OMC-Ecuador

Consider a single loan with resolution intensity according to Model I (λI) which is time constant because the linear predictor of the loan specific variables (xβ) is constant over time. In the Cox Model, the time to an event follows an exponential distribution with

rate parameterλif a constant baseline hazard rateλ0 is assumed.21 Thus, the probability

density function of the DRT in Model I is determined by

f_TI(t) = λIexp(−λIt), t ≥0. (3.7)

In contrast, the resolution intensity in Model II depends on the default time ˜t of the con- sidered loan. Therefore, the resolution intensity might be lower in recessions and higher

in expansions depending on the linear predictor of the macroeconomic variables (z˜tγ). Given the default time ˜t of the loan, the resolution intensity of Model II is fully specified because the realizations of the macroeconomic variables are known at time of default. The

DRT in Model II is, therefore, exponentially distributed with a constant rate parameter

λII_{(˜t) for a given time of default ˜t and its probability density function is}

f_{T ,}II˜_t(t) =λII(˜t) exp(−λII(˜t)t), t ≥0. (3.8)

As the resolution intensity of Model II varies over calendar time, longer DRTs might arise

during weak economic conditions and shorter DRTs in a favorable environment.

In Model III there is not such a simple expression for the probability density function

of the DRT as in Model I and II as the realization of the frailty is unknown at the time

of default. Conditioning on the frailty factor U = u, the conditional intensity of Model IIIλIII_{(˜t, u) is constant, given the quarter of default ˜t. Thus, the conditional probability} density of the DRT is determined by

f_{T ,}III˜t|U=u(t) =λ

III_{(˜t, u) exp(−λ}III_{(˜t, u) t), t≥0. (3.9)}

The unconditional probability density function can be derived by the integral of the joint

21_{To check for robustness, we derive the simulation also with the estimated time varying hazard rates}

following Bender et al. (2005) and receive similar results. We would like to thank an anonymous referee for this remark.

probability density function over the frailty realizations u

f_{T ,}III˜t(t) =

Z +∞

−∞

f_{T ,}III˜t|U=u(t) fU(u) du, t≥0, (3.10)

where fU(u) is the density of the Normal distribution with mean 0 and variance σ2 (see Equation (3.4)). Equation (3.10) can be solved by numerical integration.

As the baseline hazard rate λ0 directly impacts the distribution of DRTs and, thus,

its mean, we calibrate it on the average DRT of 1.59 years (see Table 3.1). This ensures

an average simulated portfolio DRT in accordance with the empirical data. Thus, the

average portfolio DRT corresponds to 1.59 years for Model I. Regarding Model II and III,

it amounts to 1.59 years in an average economic scenario. The simulated DRTs might be

higher relating to recessions and lower in expansions. The calibration yields in a baseline

hazard rate for Model I of λI₀ = 1.08 as well as λ₀II = 0.12 for Model II and λIII₀ = 0.07 for Model III.22

Figure 3.10 shows the probability density functions of the DRT in Model I, II, and

III as of Equation (3.7), (3.8), and (3.10) for an exemplary recession and expansion

period.23 _{The left panel of Figure 3.10 displays the probability density functions for a}

recession period. The underlying quarter (Q1 2009) is shaped by the Global Financial

Crisis and includes inter alia the crash of Lehman Brothers. Compared to Model I, the

density of Model II is lower for short DRT and higher for longer ones. The distribution is,

thus, shifted towards higher DRTs. This tendency is even more pronounced considering

Model III as the frailty intensifies the impact of the economic surrounding. Firstly, an

unobservable systematic factor widens the distribution of DRT. Secondly, impacts of the

observable systematic factors are enhanced due to the consideration of the frailty. The

right panel of Figure 3.10 shows the probability density functions for an expansion period.

Considering favorable economic surroundings, opposite effects appear. The distribution

of DRT for Model II is shifted towards lower values compared to Model I. Table 3.9

22_{The deviations in the baseline hazard rates among the models seems adequate as the difference in}

levels also emerges in the estimation of the models.

23_{The realizations of the macroeconomic variables are assumed to match their values as of Q1 2009}

summarizes the median and 95% quantile of the distributions. Whereas the difference is

less pronounced in the median, it is apparent considering the 95% quantile. In a recession

period, there is an increase of this quantile by 54% comparing Model I and III.

Figure 3.10: Density of DRT 0 2 4 6 8 recession time in years density 0.0 0.2 0.4 0.6 0 2 4 6 8 expansion time in years density 0.0 0.2 0.4 0.6

Model I Model II Model III 95% quantile

Notes: The figure illustrates the probability density function of the DRT for Model I, II, and III according to Equation (3.7), (3.8), and (3.10) in an exemplary recession (realizations of macroeconomic variables as of Q1 2009) and expansion (realizations of macroeconomic variables as of Q2 2011) period. Under the assumption of constant baseline hazard rates, the DRTs of Model I and II follows an exponential

distribution with rate parameterλI _{for Model I andλ}II _{for Model II. The density of Model III is derived}

by numerical integration.

Table 3.9: Inferences of systematic factors on the distribution of DRTs

Recession Expansion

Model I mean 1.16 1.16

95% quantile 3.48 3.48

Model II mean 1.47 0.95

95% quantile 4.42 2.84

Model III mean 1.70 0.78

95% quantile 5.35 2.45

Notes: The table summarizes the mean and 95% quantile of the DRT for Model I, II, and III

according to Equation (3.7), (3.8), and (3.10) in an exemplary recession (realizations of macroeconomic variables as of Q1 2009) and expansion (realizations of macroeconomic variables as of Q2 2011) period. The values arise from the probability density functions illustrated in Figure 3.10.

Generally, the distribution of DRTs for Model I is independent of the economic sur-

distribution towards lower values, adverse economic conditions shift it towards higher

values indicating shorter DRTs in expansions and longer ones in recessions. This effect is

enhanced in Model III.