Bankruptcy Prediction Models Based on Value Measures
Abstract
:1. Introduction
 How do we choose value measures that are intended to play the function of estimators of the bankruptcy risk of those construction companies that were listed on the Warsaw Stock Exchange during the years of 2010–2015?
 How do we build a discriminant function with the highest predictive efficiency for the aforementioned companies?
 How does the division into training and testing sets influence the predictive efficiency of the discriminant function that has been built?
 Methods for analysing and evaluating the literature that shows the current scientific achievements in the field of essence as well as using a discriminant analysis in order to predict the bankruptcy of enterprises—with a particular emphasis on the types of enterprise effectiveness measures used for this purpose.
 Statistical methods for verifying and supplementing the missing values of the variables, estimating the discriminant power of the variables, examining the information capacity of the variables, and examining the normality of the empirical distributions of the value measures.
 Statistical methods for assessing the statistical significance of the built discriminant functions as well as the diagnostic variables.
2. Discriminant Analysis Models—Theoretical Background
 Z—dependent variable;
 a_{0}—constant;
 a1, a_{2}, …, a_{n}—discriminant coefficients;
 X_{1}, X_{2}, …, X_{n}—exogenous (diagnostic) variables.
 G.L.V. Springate’s model that was developed for the Canadian economy (Talebnia et al. 2016);
 models for the Japanese economy (Takahashi et al. 1984);
 models for Asia’s emerging economies (Ashraf et al. 2019);
 H. Koh and L. Killough’s model for the American economy (Rahimipoor 2013).
 Debt and debt coverage ratios;
 Turnover ratios;
 Profitability ratios;
 Liquidity ratios;
 Asset, equity, and debt structure ratios.
3. Materials and Methods
 adoption of the output set of book measures (BM), cash measures (CM), market measures (MM), and measures based on economic profit (EPM) as potential estimators of the bankruptcies of the analysed companies;
 verification and supplementation of the missing values of the variables with the use of the median as the well as verification of the variables from the point of view of outliers (using a twoway Tukey’s criterion with α = 0.05);
 estimation of the discriminant power of the variables with the use of the classical coefficient of variation;
 examination of the information capacity of the variables on the basis of Pearson’s linear correlation level;
 examination of the normality of the empirical distributions of the value measures using the following tests: Kolmogorov–Smirnov, Lilliefors, and Shapiro–Wilk, with the assumption of α at a level of 0.05.
 Wilks’ lambda serves to determine the statistical significance of the discriminatory ability of the whole model and is calculated as the proportion of the discriminant of the variance matrix and total covariance. This takes its value from a [0; 1] range, where “0” represents perfect discriminatory power and “1” represents no discriminatory power.
 Partial Wilks’ lambda—determines the contribution of the individual variables for the discrimination of the groups. The value of this measure is determined as the quotient of Wilks’ lambda value after entering a given variable in the model and Wilks’ lambda value before adding this variable. The partial Wilks’ lambda also takes values from a [0; 1] range, where “0” represents the perfect discriminatory power of a given variable and “1” represents no discriminatory ability of a given variable.
 Standardised discriminant coefficients—these indicate the contribution of the individual variables in the discrimination of enterprises to the populations of “bankrupts” and “nonbankrupts”. A higher absolute value of the standardised discriminant coefficient means the higher contribution of a given variable with which the coefficient occurs in the division of enterprises into both of the discussed groups, as determined by the estimated model.
 Canonical correlation—provides information on how efficiently a discriminant function divides the studied businesses into groups. This takes the value from a [0; 1] range; the higher the value, the stronger the relationship is between the groups and the discriminant function.
 Tolerance of variable—this is defined as 1 minus the squared multiple correlation coefficient of that variable with all other independent variables in the regression equation. This means that the lower the tolerance of the variable, the more excessive its contribution to the regression equation is (that is, it is indispensable in light of the contribution of the remaining variables—it is redundant).
 Multiple regression (R^{2})—this is an indicator of the quality of the model fit to the data (an R^{2} close to 1.0 indicates that almost all of the variability of the dependent variable can be explained through the independent variables included in the model).
4. Results and Discussion
 five variables (62.5% of all) represented the group of market measures;
 one variable (12.5% of all) represented the groups of measures based on the economic profit concept, book, and cash measures;
 in the most efficient discriminant function, the diagnostic measures were entered in the form of the value measures of all four studied groups;
 the canonical correlation indicates a strong relationship between the discriminant function and the studied groups.
 the higher the value of the P/E or NOPATPS variables (ceteris paribus), the higher the probability is of classifying the studied company to the population of “bankrupts”;
 the lower the value of the remaining variables (P/EBIT, P/CFO, DY, CEC, EV/S, and CFROS), the higher the probability is that a studied company is a “bankrupt”;
 the higher the value of a company’s depending variable “Z” (ceteris paribus), the greater the risk of bankruptcy for the examined company;
 the greatest influence on the results of the discriminant function is exerted by the P/E variable (discriminant coefficient = 1.917), and the weakest influence is exerted by CFROS (discriminant coefficient = −0.346).
 nine variables (69% of all) represented the group of market measures;
 two variables (15% of all) represented the group of book measures, and one (8%) represented the groups of cash measures and measures based on the economic profit concept;
 in the most efficient discriminant function, the diagnostic variables were entered in the form of value measures from all four of the studied groups;
 built in the second variant of the division of companies into the training set and the testing set, the discriminant function with the highest predictive efficiency was distinguished by a higher level of efficiency than the function built in the first variant of the division;
 the canonical correlation indicates a strong relationship between the discriminant function and the studied groups (the results are better than in the case of the model in the first variant of the division).
 increases in the values of the P/CE, FCFEPS, RMVA, NOPATPS, P/FCFF, and EY variables (ceteris paribus) raised the probability of classifying a given company to the population of “bankrupts”;
 increases in the value of the other diagnostic variables (P/EBITDA, P/BVGr., ROS, EV/FCFF, P/BV, REVA, DY) reduced the bankruptcy risk;
 the higher the value of a company’s depending variable “Z” (ceteris paribus), the greater the risk of bankruptcy for the examined company;
 the greatest influence on the results of the discriminant function is exerted by the P/CE variable (discriminant coefficient = 5.269), and the weakest influence is exerted by DY (discriminant coefficient = −0.286).
5. Conclusions
 The conducted analysis has proven that shareholder value measures are a useful tool that can be applied for the needs of corporate risk management in the area of assessing a firm’s bankruptcy risk; based on these values, it was possible to build statistically significant linear discriminant functions. Based on the conducted research, it should be stated that, of the four studied groups of diagnostic variables, the market measures were characterised by the highest usefulness level in explaining the bankruptcy risks of the studied companies. These constituted 70% of the diagnostic variables. Those diagnostic variables that belong to the remaining groups of value measures constituted 14.3% of the variables that were used in the most efficient discriminant models. The selection of the most efficient estimators of bankruptcy risk was possible by considering the discriminatory power and information capacity of the individual measures.
 The selection of the most efficient discriminant functions was made by assessing the statistical significance of the discriminatory ability of the whole model. Within the two most efficient functions, 20 different variables in the form of book, cash, market measures, and measures based on the economic profit concept were used. At the same time, the obtained findings tend to claim that the presence of the value measures from all four of the studied groups in the output set of diagnostic variables is necessary for possibly building the most efficient tool for the early warning signs of bankruptcy risk.
 The research has shown that a change in the proportion of the division of objects to the learning and testing community at the stage of building and verifying the predictive effectiveness of the discriminant models affects both the general and partial predictive effectiveness of the constructed models, as well as the degree of usefulness of individual value measures in explaining the risk of bankruptcy.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Variants  Division Proportions  Division into Groups  

Training Set  Testing Set  Total  
1.  8:3 (72.737%)  104 companies, incl.:
 39 companies, incl.:
 143 companies, incl.:

2.  9:2 (81.818%)  117 companies, incl.:
 26 companies, incl.:

Measures  Calculation Formulas  Designations 

book Measures  
NOPATPS  $\frac{NOPA{T}_{t}}{{N}_{t}}=\frac{EBI{T}_{t}\xb7\left(1T\right)}{{N}_{t}}$  NOPATPS—net operating profit after taxes per share. $NOPA{T}_{t}$—net operating profit after taxes in period t. ${N}_{t}$—number of shares at end of period t. $EBI{T}_{t}$—earnings before interests and taxes in period t. T—income tax ratio. 
ROS  $\frac{N{P}_{t}}{{S}_{t}}\xb7100\%$  ROS—return on sales. $N{P}_{t}$—net profit in period t. S_{t}—net sales in period t. 
Cash measures  
FCFEPS  $\frac{FCF{E}_{t}}{{N}_{t}}=$ $\frac{NP+DEPInv\mathsf{\Delta}WC+CLRCL}{{N}_{t}}$  FCFEPS—free cash flow to equity per share. $FCF{E}_{t}$—free cash flow to equity in period t. DEP—accumulated value of depreciations. Inv—investments. ΔWC—changes in net working capital. CL—taken credits and loans. RCL—repaid credits and loans. other designations—as previously. 
CFROS  $\frac{CF{O}_{t}}{{S}_{t}}\xb7100\%$  CFROS—cash flow return on sale. CFO_{t}—operating cash flow in period t. Other designations—as previously. 
Measures based on economic profit concept  
REVA  $NOPA{T}_{t}WAC{C}_{t}\xb7\overline{I{C}_{MV,t}}$  REVA—relative market value added. $\overline{I{C}_{MV,t}}$—market value of invested capital at end of period t. WACC_{t}—weighted average cost of capital in period t. 
CEC  $\frac{EVA}{WACC\xb7IC}\xb7100\%$  CEC—cost efficiency of invested capital. EVA—economic value added. IC—invested capital. Other designations—as previously. 
Market measures  
P/BV  $\frac{{P}_{S,t}}{\left(\frac{B{V}_{t}}{{N}_{t}}\right)}$  P/BV—price to book value. P_{S,t}—market price of share at end of period t. BV_{t}—book value of equity at end of period t. other designations—as previously. 
P/BV—Gr.  $\frac{{P}_{S,t}}{\left(\frac{C{A}_{t}{L}_{t}}{{N}_{t}}\right)}$  P/BV—Gr.—price to book value by B. Graham. CA_{t}—current assets at end of period t. L_{t}—liabilities at end of period t. Other designations—as previously. 
P/E  $\frac{{P}_{S,t}}{\left(\frac{N{P}_{t}}{{N}_{t}}\right)}$  P/E—price to earnings. Other designations—as previously. 
P/EBIT  $\frac{{P}_{S,t}}{\left(\frac{EBI{T}_{t}}{{N}_{t}}\right)}$  P/EBIT—price to earnings before interest and taxes. Other designations—as previously. 
P/EBITDA  $\frac{{P}_{S,t}}{\left(\frac{EBITD{A}_{t}}{{N}_{t}}\right)}$  P/EBITDA—price to earnings before interests, taxes, depreciation, and amortization. $EBITD{A}_{t}$—earnings before interests, taxes, depreciation, and amortization in period t. Other designations—as previously. 
P/CE  $\frac{{P}_{S,t}}{\left(\frac{N{P}_{t}+DE{P}_{t}}{{N}_{t}}\right)}$  P/CE—price to cash earnings. Other designations—as previously. 
P/FCFF  $\frac{{P}_{S,t}}{\left(\frac{FCF{F}_{t}}{{N}_{t}}\right)}$  P/FCFF—price to free cash flow to firm. $FCF{F}_{t}$—free cash flow to firm in period t. Other designations—as previously. 
EV/S  $\frac{E{V}_{t}}{{S}_{t}}$  EV/S—enterprise value to sale. EV_{t}—enterprise value at end of period t. Other designations—as previously. 
EV/CFO  $\frac{E{V}_{t}}{CF{O}_{t}}$  EV/CFO—enterprise value to cash flows from operating activities. Other designations—as previously. 
EV/FCFF  $\frac{E{V}_{t}}{FCF{F}_{t}}$  EV/FCFF—enterprise value to free cash flow to firm. Other designations—as previously. 
EY  $\frac{EP{S}_{t}}{{P}_{S,t}}\xb7100\%$  EY—earnings yield. EPS_{t}—earnings per share in period t. Other designations—as previously. 
DY  $\frac{DP{S}_{t}}{{P}_{S,t}}\xb7100\%$  DY—dividend yield. $DP{S}_{t}$—dividend per share in period t. Other designations—as previously. 
DPR  $\frac{DP{S}_{t}}{EP{S}_{t}}\xb7100\%$  DPR—dividend payout ratio. Other designations—as previously. 
RMVA  $\frac{MV{A}_{t}}{I{C}_{t1}}\xb7100\%$  RMVA—relative MVA, MVA index. $MV{A}_{t}$—market value added at end of period t. Other designations—as previously. 
$\frac{MV{A}_{E,t}}{B{V}_{t1}}\xb7100\%$ 
Measures—Diagnostic Variables  “Bankrupts”  “NonBankrupts”  All Companies  

Min  Me  Max  Min  Me  Max  Min  Me  Max  
NOPATPS  −1.564  0.064  4.866  −5.619  0.096  5.702  −5.619  0.096  5.702 
ROS  −0.711  0.012  0.134  −0.218  0.031  0.289  −0.711  0.030  0.289 
FCFEPS  −41.064  0.291  51.366  −8.204  0.168  8.877  −41.064  0.168  51.366 
CFROS  −0.087  −0.016  0.152  −0.472  0.052  0.547  −0.472  0.049  0.547 
CEC  −3.665  −0.849  −0.251  −6.011  −0.880  1.102  −6.011  −0.880  1.102 
REVA  −0.542  −0.144  0.298  −0.569  −0.079  0.046  −0.569  −0.084  0.298 
RMVA  −1.842  0.353  8.988  −0.778  −0.102  4.359  −1.842  −0.102  8.988 
P/BV  0.162  1.010  13.082  0.283  0.889  5.070  0.162  0.889  13.082 
P/BVGr.  −9.889  −0.153  121.803  −33.101  1.930  37.123  −33.101  1.930  121.803 
P/E  −8.286  23.248  508.620  −51.384  10.515  74.720  −51.384  10.515  508.620 
P/EBIT  −43.262  6.002  78.995  −33.523  7.643  50.573  −43.262  7.643  78.995 
P/EBITDA  −58.570  4.672  79.349  −19.485  5.572  30.672  −58.570  5.499  79.349 
P/CE  −11.687  8.607  482.909  −24.217  7.094  38.242  −24.217  7.094  482.909 
P/FCFF  −855.401  0.334  665.674  −121.864  −2.301  123.833  −855.401  −2.301  665.674 
EY  −0.241  0.012  0.161  −0.383  0.065  0.182  −0.383  0.063  0.182 
DY  0.000  0.000  0.000  0.000  0.000  0.187  0.000  0.000  0.187 
DPR  0.000  0.000  0.000  −2.359  0.000  2.831  −2.359  0.000  2.831 
EV/S  0.320  0.991  9.870  −0.019  0.651  3.949  −0.019  0.651  9.870 
EV/CFO  −63.911  −2.226  214.365  −64.101  8.137  83.642  −64.101  8.137  214.365 
EV/FCFF  −860.017  2.975  1032.467  −154.770  −2.991  158.711  −860.017  −2.955  1032.467 
Measures—Diagnostic Variables  Coefficient of Variation (in %) 

NOPATPS  10,512.3 
ROS  553.6 
FCFEPS  761.0 
CFROS  207.6 
CEC  111.3 
REVA  122.3 
RMVA  444.9 
P/BV  125.0 
P/BVGr.  575.6 
P/E  325.2 
P/EBIT  187.1 
P/EBITDA  223.9 
P/CE  422.6 
P/FCFF  2570.8 
EY  382.8 
DY  160.7 
DPR  210.4 
EV/S  131.0 
EV/CFO  242.9 
EV/FCFF  3294.2 
Measures—Diagnostic Variables  “Bankrupts”  “NonBankrupts”  All Companies  

KS Statistics Value  p Value  KS Statistics Value  p Value  KS Statistics Value  p Value  
NOPATPS  0.35781  <0.10  0.19965  <0.01  0.19242  <0.01 
ROS  0.32387  <0.20  0.20343  <0.01  0.23437  <0.01 
FCFEPS  0.33262  <0.15  0.20358  <0.01  0.26674  <0.01 
CFROS  0.17900  >0.20  0.14068  <0.05  0.12845  <0.05 
CEC  0.23331  >0.20  0.16344  <0.01  0.16772  <0.01 
REVA  0.27069  >0.20  0.23605  <0.01  0.22309  <0.01 
RMVA  0.37894  <0.10  0.18588  <0.01  0.22479  <0.01 
P/BV  0.37226  <0.10  0.17600  <0.01  0.25141  <0.01 
P/BVGr.  0.35603  <0.10  0.18091  <0.01  0.26300  <0.01 
P/E  0.36769  <0.10  0.20231  <0.01  0.32243  <0.01 
P/EBIT  0.30432  >0.20  0.19916  <0.01  0.19368  <0.01 
P/EBITDA  0.28974  >0.20  0.21415  <0.01  0.24075  <0.01 
P/CE  0.40014  <0.05  0.21655  <0.01  0.40154  <0.01 
P/FCFF  0.29969  >0.20  0.19152  <0.01  0.30086  <0.01 
EY  0.30131  >0.20  0.22113  <0.01  0.21736  <0.01 
DY      0.26583  <0.01  0.29075  <0.01 
DPR      0.29365  <0.01  0.30333  <0.01 
EV/S  0.37114  <0.10  0.23871  <0.01  0.28445  <0.01 
EV/CFO  0.34917  <0.15  0.23092  <0.01  0.25595  <0.01 
EV/FCFF  0.22869  >0.20  0.18614  <0.01  0.31088  <0.01 
Discriminant function—Variant 1  ${Z}_{1}=1.917\xb7{x}_{1}1.084\xb7{x}_{2}0.510\xb7{x}_{3}0.519\xb7{x}_{4}+1.179\xb7{x}_{5}0.839\xb7{x}_{6}0.494\xb7{x}_{7}0.346\xb7{x}_{8}$  
where:  
x_{1}—P/E; x_{2}—P/EBIT; x_{3}—EV/CFO; x_{4}—DPR; x_{5}—NOPATPS; x_{6}—CEC; x_{7}—EV/S; x_{8}—CFROS  
Predictive Efficiency  Eigenvalue  Canonical R  Wilks’ lambda  χ^{2}  p  Means of Canonical Variables  
First Type (%)  Second Type (w%)  Overall (%)  Bankrupts  Nonbankrupts  
75  100  98.0769  1.050  0.716  0.488  70.350  0.0000  3.515  0.293 
Discriminant function—Variant 2  ${Z}_{2}=5.269\xb7{x}_{1}1.434\xb7{x}_{2}+0.944\xb7{x}_{3}0.423\xb7{x}_{4}+3.377\xb7{x}_{5}+0.484\xb7{x}_{6}1.130\xb7{x}_{7}+2.012\xb7{x}_{8}+1.766\xb7{x}_{9}$  
$1.097\xb7{x}_{10}5.052\xb7{x}_{11}1.301\xb7{x}_{12}0.286\xb7{x}_{13}$  
where:  
x_{1}—P/CE; x_{2}—P/EBITDA; x_{3}—FCFEPS; x_{4}—P/BVGr.; x_{5}—RMVA; x_{6}—NOPATPS; x_{7}—ROS; x_{8}—P/FCFF;  
x_{9}—EY; x_{10}—EV/FCFF; x_{11}—P/BV; x_{12}—REVA; x_{13}—DY  
Predictive Efficiency  Eigenvalue  Canonical R  Wilks’ lambda  χ^{2}  p  Means of Canonical Variables  
First Type (%)  Second Type (w%)  Overall (%)  Bankrupts  Nonbankrupts  
88.8889  100  99.1453  2.586  0.849  0.279  138.552  0.0000  5.523  −0.460 
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Jaki, A.; Ćwięk, W. Bankruptcy Prediction Models Based on Value Measures. J. Risk Financial Manag. 2021, 14, 6. https://doi.org/10.3390/jrfm14010006
Jaki A, Ćwięk W. Bankruptcy Prediction Models Based on Value Measures. Journal of Risk and Financial Management. 2021; 14(1):6. https://doi.org/10.3390/jrfm14010006
Chicago/Turabian StyleJaki, Andrzej, and Wojciech Ćwięk. 2021. "Bankruptcy Prediction Models Based on Value Measures" Journal of Risk and Financial Management 14, no. 1: 6. https://doi.org/10.3390/jrfm14010006