A Gaussian SVR-based intelligent framework for investment risk assessment in green finance

Establishment of risk assessment indicators

In the process of establishing indicators, we need to fully consider the characteristics of the current green finance industry. Therefore, this article builds a risk assessment model from the ESG indicators commonly used by financial experts in assessing venture capital. ESG represents Environmental, Social and Governance respectively to investigate the medium and long-term development potential of enterprises (Gillan, Koch & Starks, 2021). When selecting the company’s ESG development indicators, it is necessary to adapt to the company’s current reality, fully consider the current economic, social, environmental and management indicator systems, and include them as important references. Credit risk is the most important link in the development of green finance. Accurate review of social and environmental risks can help investment institutions better control dual risks. Specifically, the establishment of a sound risk index evaluation system can not only facilitate investors’ investment, but also escort the development of the whole green finance system.

Based on the demand for risk indicators and the connotation of ESG indicators, the ESG development indicators selected in this article are composed of three indicators, namely, environmental responsibility indicators, social responsibility indicators and management responsibility indicators. Management responsibility indicators are mainly corporate governance. The above three indicators are weighted by AHP (Yuying, 2021; Alrawad et al., 2023; Kaur et al., 2023) and expert evaluation. The calculation method is shown in Formula (1):

(1) $$Inde{x}_{E,S,G}=\sum _{i=1}^n\left[e_i,s_i,g_i\right]W^T.$$

Experts score according to the three indicators to judge the risk characteristics of their objects. Large difference will cause great difference in model training and regression. Therefore, normalize the score of the sample to form [0, 1] interval score probability, so as to ensure the accuracy of risk assessment. The data normalization is given in Formula (2)

(2) $$X^{\ast }={\displaystyle \frac{X-X_{min}}{X_{max}-X_{min}}}$$where X is the original score, Xmin is the minimum value, and Xmax is the maximum value. Through normalization, the training efficiency and accuracy of the model can be greatly improved. To mitigate the risk of information leakage arising from using the same expert pool for both features and labels, this study separates the construction of features and the determination of labels. Specifically, ESG dimension ratings are provided by Expert Group A and aggregated through the AHP, while investment outcomes are defined either by a different expert group (Expert Group B) or by ex-post performance measures (e.g., default/return after 1–2 years). If the same group of experts is retained, a temporal separation is enforced—ESG features are assessed at time T, whereas investment outcomes are validated at time T + 1. In applying the AHP weighting, the Consistency Ratio (CR) values were all below the accepted threshold of 0.1, indicating satisfactory consistency of expert judgments. In addition, robustness checks with perturbed pairwise comparisons yielded stable weight structures, confirming the reliability of the assigned weights.

Support vector regression model

Generalized support vector regression models are divided into linear separable support vector regression and nonlinear support vector regression (Kurani et al., 2023; Dash et al., 2023; Thumu & Nellore, 2024). Given a set of training samples $T=\{\left(X_i,Y_i\right){\}}_{i=1}^n$, the objective of SVR is to find a regression function $f\left(x\right)$ that approximates the underlying relationship between the input features $X$ and the output variable $Y$. The model seeks to construct an optimal hyperplane such that the predicted values remain as close as possible to their true counterparts, thereby minimizing the overall deviation. To achieve this, a tolerance margin of width $2\epsilon$ is introduced around the regression plane. If the absolute deviation between the predicted value $f\left(x\right)$ and the actual value $y$ does not exceed $\epsilon$, the prediction is considered satisfactory. This principle defines the so-called ε-insensitive loss function, which helps reduce sensitivity to small perturbations in the data and prevents overfitting.

Geometrically, SVR constructs a tube centered around the regression plane, as illustrated schematically in Fig. 2. The aim is to include as many data points as possible within this tube while keeping the model complexity low. Unlike the traditional SVM used for classification—which seeks to maximize the margin between classes—SVR attempts to minimize the deviation between the hyperplane and data samples, emphasizing prediction accuracy over separation.

For samples outside the regression plane (Huang, Wei & Zhou, 2022), the loss function is calculated as shown in Eq. (3):

(3) $$l\epsilon =\{\begin{array}{c}0\begin{array}{cc}& \end{array}\\ \left|z\right|-\epsilon \end{array}\begin{array}{c}if\left|z\right|\le \epsilon \\ \begin{array}{cc}& \end{array}otherwise\end{array}.$$

By determining the weight and compiling in the model to minimize the loss function, the optimization objective Eq. (4) can be obtained by introducing a relaxation factor $\xi i,\stackrel{\wedge }{\xi i}$ into the regression model to make the regression surface a soft interval:

(4) $$\mathrm{m}\mathrm{i}\mathrm{n}{\displaystyle \frac{1}{2}{\Vert w\Vert }^2+C\sum _{i=1}^m(\epsilon i+\stackrel{\wedge }{\epsilon i})}$$

$$\mathrm{s}.\mathrm{t}.f(x)-yi\ge \epsilon +\xi i$$

$$yi-f(x)\ge \epsilon +\stackrel{\wedge }{\xi i}$$

(5) $$\xi i\ge 0,\stackrel{\wedge }{\xi i}\ge 0,i=1,2,\mathrm{L}\dots$$

By introducing Lagrange multiplier, the original problem can be transformed into:

(6) $$\begin{array}{rl}& \mathrm{L}={\displaystyle \frac{1}{2}{\Vert w\Vert }^2+C\sum _{i=1}^m(\epsilon i+\stackrel{\wedge }{\epsilon i})-\sum _{i=1}^m\mu i\epsilon i-\sum _{i=1}^m\mu i\stackrel{\wedge }{\epsilon i}+}\\ & \sum _{i=1}^m\alpha i(f(xi)-yi-\epsilon -\xi i)+\sum _{i=1}^m\stackrel{\wedge }{\alpha i}(yi-f(xi)-\epsilon -\stackrel{\wedge }{\xi i})\end{array}.$$

Find the partial derivative of it, and when the derivative is zero, the dual problem can be obtained:

(7) $$\mathrm{m}\mathrm{a}\mathrm{x}\sum _{\mathrm{i}=1}^{\mathrm{m}}\mathrm{y}\mathrm{i}(\stackrel{\wedge }{\alpha \mathrm{i}}-\alpha i)-\sum _{\mathrm{i}=1}^m\epsilon (\stackrel{\wedge }{\alpha \mathrm{i}}+\alpha i)-{\displaystyle \frac{1}{2}\sum _{i=1}^m\sum _{j=1}^m(\stackrel{\wedge }{\alpha \mathrm{i}}-\alpha i)(\stackrel{\wedge }{\alpha \mathrm{j}}+\alpha j)\mathrm{x}{\mathrm{i}}^{\mathrm{T}}\mathrm{x}\mathrm{j}.}$$

Finally, the optimal solution of the support vector regression model can be obtained by solving the optimal solution of the dual problem

(8) $$f\left(x\right)=\sum _{i=1}^m\left(\stackrel{\wedge }{\alpha i-\alpha i}\right)x{i}^{T}x+b$$

(9) $$b=yi+\epsilon -\sum _{i=1}^m\left(\stackrel{\wedge }{\alpha i}-\alpha i\right)x{i}^{T}x.$$

SVR risk assessment based on Gaussian kernel function

To effectively model the nonlinear and abstract characteristics of risk data, this study employs a kernel-based nonlinear regression approach. Conventional statistical learning techniques are primarily designed for structured, continuous, and linearly correlated data; however, risk features often exhibit discrete distributions, complex interdependencies, and uncertain correlations even after expert quantification. Such conditions hinder the accuracy and stability of traditional regression models. To address these challenges, kernel methods are introduced to extend the data representation from the original feature space to a high-dimensional reproducing kernel Hilbert space (RKHS), where nonlinear relationships can be approximated as linear mappings. This transformation enables the regression model to capture hidden structures and nonlinear boundaries without explicitly defining transformation functions.

In this framework, the kernel function serves as the mathematical bridge that computes inner products in the transformed space while preserving computational efficiency in the original domain. Various kernel types exist—such as linear, polynomial, sigmoid, and Gaussian functions—each defining a different notion of similarity between data samples. In this work, the Gaussian kernel is selected because of its ability to provide smooth mappings, local adaptability, and strong generalization across heterogeneous datasets. The scale parameter of the Gaussian kernel controls the sensitivity of the regression surface: a smaller value enhances local fitting but increases the risk of overfitting, whereas a larger value yields smoother approximations but may reduce model precision.

To ensure reliable performance, a parameter optimization process is performed to determine the kernel width and regularization coefficients that minimize model error while preserving generalization capability. The final kernel regression model effectively captures nonlinear feature interactions, reduces bias introduced by expert scoring, and improves the accuracy and robustness of the overall risk evaluation.

SVR is adopted in this study not only as a generic regression tool but also because of its specific advantages in the context of green finance risk assessment. Compared with traditional regression approaches, SVR is well suited to handle high-dimensional, nonlinear, and relatively small-sample datasets, which are common characteristics of ESG-related financial data. The ε-insensitive loss function and kernel-based mapping allow SVR to capture complex relationships while maintaining good generalization ability and robustness against noise. These properties make SVR particularly appropriate for assessing investment risks in green finance, where data heterogeneity and limited historical records are frequent challenges.

Data acquisition and processing

In line with the industrial investment requirements of green finance and relevant green credit policies, this study selects a dataset comprising corporate green projects from the past five years (doi: 10.5281/zenodo.12563169). The sample includes 50 projects spanning multiple sectors such as green insurance, renewable energy, and other green industries, thereby ensuring industry diversity. To assess investment risks, 10 domain experts were invited to provide evaluations based on the hierarchical evaluation and weighted scoring method described in ‘Establishment of Risk Assessment Indicators’. The expert scores were then aggregated using the assigned weights (Eq. (1)). Prior to normalization (Eq. (2)), the highest and lowest scores were trimmed to mitigate the influence of outliers and reduce extreme deviations. This process ensures both the robustness of the assessment and the reproducibility of the dataset description in terms of sample size, time span, and industry coverage. The numerical features of the three categories of indicators established according to ESG indicators are processed according to the original data, and 50 sample points are plotted through three-dimensional features. The results are shown in Fig. 3. It can be found that it is difficult to form effective classification regression judgment from the perspective of their feature distribution alone.

Model establishment and training

According to the collected data, this article takes the characteristic indicators of its three dimensions as input and the normalized investment probability as output. For machine learning methods based on statistical learning, the proportion of training set, verification set and test set needs to be strictly divided in the process of model verification. According to the dimensional characteristics of the data and the size of the data sample, this article completes the training and establishment of the model through 5-fold cross validation, so as to optimize the parameters of the algorithm model and get the final model. The so-called 5-fold crossover experiment means that the 50 sample data collected in this article are divided into five parts, 10 samples for each part, one of which is used as the test set and the other four are used as the training set, and the model with the smallest loss function is selected. In the process of model training, in order to verify whether the Gaussian kernel function used has the advantage of regression accuracy, the commonly used linear kernel, that is, the kernel function, is also verified and tested. In order to ensure the rationality of algorithm comparison, the methods used in this article all use default parameters, and do not use network search, particle filter and other parameter optimization methods to improve the model. For the SVR model, a grid search procedure was conducted to tune the key hyperparameters C, $\epsilon$, and $\gamma$. Candidate values were systematically evaluated, and the combination yielding the best cross-validation performance was selected for the final model, thereby ensuring that the reported results are not based on default settings but on optimized parameters.

Experimental results and comparative analysis

The regression results of the optimal model obtained in this article according to the 50% cross validation introduced in ‘Model establishment and training’ are shown in Fig. 4. It can be found that the regression results of this model are relatively average, except for the 44th sample, which has a high investment probability due to the favorable opinion of experts and has a large gap with the predicted value. By observing the distribution of the predicted value and the true value of yellow and blue, it is not difficult to see that the overall predicted model is relatively conservative, and its value distribution is stable around 0.2, which is also the error caused by the small sample size of large deviation values in statistical regression. Although the extreme value has been removed in the initial scoring process of the data, in order to ensure the minimum loss function, it has to sacrifice some abnormal values to optimize the model in order to include more sample data into the model.

We can better understand the error distribution of the data in this article through the residual diagram of the real value of the label and the abscissa. We can see that the residual error of the sample data with the investment probability of 0.2 is relatively small, which shows that the sample advantage is a factor that can not be ignored in the prediction based on the regression method. For the samples with a small number of investment probabilities of 0.5 and 0.6, there will be a large deviation, but from the prediction results of the whole sample, that is, when only the prediction value in Fig. 4 is concerned, it still has certain advantages over others.

In order to verify the accuracy of the algorithm, the linear regression, tree regression and neural network methods commonly used in statistical regression methods are selected in this article for data regression test. The training method is also five-fold cross validation. The results are shown in Figs. 5A, 5B and 5C. In order to test the SVR method itself, the method based on linear kernel function is also selected in this article for validation test. From the way, we can see that the distributions of the predicted values in Figs. 5A and 5C are more discrete, the regression results are not ideal, and the problem of extreme values in the sample is not well solved. For the neural network (NN) method in Fig. 5B, its nonlinear characteristics make it more accurate to predict some extreme points, but it can be seen that the prediction and regression results are not stable as a whole. Although it is sensitive to extreme values, the deviation for most samples is large, and the loss is not minimized. In the SVR prediction based on linear kernel shown in Fig. 5D, it can be seen that its overall regression result is moderate, not much different from Gauss SVR, but its sensitivity to extreme values is worse. In order to more intuitively compare the differences among results, this article selects Root Mean Square Error (RMSE) as an index to compare the listed methods. Although the Gaussian kernel was adopted as the primary specification, we also conducted comparative experiments using a linear kernel. The results showed that the Gaussian kernel consistently outperformed the linear kernel in terms of prediction accuracy, suggesting that it is better suited to capture the nonlinear structures present in ESG-related features.

According to the RMSE results listed in Table 1, it can be seen that the SVR method based on Gaussian kernel function proposed in this article has the smallest regression deviation. At the same time, through the above discussion, it can be found that this method can not only meet the regression accuracy of most data, but also take into account the influence of extreme values to a certain extent, ensure the accuracy of the regression model, and provide reliable data support for risk assessment in the investment process.