**3. Marginal economic effect improving of the discrimination power of bank borrowers**

To assess the discriminating power of a rating system (model) in financial engineering, curves are traditionally used that determine its quality [15, 16], this analytics is borrowed from a well-developed theory of radio signal reception. Any rating (or scoring) system, if it confidently discriminates between "good" and "bad" borrowers, will spread the initial statistical distribution of customers by rating (scoring) score *s*. That is, two different distributions are obtained - default, with a density of *f <sup>D</sup>*ð Þ*s* , and nondefault, with a density of *f <sup>N</sup>*ð Þ*s* , which can be determined by the expiration of the term (usually one year) after the "measurement" of the rating *s* .

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The probability distribution functions of getting into the rating below s for nondefault and default clients will be expressed by the corresponding integrals<sup>2</sup>

$$F\_N(s) = \int\_{-\infty}^s f\_N(\xi) \mathrm{d}\xi , F\_D(s) = \int\_{-\infty}^s f\_D(\xi) \mathrm{d}\xi , F\_N(s) \in [\mathbf{0}, \ \mathbf{1}],\\ F\_D(s) \in [\mathbf{0}, \ \mathbf{1}].$$

ROC<sup>3</sup> end CAP curves are defined in the square of unit area on the plane (X, Y) in parametric form:

$$ROC(\mathfrak{x}) = F\_D(\mathfrak{s}), \ \mathfrak{x} = F\_N(\mathfrak{s}),$$

$$CAP(\mathfrak{x}) = F\_D(\mathfrak{s}), \ \mathfrak{x} = (\mathfrak{1} - p) \cdot F\_N(\mathfrak{s}) + p \cdot F\_D(\mathfrak{s}),$$

where *p* ¼ *DR* (likewise (3)) is the share of defaults for the period under review. From the CAP and ROC curves, the exact formula [17] can be used to express the default probability of the borrower, whose position in the rating is determined by the coordinate *<sup>x</sup>*∈½ � 0, 1 (local position in the distribution of all borrowers)<sup>4</sup> :

$$PD(\mathfrak{x}) = p \cdot \mathsf{CAP}'(\mathfrak{x}),\tag{5}$$

or

$$PD(\boldsymbol{x}) = \frac{\boldsymbol{p} \cdot \boldsymbol{ROC}'(\boldsymbol{x})}{\boldsymbol{p} \cdot \boldsymbol{ROC}'(\boldsymbol{x}) + \mathbf{1} - \boldsymbol{p}},\tag{6}$$

where х—quantile position of the borrower among nondefault one. The Gini index will be calculated using the well-known formula

$$AR = \frac{2 \cdot \int\_0^1 CAP(\varkappa) d\varkappa - 1}{1 - p}. \tag{7}$$

According to formulas (5) and (6), the dependence of the default probability on the rating will be largely determined by the behavior of the CAP (or ROC) curves of the rating model, as well as the distribution of borrowers (companies, clients) by rating score.

The average annual expected losses for the borrower EL are determined by the formula �*LGD*, so the expected loss for a borrower with a quantile *x* coordinate will be determined as follows:

$$EL(\mathbf{x}) = PD(\mathbf{x}) \cdot LGD = p \cdot LGD \cdot CAP'(\mathbf{x}) = EL \cdot CAP'(\mathbf{x}),\tag{8}$$

where *EL* ¼ *p* � *LGD* is the average market loss parameter, which is exogenous.

Suppose that the bank has an unlimited resource base and is potentially ready to lend to the entire flow of incoming applications, with a volume of "1", then it will receive a

<sup>2</sup> If the rating score or rating is not continuous (i.e., has a limited number of positions), then the integral is replaced by the sum, and the distribution over the rating is discrete.

<sup>3</sup> Receiver Operating Characteristic.

<sup>4</sup> Here and below, the prime denotes the derivative, as in the conventional notation.

*Risk Management Tools to Improve the Efficiency of Lending to Retail Segments DOI: http://dx.doi.org/10.5772/intechopen.108527*

loss, with a volume of EL. However, the rating system (i.e., the entire risk management process) rejects x (%) of the incoming flow, generating a loss,5 due to unrealized profit, determined by the credit margin M. In addition, there are credit losses among systemapproved borrowers (3) *ELx* ¼ *EL* � ð Þ 1 � *CAP x*ð Þ (**Figure 3**).

The economic effect of improving the rating system has two fundamental components. The first is the reduction in risk *ELx* among approved borrowers, which is obvious since the improved rating system will have a steeper profile of the *EL x*ð Þ schedule, which means that the level of losses will be lower. The second is a decrease in the level of deviation (cut-off), which implies an increase in the volume of the loan portfolio with a constant flow of applicants and constant lending rates.

We introduce the notation as ¼ EL*=M*.

**Marginal income theorem from increasing the discriminatory power of scoring** Let the CAP curve CAP x, AR ð Þ have a single root x AR ~ð Þ∈ ð Þ 0, 1 of the equation *∂*2 <sup>∂</sup>x∂ARCAP x, AR ð Þ¼ 0. If the business is guided by the goal of maximizing profit in its decision on the cut-off level, then there is a region in the parameter space β >0, AR∈ ð Þ 0, 1 , in which the annual return P from a marginal increase in the Gini index by ΔAR will be estimated as follows*:*

$$P > \tilde{\pi} \cdot \Delta AR. \tag{9}$$

Where *<sup>π</sup>*<sup>~</sup> <sup>¼</sup> *<sup>E</sup>*^ � *EL* <sup>2</sup> this is a parameter for a loan portfolio with a constant amount *E.* ^

In fact, this means that, given the amount of loans *E*^ (annual) and the level of expected average annual losses of applicants EL, for each percentage point of increase in

#### **Figure 3.**

*The optimal point* **X** *cut-off (cutting off "bad" from "good" borrowers) must correspond to the level of losses* EL*(*x*) that does not exceed the marginal return on the loan (margin* M*).*

<sup>5</sup> In addition to the lost profit, the loss from the deviation is assumed to be zero, although in practice this is not the case, since the "refuseniks" go through the process of internal underwriting, which is costly for the bank. These costs are a much smaller order of magnitude than credit losses, but in an accurate financial model, of course, should be taken into account.

the power of the rating system, there will be a guaranteed increase in profits of *π*~ � 0*:*01 in the parameter space estimated in the next section. That is, given that provisions on placed loans (passed by risk management) may be less than the risks of the applicants, then for each percentage point increase in the risk management Gini index, the income level is estimated as half the volume of provisions multiplied by 0.01.
