Determinants of Stochastic Distance-to-Default

Abstract

Efficient management of bankruptcy risk requires treating distant-to-default (DD) stochastically as long as historical stock prices move randomly and, thus, do not guarantee that history may repeat itself. Using long-term data that date back to 1952–2023, including the nonfinancial companies listed in the Dow Jones Industrial Average and National Association of Securities Dealers Automated Quotations indexes, this study estimates the historical and stochastic DDs via the geometric Brownian motion (GBM). The results show that (a) the association between the debt-to-equity ratio and the stochastic DD can be used as an indicator of excessive debt financing; (b) debt tax savings have a positive effect on stochastic DD; (c) bankruptcy costs have negative effects on stochastic DD; (d) in terms of the size of the company being proxied by sales revenue and the equity market value of the company, the DD is a reliable measure of bankruptcy costs; (e) in terms of macroeconomic influences, increases in the percentage change in manufacturing output are associated with lower observed and stochastic DD; and (f) in terms of the influences of industry, the stochastic DD is affected by the industry average retail inventory to sales. This paper contributes to related studies in terms of focusing on the indicators that a company’s management can focus on to address the stochastic patterns inherent in the estimation of the DD.

1. Introduction

The link between a company’s financials and bankruptcy risk requires an estimation of the stochastic distance-to-default for the company in order to adjust its financial performance to avoid or reduce expected bankruptcy risk. The same argument is true when considering the volatility of economic conditions (Hasan & Habib, 2017). Several studies have argued that factors leading to bankruptcy are either company-specific (Altman, 1968; Ohlson, 1980; Bellovary et al., 2007; Bakhtiari, 2017; Cathcart et al., 2020; Zhang et al., 2022) or country-specific (Jónsson & Fridson, 1996; Qu, 2008; Carling et al., 2007; Lando & Nielsen, 2010; Altman et al., 2016).

1.1. Stochastic Distance-to-Default Raises Management Concerns

Empirically, the DD is as vital to high creditworthy companies as it is to low creditworthy companies. The latter have an ongoing need to improve creditworthiness. At the same time, former companies have a need to either sustain or improve creditworthiness. In both cases, the DD, which is estimated via the Merton algorithm (Merton, 1974), is intrinsically stochastic as long as the firm’s equity market value is subject to a random walk.

The implementation of this model is further discussed in detail in Kealhofer and Kurbat (2001), Kealhofer (2003a, 2003b), Kliestik et al. (2015) and Hsu and Wu (2020) as follows:

$$V_E=V_{A}N\left\{d_1\right\}-X{e}^{-rt}N\left\{d_2\right\}$$

(1)

$V_E$ = company’s value of equity stocks, $V_A$ = the total value of a company’s asset = book value of debt in addition to value of equity stocks, $X$ = book value of debt, $N\{.\}$ = the value of the normal distribution (CDF) given a certain mean and standard deviation, $e$ = Euler’s mathematical constant (base of the natural logarithm), $r$ = quarterly T-bill rates, and t = time interval. The DD is referred to as $d_2$ in the Merton algorithm and is computed as follows.

$$d_2={\displaystyle \frac{\ln \left(\frac{V_A}{X}\right)+\left(r-0.5{\sigma }_A^2\right)t}{{\sigma }_A\sqrt{t}}}$$

(2)

where ${\sigma }_A^2$ is the implied volatility of a company’s assets and is computed as follows:

$${\sigma }_A={\displaystyle \frac{{\sigma }_E\times V_E}{V_A}}$$

(3)

$V_E$ = the value of the company’s stocks and ${\sigma }_E$ = standard deviation of the percentage change in value of the company’s stocks over four rolling quarters. Equation (2) $d_2$ is referred to in the literature as the DD (Duan et al., 2012; Duffie et al., 2009; Sima-Grigore & Sima, 2011; Leland, 2004).

The stochastic DDs are estimated via the GBM (Feynman, 2013; Ibe, 2013; Reddy & Clinton, 2016; Kumar et al., 2024; Sinha, 2024) as follows:

$$\Delta D{D}_{t+\Delta t}=D{D}_t\mu \Delta t+D{D}_t\sigma \epsilon \sqrt{\Delta t}$$

(4)

where $\Delta D{D}_{t+\Delta t}$ is the change in distance-to-default, $\mu$ and $\sigma$ are the mean and volatility in the percentage change in DD, respectively, and $\epsilon$ is the Weiner process that follows the normal distribution with a mean = 0 and standard deviation = 1. The number of iterations for each company = 72, which equals the number of years under examination. In this paper, the DDs are calculated for each company and every year via end-of-month stock prices.

In this case, it is possible to compare historical and stochastic DDs, as shown in Figure 1, where stochastic DDs are estimated and simulated via the GBM.1

Figure 1 indicates that the average stochastic DDs are expected to exceed the historical average DDs across the nonfinancial companies listed in the Dow Jones Industrial Average Index and the National Association of Securities Dealers Automated Quotations Index. This is an interesting outcome, since those companies are quite viable financially. Nevertheless, this stochastic trend raises interest in companies’ management in terms of the financial strategies that must be developed to achieve this expected improvement in the DD. Figure 1 raises the concern that DDs vary across financially healthy companies. These variations call for company management to examine the financial factors that help converge DDs to peers in the industry. Figure 2 shows that time-varying stochastic DDs are greater than observed DDs, which also raises the concern that companies’ management must examine the financial factors that help achieve the expected stochastic DDs.

1.2. Objectives

This paper aims to examine the financial indicators that help companies manage stochastic bankruptcy risk as measured by DDs.

1.3. Contribution

The paper extends the contributions of previous related studies as follows. First, stochastic distance-to-default treatment via the Merton algorithm (Merton, 1974) is intrinsic as long as the movements of stock prices are stochastic. Second, DD estimation is usually carried out via historical stock prices, whereas DD management involves future financial decisions. As the movements of expected stock prices are unknown, stochastic estimation of the DD remains plausible.

This paper is divided into four parts. Section 2 discusses the determinants of bankruptcy risk, especially when distance-to-default is used at three levels, namely company, industry and country. Section 3 describes the data and the variables. Section 4 discusses the results. Section 5 presents our conclusions.

2. Literature Review

The DD is a pragmatic measure of credit risk, as it offers space for adjusting company-specific financials to widen the DD (Dar & Qadir, 2019). Hotchkiss et al. (2023) stated that assessing the bankruptcy level is critical for estimating a company’s credit quality. The distance-to-default model was adopted operationally by Moody’s credit rating agency, which is referred to as the KMV model (Vasicek, 1984). A number of studies have extended the examination of the efficiency of the KMV model by comparing it to other credit risk models and have advocated the development of a new KMV model (Farmen, 2004; Bharath & Shumway, 2008; Kollar, 2014). Nagel and Purnanandam (2019) conclude that corporate ratings via the DD are more reliable than using the Leland and Toft model (Leland & Toft, 1996). Vu et al. (2019) concluded that the DD can efficiently estimate a company’s value by assessing the likelihood of a company’s failure over time.

However, previous studies suggest that insolvency is likely a multilevel phenomenon with significant factors that have cross-level links (Khoja et al., 2019). For example, company-level research suggests that insolvency can be predicted by a set of financial ratios, such as profitability, liquidity, and leverage (Vu et al., 2019; Choi & Richardson, 2016; Islam et al., 2022). In addition, macroeconomic conditions, which usually affect debt, equity, earnings and cash generation, certainly affect the risk of insolvency (Agrawal & Maheshwari, 2014).

2.1. Company-Specific Indicators and the Distance-to-Default

The business environment can be classified into two factors. The first factor is the internal environment, where company-specific factors help in estimating a company’s performance and, accordingly, the probability of default (hereinafter, PD). The other factor is the external environment, which is represented by country-specific factors. Hansen and Wernerfelt (1989) stated that company-specific factors outweigh country-specific factors in terms of effects on the PD. The company-specific factors examined in related studies are size, growth, profitability, liquidity, leverage, asset utilization, market efficiency, market capitalization, asset tangibility, the relative tax rate and financial risk (Ricca et al., 2023). As bankruptcy risk is significantly affected by the size of a company (Duan et al., 2018), sales and total assets are popular proxies for size. A larger company size results in greater economies of scale, which reduces the cost of production and consequently increases profitability and decreases the PD, and vice versa. Therefore, different authors have concluded that there is a negative relationship between a company’s size and the PD (Kalak & Hudson, 2016; Singh & Singla, 2021).

Bhutta and Hasan (2013) and Lee (2014) reported a positive relationship between a company’s growth in total assets and distance-to-default. Nevertheless, Jermias and Yigit (2019) reported the opposite results. Notably, a higher ratio of debt financing increases the PD and decreases the distance-to-default (Dwyer et al., 2004; Collin-Dufresne et al., 2001; Vassalou & Xing, 2004; Traczynski, 2017), especially when the debt ratio exceeds a certain buffer (Eldomiaty et al., 2016). Chandrapala and Knápková (2013) and Kavussanos and Tsouknidis (2016) conclude that companies with a higher debt ratio might experience increased financial distress. Nevertheless, in terms of the pecking order theory, Lozinskaia et al. (2017) document a negative relation between the debt-to-equity ratio and distance-to-default. Hamid and Siddiqui (2023) reported an inverse relationship between profitability and the PD and, thus, a positive relationship with the DD. Dwyer et al. (2004) stated that greater liquidity reduces the PD, thus reducing the DD. Han et al. (2017) reported a positive relationship between the effective tax rate and the PD, and thus a negative relationship with the DD.

2.2. Country-Specific Indicators and the Distance-to-Default

Several studies have examined the influence of several macroeconomic factors (such as GDP growth, inflation, unemployment, the T-bill rate, and industrial production) on bankruptcy risk (Virolainen, 2004; Jakubik, 2007; Simons & Rolwes, 2009; Jimenez & Mencia, 2009; Laurin & Martynenko, 2009; Bai, 2021).

A number of related studies have reported a negative relationship between GDP growth and the PD (Simons & Rolwes, 2009; Virolainen, 2004; Jakubik, 2007; Laurin & Martynenko, 2009; Jimenez & Mencia, 2009). Goodhart et al. (2023) reported that higher factor prices (e.g., inflation) decrease the PD as a company’s creditworthiness increases. Thus, there is a positive relationship between growth inflation and the distance-to-default. Furthermore, unemployment usually increases economic instability and disruptions and thus may increase the PD (Eckstein et al., 2018; Kilic & Wachter, 2017).

Risk-free rates play a central role in the estimation of the PD and, thus, the DD. Bessembinder et al. (2019) reported that treasury bill rates are positively related to a company’s PD. That is, increased T-bill issuance in the market coupled with increased volatility of the equity price index increases both interest rates and economic instability; thus, the distance-to-default decreases (Fang et al., 2021).

2.3. Industry-Specific Factors and Distant to Default

As far as the industrial production index measures the level of production in a specific country, Chi and Meng (2019) and Ni et al. (2023) reported that an increase in the industrial production index would result in a higher GDP and, accordingly, an increased distance-to-default. Stenbäck (2013) reached the same conclusion when the industry volume (manufacturing output %) was used. Qu (2008) reported that manufacturing and service and retail/wholesale are positively correlated with the PD and, thus, the negatively correlated with the DD.

3. Data

The company-specific data are for the nonfinancial companies listed in the Dow Jones Industrial Average Index and the National Association of Securities Dealers Automated Quotations Index covering the period 1952–2023 on an annual basis. The data are obtained from the Reuters Finance Center©. Descriptive statistics of the variables are reported in Appendix A.

3.1. Dependent Variable

Merton (1974) offers a mathematical algorithm that links a company’s equity market value to the book value of liability, thus capturing the extent of debt financing. The algorithm is commonly used for the estimation of the PD, which is inherently affected by the volatility of the market value of a company’s stock (Chiang & Chiang, 1996; Du & Suo, 2003; Dennis et al., 2006; Anderson & Prezas, 2003; Crosbie & Bohn, 2003).

3.2. Independent Variables

Table 1. Measurement and Hypothesis of the Independent Variables.

Variable	Measurement	Reference	Hypothesis
Debt-to-Equity ratio	Total debt/total equity.	Marsh (1982), Auerbach (1985), Bhutta and Hasan (2013), Lee (2014), Chandrapala and Knápková (2013), Lozinskaia et al. (2017).	There is a negative relation between “Debt to Equity” and the distance-to-default.
Effective Corporate Tax Rate	Tax expenses/taxable income.	Walsh and Ryan (1997), Alworth and Arachi (2003), Han et al. (2017).	There is a negative relation between “Effective tax rate” and the distance-to-default.
Bankruptcy Costs	Bankruptcy Costs = fixed charges − earnings before income and tax)/(${\sigma }_{EBIT}$).	White and Turnbull (1974), Warner (1977), Gong (2004).	There is a negative relation between bankruptcy costs and the distance-to-default.
Compound Growth Rate of Free Cash Flow (FCF)	FCF = EBIT + depreciation − tax − change in net fixed assets − change in net working capital.	Stretcher and Johnson (2011), Denis (2011).	There is a positive relation between a “Company’s Growth of free cash Flow” and the distance-to-default.
Size	Natural log of sales revenue. Natural log of equity market value.	Bhutta and Hasan (2013), Dang et al. (2018).	There is a positive relation between “Sales Growth” and the distance-to-default.
Growth of GDP	Annual compound growth of nominal GDP.	Simons and Rolwes (2009).	There is a positive relation between “Growth of GDP” and the distance-to-default.
Inflation Rate	Annual compound growth of CPI.	Qu (2008), Laurin and Martynenko (2009).	There is a negative relation between “Growth of inflation” and the distance-to-default.
Productivity Growth	Total industrial production index2.	Qu (2008), Laurin and Martynenko (2009), Figlewski et al. (2012), Boutchaktchiev (2017), Xing et al. (2023).	There is a negative relation between “percentage change in manufacturing output” and the distance-to-default.
% change in manufacturing output	Gross output by industry: manufacturing3.	Stenbäck (2013), Demirhan and Sayilgan (2021).	There is a negative relation between “percentage change in manufacturing output” and the distance-to-default.
Interest Rates	Annual T-bill rates4.	Laurin and Martynenko (2009).	There is a negative relation between interest rates and the distance-to-default.
Growth of Unemployment Rate	Annual compound growth of unemployment rates5.	Nkusu (2011), Castro (2013), Chaibi and Ftiti (2015).	There is a negative relation between “unemployment rate’s growth” and the distance-to-default.
Industry Average Retail Inventory to sales	Ratio of annual ratio of retail inventory to sales revenue6.	These variables are added by the authors to examine whether companies follow industry targets.	There is a positive relation between industry ratios and companies’ distance-to-default.
Industry Average Growth Sales (Retail)	Merchant wholesalers: inventories to sales ratio7.
Industry Average Growth Inventory (Wholesalers)	Merchant wholesalers inventories8.

includes details about the independent variables, the measurement of each variable and the relevant hypotheses, as cited in related studies.

4. Results and Discussion

The statistical estimation includes three models. Model 1 is the base estimation that includes company-specific variables. Model 2 includes company-specific, macroeconomic, and industry-specific variables. Model 3 examines the effects of company-specific variables, macroeconomic variables and industry-specific variables on the stochastic DD. The results of the RESET test (Ramsey, 1969; Thursby & Schmidt, 1977; Thursby & Schmidt, 1979; Sapra, 2005; Wooldridge, 2015) show that nonlinearity fits the data where the F statistics (Prob. F) for the three models are as follows: Model 1 [F_{2, 7833} = 31.93 (0.00)], Model 2 [F_{2, 7833} = 48.88 (0.00)], Model 3 [F_{2, 7833} = 36.24 (0.00)]. Therefore, a cubic transformation is carried out. The results of the Hausman test (Hausman, 1978; Hausman & Taylor, 1981) show that the three models are subject to fixed effects. The ${\chi }_{df}^2$ statistics (p values) for the three models are as follows: Model 1 [${\chi }_3^2$ = 36.8 (0.0000)], Model 2 [${\chi }_3^2$ = 48.21 (0.0000)], Model 3 [${\chi }_3^2$ = 39.93 (0.0000)]. The minimum and maximum VIFs are 2.637 and 4.019, respectively. The DD is estimated via Equation (2). The GBM simulation via Equation (4) is used for the estimation of the stochastic DD. The results are reported in

Table 2. Determinants of historical and stochastic distance-to-default (DD).

	Model 1: Company-Specific	Model 2: Company-Specific and Country-Specific	Mode 3: Company-Specific and Country-Specific Determinants of Stochastic DD
Debt-to-Equity	−0.0002 (−3.13) **	−0.0081 (−3.04) **	−0.004 (−2.95) **
Effective Tax Rate	−0.008 (−0.89)	0.0073 (1.70) *	0.0226 (2.69) **
Bankruptcy Costs	−0.0015 (−2.77) ***	−0.039 (−3.22) ***	−0.0568 (−3.81) ***
Growth of Free Cash Flow	−0.0072 (−3.29) **	−0.0048 (−3.37) ***	−0.00037 (−0.165)
Return on Assets	−0.0051 (−2.44) ***	−0.0051 (−2.84) **	−0.0021 (−1.093)
Price-to-Earnings	−0.0037 (−1.08)	−0.004 (−0.73)	−0.0061 (−0.48)
Size (ln Sales Revenue)	1.050 (10.68) ***	1.230 (9.93) ***	1.7296 (11.99) ***
Size (ln Market Value)	0.025 (9.01) ***	0.851 (8.41) ***	0.7052 (8.01) ***
Growth of GDP		−5.948 (−3.58) ***	−12.63 (−1.181)
Inflation Rate		−0.018 (−1.59)	−0.008 (−0.055)
Productivity Growth		1.018 (1.20)	−4.423 (−0.306)
% Change in Manufacturing Output		−0.328 (−0.011)	2.829 (1.991) **
T-Bills		4.678 (3.75) **	−4.283 (−1.940) ***
Growth Unemployment Rate		−0.996 (−1.87) *	4.122 (0.2953)
Industry Average Retail Inventory to sales	84.625 (5.85) ***	140.478 (8.61) ***	41.38 (5.621) ***
Industry Average Growth Sales (Retail)	−1.170 (−6.09) ***	−1.433 (−2.39) **	2.391 (0.994)
Industry Average Growth Inventory (Wholesalers)	3.123 (2.72) ***	2.197 (2.29) **	−10.168 (−0.228)
Type of Industry (Dummy, Binary)	Yes	Yes	Yes
Constant	4.210 (10.36) ***	5.378 (11.97) ***	−72.38 (−5.947) ***
R2	0.6654	0.8335	0.8732
F Statistic	24.75 ***	31.31 ***	22.28 ***
VIF (Max)	2.8	4.64	3.70
N	7848	7848	7848

Notes: The dependent variables are historical and stochastic DDs. The estimation equation is as follows. $y_{\mathrm{tk}}={\mathrm{\alpha }}_{\mathrm{k}}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{k}}{\mathrm{\beta }}_{\mathrm{ik}}{\mathbf{X}}_{\mathrm{itk}}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{k}}{\mathrm{\beta }}_{\mathrm{ik}}{Z}_{\mathrm{itk}}}+{\upsilon }_{\mathrm{tk}}}$, where t = 1, …, n; k = number of companies; ${\mathbf{y}}_{\mathrm{tk}}$ = ln DD; and $X_{\mathrm{itk}}$ includes three groups, namely company-specific, macroeconomic, and industry-specific variables. $Z_{\mathrm{itk}}$ = dummy variables (binary values) to capture the type of industry (Westerlund, 2007), and ${\upsilon }_{\mathrm{tk}}$ = random error. * Significant at 1%, ** Significant at 5%, *** Significant at 10%.

.

The negative estimates of the debt-to-equity ratio reflect the expected consequences of debt financing, which is usually considered alarming in regard to the use of excessive debt (Chandrapala & Knápková, 2013; Lee, 2014; Affandi et al., 2019; Issa et al., 2024).

The positive effect of the stochastic effective tax rate offers updated insights into the effects of tax savings on debt financing (Kemsley & Nissim, 2002; Barbi, 2012; Han et al., 2017). Good management of debt financing can effectively benefit from debt tax savings as long as the tax savings are associated with an increase in interest rates (Fischer & Jensen, 2024).

The negative effect of bankruptcy costs (White & Turnbull, 1974; Warner, 1977; Gong, 2004) on the DD is evidenced either by the observed estimates or the stochastic estimates. Therefore, companies consider bankruptcy costs (Altman & Saunders, 1998; Jacobson et al., 2013). The statistical significance of bankruptcy costs updates the arguments made earlier (Haugen & Senbet, 1978).

The historical estimates of the DD in Table 2 are associated with the negative effects of the growth of free cash flow. Nevertheless, the stochastic estimates of the DD show no expected statistical significance.

The positive estimates of the natural log of sales revenue and the natural log of a company’s equity market value offer further insights into the effect of size on the reliability of using the DD as a measure of bankruptcy risk. That is, in terms of sales revenue, large companies are usually associated with greater DDs as long as the equity market value is greater than the book value of debt (Bhutta & Hasan, 2013; Kuntluru et al., 2008; Kalak & Hudson, 2016; Singh & Singla, 2021; Lee, 2014; Dang et al., 2018; Black & Scholes, 1973; Merton, 1974). Therefore, the results offer a significant indication that both proxies of the size of the company are valid and reliable.

In terms of the positive effect of the percentage change in manufacturing output on the DD, Stenbäck (2013) concludes that an increase in the industrial production index (IPI) would result in a higher GDP and, accordingly, an increased distance-to-default.

The positive effect of T-bill rates on the estimation of the DD extends the findings of Bessembinder et al. (2019). Nevertheless, the negative stochastic estimates reflect the operationalization of the Merton model (Merton, 1974) and are also concluded by Laurin and Martynenko (2009) and Fang et al. (2021). That is, it is plausible to assume that increases in risk-free rates deter further corporate investments, as the former might be a viable investment, which is intrinsically associated with no credit risk.

The positive effect of the industry average retail inventory of sales on historical and stochastic DD offers an indication that companies use this ratio as a target and actually become close enough to the target, which ensures that the operating profits are enough to increase the DD (Kayo & Kimura, 2011).

Notably, the results in Table 2 reflect three perspectives. First, the trends of the estimated coefficients reflect the inherent characteristics of the DD in the BSM model. That is, the DD is able to differentiate between low and high bankruptcy risk whether in the case of historical or stochastic estimates. Second, the estimated coefficients of the GBM add to the reliability and robustness of the stochastic estimates of the DD as far as the inherent stochastic movements in stock prices affect the estimated DD, which is shown in Figure 3.

Third, the PB ratio contributes to the estimation of the historical as well as stochastic DD, as does the equity market value, as shown in Figure 4. In fact, the PB ratio outperforms the equity market value in terms of reflecting shareholder values.

5. Conclusions

As the estimation of bankruptcy risk requires historical data, the efficient management of bankruptcy risk requires stochastic simulations. Notably, the latter extends the robustness of the DD (Jessen & Lando, 2015). Although the sample in this study includes companies that are far from bankruptcy, a stochastic simulation using the GBM shows that the expected DDs are greater than historical DDs, which calls for an examination of the financial aspects that companies must follow in order to reach higher expected DDs than those observed. These financial aspects that help manage stochastic DDs are as follows:

• A reduction in debt financing proportional to equity financing;
• The inclusion of tax savings in the determination of debt financing;
• An adoption of marketing strategies that promote sales growth;
• Making investment decisions that strengthen companies’ market value in the stock market;
• An adoption of targets derived from the industry, specifically, the industry average retail inventory to sales.