Caracterización del maltrato escolar

Consider, once again, a portfolio of N obligors in which every obligor (i) has been assigned a Probability of Default (PD) over a given time horizon T (say one-year). This

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reflects the probability that the customer’s status, at the end of the one year horizon will be either defaulted or performing. This two-state scenario can be depicted through the use of a Bernoulli distributed indicator variable, ( ):

{ if firm is in default at time

otherwise ( ) ( )

In its most basic form, CreditRisk+ assumes that an obligor’s Exposure at Default (EAD) and Loss Given Default (LGD) are both given constants, and describes the loss of any obligor n through the Loss Variable:

( )

such that the Expected Loss ( ) of obligor (i) can be expressed as the expectation of its corresponding loss variable ( ):

[ ] ( ) ( )

For our portfolio of (N) obligors, this then allows for the definition of the Portfolio Loss as:

∑

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For a discrete random variable (Y), we define its probability generating function (pgf) as:

( ) [ ] ∑ [ ]

( )

For our Bernoulli distributed default indicator for obligor (i), we write:

( ) ( ) ( ) ( ) ( )

Working with this representation of ’s pgf we can move from a Bernoulli setting to a Poissonian one at the heart of the analytically tractable solution that the CreditRisk+ framework brings to the problem of credit portfolio loss distribution generation. That transition results in the rewriting of Equation (5.6) as the pgf of a Poisson variable with intensity :

( ) ( ) ∑

( )

For our portfolio of (N) independent obligors, in which the risk of default for each obligor (i) is now given by a Poisson variable , we define our loss variable as:

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where is defined as the non-random Exposure for obligor (i). Therefore, maintaining our assumption of independent defaults in our portfolio, we can construct the portfolio loss distribution by defining the pgf of :

( ) ( ) ( ) ∑ [ ]

( )

Next, the portfolio loss distribution pgf can be written as:

( ) ∏ ( ) ∏ [∑( [ ] ₎ ] ∏ [∑ ( ) ] ∏ [ _] (∑ ( ) ) ( )

We note here that the portfolio loss distribution is not Poisson distributed, due to the inclusion of the variability of (Ei), in contrast to the portfolio distribution of default events.

Recall, the simulation-based AVM model of Chapter 4 presented borrower defaults as realizations of a normally distributed latent variable below some given default barrier. In the single sector implementation, the latent variable was assumed to depend on a single systematic factor and an idiosyncratic shock, both generated from standard normal distributions. Correlations between obligors were determined through common

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dependence on the single systematic factor, and were calibrated from PD averages and volatilities observed in the Financing Company SME portfolio.

The CreditRisk+ model, in contrast, does not place assumptions on the cause of default, instead borrower default probabilities are modelled to vary over time, increasing or decreasing with gamma-distributed latent systematic factors. Borrowers’ probability of default sensitivity to, and co-movements with, the systematic factor thereby generates correlations in defaults, Gordy (2000, p. 119). Whether calibrated to a single sector or a multi-sector analysis, the mean default rate stochasticity can be attributed to one or several background factors, each associated with a given sector.

In both the Single Sector and the Multiple Sector case, the CreditRisk+ framework allows for a closed form solution for the loss distribution to be generated. In particular, let our portfolio be divided into (K) sectors, each with a Gamma distributed risk factor with a long-term mean of and a variance of , Credit Suisse (1997, p. 42):

( ) ( )

where,

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Assuming that the default rate of each obligor depends on only one factor, obligors are assigned to the sector with which they are associated. CreditRisk+ includes a more general framework in which obligors can be associated with more than one sector. Under such a generalized framework, an obligor’s dependence on a given sector is represented with a given weighting, such that the sum of an obligor’s weights across the set of sectors should be less than or equal to one. Finally, this framework allows for the inclusion of an idiosyncratic sector capturing the volatility in obligors’ default rates which may be due to idiosyncratic factors, as opposed to systematic ones; for more information on these aspects of the CreditRisk+ framework, the reader can refer to Section A.12 of Credit Suisse (1997).

For each obligor (i) we introduce a series of sector weights { } satisfying:

∑

( )

such that, These weights, act as factor loadings, measuring obligor (i)’s sensitivity to each of the risk factors, while ( ) can be viewed as assigning a weight to an idiosyncratic sector with mean one and variance zero.

For a given borrower (i) in Risk Rating (ζ), the probability of default, conditional on realizations of the systematic factor, is amplified or subdued according to a given sensitivity wi,ζ. More specifically, we write:

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( ) ̅̅̅̅ (∑ ) ( )

where ̅̅̅̅ is the unconditional long term average probability of default for a given Risk Rating (ζ); see Gundlach (2004, pp. 16-17) and Gordy (2000, pp. 121-122). In addition, we write:

[ ( )] ̅̅̅̅ [ ( )] ̅̅̅̅ ∑ ( [ ])

( )

For our portfolio of (N) obligors, we can now write our expected portfolio loss distribution conditional on the realization of our K sectors { } as:

[ | ] ∑ [ | ] ∑ [ ( )| ] ∑ [ ̅̅̅̅ (∑ )] ∑ [ ∑( ̅̅̅̅ ) ] ( )

In order to derive the portfolio loss distribution, we begin by deriving the pgf for obligor n conditional on . Analogously to equation (5.10), we write:

( | ) ( | ) ∑ [ ]

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Our assumption of conditional independence then allows us to write:

( | ) ∏ ( | ) (∑ ( ) ( ) ) (∑ ∑ ̅̅̅̅ ( ) ) ( )

Using the gamma distribution functional forms and integrating out X, we obtain:

( ) ∏ ( ) (∑ ̅̅̅̅ ( ) ) ∏ ( ∑ ̅̅̅̅ ) ( ) where, ( ) ( )

Credit Suisse (1997, pp. 48-49) and Gundlach (2004, pp. 21-23) present the Panjer recursion used in the original CreditRisk+ for generating portfolio losses from the pgf; for alternative solution schemes see, for example, Gordy (2002), Haaf, Reiss, and Schoenmakers (2004), and Merino and Nyfeler (2004). We will use the Panjer recursion in our estimation of portfolio losses according to the analytical CreditRisk+ implementations.

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What is left is the calibration of the sector factor parameters and . Kluge and Lehrbass (2004, p. 317) observe that gamma distributed sector factors can be normalized to any desired expected value. When obligor-specific default rate standard deviations are available, Credit Suisse (1997, pp. 51-52) shows that an appropriate and pragmatic calibration of the sector parameters can be undertaken as follows:

∑ ̅̅̅̅ ∑ ̅̅̅̅̅̅̅̅̅ ( )

The settings in Equation (5.21) emphasise the importance of the distribution of borrowers across various PD-calibration segments within a given sector, as opposed to other outstanding sectoral characteristics; see Credit Suisse (1997, pp. 43). When default rate standard deviations are not available, Credit Suisse (1997, pp. 44) suggests the use of a single fixed ratio for ⁄ and suggests a value of order one in accordance with historical experience. This fixed ratio setting, and specifically the unitary volatility ratio setting, is widely applied in the literature. In Chapter 6 we explore the use and impact of these two calibration methods, on the loss distribution and resultant EC.