Bank insolvency risk and Z-score measures: caveats ...

Bank insolvency risk and Z-score measures: caveats and best practice Vincent Bouvatier Laetitia Lepetity Pierre Nicolas Rehaultz Frank Strobelx January 2, 2017

Abstract We highlight caveats arising in the application of traditional Z-scores, develop alternative Z-score measures to resolve these issues, and then proceed to recommendations for best practice. Using a probabilistic approach, the best Z-scores are our novel exponentially weighted regulatory capital Z-score, and the exponentially weighted ROA-based Z-score using current values of the capital-asset ratio. In the overall preferred multivariate-model approach, the best Z-score to use is the ROA-based Z-score using current values of the capital-asset ratio, calculated either with moving or exponentially weighted moments. We also construct an "augmented" Z-score formula that can be used in out-of-sample applications. Keywords: Z-score; banking; insolvency risk

Université de Paris Ouest - Nanterre La Défense, EconomiX - CNRS, Nanterre, France. E-mail: [email protected] y Université de Limoges, LAPE, Limoges, France. E-mail: [email protected] z Université de Limoges, LAPE, Limoges, France. E-mail: [email protected] x University of Birmingham, Department of Economics, Birmingham, UK. E-mail: [email protected] (corresponding author)

1 Electronic copy available at: https://ssrn.com/abstract=2892672

1

Introduction

Given the recent …nancial crisis, analysis of the general importance, impact and measurement of bank insolvency risk has regained momentum. Banking supervisors are generally able to compute re…ned measures of default risk using a wide range of information, fully exploited through statistical techniques (discriminant analysis, logistic regression, factor analysis) and/or intelligent techniques (neural networks).1 However, these re…ned measures, or even the information used to compute them, are often not publicly available and cannot therefore be readily used by stakeholders other than supervisors. As a consequence, alternative measures of default risk which can be computed by any stakeholder have been developed; these are generally based on accounting information, market information or a combination of the two. When market data is available, market-based risk measures can be of interest as they can allow for forward-looking information re‡ected in security prices. The distance-to-default, a marketbased measure using a stylized structural model based on Merton (1974), has become a popular indicator of default risk in this context. More directly, bond yield spreads and credit default swaps can also be used to proxy insolvency risk. On the other hand, a widely used insolvency risk measure in the banking and …nancial stability literature is the Z-score; using accounting data, it essentially compares bu¤ers (capitalization and returns) with the potential for risk (volatility of returns).2 In a related recent contribution, Tsionas (2016) develops a structural model where the probability of default can also be estimated using only accounting information, based on expected utility maximization for a bank facing uncertain loan prices. Among these varying measures, the Z-score is the most widely used in the empirical banking literature to re‡ect a bank’s probability of insolvency; it is, e.g., one of the indicators used by the World Bank in their Global Financial Development Database to measure …nancial institutions’ soundness.3 In its general form, the Z-score is originally attributed to Hannan and Hanweck (1988) and Boyd et al. (1993), and was initially only used for cross-sectional studies; Lepetit and Strobel (2015) give a re…ned probabilistic interpretation of this insolvency risk measure. Starting with work by Boyd et al. (2006), the Z-score is now also commonly being implemented as a time-varying measure in panel studies; Lepetit and Strobel (2013) discuss the di¤erent approaches used in the construction of time-varying Z-scores in the literature so far. Z-scores’ widespread use can be 1

See e.g. Kumar and Ravi (2007) and Demyanyk and Hasan (2010) for more details. This methodology is di¤erent from the Altman (1968) Z-score measure common in the corporate …nance literature; see Altman (2002, ch. 1) for a discussion. It is also distinct from non-statistical Z-score models such as the DICO Z-score, which is used by the Deposit Insurance Corporation of Ontario (DICO) to rank the performance of credit unions in Ontario (see Pille and Paradi (2002)). 3 For some recent papers using this methodology, see e.g. Tsionas and Mamatzakis (2017); Berger et al. (2016), Fukuyama and Matousek (2016), Han et al. (2016); Vazquez and Federico (2015), Hakenes et al. (2015); Berger et al. (2014), Delis et al. (2014), Fang et al. (2014), Fu et al. (2014); Beck et al. (2013), Bertay et al. (2013), DeYoung and Torna (2013). 2

2 Electronic copy available at: https://ssrn.com/abstract=2892672

explained by their relative simplicity and the fact that they require only accounting information for their computation; as such, they can equally be applied to the large number of unlisted …nancial institutions. However, both their simplicity and reliance on accounting data result in potential challenges in their application; our paper highlights these caveats, develops alternative Z-score measures that resolve some of these issues, and then proceeds to develop recommendations regarding best practice in the application of Z-score measures for the measurement of bank insolvency risk. The Z-scores widely used at present could be challenged along several dimensions; we consider improvements for each of these issues. First, the traditional Z-score is ROA-based and thus involves implicit scaling by total assets; this could introduce spurious variability, which can be avoided using an alternative ROE-based measure. Secondly, both ROA- and ROE-based Z-score measures could su¤er from measurement bias introduced through banks potentially engaging in income smoothing behavior; we derive an alternative Z-score measure that corrects for this. Thirdly, returns-based Z-score measures disregard the importance of regulatory capital constraints faced by banks; we take this aspect into account by constructing an alternative "regulatory capital" Z-score. Furthermore, the construction of time-varying Z-score measures introduces questions of appropriate estimation of relevant moments (see Lepetit and Strobel (2013)); we also consider exponentially weighted moments in this paper. Lastly, both the traditional and alternative Z-score measures relate to unconditional probabilities of distress. However, the link between Z-scores and probability of distress could be conditional on observable bank characteristics (e.g. bank size) and the macroeconomic environment (i.e. a reduction in capital might be more challenging for a bank during a …nancial crisis period than during normal times). We address this issue by constructing a multivariate model which allows for heterogeneity in the relationship between the Z-score and the probability of distress, leading to an improved assessment of the probability of distress of banks and the construction of an "augmented" Z-score formula for out-of-sample use. We then move on to comparing the performance of these di¤erent Z-scores to determine the best Z-score(s) to use in practice. We pursue comparisons drawing on a range of alternative testing procedures, …rst at a model-free level, where our Z-score measures can be related to predicted probabilities of distress using sound statistical foundations. We then progress to model-based approaches using two speci…cations for the econometric model: a univariate and a multivariate speci…cation. These two model-based approaches allow additional testing procedures to assess the probabilities of distress and determine the "best" Z-score measures to use in empirical analysis. We focus for our empirical analysis on the …nancial crisis of 2007-2008, using …nancial statement data for US commercial banks for the period 2006-2014. Our analysis using the model-free, probabilistic approach shows that Z-scores computed with exponentially weighted moments are better than those with moving moments. We …nd the two best Z-scores to use in this case to be the exponentially weighted regulatory capital Z-score and the 3

exponentially weighted ROA-based Z-score using current values of the capital-asset ratio. Perhaps surprisingly (but reassuringly), income-smoothing does not seem to pose a signi…cant problem requiring correction in this context. Moving on to the model-based approach, we then demonstrate that computing augmented Z-scores using a multivariate-model approach provides better probabilities of distress to discriminate between distressed and surviving banks than the widely used model-free approach. Clearly, the additional information included in the multivariate model approach is valuable in producing Z-score measures that outperform those of the model-free (or the univariate) approach. When considering the multivariate model approach, our results overall suggest that the best Z-score to use is the ROA-based Z-score using current values of the capital-asset ratio, calculated either with moving or exponentially weighted moments. Section 2 presents the traditional and alternative Z-score measures considered and links them to predicted probabilities of distress; Section 3 presents the data and tests the relative performance of the di¤erent Z-scores using the model-free approach; Section 4 uses parametric econometric models to assess model-based probabilities of distress using the di¤erent Z-score measures, and develops an augmented Z-score formula for out-of-sample use; …nally, Section 5 concludes the paper.

2

Z-score measures and probabilities of distress

We now characterize both the traditional and alternative Z-score measures to be examined later using both model-free and model-based approaches.

2.1

Traditional Z-score

Let us …rst restate the traditional justi…cation for using Z-scores as a risk measure re‡ecting a bank’s probability of insolvency. For these purposes, bank insolvency is commonly de…ned as a state where (EQ + ROA) 0, with EQ the bank’s capital-asset ratio and ROA its return on assets. Hannan and Hanweck (1988) and Boyd et al. (1993) showed that if ROA is a random variable with …nite 2 mean ROA and variance ROA , the Chebyshev inequality gives an upper bound of the probability of insolvency as a nonlinear function of the Z-score, which is de…ned as Z (EQ + ROA ) = ROA > 0. For our purposes here, we draw on the (re…ned) probabilistic interpretation of the Z-score given in Lepetit and Strobel (2015), which we restate for convenience as Result 1 (Proposition 1 in Lepetit and Strobel (2015)) If ROA is a random variable with 2 …nite mean ROA and variance ROA , an (improved) upper bound of the bank’s probability of insol1 2 vency P is given by P(ROA EQ) (1 + ZROA ) < 1, where the Z-score ZROA is de…ned as ZROA (EQ + ROA ) = ROA > 0.

4

Result 1 links the Z-score to a probability of bank distress, giving it a solidly founded (re…ned) probabilistic interpretation; nevertheless, it has several potential limitations, which lead us to develop and examine several alternative approaches to the construction of Z-score measures.

2.2

Alternative Z-scores

We now give the potential rationales for the use of our alternative Z-score measures, and derive the corresponding theoretical probability of distress measures. 2.2.1

ROE-based Z-score

Goyeau and Tarazi (1992) had …rst advanced a return on equity (ROE) based Z-score measure, in the special context of normal return distributions, which is de…ned as4 ZROE (1 + ROE ) = ROE . Lepetit and Strobel (2015) provide a (re…ned) probabilistic interpretation for such a measure that allows for non-normal return distributions. They further argue that an ROE-based Z-score measure could be preferable to the more commonly used ROA-based one as it is (by construction) una¤ected by potentially spurious variability in total assets; this could be particularly relevant in the context of time-varying Z-score measures. We restate their …nding relating this alternative Z-score measure to the (unconditional) probability of distress as Result 2 (Proposition 2 in Lepetit and Strobel (2015)) If ROE is a random variable with 2 …nite mean ROE and variance ROE , an upper bound of the bank’s probability of insolvency P is 1 2 ) < 1, where the (alternative) Z-score ZROE is de…ned as given by P(ROE 1) (1 + ZROE ZROE (1 + ROE ) = ROE > 0. 2.2.2

Income smoothing adjusted Z-score

A potential measurement bias could arise in the construction of ROA- and ROE-based Z-score measures through the fact that banks have incentives to engage in income smoothing behavior (see e.g. Collins et al. (1995), Beaver and Engel (1996), Liu et al. (1997) and Ahmed et al. (1999) for US banks; Anandarajan et al. (2003), Perez et al. (2008), Bouvatier and Lepetit (2012) and Bushman and Williams (2012) for non-US banks). When earnings are expected to be low, loan loss provisions could be deliberately understated; when earnings are unusually high, banks could choose discretionary income-reducing accruals. Thus, banks that engage in income smoothing behavior can reduce the variance of reported earnings, making Z-score measures potentially less reliable predictors of bank distress for them. We construct income smoothing adjusted Z-score measures to allow for this possibility, following the procedure outlined next. 4

This becomes ZROE (100 + ROE ) = ROE when returns are expressed in percent. It has e.g. been applied in Barry et al. (2011) and Bouvatier et al. (2014).

5

First, we consider the following speci…cation, based on Ahmed et al. (1999) and Bouvatier and Lepetit (2012), to identify income smoothing behavior LLPi;t = +

i

+ LLPi;t

4 ERBTi;t

+

1

+

1 N P Li;t

5 T ier1i;t 1

+

+

2

N P Li;t +

6 SIGNi;t

+

t

3 Li;t

(1)

+ "i;t

where LLPi;t is the ratio of loan loss provisions to total assets, N P Li;t the ratio of non-performing loans to total assets, N P Li;t the …rst di¤erence of N P Li;t ( N P Li;t = N P Li;t N P Li;t 1 ), Li;t the ratio of net loans to total assets. This …rst set of variables controls for the non-discretionary component of loan loss provisions. The three remaining variables are associated with the discretionary component of loan loss provisions: ERBTi;t is the ratio of earnings before taxes and loan loss provisions to total assets, T ier1i;t 1 the Tier1 capital ratio and SIGNi;t the one-year-ahead change of earnings before taxes and loan loss provisions (SIGNi;t = ERBTi;t+1 ERBTi;t ). These three variables capture income smoothing, capital management and signaling behavior, respectively (Ahmed et al. (1999)). In particular, the parameter 4 is expected to be positive if loan loss provisions are used to smooth income. This speci…cation is estimated with the Blundell and Bond (1998) estimator, as equation (1) corresponds to a dynamic panel speci…cation.5 This allows us to compute the levels of loan loss provisions adjusted for income smoothing behavior, as follows6 adj = LLPi;t LLPi;t

^4 ERBTi;t

where the parameter 4 from equation (1) captures the immediate/short term e¤ect of ERBTi;t on loan loss provisions.7 These loan loss provisions are used to compute the corresponding adjusted ROA and ROE measures AdjROAi;t = ERBTi;t

adj LLPi;t

;

AdjROEi;t = (ERBTi;t

adj LLPi;t )=EQi;t

with EQi;t the bank’s capital-asset ratio as above. These are then, as a …nal step, used to compute 5

We use the forward orthogonal deviations transformation of the original equation as suggested by Arellano and Bover (1995), and report the two-step estimator. We restrict at 6 the lag range used in generating the instruments and use the collapsed matrix of instruments (Roodman (2009)) to limit the number of instruments. The validity of estimates is checked with the AR(2) test and the Hansen test. 6 The variable ERBTi;t is demeaned to ensure that the variables LLPadj i;t and LLPi;t have the same average. 7 Alternatively, one could compute loan loss provisions adjusted for the long term (LT) e¤ect of income smoothing adj_LT behavior, as LLPi;t = LLPi;t ( ^4 =(1 ^))ERBTi;t , where ^4 =(1 ^) captures the full/long term e¤ect of ERBTi;t on loan loss provisions. Results in our later analysis are qualitatively unchanged using Z-scores adjusted for long term e¤ects of income smoothing instead of short term ones; further details are available on request.

6

income smoothing adjusted Z-scores, which allow for the (short term) e¤ects of income smoothing, as ZAdjROA and ZAdjROE . 2.2.3

Regulatory capital Z-score

In analogy to the de…nition of bank distress in the development of returns-based Z-score measures above, one could argue that a bank is in "regulatory" distress when a particular regulatory capital ratio RCAR drops below a given regulatory threshold T R, so that, e.g., its total capital ratio drops below 8% or its Common Equity Tier 1 ratio drops below 4.5% (under Basel III). In analogy to the probabilistic interpretation of the traditional Z-score (discussed above), the (one-sided) BienayméChebyshev inequality then allows us to relate a "regulatory capital" Z-score, de…ned as ZRCAP ( RCAR T R) = RCAR , to an upper bound of the (unconditional) probability of regulatory distress. This holds for the most general of distributional assumptions regarding the regulatory capital ratio RCAR, and can be stated as follows Proposition 1 If the regulatory capital ratio RCAR is a random variable with …nite mean RCAR > 2 T R and variance RCAR > 0, an upper bound of the bank’s probability of not satisfying the regulatory 1 2 threshold tr is given by P(RCAR T R) (1 + ZRCAP ) < 1, where the "regulatory capital" Zscore is de…ned as ZRCAP ( RCAR T R) = RCAR > 0. Proof. This is an application of the one-sided Chebyshev inequality (see Ross, 1997, p. 414, or previously, Feller, 1971, p. 152): it states that for a random variable X with …nite mean and 2 variance 2 , it holds for any a > 0 that P fX ag 2 +a2 . Our result then follows setting X = RCAR and a = RCAR T R > 0, and dividing both numerator and denominator of the right 1 2 hand side of the inequality by RCAR ; we observe that limZ!0 (1 + Z 2 ) = 1. As the regulatory capital Z-score is not constructed using returns data, it is, by de…nition, una¤ected by the potential measurement problem associated with bank income smoothing discussed previously.

2.3

Calculation of time-varying Z-scores

The time-varying implementation of the di¤erent Z-score measures introduced above can be carried out in di¤erent ways; common approaches to this are discussed in Lepetit and Strobel (2013). We …rst calculate "rolling" versions of the di¤erent Z-scores we consider. For the ROE-based r3 r3 and regulatory capital Z-scores, we calculate ZROE and ZRCAP using moving mean and standard deviation estimates for ROE and the total capital ratio (T CR), respectively, with window widths of

7

three observations.8 For the traditional, ROA-based Z-score, we calculate two alternative versions: r3 (i) ZROA , using moving mean and standard deviation estimates for both ROA and EQ, with window r3;c widths of three observations; and (ii) ZROA using moving mean and standard deviation estimates for ROA, with window widths of three observations, and combining these with current values of EQ. exp We additionally introduce a novel approach to the construction of time-varying Z-scores ZROA , exp;c exp exp ZROA , ZROE and ZRCAP , which use exponentially weighted moments instead of moving ones. Exponentially weighted moments have the advantage of always drawing on the full data history available, while weighting more recent observations more heavily. For each of our relevant variables, the exponentially weighted moving average EWMA, and exponentially weighted moving variance EWVAR, are implemented recursively as9 mean: EWMAt (x) = xt + (1 variance: EWVARt (x) = (1

)EWMAt

)(EWVARt

1

1

(x)

(x) + (xt

EWMAt

1

(x))2 )

where the recursion for the (naive) exponentially weighted variance follows Finch (2009) and West (1979). We determine the smoothing parameter optimally, carrying out a grid search over the range 0.01 to 0.99 with increments of 0.01. This produces 99 competing exponential Z-scores; amongst those, the one maximizing the AUROC curve is then chosen (see Section 3.2 for details of our testing procedure).

3

Evaluating Z-score measures under the model-free approach

We now carry out a …rst empirical evaluation of the di¤erent Z-score measures we consider, using a model-free/probabilistic approach. 8

Z-score measures using 3-year rolling windows are most commonly used in the literature. Alternatively, one might consider moving moments using 5-year windows, which however decreases the number of usable observations (in our case, the number of observations drops from 5823 to 5769, with a lower number of distressed banks of 489 instead of 543 before). Furthermore, we con…rmed that Z-score measures based on 3-year windows outperform the Z-scores using 5-year windows for all the criteria we use to compare their performance (see Section 3.2); detailed results from these tests are available on request. 9 Stata code implementing these exponentially weighted moments and associated Z-score measures is available from the corresponding author on request.

8

3.1

Data

In order to try and determine which one of the di¤erent Z-score measures introduced in Section 2 is best, we focus on the …nancial crisis of 2007-2008. Z-score measures should allow us to discriminate between banks that survived the …nancial crisis and banks that fell into distress during it. Using …nancial statement data for US commercial banks drawn from SNL Financial LC, we consider for our empirical investigation banks existing in 2006 and still operating in 2014, and banks existing in 2006 but falling into distress between 2007 and 2014. Thus, we exclude non-distressed banks existing in 2006 but disappearing between 2007 and 2014 due to mergers & acquisitions, and all banks created between 2007 and 2014. We classify a bank as distressed if: (i) it failed according to the FDIC; or (ii) it asked for Troubled Asset Relief Program (TARP) funds; or (iii) it was "weakly" capitalized and was subsequently acquired by a safer bank. We consider a bank as weakly capitalized if the tangible common equity/tangible assets ratio is lower than 2%, or the Tier 1 capital ratio is lower than 4%, or the total capital ratio is lower than 8%. When a bank is classi…ed as distressed during a given year, it is dropped from the sample in the following and all subsequent years. We use SNL Financial LC to identify banks which asked for TARP funds, or were weakly capitalized and subsequently acquired by another bank. Our initial sample consists of 5931 banks; amongst those, 545 fell into distress between 2007 and 2014 (i.e. 5386 survived). Dropping banks with a net loans to total assets ratio lower than 5%, we end up with a …nal sample of 5823 banks, of which 543 fell into distress between 2007 and 2014 (i.e. 5280 survived). Table 1 provides summary statistics for our sample. As expected, the unconditional probability of distress shows considerable ‡uctuation, and is naturally highest in 2008 and 2009 (at 3.30% and 2.36%, respectively). Table 2 presents summary statistics for the main Z-score measures computed for our samples of distressed and non-distressed banks. As expected, Z-score measures of distressed banks are signi…cantly lower than those of non-distressed banks.10 Table 3 further gives correlation matrices for the main Z-score measures, in both levels and logs. Returns-based Zscores are moderately correlated in levels; they are also moderately correlated with their equivalent income smoothing adjusted versions. However, the di¤erent returns-based Z-scores, whether income smoothing adjusted or not, are strongly correlated in logs. We observe that the regulatory capital Z-score is only weakly correlated with the total capital ratio (TCR), indicating that it is indeed relevant and interesting to consider our novel regulatory capital Z-score as a measure of default risk, rather than simply the TCR itself. 10

Mean tests are available on request.

9

3.2

Testing procedure

The di¤erent Z-scores presented in Section 2 are considered as rival binary classi…ers. Each classi…er provides predicted probabilities that banks become distressed.11 These predicted probabilities are then compared with the observed discrete outcomes of distressed banks, as de…ned in Section 3.1. A large set of criteria have been developed to compare formulas that predict the probabilities of having observed a positive outcome (Yi;t = 1) of a corresponding binary response (Yi;t = f0; 1g). We consider four complementary criteria to compare the performance of the di¤erent rival classi…ers; they can be split into two categories: (i) criteria that focus on the relative ranking of the predicted probabilities provided by each classi…er (AUROC curve, AUPR curve and H measure); and (ii) a criterion that takes into account the numerical values of the predicted probabilities (Tjur R2 ).12 The …rst category of criteria are de…ned and computed from the confusion matrix (Hand (2012)).13 We …rst consider the area under the receiver operating characteristic (AUROC) curve. The receiver operating characteristic (ROC) curve plots the true positive rate against the false positive rate at various threshold settings. Therefore, the area under the ROC curve, considering all possible thresholds, has the attractive property of providing a summary measure of classi…cation ability. In addition, the AUROC curve has an intuitive interpretation: it equals the concordance probability (also called the c index) that is equivalent to the probability that a randomly chosen positive outcome is ranked higher than a randomly chosen negative outcome. Realistic values for the AUROC curve range thus from 0.5 (random ranking) to 1 (perfect ranking). The AUROC curve is a widely used measure of classi…er performance, but can be a misleading indicator when positive outcomes are sparse in the dataset (i.e. for imbalanced datasets). This issue is related to the fact that the ROC curve reports the false positive rate on the x-axis, calculated as the ratio of false positives to overall negatives (i.e. false positives and true negatives). When positive outcomes are sparse, it becomes easier to predict negative outcomes and, consequently, true negatives are much more numerous than false positives (i.e. the false positive rate is generally weak). Therefore, if we care more about the positive outcomes, the AUROC curve can overstate the overall performance of the classi…er. The area under the precision-recall (AUPR) curve can be considered as a more appropriate evaluation metric in such situations (Saito and Rehmsmeier (2015)). The precision-recall (PR) curve plots precision against recall at various threshold settings, 11 In order to retain observations with negative Z-scores in the calculation of predicted probabilites, negative Zscores are rescaled to lie between zero and the minimum observed Z-score in the sample. 12 These criteria not based on an econometric approach are not commonly used in economics, but are more widely used in di¤erent scienti…c disciplines such as biostatistics and machine learning. 13 The confusion matrix reports the number of false positives (FP), false negatives (FN), true positives (TP), and true negatives (TN) for a given threshold. For instance, for a given threshold t 2 [0; 1], the number of positive outcomes with predicted probabilities higher than t gives the TP. Regarding the criteria computed from the confusion matrix, some simpler …t statistics like accuracy (i.e., (T P +T N )=(F P +T P +F N +T N )) require specifying a particular threshold, while some more complex measures provide an average performance covering all possible thresholds.

10

where precision is de…ned as the ratio of true positives to overall positives, and recall is a synonym for the true positive rate. As a consequence, the PR curve and the ROC curve both use the true positive rate, but the PR curve considers precision instead of the false positive rate. In other words, in contrast to the ROC curve, the PR curve does not consider the true negatives; this gives a more informative picture of the performance of the classi…er in an imbalanced dataset. However, the AUPR curve does not have an intuitive interpretation, unlike the AUROC curve. A more fundamental criticism has been expressed towards the AUROC curve by Hand (2009, 2010). Considering optimal thresholds, these papers demonstrate that optimising the AUROC curve is equivalent to minimising expected minimum loss (resulting from misclassi…cation costs), showing that the latter depends on the score densities, and hence on the classi…er. Therefore, the interpretation of the AUROC curve is model-speci…c, leading to the conclusion that the AUROC curve is not a coherent measure to compare rival classi…ers. Hand (2009) proposes the H measure, which is not classi…er-dependent, as a coherent alternative to the AUROC curve.14 However, Flach et al. (2011) put into perspective the argument developed by Hand (2009, 2010), contending that considering only optimal thresholds is over-optimistic from a practical perspective. When all scores that have been assigned to data points are considered as possible thresholds, the AUROC curve is not model-speci…c, and thus remains a coherent measure of classi…cation performance. The AUROC curve, AUPR curve and H measure focus only on the relative ranking of the predicted probabilities. However, the numerical values of the predicted probabilities can also be considered as meaningful. As a result, we use an additional criterion in order to also measure the accuracy of the predicted probabilities, the Tjur (2009) R2 , which is de…ned as the di¤erence between the mean of the predicted probabilities of positive outcomes and the mean of the predicted probabilities of negative outcomes. These four criteria are used to empirically compare the performance of the di¤erent Z-scores as classi…ers providing predicted probabilities that banks become distressed. A "best" Z-score would ideally present the highest values for the AUROC curve, the AUPR curve, the H measure and the Tjur R2 . These criteria assess di¤erent aspects of performance, hence it is unclear whether one should expect one of the alternative Z-score measures to dominate all others for the entire set of criteria. Rather, we aim to identify a hierarchy in which a subgroup of Z-score measures is dominated on average by other more e¢ cient ones.

3.3

"Best" Z-score measure

We …rst evaluate the performance of the di¤erent Z-scores when calculated using moving moments. r3;c r3 The two traditional, widely used ROA-based Z-scores, ZROA and ZROA , computed with either 14

Flach et al. (2011) show that the H measure is actually a variation of the area under the cost curve developed by Drummond and Holte (2006).

11

current values or moving mean of the capital-asset ratio, are compared with those of the ROEr3 r3 based Z-score ZROE , and our novel regulatory capital Z-score ZRCAP (see Table 4). As expected, none of these Z-scores dominates the others for all four criteria we use to compare their performance. r3 Our results seem to indicate that the ROE-based Z-score and the ROA-based Z-score ZROA (with moving mean of the capital-asset ratio) are dominated by the regulatory capital Z-score and the r3;c alternative ROA-based Z-score ZROA (with current values of the capital-asset ratio). The regulatory capital Z-score is ranked …rst for the AUROC curve, the H measure and the Tjur R2 , whereas the r3:c ROA-based Z-score ZROA is ranked …rst for the AUPR curve and in second position for the three other criteria. Our …ndings therefore indicate that when time-varying Z-scores are computed using 3-year rolling windows, the regulatory capital Z-score seems to dominate the returns-based Z-scores, and if choosing narrowly between the two traditional ROA-based Z-scores, the one computed with current values of the capital-asset ratio is strictly preferable. If ROA- and ROE-based Z-scores are potentially biased due to possible income smoothing behavior by banks, the superior performance of the regulatory capital Z-score might be related to its independence from earnings. We con…rm that banks do engage in income smoothing in our sample: estimation of equation (1) in Section 2.2.2 shows that the coe¢ cient associated with the ratio of earnings before taxes and loan loss provisions to total assets (ERBTi;t ) is signi…cant and positive (see Table A1 in the Appendix). In light of this result, we compare the performance of the three ROA- and ROE-based Z-scores with their equivalent versions adjusted for income smoothing. The results presented in Table 5 provide somewhat inconclusive results as to whether the income smoothing adjusted Z-scores dominate the non-adjusted ones. However, even if we were to argue for a need to more widely adjust for income smoothing behavior in the construction of Z-score measures, three important caveats should be noted. First, as the income smoothing component of loan loss provisions used to adjust earnings depends on an ad hoc econometric model, income smoothing adjusted Z-scores could possibly be a¤ected by measurement error. Secondly, any such estimation to determine income smoothing behavior may impose further data restrictions; in our case, e.g., the available number of banks for our comparisons drops from 5823 to 5816, with a lower number of distressed banks of 515 (compared to 543 before). Lastly, from a more practical perspective, Z-scores have been widely popular in the literature due to their simplicity and ease of calculation; adding steps in their construction requiring estimation would go against this spirit for many users. Finally, we investigate whether time-varying Z-scores computed using exponentially weighted moments perform better than those using simple moving moments. Results in Table 6 indicate that exponentially weighted Z-scores dominate moving moments Z-scores for all the di¤erent Z-scores we consider. Our results also demonstrate that the exponentially weighted ROA-based Z-score using the current values of the capital-asset ratio and the exponentially weighted regulatory capital Z-score each perform well for certain criteria. 12

Overall, our analysis shows that it is better to use Z-scores computed with exponentially weighted moments rather than those with moving moments. The two best Z-scores to use are the exponentially weighted regulatory capital Z-score and the exponentially weighted ROA-based Z-score using current values of the capital-asset ratio . If simple moving moments are applied, it is better to use the regulatory capital Z-score than returns-based Z-scores. Note that whereas our analysis allows the comparison of these di¤erent Z-scores measures, carried out in the form of the predicted probabilities that banks become distressed they imply, it does not o¤er guidance for what particular transformation of those Z-score measures is the most appropriate to use in applied empirical work. As Lepetit and Strobel (2015) emphasize, log-transformed Z-scores may be more appropriate in applied work due to the skewness of Z-scores in levels; the log of the Z-score can additionally be shown to be negatively proportional to the log odds of insolvency, giving it a sound probabilistic foundation. A practical inconvenience of log-transformed Z-scores is, however, the fact that calculated Z-scores can be negative by construction. This could be resolved either through appropriate rescaling of negative Z-scores into the positive domain (as detailed in footnote 11), or by resorting to alternative transformations such as the log-modulus transformation, which we apply in the model-based approach of Section 4. Next, we examine whether the probabilistic approach that links the Z-score to the probability of bank distress discussed in Section 2 can be outperformed by an alternative approach based on a parametric econometric model.

4 4.1

A model-based approach for an augmented Z-score Econometric model speci…cation

We now use a parametric econometric model to assess model-based probabilities of distress. More precisely, we consider two speci…cations for the econometric model: a univariate speci…cation and a multivariate speci…cation. These two model-based approaches allow additional procedures to assess the probabilities of distress. 4.1.1

Univariate-model approach

We consider a logit model to link the Z-score to the probability of distress. In the univariate-model approach, by de…nition, we do not include any control variables in the speci…cation. The baseline model for the univariate-model approach corresponds to a pooled logit speci…cation P Yi;t = 1 p ZK i;t = P Yi;t = 0 p ZK i;t = 1 13

ZK i;t ZK i;t

(2)

where the subscripts refer to bank i (i = 1; :::; N ) in period t (t = 1; :::; T ). The variable Yi;t is a binary variable that is equal to 1 if the bank falls into distress and 0 otherwise, ZK i;t is a Z-score measure of type K, is a parameter estimate and (:) is the cumulative density function of the logistic distribution given by ZK i;t =

1 1 + exp

ZK i;t

c

(3)

where c is the intercept. The estimated model allows us to compute the probabilities of distress for the univariate-model approach denoted P ZK i;t . The Z-score measure (ZK i;t ) used here is the log-modulus transformation (John and Draper (1980)) K of the Z-score variable Zi;t de…ned as K K ZK i;t = sign Zi;t : ln Zi;t + 1 K K is negative and 1 otherwise. The log-modulus is equal to -1 if the variable Zi;t where sign Zi;t transformation allows us to both handle the skewedness of the distribution of the Z-score and the potential negative values taken by the Z-score when returns are negative and equity turns negative. The latter aspect may be particularly relevant in studies like ours dealing speci…cally with bank distress, as it would lead to exclusion of such observations, or require some ad-hoc rescaling, when applying a simple log-transformation.15 The univariate-model approach and the model-free approach use the same information. The main di¤erence between these two approaches is the link function that computes the probability of distress from the Z-score.16 In other words, the functional forms are di¤erent and thus the probabilities of distress are numerically di¤erent, but the probabilities of distress are ordinally equivalent. Consequently, for a given Z-score, the relative ranking of the predicted probabilities obtained with these two approaches are the same. Similarly, the criteria that only depend on relative ranking of the predicted probabilities (i.e. AUROC curve, AUPR curve and H statistic) are the same for these two approaches. Therefore, one can consider that the univariate-model approach and the model-free approach are, to a certain extent, equivalent. However, the univariate-model approach will be valuable in the context of comparison because this approach will allow us to consider additional testing criteria, i.e. some model-based criteria (see Section 4.2). 15

Using a log transformation, the number of distressed banks considered decreases dramatically if we exclude negative Z-scores (e.g. 413 compared to 543 for the regulatory Z-score). We did, however, ascertain that we obtain similar results when using log-transformed Z-scores instead of applying the log-modulus transformation; details are available on request. 16 Note that the treatment of potential negative Z-scores also di¤ers in the two approaches.

14

4.1.2

Multivariate-model approach

The multivariate-model approach includes selected control variables in the speci…cation, with the objective to take into account additional information in computing the probability of distress. The baseline model for the multivariate-model approach also corresponds to a pooled logit speci…cation given by ZK P Yi;t = 1 p ZK i;t ; Xi;t i;t ; Xi;t = (4) K P Yi;t = 0 p Zi;t ; Xi;t = 1 ZK i;t ; Zi;t where Xi;t is a set of control variables and density function (:) is given by ZK i;t ; Zi;t

=

is a vector of parameter estimates. The cumulative

1 + exp

1 ZK i;t

Xi;t

c

(5)

The multivariate-model approach allows us to compute alternative augmented (model-based) probabilities of distress, denoted P ZK i;t ; Xi;t . The control variables take into account selected observable characteristics at the bank level and changes in economic conditions. We select bank-level control variables that are publicly available and provided by most, if not all, banks to ascertain that they can be used universally to compute the augmented Z-scores. More precisely, we control for bank size by including the log of total assets (Sizei;t ) and the square of the log of total assets ((Sizei;t )2 ). Economies of scale and scope in information production are highlighted as theoretical rationales for the special role of banks (e.g. Diamond (1984), Williamson (1986), and Boyd and Prescott (1986)), with the direct implication that larger banks should be less prone to falling into distress and be more e¢ cient than smaller ones. However, according to a competing view (e.g. Farhi and Tirole (2012)), large banks might also have incentives to have higher asset risk relative to smaller banks in response to "too-big-to-fail" subsidies and government bailouts. We additionally take into account if a bank is subject to market scrutiny by including the dummy variable Listedi;t equal to 1 if bank i is listed on a public stock market, and 0 otherwise. Market discipline has the potential to curb incentives for excessive risk taking, by rendering risk-taking more costly (Kwan (2004), Nier and Baumann (2006)). Lastly, we include the VIX (V IXt ) to account for changes in economic conditions.17 Controlling in this way for selected observable characteristics at the bank level and changes in economic conditions, the marginal e¤ect of the Z-score on the probabilities of distress varies across banks and over time. Therefore, one can expect that including meaningful information to produce an augmented Z-score will allow us to compute more accurate probabilities of distress to discriminate between distressed banks and survivor banks. 17

The variable V IXt is preferred to time dummies to allow for changes in economic conditions, as time dummies would not be appropriate for out-of-sample use of the estimates.

15

4.2

Model comparisons and "best" Z-score

The testing procedure we implement has two steps. The …rst step compares the performance of the three approaches considered to compute the probabilities of distress using a variety of criteria. The second step compares the performance of di¤erent Z-scores when the best model-based approach is applied. 4.2.1

Multivariate-model approach vs. univariate/model-free approach

We …rst use several model-based criteria to disentangle whether the multivariate-model approach provides a better signal to identify distressed banks than the univariate-model approach. The log-likelihood, McFadden R2 or AIC for instance are familiar model selection criteria assessing goodness of …t and can be considered as the "gold standards" for comparing models. In addition, since the comparison between these two approaches is equivalent to the comparison of nested logit models, we can also implement a likelihood ratio (LR) test to get a probabilistic statement regarding model selection. The model-free approach cannot be explicitly considered when using these modelbased criteria as the predicted probabilities do not come from an econometric model. However, as stated previously, the univariate-model approach and the model-free approach are, to a certain extent, equivalent. Therefore, if the multivariate-model approach outperforms the univariate-model approach, one can deduce that the multivariate-model approach also outperforms the model-free approach. Table 7 presents the results of the univariate and multivariate regressions for the best Z-scores we identi…ed in Section 3.3: the regulatory capital Z-score and the traditional ROA-based Z-score using current values of the capital-asset ratio, calculated alternatively with moving moments or exponentially weighted moments. The results show that the addition of control variables in the four multivariate models we consider results in a statistically signi…cant improvement in model …t, obtaining higher Log-Likelihoods than with univariate models, with large LR test statistics associated with very low p-values. The results further show that the di¤erent multivariate models have larger McFadden R2 and lower values for AIC and BIC compared to the univariate models. All the criteria used therefore point to a preferred model, with the multivariate model outperforming the univariate one for the identi…cation of distressed banks for the four Z-score measures we consider.18 We further compare the performance of the multivariate-model approach with that of both the univariate-model and the model-free approach, using the same criteria as in Section 3.3 (AUROC curve, AUPR curve, H statistic and Tjur R2 ). The results displayed in Table 8 con…rm that the multivariate-model approach outperforms the univariate-model approach and also the model-free 18

We similarly con…rmed that the preferred model is also the multivariate model for all the other Z-score measures we considered in Section 3; results are available on request.

16

approach, for three of the four criteria considered (AUROC curve, AUPR curve and H measure). The AUROC test further shows that the multivariate-model approach signi…cantly increases the value of the AUROC curve compared to the two other approaches. As could be expected, the performance of the univariate-model and the model-free approach are very similar for all the criteria, except for the Tjur R2 . However, the …nding that the model-free approach outperforms the two model-based approaches according to the Tjur R2 should be treated with caution, as it is mainly driven by the properties of the link functions between the Z-score measure and the probability of distress.19 4.2.2

Evaluating Z-score measures under the multivariate-model approach

We then compare the performance of di¤erent Z-scores when the preferred, multivariate-model approach is considered. This comparison of di¤erent Z-score measures (for a given model-based approach) is equivalent to the comparison of non-nested logit models. We rely on the Vuong (1989) test and the Clarke (2003) test. These two procedures focus on the relative strength of two rival models and exploit the di¤erences in the log-likelihoods of two competing models. The non-nested tests are more appropriate than familiar model selection criteria such as the McFadden R2 or the AIC and BIC criteria because non-nested tests provide probabilistic statements regarding model selection. In addition, the non-nested tests include information from the rival model in the selection procedure, unlike the familiar model selection criteria. More precisely, Vuong (1989) proposes a procedure to discriminate between two rival models. He puts forward that the …rst model should be selected over the second model if the average loglikelihood of the …rst model is signi…cantly greater than the average log-likelihood of the second model, and conversely. Thus, the Vuong test uses the average di¤erence in the log-likelihoods of two competing speci…cations. The null hypothesis is that the average di¤erence is zero, suggesting that the two rival models are equally close to the true model. The test statistic, based on an appropriate normalization of the estimated average di¤erence, is normally distributed under the null hypothesis. Furthermore, when the null hypothesis is rejected, the …rst model should be selected if the test statistic is positive, while the second model should be selected if the test statistic is negative. Additionally, Clarke (2003) shows that some conditions a¤ect the performance of the Vuong test. In particular, a small sample size, a strong canonical correlation between the sets of covariates in the two rival models and the presence of outliers in the (individual) log-likelihood ratios reduce 19

Note that the probability of distress increases faster with decreasing Z-scores with the link function used in the model-free approach than with the one used in the model-based approach (see Figure A1 in the Appendix). As a result, if a given Z-score performs well in discriminating between distressed and surviving banks, the mean of the predicted probabilities of positive outcomes will be higher under the model-free approach than the model-based approach; hence, the Tjur R2 will conclude that the model-free approach outperforms the two model-based approaches. This result highlights that for users that are sensitive to the numerical values of the predicted probabilities, the model-free approach has the interesting feature of generating on average a large gap between the respective predicted probabilities of distressed and surviving banks.

17

the performance of the Vuong test. Clarke (2003) proposes a more robust selection procedure in such conditions. The Clarke test, also called the distribution-free test, is based on the median di¤erence in the log-likelihoods of two competing speci…cations. The null hypothesis is that the two speci…cations are equally close to the true model. Therefore, under the null hypothesis, half of the di¤erences of the log-likelihoods should be greater than zero. The number of positive di¤erences is used to compute the test statistic that is distributed binomial (n; 0:5), where n is the number of observations.20 Furthermore, when the null hypothesis is rejected, the number of positive di¤erences is signi…cantly di¤erent from 0.5, suggesting that the median log-likelihood ratio is statistically di¤erent from zero. A number of positive di¤erences signi…cantly larger than 0.5 means that the …rst model should be selected while a number of positive di¤erences signi…cantly lower than 0.5 means that the second model should be selected. Finally, we also rely on the AUROC curve which is reliable for assessing whether one of the Z-score measures performs better for the detection of distressed banks. More precisely, we consider the statistical test proposed by DeLong et al. (1988) to compare two AUROC curves. Results are presented in Tables 8 and 9 when we compare the performance of the ROA-based Zscore using the current values of the capital-asset ratio and the regulatory capital Z-score. Perhaps surprisingly, the ROA-based Z-score appears to perform better in the multivariate context than the regulatory capital one based on most of the criteria, in contrast to the model-free approach.21 Our results further show that, in the multivariate context, time-varying Z-scores computed with exponentially weighted moments do not systematically perform better than those using simple moving moments. The AUROC tests seem to indicate that the di¤erent Z-score measures considered are not signi…cantly di¤erent, indicating that the risk classi…cation is not signi…cantly di¤erent between them. However, given the characteristics of the AUROC test, this is not surprising as these Z-score measures are related. Overall, our …ndings thus show that augmented Z-scores computed using a multivariate-model approach provide better probabilities of distress to discriminate between distressed banks and surviving banks than the widely used model-free approach. These results highlight that the additional information included in the multivariate model approach is meaningful to produce Z-scores that outperform those of the model-free approach (or the univariate approach) for the di¤erent Z-score measures considered. Our results point to the best Z-score to use being the ROA-based Z-score using current values of the capital-asset ratio, as implemented in the multivariate model approach, calculated either with moving or exponentially weighted moments. 20

The competing speci…cations we consider have the same number of coe¢ cients; hence, we do not need to rely on a correction for the degrees of freedom. 21 We also …nd that the ROA-based Z-score using the current values of the capital-asset ratio performs better than the other Z-score measures we considered in Section 3; results are available on request.

18

4.3

Augmented Z-score formula for out-of-sample use

An additional interesting aspect of our model-based approach is that it allows the construction of a formula de…ning an "augmented" Z-score that can be used in out-of-sample applications. i.e. as an alternative to traditional Z-scores in general empirical work in the banking and …nancial stability related literature. Generally, the multivariate model framework allows the computation of a formula ^ K , de…ned as for an augmented Z-score measure Z i;t ^K = Z i;t

^ ZK i;t

Xi;t ^

For the best Z-score measure we identi…ed in the multivariate model approach, i.e. the ROAbased Z-score using current values of the capital-asset ratio, we can retrieve the relevant estimated parameters ^ and ^ from Table 7. The corresponding augmented ROA-based Z-score measure r3;c ZROA;i;t , computed using 3-year rolling windows (for simplicity) and current values of the capitalasset ratio, can then be given as r3;c ^ r3;c Z ROA;i;t = 1:6181ZROA;i;t

where

0:4014Listedi;t

1:1543Sizei;t + 0:0432 (Sizei;t )2

0:1178V IXt

r3;c r3;c Zr3;c ROA;i;t = sign ZROA;i;t : ln ZROA;i;t + 1

The associated probability of distress for this augmented Z-score is given by the logit function 1 ^ r3;c [1 + exp(Z ROA;i;t + 11:1360)] : Thus, the estimated coe¢ cients can be used as optimal weights in an out-of-sample perspective that make the Z-score measure conditional on economic conditions and banks’observable characteristics, and thereby provides an improved assessment of a bank’s risk of distress. We can use two graphical representations to illustrate the main properties of this augmented Z-score. First, Table 8 shows that the AUROC curves are higher with the multivariate model compared to both the univariate model and the model-free approach. This di¤erence is illustrated in Figure 1. The Z-score measure is on the horizontal axis, whereas the vertical axis gives the probability of distress in the univariate model (Figure 1-a) and in the multivariate model (Figure 1-b), respectively. Figure 1-b shows that, for a given Z-score, the probability of distress is not unique in the multivariate model as economic conditions and banks’observable characteristics change for each observation. In particular, Figure 1-b shows that the di¤erences in the probabilities of distress for a given Z-score can be substantial when Z-scores are low, i.e. when banks face a greater risk of distress. Conversely, for a given Z-score, Figure 1-a shows that the probability of distress is (by de…nition) unique in the univariate model (as in the model-free approach). Therefore, the augmented Z-score measures computed from the multivariate model allow the introduction of heterogeneity in the relationship between the Z-score and the probability of distress. This heterogeneity between 19

banks and over time ensures an improved assessment of the probability of distress of banks. Secondly, we can use a graphical representation to highlight the main advantage of the augmented Z-score measure in the time dimension. More precisely, we can use the multivariate model to construct a Z-score map that illustrates how the relationship between the Z-score and the probability of distress is a¤ected by changes in economic conditions over time (i.e. the VIX). Figure 2 shows to what extent the probability of distress changes when both the Z-score measure and the VIX vary. In other words, the non-linearity of the logit model implies that the marginal e¤ect of the Z-score on the probability of distress is conditional on the level of the control variables, including the economic conditions proxied by the VIX. Thus, Figure 2 shows that in a high volatility environment (i.e. V IX > 40) the probabilities of distress start to increase when the Z-score measure goes below 5, whereas this increase only starts when the Z-score measure drops below three in a low volatility environment (i.e. V IX < 15). In addition, the slope of the surface in Figure 2 shows that a decline in the Z-score measure can lead rapidly to a high probability of distress in a high volatility environment, whereas a similar decline will not have a substantial e¤ect on the risk of distress in a low volatility environment.

5

Conclusion

We examine traditional and alternative versions of Z-score measures, a commonly used, accounting data based bank insolvency risk measure. After highlighting various caveats arising in the practical application of traditional Z-scores, we develop alternative Z-score measures that aim to resolve these issues, and then proceed to recommendations for best practice in the application of Z-score measures for the measurement of bank insolvency risk in the empirical banking and …nancial stability related literature. Our empirical analysis is focussed on the …nancial crisis of 2007-2008, using …nancial statement data for US commercial banks for the period 2006-2014. We carry out comparisons of the di¤erent Z-score measures considered using a range of alternative testing procedures, drawing on both a model-free/probabilistic approach as well as on model-based approaches. The latter use a univariate speci…cation as well as a multivariate one which also conditions on observable bank characteristics and the macroeconomic environment. Our analysis using the model-free/probabilistic approach suggests that the two best Z-scores to use in this case are our novel exponentially weighted regulatory capital Z-score, and the exponentially weighted version of the more traditional ROA-based Z-score using current values of the capital-asset ratio. Potential income-smoothing by banks which might bias these accounting based insolvency risk measures does not seem to pose a signi…cant problem requiring correction. Moving on to the model-based approach, we then demonstrate that computing augmented Z-scores using a multivariate-model approach provides better probabilities of distress to discriminate be20

tween distressed and surviving banks than the widely used model-free/probabilistic approach. The multivariate-model approach suggests that the best Z-score to use in this context is the ROA-based Z-score using current values of the capital-asset ratio, calculated either with moving or exponentially weighted moments. This approach furthermore allows construction of a formula de…ning an "augmented" Z-score that can be used in out-of-sample applications as an alternative to traditional, unconditional Z-scores.

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25

Figure 1: Probabilities of distress in the univariate and multivariate models b- Multivariate model 1

0.9

0.9

0.8

0.8

0.7

0.7

Prob. of distress

Prob. of distress

a- Univariate model 1

0.6 0.5 0.4

0.6 0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

-2

0

2

4

6

8

10

Z-score (log-modulus transformation)

0

-2

0

2

4

6

Note: The univariate and the multivariate models correspond to the estimates reported in columns (1) and (2) of Table 7. Therefore , the ROA-based Z-score (with 3-year rolling windows) computed with current values of the capital-asset ratio is used.

26

8

Z-score (log-modulus transformation)

10

Figure 2: Z-score map: VIX, Z-score and probability of distress

27

Table 1: Sample characteristics Year 2007 2008 2009 2010 2011 2012 2013 2014 All sample Number of observations 5,692 5,780 5,589 5,447 5,373 5,327 5,304 5,285 43,806 Number of distressed banks 43 191 132 84 46 23 19 5 543 Percentage of distressed banks 0.76 3.30 2.36 1.54 0.86 0.43 0.36 0.09 1.24

Table 2: Descriptive statistics of Z-score measures Z-score r3 ZROA r3;c ZROA r3 ZROE r3 ZRCAP r3 ZAdjROA r3;c ZAdjROA r3 ZAdjROE

Banks

Obs.

Mean

Median

Std. Dev.

Min

Max

Non-dist. Distressed Non-dist. Distressed Non-dist. Distressed Non-dist. Distressed Non-dist. Distressed Non-dist. Distressed Non-dist. Distressed

43263 543 43263 543 43263 543 43263 543 42815 515 42815 515 42815 515

102.39 9.57 102.68 7.56 103.18 10.29 13.44 1.18 76.17 7.72 76.36 5.68 76.70 7.80

55.03 1.93 55.27 0.69 55.60 0.91 8.15 0.56 42.68 1.78 42.71 0.68 43.31 0.80

234.25 27.05 236.54 26.33 213.87 38.66 21.76 4.30 157.11 18.00 158.72 16.64 153.14 23.22

-3.26 -5.23 -7.76 -9.18 -1.19 -5.81 -47.45 -23.55 -1.95 -5.69 -9.24 -8.96 -1.09 -6.25

20785.18 354.75 21610.53 337.75 12961.08 651.24 1015.53 65.97 11457.04 205.85 11892.50 212.49 10529.38 326.15

Variable de…nitions, Z-scores computed with 3-year rolling windows (see Section 2 for more details): r3;c r3 = ROA-based Z-score computed with moving mean of the capital-asset ratio; ZROA = ROA-based ZROA r3 r3 = Z-score computed with current values of the capital-asset ratio; ZROE = ROE-based Z-score; ZRCAP r3 Regulatory capital Z-score (using TCR and 8% threshold); ZAdjROA = ROA-based Z-score computed r3;c with moving mean of the capital-asset ratio, and adjusted for income smoothing behavior; ZAdjROA = ROA-based Z-score computed with current values of the capital-asset ratio, and adjusted for income r3 smoothing behavior; ZadjROE = ROE-based Z-score adjusted for income smoothing behavior.

28

Table 3: Correlation of Z-score measures A. Z-scores (levels) and total capital ratio r3;c r3 r3 r3 ZROA ZROA ZROE ZRCAP r3 ZROA 1 r3;c ZROA 0.9986 1 r3 ZROE 0.3373 0.3345 1 r3 0.0846 0.0837 0.0987 1 ZRCAP T CR 0.1187 0.1192 0.1280 0.1457 r3 0.4808 0.4779 0.3497 0.0937 ZAdjROA r3;c ZAdjROA 0.4777 0.4765 0.3467 0.0927 r3 ZAdjROE 0.3051 0.3020 0.5298 0.1015 B. Z-scores (log transformations) and total r3;c r3 r3 r3 ZROA ZRCAP ZROA ZROE r3 ZROA 1 r3;c ZROA 0.9894 1 r3 0.9376 0.9368 1 ZROE r3 0.4681 0.4687 0.4734 1 ZRCAP T CR 0.2622 0.2532 0.2564 0.2437 r3 0.9404 0.9517 0.8961 0.4643 ZAdjROA r3;c ZAdjROA 0.9432 0.9544 0.8985 0.4656 r3 0.9050 0.9034 0.9545 0.4690 ZAdjROE

T CR

r3 ZAdjROA

r3;c ZAdjROA

r3 ZAdjROE

1 0.1137 0.1139 0.1238

1 0.9987 0.3908

1 0.3863

1

r3;c ZAdjROA

r3 ZAdjROE

1 0.9378

1

capital ratio r3 T CR ZAdjROA

1 0.2514 0.2538 0.2530

1 0.9987 0.9364

r3 Variable de…nitions, Z-scores computed with 3-year rolling windows (see Section 2 for more details): ZROA r3;c = ROA-based Z-score computed with moving mean of the capital-asset ratio; ZROA = ROA-based Z-score r3 r3 computed with current values of the capital-asset ratio; ZROE = ROE-based Z-score; ZRCAP = Regulatory r3 capital Z-score (using TCR and 8% threshold); TCR = Total capital ratio; ZAdjROA = ROA-based Z-score r3;c computed with moving mean of the capital-asset ratio, and adjusted for income smoothing behavior; ZAdjROA = ROA-based Z-score computed with current values of the capital-asset ratio, and adjusted for income smoothing r3 behavior; ZadjROE = ROE-based Z-score adjusted for income smoothing behavior.

29

Table 4: Evaluation of Z-scores (moving moments, model-free approach)

AUROC curve AUPR curve H measure Tjur R2 Observations Number of banks Number of distressed banks

r3 ZROA

r3;c ZROA

r3 ZROE

r3 ZRCAP

0.9256 0.3025 0.6341 0.3171 43,806 5,823 543

0.9376b 0.4250a 0.6682b 0.5456b 43,806 5,823 543

0.9238 0.3808b 0.6464 0.5016 43,806 5,823 543

0.9379a 0.2668 0.6760a 0.5875a 43,806 5,823 543

r3 Variable de…nitions, Z-scores computed with 3-year rolling windows (see Section 2 for more details): ZROA r3;c = ROA-based Z-score computed with moving mean of the capital-asset ratio; ZROA = ROA-based Z-score r3 r3 = Regulatory = ROE-based Z-score; ZRCAP computed with current values of the capital-asset ratio; ZROE capital Z-score (using TCR and 8% threshold). Four criteria are used to compare the performance of the di¤erent rival classi…ers (AUROC curve, AUPR curve, H measure and Tjur R2 ). a and b indicate, respectively, the Z-score with the highest and the second highest value for each criterion.

Table 5: Evaluation of income smoothing adjusted Z-scores (moving moments, model-free approach)

AUROC curve AUPR curve H measure Tjur R2 Observations Number of banks Number of distressed banks

r3 ZROA

r3 ZAdjROA

r3;c ZROA

r3;c ZAdjROA

r3 ZROE

r3 ZAdjROE

0.9249a 0.2829 0.6298a 0.2953 43,330 5,816 515

0.9210 0.2881a 0.6238 0.3180a 43,330 5,816 515

0.9375a 0.4105 0.6654a 0.5333 43,330 5,816 515

0.9357 0.4188a 0.6632 0.5557a 43,330 5,816 515

0.9232a 0.3756 0.6417 0.5005 43,330 5,816 515

0.9210 0.3789a 0.6442a 0.5198a 43,330 5,816 515

r3 Variable de…nitions, Z-scores computed with 3-year rolling windows (see Section 2 for more details): ZROA = r3 ROA-based Z-score computed with moving mean of the capital-asset ratio; ZAdjROA = ROA-based Z-score r3;c computed with moving mean of the capital-asset ratio, and adjusted for income smoothing behavior; ZROA = r3;c ROA-based Z-score computed with current values of the capital-asset ratio; ZAdjROA = ROA-based Z-score r3 computed with current values of the capital-asset ratio, and adjusted for income smoothing behavior; ZROE = r3 ROE-based Z-score; ZadjROE = ROE-based Z-score adjusted for income smoothing behavior. Four criteria are used to compare the performance of the di¤erent rival classi…ers (AUROC curve, AUPR curve, H measure and Tjur R2 ). a indicates the higher criterion for Z and ZAdj .

30

Table 6: Evaluation of Z-scores (exponentially weighted moments, model-free approach)

AUROC curve AUPR curve H measure Tjur R2 Observations Number of banks Number of distressed banks

exp ZROA

exp;c ZROA

exp ZROE

exp ZRCAP

0.9399+ 0.4384+ 0.6614+ 0.5597+ 43,806 5,823 543

0.9438a+ 0.4400a+ 0.6709+ 0.6257+ 43,806 5,823 543

0.9282+ 0.4109+ 0.6591+ 0.7067+ 43,806 5,823 543

0.9371 0.2728+ 0.6867a+ 0.7418a+ 43,806 5,823 543

Variable de…nitions, Z-scores computed with exponentially weighted moments (see Section 2 for more details): exp ZROA = ROA-based Z-score computed with exponentially weighted mean of the capital-asset ratio with =0.77; exp;c exp ZROA = ROA-based Z-score computed with current values of the capital-asset ratio with =0.39; ZROE = exp ROE-based Z-score with =0.85; ZRCAP = Regulatory capital Z-score (using TCR and 8% threshold) with =0.69. The parameter is provided by a grid search and maximizes the AUROC curve. Four criteria are used to compare the performance of the di¤erent rival classi…ers (AUROC curve, AUPR curve, H measure and Tjur R2 ). a indicates the Z-score with the highest value among the four Z-scores, for each criterion. += indicate if the Z-score computed with exponentially weighted moments has higher/smaller criterion compared to its equivalent computed with moving moments (as reported in Table 4).

31

32

(0:0489)

(0:0528)

0.1178 -11.1360

-0.3781

43,806 5,823 543 -1627.61

0.4433 3259.23 3276.60

V IX t

Intercept

No. obs. No. banks No. failures. Log-likelihood LR test

McFadden R2 AIC BIC

[0.0000]

0.4786 3060.60 3112.73

43,806 5,823 543 -1524.30 206.63

(2:8142)

(0:0102)

(0:0173)

(0:1528)

0.3333 3902.27 3919.64

43,806 5,823 543 -1949.13

-1.2692

(0:1138)

-2.0496

Univariate

0.4175 3418.01 3470.13

[0.0000]

(2:5573)

43,806 5,823 543 -1703.00 492.26

-11.4991

(0:0123)

0.1886

(0:0140)

-0.0293

(0:3773)

0.8073

(0:1424)

0.4429

(0:1177)

-2.3353

Multivariate

r3 ZRCAP

(0:0866)

0.4109 3448.70 3466.07

43,806 5,823 543 -1722.35

-0.7076

(0:0793)

-1.6727

Univariate

0.4506 3224.27 3276.40

[0.0000]

(3:2416)

43,806 5,823 543 -1606.13 232.42

-14.5717

(0:0082)

0.1127

(0:0201)

-0.0648

(0:5103)

1.6876

(0:1190)

0.3613

(0:0800)

-1.6200

Multivariate

exp ;c ZROA

(0:0916)

0.3485 3813.48 3830.85

43,806 5,823 543 -1904.74

-2.0383

(0:0664)

-1.2339

Univariate

[0.0000]

0.4094 3465.61 3517.74

43,806 5,823 543 -1726.80 355.86

(2:7231)

(0:0103)

-15.1030

0.1420

(0:0153)

(0:4085)

-0.0440

1.3042

(0:1299)

(0:0615)

0.3345

-1.2787

Multivariate

exp ZRCAP

r3;c Variable de…nitions, Z-scores computed with 3-year rolling windows (see Section 2 for more details): ZROA = ROA-based Z-score computed r3 with current values of the capital-asset ratio; ZRCAP = Regulatory capital Z-score (using TCR and 8% threshold). Z-scores computed with exp;c exp exponentially weighted moments: ZROA = ROA-based Z-score computed with current values of the capital-asset ratio with =0.28; ZRCAP = Regulatory capital Z-score (using TCR and 8% threshold) with =0.82. The parameter is provided by a grid search and maximizes the AUROC curve in the multivariate model. Control variables: Listedi = 1 if bank i is listed on a public stock market, and 0 otherwise; Size = log of total assets; V IX = the CBOE Volatility Index. ***, ** and * indicate signi…cance respectively at the 1%, 5% and 10% levels. Cluster robust standard deviations are in brackets.

[p-value]

-0.0432

(Sizei;t )2

(0:4394)

1.1543

Sizei;t

(0:1421)

Multivariate -1.6181

0.4014

(0:0872)

Univariate -1.6162

r3;c ZROA

Listedi;t

Zexp RCAP;i;t

;c Zexp ROA;i;t

Zr3 RCAP;i;t

Zr3;c ROA;i;t

Z-score : Model :

Table 7: Comparison of univariate and multivariate models (model-based criteria)

Table 8: Comparisons between the three approaches (non model-based criteria) Criteria AUROC curve

Approach

r3;c ZROA

r3 ZRCAP

exp;c ZROA

exp ZRCAP

Model-free Univariate-model Multivariate-model

0.9376 0.9376 0.9499a 35.59

0.9379 0.9379 0.9514a 16.23

0.9438 0.9434 0.9532a 51.42

0.9371 0.9343 0.9479a 23.71

0.4250 0.4250 0.4693a 0.6682 0.6682 0.6998a 0.5456a 0.2802 0.3102

0.2668 0.2668 0.3389a 0.6760 0.6760 0.6993a 0.5875a 0.1541 0.2181

0.4404 0.4306 0.4306a 0.6709 0.6650 0.7072a 0.6257a 0.1996 0.2387

0.2728 0.2840 0.2890a 0.6867 0.6954 0.7124a 0.7418a 0.1431 0.1870

AUROC test AUPR curve

H measure

Tjur R2

[0.0000]

Model-free Univariate-model Multivariate-model Model-free Univariate-model Multivariate-model Model-free Univariate-model Multivariate-model

[0.0001]

[0.0000]

[0.0000]

Variable de…nitions, Z-scores computed with 3-year rolling windows (see Section 2 for more details): r3;c r3 = Regulatory ZROA = ROA-based Z-score computed with current values of the capital-asset ratio; ZRCAP capital Z-score (using TCR and 8% threshold). Z-scores computed with exponentially weighted moments: exp exp;c = ROA-based Z-score computed with current values of the capital-asset ratio with =0.28; ZRCAP ZROA = Regulatory capital Z-score (using TCR and 8% threshold) with =0.82. The parameter is provided by a grid search and maximizes the AUROC curve in the multivariate model. Four criteria are used to compare the performance of the di¤erent rival classi…ers (AUROC curve, AUPR curve, H measure and Tjur R2 ). a indicates the approach with the highest value for each criterion and each Z. The null hypothesis of the AUROC test is that the AUROC curves are equal in the multivariate and univariate approaches.

33

Table 9: Performance of traditional ROA-based Z-score and regulatory capital Z-score (model-based approach) Model 1 vs Model 2

No. obs.

Vuong test

Clarke test [p-value]

[p-value] r3;c ZROA

vs

r3 ZRCAP

43,806

r3;c exp ;c ZROA vs ZROA

43,806

exp r3 ZRCAP vs ZRCAP

43,806

exp;c exp ZROA vs ZRCAP

43,806

5.23

0.54

[0.0000]

[0.0000]

3.54

0.25

[0.0003]

[0.0000]

1.23

0.51

[0.2152]

[0.0334]

3.57

0.68

[0.0003]

[0.0000]

AUROC test [p-value]

0.14

[0.7119]

0.98

[0.3231]

1.64

[0.2006]

1.70

[0.1922]

Note: The null hypotheses of the Vuong test and the Clarke test indicate that the two Z-score measures perform similarly. Positive and signi…cant values for the Vuong test indicate that the …rst measure performs better than the second measure; and conversely for negative and signi…cant values. Values signi…cantly higher than 0.5 for the Clarke test indicate that the …rst measure performs better than the second measure; and conversely for values signi…cantly lower than 0.5. The null hypothesis of the AUROC test is that the AUROC curves are equal in the two speci…cations. Variable de…nitions, Z-scores computed with 3-year rolling windows (see Section 2 for more details): r3;c r3 = ROA-based Z-score computed with current values of the capital-asset ratio; ZRCAP = Regulatory ZROA capital Z-score (using TCR and 8% threshold). Z-scores computed with exponentially weighted moments: exp;c exp ZROA = ROA-based Z-score computed with current values of the capital-asset ratio with =0.28; ZRCAP = Regulatory capital Z-score (using TCR and 8% threshold) with =0.82. The parameter is provided by a grid search and maximizes the AUROC curve in the multivariate model.

34

Appendix Table A1. Loan loss provisions and income smoothing behavior

LLP i;t 1 N P Li;t N P Li;t Li;t ERBT i;t T ier1i;t 1 SIGN i;t Intercept No. obs. No. banks No. instruments GMM instruments AR(1) stat (p value)

AR(2) stat (p value)

AR(3) stat (p value)

Hansen stat (p value)

Coef.

Std. Err.

Coef.

Std. Err.

0.4146 0.0515 0.0492 0.0051 0.0564 0.0027 -0.0125 -0.4639 94,158 5,820 31 lags 2-6 -16.85 (0.0000) 3.66 (0.0002) 0.40 (0.6829) 4.51 (0.6067)

(0.0254)

0.4506 0.0492 0.0515 0.0049 0.0786 0.0029 0.0010 -0.4834 94,158 5,820 30 lags 3-6 -14.15 (0.0000) 3.04 (0.0024) 0.59 (0.5539) 9.19 (0.1017)

(0.0285)

(0.0027) (0.0036) (0.0002) (0.0175) (0.0002) (0.0117) (0.0236)

(0.0036) (0.0038) (0.0002) (0.0132) (0.0002) (0.0095) (0.0227)

Variable de…nitions: LLP = ratio of loan loss provisions to total assets: N P L = ratio of non-performing loans to total assets; N P L = the …rst di¤erence of N P L; L = ratio of net loans to total assets; ERBT ratio of earnings before taxes and loan loss provisions to total assets; T ier1 = Tier1 capital ratio; SIGN = the one-year-ahead change of earnings before taxes and loan loss provisions. ***, ** and * indicate signi…cance respectively at the 1%, 5% and 10% levels. All regressions include time …xed e¤ects.

35

Figure A1. Link function and probabilities of distress 1 0.9

Probability of distress

0.8 Model-free approach Univariate model approach

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

11

Z-score

12

2 Note: The probability of distress is given by 1/(1+Z-score ) in the model-free approach. The probability of distress is given by 1/(1+exp(1.6162*ln(Z-score+1)+0.3781)) in the univariate-model approach. The parameters used in the univariate-model approach to compute the probabilities are taken from the estimates reported in column 1 of Table 7.

36