**2. Method for evaluating effectiveness indicators of risk management in retail lending business**

Let's simulate the situation, assuming that a fixed number of applicants applied for a loan to the bank *B* . At the same time, a number *A* of them passed the procedure of credit risk management and received a loan. We also know the number *D* of defaults among those who received the loan. Let's also assume that we can evaluate the situation in the credit market and know which share *DR* would have defaulted if it had not gone through our credit risk management procedure but would have received a loan as soon as it asked for it.

The entire population of applications can be presented in the form of **Table 1**, in which all values are given to the results of the bank's risk management procedures. The values in the lighter cells are the result of the calculation, while the values in the remaining cells are objective data.

From **Table 1**, you can see the classification errors (Type I errors, Type II errors [12]):


Parameters *A* end *D* are known exactly after the choice of the period for which the effectiveness of risk management is assessed. However, the parameters *B* end *DR* require additional calculation. In [13], a method for estimating these parameters and substantiating the corresponding formulas. The parameter *B* (the number of applicants who applied to the bank) should be adjusted taking into account the number of borrowers who were approved, but for some reason did not take a loan. This parameter will be less than the number of personal applications that have been considered. The parameter *DR* is also not equivalent to the share of defaulted borrowers for the selected period, which can be obtained from the standardized credit history bureau bulletin for the lending segment of interest. Because if a client comes to the bank and is denied a loan application, then there is a probability not equal to one that this client will receive a loan from another bank. Therefore, the population of applicants coming to the bank is not equivalent to the population of borrowers receiving a loan in the market, which is monitored by the Credit Bureau, it is worse. To assess the scale of this phenomenon, a specialized report of the Credit Bureau helps to find out the share of such applicants among the "refuseniks" of the bank, as well as the quality of servicing the loans they received. This requires additional research, which is practiced, and it is quite legal.


**Table 1.**

*Segmentation of the applicant population in terms of risk management.*

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

After evaluating the above parameters, **Table 1** will give the coordinates of the fat dot **Figure 1**.

Through this bold point, you can draw a CAP<sup>1</sup> curve, according to which you can evaluate the Gini index, which is a generally recognized measure of the discriminatory power of the rating system, equivalent to the work of the entire risk management (more precisely, the quasi-Gini index).

To restore the CAP curve, the well-known Van der Burgt model [14] is used, which has an independent variable *k* that is a solution to the equation:

$$\text{CAP}(\mathbf{x}) = \frac{\mathbf{1} - e^{-k\mathbf{x}}}{\mathbf{1} - e^{-k}},\tag{1}$$

where *k* is a parameter showing the effectiveness (power) of risk management decisions.

The constructed curve includes a point known to us, the coordinates of which we know:

$$\infty = \frac{B - A}{B}, \text{CAP}(\infty) = \frac{B \times \text{DR} - D}{B \times \text{DR}}$$

#### **Figure 1.**

*Reconstruction of the CAP-curve of discriminatory accuracy of risk management procedures.*


**Table 2.**

*Stereotypical recommendations of zonal assessments of the Gini metric.*

<sup>1</sup> Cumulative Accuracy Profile.

The Gini curve index (1) is calculated by the formula:

$$Gini(k, DR) = \frac{2}{1 - DR} \times \left(\frac{1}{1 - e^{-k}} - \frac{1}{k} - \frac{1}{2}\right),\tag{2}$$

which sets an objective metric of the power of discriminatory risk management procedures.

Obviously, the requirements for this metric may or may not be very strict, but the widely used recommendations of zonal estimates can be offered as a baseline for retail lending banking practice (**Table 2**).

Each obtained value of the Gini index of all cumulative risk management procedures can be attributed to one or another zone. The concept of totality means that not one internal procedure, for example, a scoring model, is evaluated, but the whole set of rules and procedures is used by risk management to make a decision on a loan application. The complex uses, among other things, antifraud, manual underwriting tools, brake lights, etc.

The next tool for assessing the effectiveness of risk management should be the assessment of commercial effectiveness. To what extent is the point of "refusal" justified from the point of view of the economics of lending to the retail product under study in a bank? It is clear that the optimal discrimination point for "bad" and "good" borrowers should correspond to the level of losses EL xð Þ that do not exceed the marginal return (M) on the loan product.

The level of expected losses EL will be determined by the level of default of borrowers who have passed the approval procedure above the level of the quantile position x of the entire population of applicants

$$\text{EL}(\mathbf{x}, \text{Gini}) = \text{DR} \times (\mathbf{1} - \text{CAP}(\mathbf{x}, \text{Gini})) \times \text{LGD}.\tag{3}$$

This level is determined by Type II errors and the level of losses given default (LGDÞ*:* Where CAP x, Gini ð Þ) will be determined through the expressions (1) and (2), which are defined in the previous step of estimating the discriminating power.

**Figure 2.** *Zonal representation of approval levels relative to the optimal profit level.*

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

Assuming that the value M is given as NPV rate of the income of the credit product under study for the vintage period, taking into account all costs and terms of amortization of loans (credits), then we can propose a simple formula for the profit from a unit volume of all incoming applicants:

$$\mathbf{P(x, Gini)} = \mathbf{M} - \mathbf{x} \times \mathbf{M} - \text{EL}(\mathbf{x, Gini}) \tag{4}$$

Formula (4) simultaneously depends on Type I/II errors. Moreover, their balance is determined by external factors—the level of the market default rate and the profitability of the product. Gross profit (4) will have a maximum at a certain level of approval (optimal rejection rate) because with complete rejection, all applicants' P 1, Gini ð Þ¼ 0 and vice versa, if you approve everyone, then you can have a loss if DR � *LGD* > *M* .

The question of determining the parameter regions where the maximum exists will be considered in the next section. But it can be argued that in the condition of equilibrium activity of the credit market (there is no excess profit and excessive demand, there is no global depression and catastrophic risks, etc.), such a maximum takes place. Therefore, it is logical to formulate the efficiency metric of commercial approval/rejection decisions in terms of the levels of deviation of these decisions from the maximum efficiency. And both in the direction of more approval, and less. **Figure 2** shows the profit curve.

Typical evaluation zones are indicated. Typical zonal levels are not the most stringent, but each bank can zone this metric for itself based on its own experience and goals.

The allowable level *α* of failure range *X*� *<sup>α</sup>* , *X*<sup>þ</sup> *α* is calculated quite simply:


As a result, we get two tools for evaluating the effectiveness of risk management in a given segment of the retail lending business. The first is an assessment of the discriminating power of risk management, and the second is the economic efficiency of the credit policy, considering risks (the cut-off level).
