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By Stefan Stoyanov, Solutions Manager, Experian Decision Analytics

Both linear and logistic regression models were built in order to determine which of them
provides the best estimate of the observed LGD.
•    Linear regression - continuous dependent variable
In order to meet the statistical requirements of the OLS regression an attempt was made to
transform the LGD to normality or at least symmetry by using Box – Cox type of transformation function
•    Logistic regression – the LGD was transformed to a binary dependent variable using two methods:
o    Uniform random number: if LGD > random number then LGD_ Binary = 1 (Bads) ; else LGD_Binary = 0 (Goods)


Set import_data;
Bin_lgd=1;
_freq=LGD;
Output;
Bin_lgd=0;
_freq=1-LGD;
Output;
Run;

•    Manual Cut-Off: if LGD > 0.2 then LGD_Binary = 1 (Bads) ; else LGD_Binary = 0

(Goods)

Power transformation of the LGD distribution
•    Usually the LGD has highly non-normal distribution, often with an U shape and spikes at the two tails of the distribution
•    However, the basic properties of the least squares regression do not require normality
•    Non-normality does not affect the estimation of the regression parameters. The least 
squares estimates are still BLUE (best linear unbiased estimates) if the other regression assumptions are met
•    Non-normality affects the tests of significance and the confidence interval estimates of the regression parameters

Binary transformation of the LGD using uniform random Numbers
Theoretical Beta distribution parameters:
The estimation of the alpha (a) and beta
(b) parameters of the theoretical beta distribution was based on the first two
moments of the observed LGD distribution
 


Functional calibration of the logistic regression scores to estimated LGD

o    The OLS regression model provides direct LGD estimates whereas the
logistic regression models provide indirect LGD estimates. Hence, it is
necessary to calibrate the logistic regression scores to direct LGD estimates in
order to be able to compare the two types of models
o    After obtaining the functional relationship between the logistic regression
scores and LGD it is possible to assign an estimated LGD to each individual
score
Due to the small sample size all data were used for model development. The
models were validated using bootstrap techniques
o    The following calibration tests were used to validate the LGD models:
–Spearman’s Rank Correlation
–Mean Squared Error
–R-square

Main objectives:
  discriminatory power: relative assessment of the PDs
  calibration:  absolute assessment of the PDs
  Other:
– stochastic dominance
– monotonicity

Model Performance

Model performance is important for ensuring high-quality pooling and approval decisions.

•     Do the bad customers have low scores?
•     Do the good customers have high scores?
•     To what degree do the score distributions of good and bad customers overlap?
•     How well does the model separate the good from the bad customers?
•     How many of all bad customers can you find within the low-scoring customers?

Calibration of PDs

Are the estimated PDs consistent with the observed default rates?

Backtesting and Stresstesting

•  is an internal rating system that has been developed several periods ago still applicable for todays’ data?
•  how stable is the rating system?
•  simulations of different scenarios

Since the 90s, most of the large international banks have set up heavy credit risk management systems, and in     particular in order to measure and monitor the risks they hold on each business line. One of the goals of theses systems is to allocate capital to each business line and to compute the overall capital of the bank. All these

techniques are known under the generic acronym RAROC methodology(Risk Adjusted Return On Capital) ;

implicitly, this methodology focuses on economic based estimations of credit risk, taking into account both all the individual risks and the portfolio view of the bank. The aim of the RAROC methodology is twofold:

1. Risk management: in the financial theory, the bank aims at reaching its optimal capital structure and finding the proportion of equity to assets that minimizes the cost of funding. The RAROC methodology is used for determining the overall capital requirement of the bank and the contribution of each business line to the total risk of the bank. This process is called capital allocation.

2. Performance measurement: the RAROC device computes the profitability of each transaction or business line for the shareholder. The performance measurement is the result of the interplay between revenues on one hand, and risk components on the other hand.

 Let’s consider the case of a AA rated bank that wants to capitalize its portfolio in a manner consistent with a AA rating target. This amount of capital is of course driven by all the stand alone risks included in the portfolio, but also benefits the internal diversification of this portfolio. In this sense, this equity capital requirement is called

economic capital.

 

Whereas the expected loss is the average loss that the bank anticipates to loose on its portfolio, the economic capital refers to the unanticipated losses that occur in extreme situations or market conditions. The economic capital is the cushion required above the expected loss for the bank to remain solvent in the event of extreme losses on the bank’s portfolio.

There are many available criteria for defining economic capital, but generally, economic capital is defined as the amount required to cushion the portfolio up to a given confidence level. The required confidence level depends on the target rating of the bank. For instance, if the bank’s portfolio has an average maturity of 2.5 years, the confidence level is around 99.9% for a AA- target rating. From a mathematical

viewpoint, the economic capital is linked to Credit Value at Risk (CvaR) of the portfolio and to the expected loss

of the portfolio by the relationship:

EC = VaR _99.9% − EL

 

The portfolio loss distribution is generally obtained by Monte-Carlo simulations.

 

Portfolio managers make the distinction between marginal capital and incremental capital of a transaction. The incremental capital is the additional amount of equity capital required when the transaction is added to the portfolio, whereas the marginal capital is equal to the contribution of the transaction to the total capital once this

transaction is included inside the portfolio.

 

To make this more precise, we call P the reference credit portfolio

and Mx a marginal transaction with nominal amount x. Finally, we call EC(A) the economic capital of portfolio

A. The incremental capital of the transaction M is equal to:

ECi = EC(P+ Mx )- EC(P)

 The main property of the marginal capital compared to incremental capital is that the sum of the marginal capital over all the transactions of the portfolio is equal to the economic capital of the portfolio. This property leads to an easy capital allocation on condition that we are able to compute accurately marginal capitals.

Bank needs is to include risk adjustment functions into the traditional performance

measures. There are many ways of doing that. One of them is to introduce risk elements into the traditional

Return On Capital (ROC) ratio defined by:

 

ROC= ^REVENUES

            ^{allocated capital}

 

 

Allocated capital is the regulatory capital that the bank has to allocate to the transaction of interest. Both revenues and allocated capital don’t take into account any risk sensitivity. There are several possibilities to introduce risk sensitivity in this equation, at the numerator and the denominator. According to where we include

a risk adjustment, we are led to different ratios such as RAROC, RORAC and RARORAC (RISK ADJUSTED PERFORMANCE MEASURES –RAPM) . The most popular performance measure is the Risk Adjusted Return On Risk Adjusted Capital (RARORAC), obtained by correcting the revenues by the anticipated losses on the transaction, and by replacing the allocated capital by the

marginal economic capital of the transaction:

• RORAC= (financial income – financial costs)/ scaled economic capital allocation
• RAROC=(financial income – financial costs – expected losses)/ scaled economic capital alocation

The management may also be interested in the value created by a marginal transaction or a business line. EVA (Economic Value Added) is the relevant performance measure for value creation. It is equal to the revenues less the cost of capital.

 

 

from Vivien BRUNEL