Multi-criteria ranking of corporate distress prediction models: empirical evaluation and methodological contributions

Abstract

Although many modelling and prediction frameworks for corporate bankruptcy and distress have been proposed, the relative performance evaluation of prediction models is criticised due to the assessment exercise using a single measure of one criterion at a time, which leads to reporting conflicting results. Mousavi et al. (Int Rev Financ Anal 42:64–75, 2015) proposed an orientation-free super-efficiency DEA-based framework to overcome this methodological issue. However, within a super-efficiency DEA framework, the reference benchmark changes from one prediction model evaluation to another, which in some contexts might be viewed as “unfair” benchmarking. In this paper, we overcome this issue by proposing a slacks-based context-dependent DEA (SBM-CDEA) framework to evaluate competing distress prediction models. In addition, we propose a hybrid cross-benchmarking-cross-efficiency framework as an alternative methodology for ranking DMUs that are heterogeneous. Furthermore, using data on UK firms listed on London Stock Exchange, we perform a comprehensive comparative analysis of the most popular corporate distress prediction models; namely, statistical models, under both mono criterion and multiple criteria frameworks considering several performance measures. Also, we propose new statistical models using macroeconomic indicators as drivers of distress.

Appendix: Statistical models of corporate distress prediction

Framework	Model	Explanation
Multiple discriminant analysis (MDA)	Altman (1968) $Z = 1.2 WCTA + 1.4 RETA + 3.3 EBITTA + 0.6 MVTL + 0.999 STA$ WCTA: Working capital/Total Assets; RETA: Retained Earnings/Total Assets; EBITTA: Earnings before interest and taxes/Total assets; METL: Market value of equity/Total Liabilities; STA: Sales/Total assets	Assuming there are $n$ groups, the generic form of DA model for group $k$ is: $z_{k} = f\left( {\mathop \sum \limits_{j = 1}^{p} \beta_{kj} x_{j} } \right)$ where $x_{j}$ is discriminant feature $j$, $\beta_{kj}$ is the discriminant coefficient of feature $j$ in group $k$, $z_{k}$ represents the score of group $k$, and $f$ is the linear or non-linear classifier that maps the scores, say $\beta^{t} x$, onto a set of real numbers. To compare DA models to other statistical models, we need to estimate the probability of failure, which is used as an input for estimating many measures of performance. For this, we follow Hillegeist et al. (2004) in using a logit link to calculate the probability of failure for companies: $P\left( {distress} \right)_{i} = \frac{{e^{z} }}{{1 + e^{z} }}$
Multiple discriminant analysis (MDA)	Altman (1983) $Z = 0.717 WCTA + 0.847 RETA + 3.107 EBITTA + 0.42 BVTL + 0.998 STA$ WCTA: Working capital/Total Assets; RETA: Retained Earnings/Total Assets; EBITTA: Earnings before interest and taxes/Total assets; BVETL: Book value of equity/Total Liabilities; STA: Sales/Total assets
Multiple discriminant analysis (MDA)	Lis (1972) $Z = 0.063 WCTA + 0.092 RETA + 0.057 EBITTA + 0.0014 NWTL$ $WCTA$: Working capital/Total assets; $EBITIA$: Earnings before interest and taxes/Total assets; $METL$: Market value of equity/Total liabilities; $NWTA$: Net wealth/Total assets
Multiple discriminant analysis (MDA)	Taffler (1984) $Z = 3.2 + 2.5 CATL + 12.18 PBTCL + 0.029 NCI - 10.68 CLTA$ $CLTA$: Current liabilities/Total assets; $PBTCL$: Profit before tax/Current liabilities; $NCI$: Number of credit intervals as (quick assets − current liabilities)/((sales − PBT − depreciation)/365); $CATL$: Current assets/Total liabilities
Linear probability model (LPA)	Theodossiou (1991) $Z = - 0.075 + 0.51 WCTA - 0.21 TDTA + 0.449 NITA + 0.663 RETA {-} 0.446 LTDTA$ $WCTA$: Working capital/Total assets; $TDTA$ = Total debt/Total assets; $NITA$: Net income/Total assets; $RETA$ = Retained earnings/Total assets; $LTDTA$ = Long term debt/Total assets	The generic linear probability model (LPA) is a particular case of OLS regression and results in an estimate of probability of distress, the formula for which is as follows; $P\left( {distress} \right)_{i} = \beta_{\text{o}} + \mathop \sum \limits_{j = 1}^{p} \beta_{j} x_{ij}$
Logit analysis (LA)	Ohlson (1980) $\log \left[ {\frac{Pi}{1 - Pi}} \right] = - 1.32 - 1.43 WCTA + 6.03TLTA - 2.37NITA{-}0.407OSIZE{-}1.83FUTL + 0.0757CLCA + 0.285\,INTWO{-}1.72OENEG{-}0.521CHIN$ $WCTA$: Working capital/Total assets; $TLTA$: Total liabilities/Total assets; $NITA$: Net income/Total assets; $OSIZE$ = log (Total assets/GNP price-level index); $FUTL$: Funds from operations (operating income minus depreciation)/Total liabilities; $CLCA$: Current liabilities/Current assets; $INTWO$ = 1 if net income has been negative for the last 2 years, 0 otherwise; $OENEG$ = 0 if total liabilities exceed total assets, 1 otherwise; $CHIN = \left( {NI_{t} - NI_{t - 1} } \right)/\left( {\left| {NI_{t} } \right| + \left| {NI_{t - 1} } \right|} \right)$, where $NI_{t}$ is the net income for the last period. The variable is thus a proxy for the relative change in net income	The generic model for binary variables could be stated as follows: $\left\{ {\begin{array}{*{20}c} {P\left( {distress} \right)_{i} = P\left( {Y = 1} \right)} \\ {P\left( {distress} \right)_{i} = G\left( {\beta ,X} \right) } \\ \end{array} } \right.$ where $Y$ denotes the binary response variable, $X$ denotes the vector of features, $\beta$ denotes the vector of coefficients of $X$ in the model, and $G\left( . \right)$ is a link function that maps a score $\beta^{t} x$ onto a probability. In practice, depending on the choice of link function, the type of probability model is determined. For example, the logit model (respectively, probit model) assumes that the link function is the cumulative logistic distribution, say $\varTheta$ (respectively, cumulative standard normal distribution, say $N$) function
Probit analysis (PA)	Zmijewski (1984) ${\text{log }}\left[ {{\text{Pt}}/\left( {1 - {\text{Pt}}} \right)} \right] = 4.336 - 5.769 TLTA + 4.513 NITA {-} 0.004 CACL$ $NITA$: Net income/Total assets; $TLTA$: total liabilities/Total assets; $CACL$: Current assets/Current liabilities
Contingent claim analysis (CAA): Black–Scholes-Merton (BSM) Based Models	Hillegeist et al. (2004), Bharath and Shumway (2008) $P\left( {distress_{i} } \right) = N\left( { - \frac{{\ln \left( {\frac{{V_{a} }}{L}} \right) + \left( {\mu - \delta - 0.5\sigma_{a}^{2} } \right) \times T}}{{\sigma_{a} \sqrt T }}} \right)$ $N\left( . \right): {\text{the cumulative normal distribution function}}$,$V_{a}$:the value of the company’s assets; $L:$ total liabilities; $\mu :$ the expected return of the firm; $\sigma_{a}$: volatility of the company’s asset; $\delta$ is the divided rate; which is estimated by the ratio of dividends to the sum of $L$ and $V_{e}$ (market value of common equity); $T$ is time to maturity for both of call option and liabilities	The probability of failure is extracted as the probability that call option expires worthless at the end of maturity data—i.e. the value of the company’s assets ($Va$) be less than the face value of its debt liabilities ($L$) at the end of the holding period [$P(Va < L)$] In Hillegeist et al. (2004), $V_{a}$ and $\sigma_{a}$ are estimated by solving a system of equations; i.e. the call option Eq. (1) and the optimal hedge Eq. (2). $\left\{ {\begin{array}{*{20}l} {V_{e} = V_{a} e^{ - \delta T} N\left( {d_{1} } \right) - Le^{ - rT} N\left( {d_{2} } \right) + \left( {1 - e^{\delta T} } \right)N\left( {d_{1} } \right)V_{a} } \hfill & {(1)} \hfill \\ {\sigma_{e} = \frac{{V_{a} e^{ - \delta T} N\left( {d_{1} } \right)\sigma_{a} }}{{V_{e} }}} \hfill & {(2)} \hfill \\ \end{array} } \right.$ where $V_{\text{e}}$ is the market value of common equity at the time of estimation, $\sigma_{\text{e}}$ is the annualized standard deviation of daily stock returns over 12 months prior to estimation, $r$ is the risk-free interest rate, and $d_{1}$ and $d_{2}$ are calculated as follows; $d_{1} = \frac{{\ln \left( {\frac{{V_{a} }}{L}} \right) + \left( {r - \delta - \frac{1}{2}.\sigma_{e}^{2} } \right) \times T}}{{\sigma_{e} \sqrt T }}$; $d_{2} = d_{1} - \sigma_{e} \sqrt T$ where $V_{a,t}$ is the value of the company’s assets in year $t$ and $V_{a,t - 1}$ is the value of the company’s assets in year $t - 1$ Bharath and Shumway (2008) proposed a naïve approach to estimate $V_{a}$ and $\sigma_{a}$ as follows; $V_{a} = V_{e} + D ;\sigma = \frac{{V_{e} }}{{V_{a} }}\sigma_{e} + \frac{D}{{V_{a} }}\sigma_{d}$ where $\sigma_{d} = 0.05 + 0.25\sigma_{e}$. Further, the firm’s expected return $\mu$ is peroxided by the risk-free rate, $r$ or the stock return of previous year restricted to be between $r$ and 100%
Contingent claim analysis (CAA): Down-and-Out Call (DOC) Barrier Option Model	Jackson and Wood (2013) $P\left( {distress} \right)_{i} = N\left[ {\frac{{\ln \left( {\frac{L}{{V_{a} }}} \right) - \left( {\mu - \frac{1}{2}\sigma_{e}^{2} } \right)T}}{{\sigma_{e} \sqrt T }}} \right] + \left( {\frac{L}{{V_{a} }}} \right)^{{\frac{2\left( \mu \right)}{{\sigma_{e}^{2} }} - 1 }} N\left[ {\frac{{\ln \left( {\frac{L}{{V_{a} }}} \right) - \left( {\mu - \frac{1}{2}\sigma_{e}^{2} } \right)T}}{{\sigma_{e} \sqrt T }}} \right]$	A naïve DOC barrier option as an extension of BSM model, which assumes that debt holder’s position in the firm is like holding a portfolio of risk-free debt and a DOC option with a strike price (or Barrier) equal to total liabilities (L). The model rests on the assumptions of no dividends, zero rebate, costless failure proceedings, and set return on asset equal to the risk-free rate
Discrete time hazard model (Duration dependent hazard model)	Shumway (2001) $log \left[ {p_{i,t} /\left( {1 - P_{i,t} } \right)} \right] = - 13.303 - 1.983 NITA + 3.593 TLTA - 0.467 R.size - 1.809 LAGEXRET + 5.791 SIGMA$ $NITA$: Net income/Total assets; $TLTA$: Total liabilities/Total assets; $RSIZE$: Relative size; $LAGEXRET$: Lag of excess return ($r_{it - 1} - r_{mt - 1} )$	Shumway proposed a discrete time hazard model using an estimation procedure like the one used for estimating the parameters of a multi-period logit model. $P\left( {y_{i,t} = 1 |x_{i,t} } \right) = h\left( {t |x_{i,t} } \right) = \frac{{{ \exp }^{{\alpha \left( t \right) + X_{i,t} \beta }} }}{{1 + { \exp }^{{\alpha \left( t \right) + X_{i,t} \beta }} }}$ where $h\left( {t |x_{i,t} } \right)$ represent the individual hazard rate of firm $i$ at time $t$, $X_{i,t}$ is the vector of covariates of each firm $i$ at time $t$ Shumway employed a constant time invariant term, say $ln \left( {age} \right),$ as a proxy of the baseline rate
Duration-independent hazard model	$\begin{aligned} & h\left( {t |x_{i,t} } \right) = h_{0} . e^{{x_{i,t} .\beta }} \\ & p\left( {y_{i,t} = 1} \right) = \frac{1}{{1 + e^{{ - x_{i,t} .\beta }} }} \\ \end{aligned}$	where, $\alpha_{t}$ is the time-varying baseline hazard function related, which could be relate to firm, e.g. ln(age) or related to macroeconomic variables, e.g. foreign exchange rate
Duration-dependent hazard model	$\begin{aligned} & h\left( {t |x_{i,t} } \right) = h_{0} \left( t \right) \cdot e^{{x_{i,t} .\beta }} \\ & p\left( {y_{i,t} = 1} \right) = \frac{1}{{1 + e^{{ - \left( {\alpha_{t} + x_{i,t} .\beta } \right)}} }} \\ \end{aligned}$
Cox hazard framework	$PL \left( \beta \right) = \mathop \prod \limits_{i = 1}^{m} \left[ {\frac{{exp\left( {\mathop \sum \nolimits_{j = 1}^{p} \beta_{j} x_{j}^{i} \left( t \right)} \right)}}{{\mathop \sum \nolimits_{{k \in R_{t} \left( t \right)}} exp\left( {\mathop \sum \nolimits_{j = 1}^{p} \beta_{j} x_{j}^{k} \left( t \right)} \right)}}} \right]$ where $i$ is the firm in distress, $k$ is the firm in the risk set at time $t$, and $p$ is the number of features	A partial likelihood function on the training sample is used to estimate the coefficients $\beta$. This equation estimates $\beta$ without considering the baseline hazard rate (Hosmer and Lemeshow 1999). However, to use the developed model for estimation of distress probabilities, the baseline hazard rate is required. We follow Chen et al. (2005) in estimating the integrated baseline hazard function with time-varying covariates based on Andersen (1992) as follow: $\hat{H}_{0} \left( t \right) = \mathop \sum \limits_{{\hat{T}_{i} \le t}} \frac{{D_{i} }}{{\mathop \sum \nolimits_{{j \in \left( {\hat{T}_{i} } \right)}} { \exp }\left( {\hat{B}^{{\prime }} \cdot x_{j} \left( {\hat{T}_{i} } \right)} \right)}}$ where $D_{i}$ is a dummy variable representing whether firm $i$ faces distress or not, i.e. $D_{i} = 0$ for non-distress and $D_{i} = 1$ for distress; $\hat{T}_{i}$ is the distress time for the $i$ th firm; $\hat{\beta }$ is the vector of estimated coefficients; and $\hat{T}_{i}$ is the distress time for the $i$ th firm. Using Eqs. (4-29) and (4-30), we estimate the probability of distress for individual firms in Equation