## 金融代写|金融风险管理代写Financial Risk Management代考|Methods of Mathematical Programming

Mathematical programming methods are an important group of methods that are used in predicting the risk of corporate bankruptcy. With regard to mathematical programming, the problem of linear programming stands out. The goal of linear programming is to find the optimal value so that all ratios are based on the optimal solution. When creating a model, it is necessary to pay close attention to the accuracy of determining individual limiting conditions. In the practical solution of a given task, it is necessary to proceed from its simplification. The most common algorithm for solving linear programming problems is the simplex method. Mangasarian (1965), Hand (1981) and Nath et al. (1992) were the first to apply linear programming to predict financial health (Dipak and Purnendu, 2007).

One of the methods which fits the issue of mathematical programming, and which we will describe in more detail, is the Data Envelopment Analysis (DEA) method (Horváthová and Mokrišová, 2018). Compared to statistical methods, DEA is a relatively new, non-parametric method, which represents one of many approaches to assessing the financial health of companies and the risk of their bankruptcy. Charnes, Cooper and Rhodes first used this method in 1978. It is based on the idea presented in the paper “Measuring efficiency of decision making units”, published by Farrell in 1957, whose work was based on the work of Debreu (1951) and Koopmans (1951). Farrell (1957) proposed a new approach to measuring efficiency based on a convex efficient frontier and the use of functions to measure the distance between the observed enterprise and the projected point on the efficiency curve. In this way, he proposed a new measure of enterprise efficiency based on the calculation of two components of overall enterprise efficiency: technical and allocative efficiency. Farrell’s approach was based on measuring the ability of the observed company to transform inputs into outputs, and is therefore called input-oriented. Charnes, Cooper, and Rhodes (1978) applied a multiplicative input/output model to measure business efficiency.

The approach of these authors is described as a two-step calculation of efficiency. The first step is to identify the efficiency frontier. If the combination of inputs and outputs of the production unit lies on the frontier, it is efficient and financially healthy production unit. If the production unit is inefficient, it does not lie on the efficiency frontier. This unit need to reduce inputs or increase outputs. In the second step, the efficiency score for the analyzed companies and their distance from the efficiency frontier is calculated. DEA models can be divided into DEA CCR (Charnes, Cooper and Rhodes, 1978) and DEA BCC (Banker, Charnes and Cooper, 1984) in terms of whether each unit of input yields the same amount of output or a variable amount of output. The CCR and BCC models have been proposed in the literature in multiplier or envelopment (dual) form. From a practical point of view it is more appropriate to work with dual form of the models. (Jablonský, Dlouhý, 2015). The CCR and BCC model can be computationally oriented on inputs (input-oriented) or outputs (output-oriented) (Klieštik et al., 2019). Formulas for input-oriented and out-put-oriented dual models are stated in Table 1. Simak (1997) was the first who thought of using the DEA method to predict bankruptcy by comparing its results with the results of Altman’s Z-score. In 2009, Premachandra, Bhabra and Sueyoshi used the additive DEA model and compared its results with the results of logistic regression. The research yielded a satisfactory level of correct prediction of bankruptcy.

## 金融代写|金融风险管理代写Financial Risk Management代考|Estimation Accuracy of the DEA Model

The estimation accuracy of the DEA model was compared at different cut-offs. Figure 4 shows the estimation accuracy of the DEA model for different cut off values.

It is clear from Figure 4 that the highest estimation accuracy of the DEA model, in the case of bankrupt companies, is already at the cut-off $0.78$ and remains at approximately the same level up to the cut-off $0.5$, and for non-bankrupt companies it is at the cut-off 1 . Based on the fact that bankrupt companies are those with a score of 1 , the estimation accuracy of the CCR DEA model for bankrupt companies is only $14 \%$. It should be pointed out that $14 \%$ of companies are located on financial distress frontier, but it can be assumed that there are more bankrupt companies than the number suggests. Based on this fact, it is necessary to lower the cut-off and find the optimal cut-off at which the highest possible number of bankrupt and non-bankrupt companies will be identified. This optimal limit is at the level of $0.83$, at which the sum of sensitivity and specificity is the highest. This means that if the probability of bankruptcy is equal to or higher than $0.73$, then the company has a high probability of going bankrupt.
The ROC curve was constructed to assess the estimation accuracy of the DEA model (Figure 5). This curve captures the relationship between sensitivity and specificity. The more convex the ROC curve and approaches the upper left corner, the better the discriminative ability of a particular model (Gajowniczek, Zabkowski and Szupiluk, 2014).

The area under the ROC curve (AUC) reaches a value of $0.85$, which can be evaluated positively and it can be stated that the model has a very good estimation accuracy.

If we compare the results of this study with the results of some other authors, the model achieves approximately the same estimation accuracy. Mendelová, Bieliková (2017) achieved $24 \%$ accuracy with the DEA model in the identification of eompanies in financial distress and $96.7 \%$ ateuracy in the classification of financially sound companies. In the case of some authors, the estimation accuracy of the DEA model was higher, in particular Premachandra et al. (2009) achieved $84.89 \%$ accuracy in predicting failing companies, Mendelová and Stachová (2016) 10-42.86\% and Cielen et al. (2004) 74.4-75.7\%.
However, these authors do not indicate a cut-off for the probability of bankruptcy. This low estimation accuracy of the DEA model in the area of bankruptcy prediction is due to the methodology of creating the financial distress frontier. Only a small number of companies fall within this limit – these are the companies that achieve extreme values in the analyzed indicators.

## 金融代写|金融风险管理代写Financial Risk Management代考|Estimation Accuracy of the DEA Model

ROC曲线下面积（AUC）达到0.85，可以正面评价，可以说该模型具有很好的估计精度。

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