**4. The area of risk-return and the Gini index, in which the margin effect works**

This section presents the result of modeling the marginal profit of a unit amount of placed funds *E*^ in relation to the marginal profit guaranteed by formula (9) using the Van der Burgt CAP-curve model (1).

We define the normalized marginal return from an increase in discriminatory power AR as the ratio *<sup>π</sup>=<sup>π</sup>*~. The profitability level *π* is determined by formula (12), guaranteed by *π*~ formula (9). In the Van der Burgt CAP-curve model, the function *k AR* ð Þ , *p* is implicitly defined (2). In addition, as it is easy to see, the condition of the Theorem on the existence of a unique root <sup>~</sup>*x AR* ð Þ∈ð Þ 0, 1 of equations *<sup>∂</sup>*<sup>2</sup> *<sup>∂</sup>x∂AR CAP x*ð Þ¼ , *AR* 0 is true. This means that there is a single segment belonging to the cut-off level of

½ � *x*1, *x*<sup>2</sup> ∈½ � 0, 1 , in which inequality (14) is satisfied and there is a connected domain of parameters in which the conservative estimate (9) is true. Let's show it.

The normalized marginal return will be calculated by the formula as shown below:

$$\pi\_{\hat{\pi}} = \frac{2 \cdot CAP\_k'(\hat{\pi}, k, p)}{(1 - \hat{\pi}) \cdot \frac{dAR}{dk}},$$

where *k AR* ð Þ , *p* is the solution of the transcendental equation (2). After simple transformations, the formula for the normalized marginal return for the model under consideration is obtained:

$$\sigma\_{\hat{\pi}} = \frac{1 - p}{1 - \hat{\pi}} \cdot \frac{(\hat{\mathbf{x}} - \mathbf{1}) \cdot e^{-k(\hat{\mathbf{x}} + \mathbf{1})} - \hat{\mathbf{x}} \cdot e^{-k\hat{\mathbf{x}}} + e^{-k}}{e^{-k} - \frac{\left(1 - e^{-k}\right)^2}{k^2}}.$$

.

<sup>6</sup> Obviously, the function *CAP*<sup>0</sup> *AR*ð Þ *x*^, *AR*, *p* has zero values at the boundaries of the interval x^ ∈ [0,1], as well as the function *CAP*ð Þ� *x*^, *AR*, *p x*^. Therefore, it is sufficient to have a unique extremum x of the ^ function *CAP*<sup>0</sup> *AR*ð Þ *x*^, *AR*, *p* for inequality (14) to give a unique segment x½ � 1, x2 ∈ ½ � 0, 1 as a solution.

#### **Figure 4.**

*Level lines of the normalized marginal return in the range of parameters* **∈** ½ � **0***:***2**,**2** *, AR*∈ ½ � **0***:***2**, **0***:***8** *. In the blue area, the normalized marginal return does not reach unity (guaranteed return (9) is not achieved).*

Numerical study of the Van der Burgt CAP curve model for a realistic set of parameters EL <sup>M</sup> ∈½ � 0*:*2,2 , AR∈½ � 0*:*2,0*:*8 ,p ¼ 4% (the influence of parameter p is small) presented in **Figure 4**.

**Figure 4** shows the normalized marginal return level lines in a two-dimensional range of parameters. In the blue (close to triangular) area of low risk and the power of the rating system, the guaranteed marginal return is not achieved. In the upper part of the parameter area, the marginal return from improving the rating system already exceeds the guaranteed value by 2–3 times, especially in reddening zones.

The presented case shows the practical reliability of a conservative estimate of marginal return when the rating system is improved in the range of risk parameters that are most interesting for the use of rating models in decision making.
