable to the econometrician. Similarly, we adopt an industry-specific hidden factor into our model. In the financial market it is difficult to find the coordinate variables related to each industry. For this reason, we define an industry-specific factor as a hidden factor. By doing so, not only can we evenly incorporate industry factors into our model, but we can also capture unobservable risk

factors into our model.

In order to carry out the statistical analysis, we discretized time as∆t=1/12 (monthly data). The probability that theithfirm’s default event happens during the time interval[t−∆t,t]is

P(Di[j],t) =λi[j],t∆t. (2.1)

Under the factor structure, we model the conditional mean arrival rate of default, λi[j],t, with a mixed effect model, as given below:

λi[j],t =exp{a0+a′1Ui,t+a′2Xt+Yj,t}, (2.2) ⃗ Yt = KYt⃗−∆t+⃗εt, (2.3) εt ∼ N(0,Σ)J, (2.4) where ⃗ Yt = (Y1,t,Y2,t, ...,YJ,t)′ ⃗εt = (ε1,t,ε2,t, ...,εJ,t)′.

The indexirefers to the firmi, wherei=1, ...Iand the indexjis for industry, where j=1, ...J.

t is for time, where t=1, ...Ti and Ti is min(Exit time ofithfirm,T), where T is the data end- period. Ui,t denotes the vector of variables that are specific toith firm, andXt denotes common macroeconomic variables vector. Yj,t denotes jth industry-specific factor at timet. We use the index i[j]. This means that theith firm belongs to the jth industry, where there are J industries in total. We use the firm’s distance to default and trailing one-year stock return as firm-specific variablesUit and the three-month Treasury bill rate and the trailing one-year return on S&P500 index as common macroeconomic variablesXt. a′1 anda′2are corresponding coefficient vectors.

Our model is summarized as a mixed effect model with the following random and fixed effects. 1. Random effect (Hidden factor effect)

a. Yjt: jthindustry effect at timet.

2. Fixed effect (Observed factor effect) a. a1: Effect of firm-specific variables.

b. a2: Effect of macroeconomic variables.

The description of each effect is provided in the next two subsections.

2.3.2 Observed Factors:Ui,t,Xt

The observed factors used in this essay are again divided into two categories. One is a firm- specific factor (Ui,t), and the other is a macroeconomic factor (Xt). We follow Duffie, Eckner, Horel, and Saita (2009) for selecting these factors as shown below:

1. Firm-specific variables (Ui,t)

1.1. Distance to default (DTD) : Distance to default means the distance of a firm’s asset growth from its liability. This variable is not directly observed, so it needs to be calculated along with other financial variables as follows:

DT Dt=

ln(Vt/Lt) + (µA−0.5σ2A)T

σA

√

T , (2.5)

whereVt is the firm’s market asset value, Lt is its liability andµA,σA are the firm’s mean rate of asset growth and asset volatility. Because asset value is not an observable variable in the market, we construct this with an iterative method described in Duffie and Wang (2007). In order to construct distance to default, we use stock prices, number of outstanding, short-term debt, long-term debt, and the one-year Treasury-bill rate. 1.2. The firm’s trailing one-year stock return (Lag12) : This covariate suggested by Shumway

2. Macroeconomic variables (Xt)

2.1. The three-month Treasury bill rate (Tbill3): This covariate plays a role as an effect of monetary policy.

2.2. The trailing one-year return on the S&P 500 index (SP500): This covariate measures the market return.

Duffie and Wang (2007) and Duffie, Eckner, Horel, and Saita (2009) give a detailed description of these covariates and discuss their relative importance.

2.3.3 Hidden Factor (Yj,t)

The need and importance of the hidden factor in a default model are discussed in several recent studies. Vasicek (1991) and Gordy (2003) showed that defaults are more heavily clustered in time than currently captured in models with observed covariates. Duffie, Eckner, Horel, and Saita (2009) found that, after controlling for observed covariates, defaults were persistently higher than expected during a lengthy period of time from 1986 to 1991 and persistently low during the mid-90s. To capture this un-captured, time-varying default correlation, Duffie, Eckner, Horel, and Saita (2009) introduced the idea of a common, dynamic, latent factor named frailty, which was driving the default.

By extending the frailty model, we set the industry-specific hidden factor as a random ef- fect with a probability distribution. Different time patterns of defaults (Figure 2.2) show that an industry-specific default analysis is therefore required. However, this is not enough evidence for the necessity of incorporating industry-specific hidden factors, because we also use the firm- specific variables for the intensity model. It is possible that these firm-specific observed variables are enough to resolve industry-specific default time patterns. We thus conduct a simple test with the residuals of the Generalized Linear Model (GLM).

First we fit following GLM only with observed factors as predictors:

P(Di[j],t=1) = λit∆t, (2.6)

λi[j],t = exp{a0+a1Ui,t+a2Xt},

where Di[j],t is the default indicator of ith firm which belongs to jth industry, between the time interval[t−∆t,t]. We then extract the following Pearson residuals:

Presii[j],t=

D_i[j],t−ˆλi[j],t ˆ

λi[j],t(1−λˆi[j],t)

. (2.7)

Figure 2.3 shows the mean of Pearson residuals for each year within each industry. Even after controlling all observed factors, the energy industry shows a fairly unique pattern compared with others. The health care and the “others” industries show similar Pearson residual patterns. Other than these two, all residual patterns are considerably different from each other. These different residual series imply that the different default time patterns are not resolved even after controlling firm-specific factors; therefore additional industry-specific information is needed.

This section provides a discussion on our industry-specific hidden factor model and common factors that will be applied to our model. A simple preliminary study based on Pearson residual after GLM fitting shows the necessity of industry-specific hidden factors. We can incorporate observed common factors into our model as in previous studies, but will still need to decide on how to incorporated hidden factor into the model. The following sections discuss this issue.