Interpretación de los Estados Financieros Vega Flor S.A

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 rateλ0is assumed.21 Thus, the probability density function of the DRT in Model I is determined by

f_TI(t) =λIexp(−_λI_{t), t}≥_{0. (2.7)}

In contrast, the resolution intensity in Model II depends on the default time ˜tof the considered loan. Therefore, the resolution intensity might be lower in recessions and higher in expansions depending on the linear predictor of the macroeconomic variables (zt˜γ). Given the default time

˜

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

exponentially distributed with a constant rate parameterλII(˜t) for a given time of default ˜tand its probability density function is

f_{T ,}II_t˜(t) =λII(˜t) exp(−_λII_{(˜t)t), t}≥_{0. (2.8)}

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 III λIII(˜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. (2.9)}

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

f_{T ,}III_t˜(t) =

Z+∞

−∞

f_{T ,}III_t˜|_U=u(t)fU(u)du , t≥0, (2.10)

wherefU(u) is the density of the Normal distribution with mean 0 and varianceσ2(see Equa-

tion (2.4)). Equation (2.10) can be solved by numerical integration.

As the baseline hazard rateλ0directly impacts the distribution of DRTs and, thus, its mean, we calibrate it on the average DRT of 1.59 years (see Table 2.1). 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 ofλI₀= 1.08 as well asλII₀ = 0.12 for Model II andλIII₀ = 0.07 for Model III.22

Figure 2.10 shows the probability density functions of the DRT in Model I, II, and III as of Equation (2.7), (2.8), and (2.10) for an exemplary recession and expansion period.23 The left panel of Figure 2.10 displays the probability density functions for a recession period. The 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 for the recession}

Figure 2.10:Density of DRT 0 2 4 6 8 recession time in years density 0.1 0.4 0.7 0 2 4 6 8 expansion time in years density 0.1 0.4 0.7 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 (2.7), (2.8), and (2.10) 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 parameterλIfor Model I andλII for Model II. The density of Model III is derived by numerical integration.

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 2.10 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 2.9 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.

Table 2.9: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 Equa- tion (2.7), (2.8), and (2.10) 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 2.10.

Generally, the distribution of DRTs for Model I is independent of the economic surrounding at the time of default. In Model II, favorable economic conditions shift the 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.