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The probability of default of a particular obligor is a forward-looking assessment of the likelihood that the obligor will fail to meet its contractual obligations or file for bankruptcy during a fixed time interval which is conventionally set at one year. A default rate is defined as the number of defaults in a risk class within a specified time interval of one year, in general, divided by the total number of obligors in the class at the beginning of the interval. Unlike the PD, the default rate is an ex post measure of the number of actual default events and refers to a set of obligors rather than to a single obligor.20 The definition of default is typically based on subjective conditions established in the loan agreements for corporate portfolios, whereas objective conditions are predominant in retail portfolios. A credit default of a borrower is triggered, if either or both of the following apply: (1) it is unlikely that the obligor will pay its obligations in full, or (2) any material financial obligation owed by the borrower is more that 90 days past due. Indicators for an unlikely payment are the borrower’s filing for bankruptcy, non-accrued status of debt, charge-offs or specific provisions, sale of the credit obligation at a material credit-related economic loss, or consent of banks to a distressed restructuring. Typically, cross-default clauses of credit arrangements synchronize the default events of all exposures against a particular counterparty, so that the default probabilities of an obligor and its exposures are considered to be equal. The definition of default must be used consistently when PD, LGD and EAD are being estimated.

Transition or migration probabilities designate the likelihood that an obligor or expo- sure will migrate from one class of a rating system to another within one year’s time. Probabilities of default and transition probabilities can refer to a risk-neutral probabil- ity measure or to a real-world probability measure. Risk-neutral default or transition probabilities can be derived from no-arbitrage credit pricing models that are calibrated using time series of cross-sectional price data of defaultable securities from efficient credit markets. In estimating the real-world default and migration probabilities of obligors, one can distinguished between direct and indirect methods. Direct methods provide a credit score that represents a single-obligor PD and includes statistical default prediction mod- els such as Logit, Probit or Hazard Rate models. If credit scores of a rating model do not represent default probabilities as in the case of the default-prediction by discriminant analysis, indirect methods estimate the pooled PD of risk classes and use either histori- cal migration rates and the default experience of obligors or external risk class mapping. Direct methods are typically connected to PIT ratings, whereas indirect methods mostly refer to TTC rating systems.

Methodologically, stressed and unstressed PD can be differentiated according to the eco- nomic state and the time period on which the credit risk assessment is conditioned. Stressed PD indicate the default probability of an obligor, conditioned on a specified stress scenario of unfavorable economic conditions, whereas unstressed PD comprise probabili- ties of default which are either conditional or unconditional on actual economic conditions.

Conditional PD indicate the likelihood that an obligor will default, assuming an extrap- olation of current economic conditions. Conditional PD incorporate static and dynamic credit-quality characteristics of the obligor and current aggregate information. By virtue of the dependence on macroeconomic variables, conditional PD are negatively correlated to the credit cycle and tend to fall (rise) during economic upturns (contractions), so that, in principle, the deviations of an obligor’s conditional PD from its long-run average are exclusively caused by the economic cyclic only. Unconditional PD indicate the likelihood of default under long-run average economic conditions, incorporate only obligor-specific information, are expected to remain stable throughout the business-cycle, and do not show a significant correlation with the economic cycle.

Pooled PD reflect the central tendency (mean or median) of the individual PD of obligors assigned to a risk class. The pooled PD of a risk class change during a business-cycle, where the dynamic properties of the fluctuation depend on methodology and stress char- acteristics of the PD considered. The utilization of the direct method of PD estimation dominates, if pooled PD condition on the business-cycle or are intended to incorporate stress scenarios. The estimation of pooled PD by statistical default prediction models involves (1) the estimation of individual default probabilities for each obligor, (2) the classification of obligors into segments of homogenous credit risk, and (3) the derivation of the pooled PD of a risk class that reflects the individual PD of all obligors in the class. Single-obligor PD can be estimated regardless of the rating methodology applied. Accord- ing to the minimum requirements of the IRB approach, banks are allowed to calculate a simple average of single-obligor PD estimates to estimate the pooled PD of a class. Testing the accuracy of pooled PD estimates involves validating the risk class assignment as well as the estimation model of single-obligor PD.

The estimation of single-obligor PD by statistical default risk models incorporates obligor- specific information as well as aggregate information on the economic environment the obligor operates in to assess the obligor’s ability and willingness to repay its debt. Obligor- specific information is unique to a particular obligor and can be static or dynamic, such as economic sector affiliation or financial leverage. Aggregate information refers to the time a PD is estimated, affects many obligors jointly and typically includes macroeconomic variables such as exchange rates, unemployment rates or GDP growth. Aggregate and dynamic obligor-specific information are often highly correlated, such as GDP growth and increased of revenues.

The default experience of proprietary corporate loan portfolios or public credit markets is used to estimate pooled unstressed PD of PIT risk classes. Default events in a risk class are typically correlated to joint background factors, so that the default rates of risk classes deviate from the pooled unstressed PD due to unexpected changes in economic conditions. Averaging one-year default rates of a risk class gives the long-run default frequency. Over time, differences between pooled unstressed PD and observed default rates cancel out and the long-run average default rate is expected to converge toward the average pooled unstressed PD of the class. In the calculation of default rates, changes in rating methodology, underwriting standards and default definitions must be reflected and, according to the minimum requirements of the IRB approach, the length of the observed default experience must cover all obligors of a risk class for a period of at least five years. In principle, the pooling of data across institutions is permitted.

Finally, the pooled PD of risk classes can be determined from external risk class map- pings. Using a mapping of the internal and external rating scale, the pooled PD estimates of external risk classes are assigned to obligors of the corresponding internal risk class. However, the suitability of the external mapping relies on the consistency of the internal and external rating methodology. The mapping of rating systems must involve a compar- ison of the internal and external rating criteria and a comparison of assigned internal and external rating classes of any common borrower. Rating agencies mostly derive estimates of the pooled PD from public credit markets, so that the conformity of PD methodol- ogy, sample population and default definition with internal risk classes must explicitly be ensured.

Despite its simplicity, the external mapping poses some difficulties with respect to rating validation. The validation of pooled PD estimates from an external risk class mapping involves the validation of the accuracy of pooled PD of the external rating system and the validation of the mapping itself. For external rating systems, the estimation of pooled PD pose the same challenges as it does for internal rating systems. If historical default experience is used, pooled PD must be checked against long-run default rates, and for an external statistical default risk model the same validation procedures apply as for an internal model, so that the main benefit of using external PD estimates is the availability of a more extensive data set in the time- and cross-sectional dimension. The mapping of risk classes is stable in time, if the bank and the external rating provider use unchanged rating methodologies. If this is not the case, the mapping is time-inhomogenous and eventually the pooled PD estimates of external risk classes will need to be adapted to the internal rating system.

In a PIT risk class, obligors share similar conditional PD, and the PIT ratings of obligors change if obligors’ conditional PD change, so that the volatility of obligor ratings is higher than in a TTC rating system. Conditional PD decrease if business conditions improve and

obligors tend to migrate upwards out of a PIT risk class, whereas previously lower-quality obligors migrate into that class. The reverse applies for adverse business conditions. In contrast, the variation in the pooled unconditional PD of a PIT risk class is positively correlated to changes in economic conditions. This seemingly paradoxical effect is caused by obligors with currently improved conditional PD and high unchanged unconditional PD who migrate into a PIT risk class, while obligors with lower unconditional PD migrate upwards. In summary, the pooled conditional PD of a PIT risk class remain stable throughout the economic cycle, while pooled unconditional PD tend to rise as business conditions improve and slump during recessions.

In a TTC risk class, all obligors share a similar unconditional PD. The TTC ratings as well as the unconditional PD of obligors can fluctuate over time, however, changes are uncorrelated to the economic cycle. Obligors migrate in and out of the TTC risk class as their particular business prospects change beyond current economic conditions, so that strong cyclical migration patterns do not occur. In contrast, the conditional PD of obligors in a TTC risk class decline (rise) when business prospects improve (deteriorate). In summary, the pooled unconditional PD of a TTC risk class remain stable as economic conditions change, whereas the variation of pooled conditional PD is negatively correlated to changes in the economic cycle and pooled conditional PD of TTC rating classes rise (decrease) during economic recessions (upswings).

Methods used for the estimation of single-obligor PD must reflect the type of rating system and the usage of PD estimates. Econometric models based on macroeconomic, statistic or fundamental factor models are prevalent in estimating conditional PD.21Since conditional PD are not suited to determine capital requirements under the IRB approach, further considerations are restricted to the estimation and validation of unconditional PD estimates from the historical default rates of risk classes.

A natural estimator for the pooled conditional default probability pt+1 of a risk class in

periodt+ 1 is the default rate ˆpt =dt/nt of nt obligors at the beginning of the preceding period t with dt defaults observed. However, as pointed out by B¨uhler, Engel, Korn and Stahl (2002), the default rate ˆpt of a risk class is a biased estimator for the pooled unconditional PDpwith strictly positive approximative variancep(1−p)ρdef _forn

t → ∞, if a positive homogenous correlation ρdef _{> 0 is present in the class, so that the central} limit theorem does not apply and the default rate ˆpt is not a consistent estimator of

p. An unbiased estimator of p is the mean default rate ˆp = PT

t=1pˆt/T of the risk class

acrosst= 1, ..., T periods.22 Assuming a time-invariant default correlationρdef, the mean default rate is consistent if n = PT

t=1nt → ∞ and nt/n → 0,∀t, which is equivalent to

21_{Cf. Hamerle, Liebig and Scheule (2002, 2004).} 22_{Cf. Huschens and Stahl (2004), p. 6.}

T → ∞.23 _{In most practical applications, however, the number of observation periods}

is typically small, and due to the default correlation ρdef _{unconditional PD estimates} include large estimation errors if the default rates of only a few periodic observations of a credit portfolio are available. To cope with this problem, Koyluoglu and Hickman (1998b) as well as Gordy and Heitfield (2000) additionally account for the correlation of credit defaults in the estimation of pooled unconditional PD of a single risk class. With default probabilities and inner-class default correlations being unobservable, Gordy and Heitfield estimate both parameters simultaneously using a multi-period maximum- likelihood-estimator (MLE). Using a one-factor asset value default model with a standard- Gaussian single factor zt and time-invariant asset correlation ρa in a risk class with nrct obligors, the pooled conditional probability of default pt|zt is calculated as follows:

pt|zt =pt(zt;p, ρ a_{) = Φ} Φ−1_(p)−√_ρa_z t √ 1−ρa . (2.1)

Conditional on the factor returnzt, default events in periodt are stochastically indepen- dent and the number of defaultsdtis binomially distributed with probability pt|z(t). This

conditional independence of defaults results in the likelihood function

L(p, ρa;dt, nt) = P(ˆpt=dt/nt) = ∞ Z −∞ nt dt pdt t|zt 1−pt|zt nt−dt ϕ(zt)dz (2.2)

of the unconditional default probability p, which is equal to the probability function of the default rate ˆpt = dt/nt and constitutes a mixture of a binomial distribution with probability pt|zt = pt(zt;p, ρ

a_{) and the standard normal distribution of the factor z} t. Enhancements of the basic likelihood function in (2.2) incorporate multiple time periods, risk classes and refined correlation structures into the estimation of unconditional PD.24 For example, Huschens, Vogl and Wania (2003) propose a simultaneous multi-period MLE with the likelihood function

L(p) = T Y t=1 ∞ Z −∞ nrc Y rc=1 nrc_t drc t pdrct t,rc|zrc t 1−pt,rc|z rc t nrct −drct ϕ(z_trc)dz_trc (2.3)

being a convolution of mixture distributions for the vectorp= (p1_{, ..., p}nrc_{) of the uncon-}

ditional default probabilities of nrc risk classes with the conditional default probability

pt,rc|zrc

t =p(z

rc

t ;prc, ρarc) and given the number of defaults drct and the number of obligors

nrc

t of risk class rc in period t = 1, ..., T. Note that the risk class factors ztrc in (2.3) are assumed to be independent. Estimation results for (2.2) and (2.3) are not satisfying,

23_{Cf. H¨ose and Huschens (2003a), p. 158.}

however, as the identification of parameters is insufficient, especially if different factors are involved and the asymptotic properties of the MLE cannot be determined analytically. Simulation results reveal that the MLE perform well in reproducing unconditional PD, whereas estimated asset correlations are biased downward and produce large standard errors.25

In summary, the single-period default rate ˆpt may be used to estimate pooled conditional PD of a PIT rating system, whereas the average default rate of risk classes in time is more suited for estimating unconditional PD of TTC ratings systems. In a situation in which historical default experience is scarce or there are structural breaches in the rating methodology or default definition, the unconditional PD of risk classes can be estimated from the default rates of risk class using the MLE estimator in (2.2) and (2.3).