Company Cost of Capital and Leverage: A Simplified Textbook Relationship Revisited

For company valuation purposes, the weighted average cost of capital (WACC) approach is well-established and a frequently used tool. According to this discounted cash flow (DCF) method, the firm value results from discounting the company’s unlevered free cash flows net of taxes with the prominent WACC discount rate: Once the debt ratio \(\frac{D}{V}\) (and correspondingly the equity ratio \(\frac{E}{V}=1-\frac{D}{V}\)), the company cost of capital \(k_{V}\) (or alternatively the cost of equity \(k_{E}\)), the cost of debt \(k_{D}\), and the corporate tax rate \(\tau\) are known, we can easily determine the WACC in order to discount the unlevered free after tax cash flows of a levered company.

A typical case in business practice, however, is that a company might want to substantially change its debt ratio once such as e.g., after an acquisition, a private equity investment, or a leveraged buyout. Clearly, the new intended debt ratio \(\frac{D}{V}\) should be taken into account for the present value of future cash flows so that the WACC discount rate needs to be adjusted, accordingly. While the direct effect from the debt ratio \(\frac{D}{V}\) in Eq. (1) is trivial, the major issue still concerns the question whether the company cost of capital \(k_{V}\) does change with the debt ratio \(\frac{D}{V}\) in both an economic and a mathematical sense.

Even though the total cash flow approach (suggested by practically oriented financial auditors such as IDW (2014) on page 56) regards \(k_{V}\) equal to the cost \(k_{U}\) of the unlevered firm and therefore ignores any impact from the debt ratio \(\frac{D}{V}\) on company cost of capital \(k_{V}\), we find a mathematical effect when referring to some more sophisticated references. Already in Miles and Ezzell (1980) and accordingly in Sick (1990), we can find the following representation translated to our notation: Equating both Eqs. (1) and (2), we obtain a simplified relation between \(k_{V}\) and \(k_{U}\): We can easily see that there is a mathematical difference between \(k_{V}\) and \(k_{U}\) so that the debt ratio \(\frac{D}{V}\) does impact \(k_{V}\). Given a positive risk premium \(k_{U}-k_{D}> 0\), the company cost of capital \(k_{V}\) must be below the unlevered cost \(k_{U}\). The numerical magnitude of this effect is, however, negligible. Even in an extreme case, with a risk premium \(k_{U}-k_{D}\) equal to \(8.00\%\), cost of debt \(k_{D}\) equal to \(2.00\%\), a debt ratio \(\frac{D}{V}\) equal to \(90\%\), and a tax rate equal to \(35\%\), the company cost of capital \(k_{V}\) for unlevered cost equal to \(10.00\%\) amounts to \(9.951\%\) and is nearly the same. This deviation of \(0.049\%\) is usually far below the estimation accuracy of the cost of risky assets. Hence, this finding economically justifies the approach followed by IDW (2014) which does not regard any effect from the debt ratio on the cost \(k_{V}\), on the one hand side, and still accounts for a marginal mathematical impact on the company cost of capital \(k_{V}\) once the debt ratio changes, on the other side.

However, this practically helpful outcome bases on a model framework without any default risk. Therefore, the important question still remains whether the cost of capital \(k_{V}\) might substantially change with the debt ratio \(\frac{D}{V}\) if the firm is subject to default risk. Credit risk might be a crucial aspect for this relationship, because a higher debt ratio impacts the company in multiple ways such as the magnitude of the default probability, the amount of tax shields realized each period, and the triggering of additional losses due to default, also known as bankruptcy costs. For this reason, it is no longer obvious whether the company cost of capital \(k_{V}\) is still sufficiently close to the unlevered cost \(k_{U}\) for practical purposes and invariant of the debt ratio. Since the approximate equality between \(k_{V}\) und \(k_{U}\) is still frequently used in business practice, we need to investigate this relation in detail in order to justify or reject it for arbitrary cases.

The Covid-19 crisis has shown that the risk of a default is a highly relevant issue and additionally might change the established conditions. For example, under the German national bankruptcy regulations, the rules for filing for bankruptcy have been relaxed. Hence, our expectation is that the probability of default throughout the Covid-19 crisis was supposed to be reduced due to a postponed insolvency process. On the contrary, those companies, who had ultimately filed for bankruptcy, should have exhibited higher bankruptcy costs because of the additional time before formal bankruptcy in which those adverse effects took place.

It is astonishing, that there are only few recent developments of DCF valuation approaches that account for default risk. Kruschwitz et al. (2005) analyze the impact of a potential bankruptcy on the value of a firm under conditions of uncertainty. They deliberately assume identical gross cash flows, regardless of whether the company is more or less exposed to default risk and stress the difficulty concerning the quantification of the present value of bankruptcy costs. Cooper and Nyborg (2008) examine the impact of an investor’s personal taxes on the valuation of tax shields in the case of default-risky debt and compare their results with the adjustment formula of Miles and Ezzell (1980). Molnár and Nyborg (2013) expand this approach by the inclusion of possible recovery effects on the tax shields of a default-risky company. Koziol (2014) provides a closed-form solution for adjusting the WACC discount rate to account for default risk and bankruptcy costs. Koziol’s paper proposes a simple adjustment of the WACC rate to include both default risk and bankruptcy costs in a consistent firm valuation. Krause and Lahmann (2016) deal with the prioritization of principal or interest payments in the event of a default and evaluate the potential differences. Baule (2019) proposes a continuous-time model to illustrate the impact of default risk and bankruptcy costs on a firm’s cost of debt.

The aim of this paper is (i) to analyze the relation between company cost of capital \(k_{V}\) and the debt ratio \(\frac{D}{V}\) in the presence of default risk, (ii) to provide an economic understanding for its drivers, and (iii) to quantify potential pricing errors using empirical cases. When evaluating this question within the continuous-time framework by Leland (1994), we can confirm a relation similar to Eq. (3) that the company cost of capital \(k_{V}\) is marginally below the unlevered cost \(k_{U}\) unless default risk matters. However, the characteristics of \(k_{V}\) dramatically change, once default risk is an issue for the firm. For those debt ratios, \(k_{V}\) can strongly increase with \(\frac{D}{V}\) and exceeds the cost of an unlevered firm excessively. After introducing a simple perodic DCF pricing model under default risk, we calibrate it to 29 companies from the German stock market to estimate the difference \(k_{V}-k_{U}\) between the two different costs of capital and to evaluate its practical significance. Our empirical analysis reveals that the differences \(k_{V}-k_{U}\) can be clearly depending on the size of bankruptcy costs. Even for moderate bankruptcy costs, we obtain a mean difference between \(k_{V}\) and \(k_{U}\) equal to 1.88 percentage points which creates a company mispricing of nearly 100%. In the special case of high bankruptcy costs resulting in a worthless firm after the default process, the average difference between \(k_{V}\) and \(k_{U}\) rises to 3.67 percentage points associated with arbitrarily increasing company mispricing errors. As a result, the company cost of capital \(k_{V}\) does practically not depend on the debt ratio if the firm is not subject to default risk or if bankruptcy costs are negligible. Otherwise, for companies with economically significant default probabilities and bankruptcy costs, changes of the debt ratio strongly impact the company cost of capital and necessarily need to be considered.

The paper is structured as follows: In Sect. 2, we endogenously determine the company cost of capital in the prominent time-continuous model framework by Leland (1994) to demonstrate the fundamental effects of default risk and bankruptcy costs. In Sect. 3 we switch to a discrete time environment allowing for default risk and bankruptcy costs. Subsequently, we calibrate the theoretical model results with empirical data in Sect. 4. The paper concludes in Sect. 5. Technical developments are in the appendix.

In this section, we regard a well-established model framework in continuous time, i.e. the Leland (1994) framework, in order to endogenously determine the company cost of capital for a firm subject to default risk. From Berk and DeMarzo (2014) on page 652 and Miles and Ezzell (1980) in Eq. (20), we can directly see that the company cost of capital does change when the firm changes its debt ratio. Still, as obtained from formula (3), the numerical differences between the company cost of capital \(k_{V}\) and the cost \(k_{U}\) of an otherwise identical but unlevered firm are negligible for typical parameter values. This observation justifies the usual practice to keep the company cost of capital \(k_{V}\) constant even if the firm follows a change in its leverage policy.

However, this simple and fortunate conclusion must not necessarily hold for every particular firm. This is because Eq. (20) from Miles and Ezzell (1980) was explicitly derived for firms not exposed to default risk. In order to get an intuitive understanding how default risk impacts the relationship between company cost of capital \(k_{V}\) and debt ratio and whether the practical rule that \(k_{V}\) does not (considerably) change once the firms alters its debt ratio, we derive the instantaneous cost of capital within the Leland framework.

The Leland model considers a firm with a stochastic asset value \(U\) that has the character of an unlevered firm value. Default risk comes from an outstanding debt obligation in form of a coupon stream \(c\). Once the asset value \(U\), that follows a geometric Brownian motion with standard deviation \(\sigma\) of the return, hits the endogenous barrier \(U_{B}\), the firm defaults. In this case, the firm is liquidated so that the debtholders obtain \((1-a)\cdot U_{B}\) taking proportionate bankruptcy costs \(a\) into account and the equityholders are left with nothing. Otherwise, it is optimal for the firm/equity holders to pay the coupon which creates a tax shield equal to \(\tau\) per unit of coupon paid. With the typical valuation assumptions and methods in a Black-Scholes world, the endogenous value \(V\left(U\right)\) of a firm with unlevered firm value \(U\) accounting for tax shields and bankruptcy costs results in where \(r\) stands for the risk-free rate and the default barrier amounts to \(U_{B}=\frac{c\cdot\left(1-\tau\right)}{r+\frac{1}{2}\sigma^{2}}\). The firm value can be represented by a replicating portfolio consisting out of units of the unlevered firm \(U\) and a risk-free asset. In Appendix I, we show that the dynamic, self-financing, replicating portfolio \(RP\) exhibits the following portfolio weights \(w_{U}\) and \(w_{f}\) for holdings of the unlevered firm value \(U\) and the risk-free asset, respectively: Since the company cost of capital coincides with the expected return obtained with the firm value, we endogenously determine the instantaneous return \(\mu_{V}\) of the firm value (which has the character of the company cost of capital) for an instance of time. Once, the instantaneous return \(\mu_{U}\) of the unlevered firm is given, the time-continuous company cost of capital \(\mu_{V}\) reads: The debt ratio \(\frac{D\left(U\right)}{V\left(U\right)}\) defined as the debt value over the firm value depends on the asset value \(U\) and is given by: A crucial characteristic of the Leland model is that the asset value \(U\) in relation to the default barrier \(U_{B}\) determines the so called distance to default and therefore the default risk of the company as well as the debt ratio.

For \(U\to U_{B}\), the firm is close to a default, the equity value is almost worthless and the debt ratio tends to one. On the other hand, for \(U\to\infty\), the debt ratio approaches zero as the debt value cannot exceed its default-free value equal to \(\frac{c}{r}\) and the firm value becomes arbitrarily large. Fig. 1 plots the company cost of capital \(\mu_{V}\) for different debt ratios \(\frac{D\left(U\right)}{V\left(U\right)}\). For debt ratios below 0.75, the firm exhibits a rather low default risk and therefore bankruptcy costs do not matter. In this region, the company cost of capital \(\mu_{V}\) is (marginally) below the cost \(\mu_{U}\) of an unlevered firm.

When the debt ratio, however, tends to one so that default risk as well as bankruptcy costs are an issue, the cost of capital \(\mu_{V}\) increases heavily up to a limit equal to \(55.37\%\). Formally, the limit can be written as:

The conclusion from this example in Fig. 1 is twofold: First, it confirms the findings from Berk and DeMarzo (2014) and Miles and Ezzell (1980) for the case without (or negligible default risk), i.e., debt ratios below 0.75. The intuition for why the company cost of capital \(\mu_{V}\) is marginally below \(\mu_{U}\) is because the levered firm can be represented by the unlevered firm and a risk-free position stemming from the (almost) risk-free tax shields. Thus, the levered firm is like a portfolio out of both positions (with positive holdings) so that the expected return, i.e., \(\mu_{V}\), needs to be between the expected returns \(\mu_{U}\) and \(r\) of both particular positions. Due to the usual parameter values, the tax shields attribute to a minor part relative to the assets \(U\), so that the company cost of capital \(\mu_{V}\) is particularly close to the unlevered cost \(\mu_{U}\).

Second, Fig. 1 also reveals that the company cost of capital is highly sensitive to changes in the debt ratio for default risky firms. The notion behind this important outcome from the Leland model is as follows: As a result of tax shields, which primarily matter in favorable states of the asset value \(U\), and bankruptcy costs, which rise for low asset values \(U\), the risk of the firm value \(V\left(U\right)\) is higher than that of the unlevered firm value \(U\). Illustratively speaking, when \(U\) increases the firm value additionally benefits from tax shields, but a reduction of \(U\) is additionally associated with a loss of the firm value \(V\left(U\right)\) due to bankruptcy costs. Technically speaking, the replicating portfolio \(RP=V\left(U\right)\) consists of a positive holding in the asset value \(w_{U}\cdot RP> 0\) but the risk-free part of the replicating portfolio \(w_{f}\cdot RP<0\) has a negative value (short position). Due to the prominent leverage effect, the expected return \(\mu_{V}\) of the replicating portfolio, i.e., the company cost of capital, needs to exceed the expected return \(\mu_{U}\) of the unlevered firm as long as the usual relation \(\mu_{U}> r\) applies.

A further important outcome from the company cost of capital \(\mu_{V}\) in formula (4) with severe default risk is that bankruptcy costs play a major role for the value of \(\mu_{V}\). When the relative bankruptcy costs \(a\) increase to its maximum value one, the company cost of capital \(\mu_{V}\) can become arbitrarily large. Still, even without bankruptcy costs, \(a=0\), the company cost of capital \(\mu_{V}\) can deviate substantially from \(\mu_{U}\) due to tax shields. In this case for \(a=0\), the limit simplifies to \(\widehat{\mu}_{V}=\mu_{U}+\left(\mu_{U}-r\right)\cdot\frac{2r+\sigma^{2}}{\sigma^{2}}\frac{\tau}{1-\tau}\). In line with Fig. 2, the limit \(\widehat{\mu}_{V}\) of the company cost of capital increases with the tax rate. While in the absence of taxes \(\tau=0\), \(\widehat{\mu}_{V}\) coincides with \(\mu_{U}=10\%\), it is fundamentally higher for typical tax rates. For \(\tau=25\%\), the limit \(\widehat{\mu}_{V}\) amounts to \(19.1\%\) and it even obtains a value equal to \(37.2\%\) when the tax rate is \(50\%\).

In this chapter, we document the economic significance of the difference between \(k_{V}\) and \(k_{U}\) using real firms. The intention of this chapter is not to provide general results for companies with different levels of debt or default risk. This would be nearly impossible due to the heterogeneity of companies in terms of financing policy, industry or operating activities. Rather, the objective is to show that real companies exhibit economically significant differences in \(k_{V}\) as soon as bankruptcy costs are considered.

Therefore, we will firstly demonstrate the calibration of the model using an example of the US capital market with severe default risk, and then secondly apply to a larger data set to demonstrate that each firm can be valued within the existing model framework.

With „Range Resources Corporation“, we first choose a company in the US American oil and gas production sector that exploits conventional energy sources, particularly by using „fracking“ technology in the USA. Especially, the drop in oil prices in the years 2014–2017 had a negative impact on the credit rating of Range Resources. With Caa1 (Moody’s) and BB\(+\) (StandardPoors) the company exhibits rating at non-investment grade. Further, we choose January 1, 2018 as the valuation date. To calibrate the model analogous to the theoretical considerations in Chapt. 3, the following exogenous parameters are essential:

The risk-free interest rate \(r_{f}\) is derived from the yields of 30-year US American government bonds and amounts to 2.82% as at January 1, 2018. The Company reports total liabilities of USD 5,954.6 million at December 31, 2017. The market capitalization amounts to USD 4,233.3 million, resulting in a debt ratio \(\frac{D}{V}\) of 58.4%, which we assume to be constant for the future. The US corporate tax rate of 35% is supposed to be constant as well. We also assume that the company will continue at a moderate growth rate of \(u=1.02\) in the case of solvency.

There are different approaches and models in financial literature to estimate the probability of default of firms with the help of scoring and rating models. Still, a simple and objective way to derive a firm’s probability of default \(p\) is to refer to external rating data of the firm and to use these rating scores to derive the probability of default from historical default frequencies of firms from the corresponding rating class. To get a value as stable as possible for the one-year probability of default \(pd_{1}\), we use the 10-year default probability \(pd_{10}\) and convert this into the corresponding one-year probability of default \(pd_{1}\) as exemplarly shown at Hartmann-Wendels et al. (2019), page 439: The rating grades from Moody’s and StandardPoors mentioned at the beginning of this section result in a weighted one-period probability of default of 5.37%.

Concerning the bankruptcy costs \(\alpha\), practical examples show that in case of a post-bankruptcy liquidation, up to 100% of the assets of a firm before bankruptcy can be completely lost. In scientific discourse a distinction is made between direct and indirect bankruptcy costs, which affect companies in total in the event of a default. Direct bankruptcy costs comprise legal expenses, court costs or advisory fees. The indirect bankruptcy costs are apparent in the loss of reputation, important customers and employees as well as the potential loss due to a fire sale of assets or an inefficient liquidation process. Various empirical studies concerning direct bankruptcy costs have been published, for example Baxter (1967); Warner (1977); Altman (1984); Weiss (1990); Betker (1997); Lubben (2000); Thorburn (2000) and LoPucki and Doherty (2004). These studies show average values between 2 and 7% for large companies, depending on the sample examined. Most studies only publish mean values, whereas the outer maximum percentiles are also of major interest. Betker (1997) mentions here, for example, a maximum value of 14%. The values also depend very strongly on the size of the firms within the sample. In the case of Chapt. 7 liquidations, Lawless and Ferris (1997) report a maximum loss of 96.1% close to a total loss of assets due to bankruptcy. This case corresponds to bankruptcy costs equal to \(\hat{\alpha}\) in our framework. Indirect bankruptcy costs are typically higher than direct bankruptcy costs and more difficult to quantify accurately. Therefore, the number of empirical studies is also smaller, among which, for example, Kwansa and Cho (1995); Andrade and Kaplan (1998) and Bhabra and Yao (2011) should be mentioned. Bhabra and Yao (2011) differentiate between different sectors and show relatively high indirect bankruptcy costs for very research-intensive and personnel-dependent technology firms, averaging up to 27%. Kwansa and Cho (1995) document maximum values of 43.2%. In total, i.e. direct and indirect bankruptcy costs combined, large firms across all sectors can exhibit severe magnitudes and very strongly among sectors. Due to the fact that up to 100% of the assets can be lost in the event of a liquidation following bankruptcy, we present the full range of bankruptcy costs in this model calibration.

Both, the cost of capital \(k_{U}\) of fictitiously unlevered firms and the down factor \(d\) are not intuitively observable on the capital market. Thus, we consider two proxies, \(k_{E}\) and \(c\), to calibrate \(k_{U}\) and \(d\) by using two conditions and Eq. (9). In particular, we apply the interest rate \(c\) as calibration constraint (A) and the cost of equity \(k_{E}\) as calibration constraint (B) to simultaneously estimate \(k_{U}\) and \(d\).

For the calibrations constraint (A), we empirically estimate the interest rate \(\hat{c}\) using credit spreads from bonds issued by Range Resources. Since in the present case a total of 13 bonds with different maturities were issued on the valuation date, we use these bonds to determine a maturity-weighted credit spread and thus determine the company’s interest rate on debt. This procedure results in an empirical observed interest rate \(\hat{c}\) of 5.79%.

In order to use \(k_{E}\) within the framework of calibration constraint (B) for \(k_{U}\), we determine the estimate \(\hat{k}_{E}\) using the well-known capital asset pricing model and use the firm’s beta factor \(\beta_{E}\) and the market risk premium \(\mu_{\text{BM}}-r_{f}\) in the following well-known way: We calculate the beta factor \(\beta_{E}\) using historical share prices for the three years 2015, 2016, and 2017 and regress daily returns of the single shares against the daily returns of the broadly diversified SP 500 index, which amounts to 1.17. Subsequently, we calculate the market return \(\mu_{\text{BM}}\) using the historical average of 1‑year returns of the MSCI World from 2000-2017 resulting in a market return of 6.92% and a market risk premium of 4.10%. Thus, the empirically measured cost of equity \(\hat{k}_{E}\) amounts to 7.62%.

To relate the empirical estimate \(\hat{k}_{E}\) to the endogenous outcome \(k_{E}\) from our model, we can use Eq. (20) for the cost of equity \(k_{E}\) as expected equity return. From the multiplier \(f_{V}\) in Eq. (18), with that the firm value is related to the cash flows, and the target debt ratio \(\frac{D}{V}\) of the respective firms, we can also determine \(f_{D}=\frac{D}{V}\cdot f_{V}\) and \(f_{E}=(1-\frac{D}{V})\cdot f_{V}\). Therefore, we can write for \(k_{E}\):

We simultaneously adjust both the factors \(d\) and \(k_{U}\) until the endogenous interest rate \(c\) and the endogenous cost of equity \(k_{E}\) match the empirically observed interest rate \(\hat{c}\) and cost of equity \(\hat{k}_{E}\). We perform this two-condition-approach for different levels of bankruptcy costs relative to the maximum value \(\hat{\alpha}\) of the respective firms. As a final step, capital cost \(k_{V}\) of the levered firms is calculated using Eq. (17).

In Table 4 we present the results of the calibration of Range Resources for different levels of bankruptcy costs \(\alpha\). The maximum bankruptcy costs \(\hat{\alpha}\), which can be calculated from the given parameters \(\hat{k}_{E}\) and \(\hat{c}\), are 61.0%. Since the cost of equity \(k_{E}\) and the interest rate \(c\) within the scope of this calibration have the same level for different bankruptcy costs \(\alpha\), \(k_{V}\) with a value of 5.6% is also the same throughout. The higher the bankruptcy costs \(\alpha\), the lower \(k_{U}\) must be chosen to ensure that \(\hat{k}_{E}\) is equal to \(k_{E}\). Likewise applies to the factor \(d\) that this also rises with rising bankruptcy costs, in order to ensure \(\hat{c}\) equals \(c\) due to the calibration.

In general, it can be seen that the differences between \(k_{V}\) and \(k_{U}\) increase significantly with rising bankruptcy costs \(\alpha\). This is also reflected in increasing pricing errors PE. With bankruptcy costs of 0%, the pricing error PE with 0.3% is economically negligible. However, even if bankruptcy costs of 40% are taken into account, equating \(k_{U}\) and \(k_{V}\) would lead to a pricing error of 58%. With the maximum bankruptcy costs of 61.0%, the pricing error strongly rises above 200%.

As described in assumption 9, at any point in time, the residual value \(d\cdot X_{t}+d\cdot V_{t}-\alpha\cdot V_{t}\) in the down state must be less than interest and repayments \(c\cdot D_{t}+D_{t}\) to satisfy the default condition. For this reason, the following inequality must be met:

Dividing inequality (22) by \(V_{t}\) gives the following simplification:

In case of Range Resources Corp., we illustrate the validity of this inequality by introducing the parameter Distance to Solvency (DtS). Thereby, we set the residual value \(d\cdot X_{t}+d\cdot V_{t}-\alpha\cdot V_{t}\) in proportion to the value of interest and repayments \(c\cdot D_{t}+D_{t}\). Table 4 demonstrates that in the case of Range Resources Corp., the residual value is 29.3% lower than the required value for repayment and interest to achieve solvency of the firm.

In order to show that these results are not only valid for firms with a severe probability of default, we calibrate the present model using large German stock corporations. The general scope of this study comprises 130 companies from the German DAX, MDAX and SDAX indices, of which we can finally use the data of 29 companies to conduct the calibration of our model as of January 1, 2018. However, we do not expect any major fluctuations in results depending on the valuation date concerning the historical input data. We obtain key corporate data, such as market capitalization, balance sheet information and rating grades from the standard information data service Thompson Reuters Eikon, subtracted on March 14, 2019. Furthermore, we use publicly available information from the Deutsche Bundesbank for interest rates as well as the rating agencies Moody’s, Standard Poors and Fitch.

We first take the risk-free interest rate \(r_{f}\) from the implicit spot rate with a maturity equal to the time to maturity of that German sovereign bond with a maximum lifetime of 30 years. The necessary parameters for this are provided by Deutsche Bundesbank, calculated using the approach by Svensson (1994) and result in a value of 1.29%. The debt ratio \(\frac{D}{V}\) results from historical balance sheet data for the years 2014 to 2017. From this, we form the arithmetic mean as a proxy for the target debt ratio of the firms (see Appendix IV, Table 7). Additionally, for the conditional growth rate of cash flow and firm value \(u\) in case of surivorship we use the target inflation rate of the European Central Bank of 2% per period. As a result, we fix the value of the factor \(u\) at 1.02 for all firms. For the corporate tax rate \(\tau\), we choose a fixed tax rate of 30%, since this tax rate is close to the average actual tax burden from corporate tax in the German tax system. Moreover, all firms are supposed to be subject to similar tax rates in the long run. In line with the procedure above external rating information available from the well-known rating agencies Moody’s, SP, and Fitch at the time of valuation. Among our basic scope of 130 companies, 45 companies have an external rating. Since all rating agencies publish historical default frequencies to indicate the rating scores, we use these percentages as input for the derivation of a default probability in our model. The respective rating grades and corresponding probabilities of default are listed in Table 8 in Appendix IV. Moreover, we obtain the interest rate \(c\) from the bond yields of the respective debt capital bonds issued. If a company has issued several bonds, the spreads are weighted by volume and maturity. As we find bonds issued by 29 of the 45 companies which have an external rating, we use these 29 firms as final scope of our study. The empirically observed values for \(\hat{c}\) are listed in Table 7 in Appendix IV.

Table 5 shows the difference between the cost of capital of the levered companies \(k_{V}\) and the unlevered companies \(k_{U}\) depending on the respective level of \(\alpha\). In column 2, we represent the maximum bankruptcy costs \(\hat{\alpha}\) of the respective firms of which we present different levels in columns 3 to 7, ranging from no bankruptcy costs \(0\cdot\hat{\alpha}\) to maximum bankruptcy costs \(\hat{\alpha}\).

All companies within the scope of this study also confirm the results found at Range Resources Corp: The difference between \(k_{V}\) and \(k_{U}\) increases with rising bankruptcy costs \(\alpha\). Without bankruptcy costs, as shown in column 3, the deviations are marginal but still observable.⁷ However, if we consider relatively low bankruptcy costs of \(\frac{1}{4}\cdot\hat{\alpha}\), we already see remarkable deviations in cost of capital of 0.95% on average with a maximum of 1.57% for Infineon. In case of the maximum bankruptcy costs at \(\hat{\alpha}\), we notice that the deviations increase on average to 3.67%, with a maximum value of 6.03%. The deviations become even more conspicuous when we consider the pricing error in relation to the firm value. In Table 6, we represent the pricing error analogous to the approach of Eq. (21). The higher the percentage value of the pricing error, the higher the company would be valued if \(k_{U}\) was used rather than \(k_{V}\). With bankruptcy costs of 0%, the pricing error is also marginal. However, if a quarter of the maximum bankruptcy costs affects the companies, we can already notice a pricing error of 32.1% on average. At 50% of the maximum bankruptcy costs, companies are remarkably overpriced by an average of 95.5%. At 75% of the maximum bankruptcy costs, the mispricing even ranges between 104.9 and 309.5%. With maximum bankruptcy costs \(\hat{\alpha}\), the pricing error reveals tremendous values.

The practical treatment that the company cost of capital does not change (approximately) when the firm follows a one-time, substantial change in the debt ratio is not an analytical relationship but a simplification. Numerical inspections justify equating the cost \(k_{V}\) of a levered company and the cost \(k_{U}\) of an unlevered one due to extremely small deviations in case the company has no default risk.

However, in the case of default risk and with bankruptcy costs, the cost of capital \(k_{V}\) of a levered company is significantly higher than the cost of capital \(k_{U}\) of an identical unlevered firm. In this context, \(k_{V}> k_{U}\) applies to any level of bankruptcy costs and can lead to significant pricing errors. Calibrating the cost of capital of real listed firms within the presented DCF model framework confirms, on the one hand, the general applicability and, on the other hand, the high pricing errors even for firms with relatively good rating grades. Especially the level of bankruptcy costs significantly affects the difference between \(k_{V}\) and \(k_{U}\). Accordingly, in the event of bankruptcy in which no residual value of the firm remains, the deviations between \(k_{V}\) and \(k_{U}\) amount to 3.67 percentage points on average with serious pricing errors.

The consideration of bankruptcy costs and default risk provides specific challenges in practical applications. Since any given firm usually has no observable bankruptcy costs – while it is alive – a reference to experiences of historical default must be applied. However, empirical studies concerning direct and indirect bankruptcy costs exhibit highly heterogeneous results making it necessary to prudently treat this variable.

A further main driver of the difference between \(k_{V}\) and \(k_{U}\) is the risk premium of \(k_{U}\) relative to the risk-free rate \(r_{f}\). In the present case, the observed cost of equity \(\hat{k_{E}}\) was used to indirectly determine \(k_{U}\). Again, the estimates are primarily sensitive in terms of the magnitude of the market risk premium and to a minor extent to the correct beta factor of the particular firm.