The Cost of Debt Capital Revisited

We propose a method to consistently estimate the cost of debt in a continuous-time framework with an infinite time horizon. The approach builds on the EBIT-based model of Goldstein et al. (2001). The model is capable of splitting the observed yield spread of a corporate bond into the risk premium, which adds to the expected return of bondholders, and the default premium, which accounts for expected losses. The model can easily be calibrated for non-public firms. The model-implied cost of debt proves to be very insensitive with respect to non-observable parameters. Analyzing the weighted average cost of capital (WACC) in the model, we demonstrate potential errors from using the textbook formula for the WACC.


Introduction
Cost of capital is one of the central issues in corporate finance. For a company's management, the cost of capital is an important benchmark for capital budgeting and performance measurement.
For external investors, the cost of capital is the appropriate discount rate for future cash flows to determine the value of a company. Estimating this key figure comes down to estimating its components, cost of equity and cost of debt.
While there is a large amount of literature concentrating on the estimation of the cost of equity, little attention has been focused on the cost of debt. It is common practice to simply use the notional yield of the company's debt securities. 1 However, using the notional yield neglects the risk of default, and is thus not adequate. Instead, cost of capital, defined as opportunity costs, is the required expected return of capital suppliers. 2 While the cost of equity is the expected return of stockholders, the cost of debt is the expected return of bondholders. When the debt is risky, meaning, when there is a non-zero probability of default, the expected return is not identical to the notional return because the risk of possible loss in the event of a default lowers the expected return.
In particular, during the recent financial crisis, the failure of the notional yield approach became obvious. With dramatically increasing default risk, the notional yields of many companies have risen to very large values. It is obvious that these large yields reflect the increased probability of default, especially in cases where the promised yield exceeds the cost of equity. To give an illustrative example, in September 2008, one year before maturity, the 4.75% General Motors medium term note traded at 80%, resulting in a yield to maturity of 104.75/80 − 1 = 31%.
Obviously, investors could not expect to actually receive a 31% return, but rather a lower 1 See, for example, the textbooks of Damodaran (2002) and Ross et al. (2008).
2 This correct approach is described in the textbook of Brealey et al. (2008), for example. expected return in line with the default risk.
To calculate the expected return, the probability of default and the recovery rate must be considered. If we assume a probability of default of 20% and a recovery rate of 30% of face value, the expected return calculation would read (0.8⋅104.75+0.2⋅30)/80−1 = 12.25%. In the general case, for multi-period or continuous-time settings, time-dependent (marginal) probabilities of default or hazard rates are required. 3 However, estimations of default probabilities are difficult to obtain and suffer from a large estimation error.
As an alternative, the expected return on debt could be calculated analogously to the common approach of calculating the expected return on equity, using the capital asset pricing model (CAPM). According to the CAPM, the expected return of any security is determined by its systematic risk, measured by the beta factor. To reasonably estimate the beta factor of corporate debt, a reliable and stable time series of bond values is required. Unfortunately, many bonds are not exchange-traded at all, and for those that are, trading volume is often so low that observed bond quotes are not accurate. Thus, this approach is only feasible for a small number of companies with outstanding bonds that are frequently traded. Hsia (1981) demonstrates the consistency of the CAPM with option pricing theory. On this basis, Hsia (1991) suggests calculating the cost of capital in an option pricing framework. 4 Cooper and Davydenko (2007) pick up this idea and propose a method of calculating the expected return of a debt security based on the Merton (1974) model. 5 They use the model in order to split the observed yield spread of a bond, specifically, the difference between the bond's notional yield to maturity and the corresponding default-free rate, into two components: the 3 See, among others, Schönbucher (2003) for an overview of default risk modeling. 4 The approach has found its way into the popular textbook of Copeland et al. (2005). 5 The Merton model is also applied by Husmann and Schmidt (2008), who derive the "incremental borrowing rate" according to International Financial Reporting Standards (IFRS 36). default premium (expected default effect), which reflects the probability of default, and the risk premium (expected excess return), which reflects the bondholders' surplus on the expected return for bearing additional risk compared to a risk-free bond. The actual cost of debt is the risk-free rate plus the second component, the risk premium.
The great advantage of the approach is that the Merton model is not needed to estimate the absolute yield spread, which is already known as an input parameter. Instead, the model is applied to calculate the relative proportion of the default premium and the risk premium. The approach proves to be very robust with respect to the debt-to-equity ratio and the equity premium, but it is rather sensitive with respect to equity volatility. This is not a serious concern when the equity volatility can be estimated from stock-price time series. For the purpose of capital budgeting, goal-setting, performance measurement, etc. in an exchange-traded company, the approach is easily applicable. However, for the purpose of firm valuation, an equity value, as well as its volatility, is not available, as it is the very objective of the valuation process to calculate this value.
As a second drawback, the approach proves to be theoretically inconsistent when applied for the analysis of long-term valuation problems-either internally, such as for capital budgeting of long-term projects, or externally, for firm valuation. The Merton model is a single-period model by nature, so any conjunction of the Merton model and multi-period capital budgeting, or firm valuation techniques, suffers from a kind of inconsistency.
In this paper we propose a method to calculate the cost of debt capital which is theoretically consistent, and which can be used for firm valuation, as well as in cases where the equity value and its volatility are not known a priori. Our approach is based on a structural asset-value model, like the Merton model. But we do not follow Merton's restrictive assumption of modeling debt as a single zero bond with finite maturity. Instead, debt is modeled as a perpetual bond which pays a continuous coupon. This approach is consistent with the common method of discounted cash flows for firm valuation, where an infinite time horizon is considered. Furthermore, our approach has the advantage of no debt maturity being necessary as a model parameter. The approach basically builds on the model of Leland (1994). As in Goldstein et al. (2001), we do not model the asset value directly, but instead consider the flow of earnings as the source of firm value. Within a structural model, the cost of debt can be deduced from the yield spread based on market risk aversion. When risk aversion is zero, the total yield spread refers to the default premium; the risk premium is zero, and the cost of debt equals the risk-free rate. When risk aversion is greater than zero, the model can be used to split the yield spread into the default premium and the risk premium: Using risk-neutral valuation, the debt value is the sum of expected payments to bondholders under the risk-neutral measure, discounted by the risk-free rate. On the other hand, given the expectation of payments under the physical measure, the expected return of bondholders is defined by the discount rate which gives the current debt value when applied to expected payments. This expected return must be larger than the risk-free rate, but smaller than the notional yield. The relative proportion of default premium (notional yield minus expected return) and risk premium (expected return minus risk-free rate) depends on the difference between physical and risk-neutral measure, hence the market risk aversion.
In Section 2 of this paper we describe the approach in more detail. Based on our structural asset-value model, we show how to calculate the cost of capital (debt and equity) within this framework. Section 3 demonstrates how the model is calibrated to input data and provides a quantitative analysis of the model results for a typical investment-grade and for a typical highly leveraged company. Section 4 gives attention to the weighted average cost of capital (WACC).
We show the applicability of the textbook formula for instantaneous returns, but demonstrate its inconsistencies for long-term returns. Section 5 is the conclusion.

Model Setup
The basis of our model is earnings before interest and taxes (EBIT). In contrast to classical firm valuation methods based on free cash flows after taxes, such an approach allows us to treat all claimants to EBIT consistently, namely the government (taxes), bondholders (interest), and shareholders (dividends). 6 The model is very similar to Goldstein et al. (2001). 7 EBIT is assumed to follow a continuous stochastic process. To be more precise, the EBIT flow grows on average at a growth rate and is subject to random fluctuations, modeled by a geometric Brownian motion with variance rate , 8 where is the differential of a Wiener process. In contrast to accounting practice, EBIT is not a discrete figure which is calculated periodically, but is rather modeled as a continuous flow. This means the flow of an infinitesimal time interval is given by . In the case of an all-equity firm, this EBIT flow is distributed by a fraction , the corporate tax rate, to the 6 As a further difference, we consider earnings instead of cash flow. The difference is the net effect of depreciation, which lowers earnings but not cash flow, and capital expenditure, which lowers cash flow but not earnings, and changes in net working capital. We assume that these effects add up to zero, in other words, depreciation equals capital expenditure, and net working capital is kept constant.
7 In contrast to the Goldstein et al. approach, we refrain from modeling personal taxes raised on payments to investors.
8 Thus, EBIT cannot become negative. As a consequence, there is no bankruptcy risk for an all-equity firm.
government, and the rest to shareholders. 9 In the general case, the company has issued a certain amount of debt. To be consistent with the firm valuation approach, we need a debt structure that is not redeemed after one period, as it is in the Merton model. In line with Leland (1994) and Goldstein et al. (2001), we consider a perpetual coupon bond. The bond pays continuous interest at rate with respect to face value ; thus, the amount paid within time interval Δ is Δ . The interest rate can be decomposed into the risk-free rate plus a spread .
As long as the firm is not bankrupt, bondholders receive continuous payments at a rate . The remainder of the EBIT flow is distributed among the government and shareholders, as above, i.e., the government receives taxes at a rate ( − ) and the rest goes to shareholders. The total claim to the EBIT flow will be called the asset value in the following. This asset value is the sum of the claims of shareholders ( ), bondholders ( ), and the government ( ), along with the present value of bankruptcy costs ( ). 10 Using risk-neutral valuation, the asset value is given by (see Goldstein et al., 2001) where is the risk-neutral drift of the EBIT flow. The relation between the actual drift, , and the risk-neutral drift, , depends on market risk aversion. In line with Goldstein et al. (2001), we model market risk using the continuous-time capital asset pricing model (see Merton, 1973), considering the dynamics of the asset value . As and are constant, follows the same geometric Brownian motion as : On the other hand, we can write the asset dynamics using an expected (instantaneous) asset return, , and the asset payout rate. As the total EBIT flow is immediately paid out to the claimants, the payout rate on asset value equals / . Thus, using (2). Based on the continuous-time CAPM, the expected instantaneous asset return is given by the instantaneous security market line where is the market price of risk, defined as the market excess return per one unit of market risk, and is the (instantaneous) correlation between asset return, , and market return, .
From the identity of (3) and (4), using the security market line (5), it follows The difference between actual and risk-neutral EBIT drift is given by the product of the market price of risk and the systematic part of EBIT volatility. As indicated in the introduction, and as we discuss in greater detail below, this difference is also crucial for splitting the notional yield spread into the default premium and the risk premium, and thus for estimating the cost of debt.
We now model default risk. When the EBIT flow falls below the claim of the bondholders, , the company does not necessarily default, as shareholders can opt to infuse payments to avoid bankruptcy. Goldstein et al. (2001) show that it is optimal for shareholders to do so as long as the asset value does not fall below a threshold , which is specified below. In this case, the remaining asset value (which is identical to the threshold ) goes to bondholders. However, bankruptcy costs occur, and the bondholders receive only a fraction 1 − of the remaining asset value.
The bankruptcy threshold is given by (see Goldstein et al., 2001) In the following, we determine the value of the different claims, in particular, the value of debt, the value of equity, the present value of taxes (government's claim), and the present value of bankruptcy costs. The debt value is the sum of discounted interest payments until a potential bankruptcy date * plus the present value of the recovery payment at default. 12 Using riskneutral valuation techniques, the debt value is given by (see Leland, 1994, andGoldstein et al., 2001) = where is the present value of receiving 1 at bankruptcy, given by The present value of bankruptcy costs is given by The remaining asset value after bankruptcy costs and the claim of bondholders is distributed among the government and shareholders:

The Cost of Capital
Once we know the current values of debt and equity, we can use the expected payments under the physical measure to calculate the cost of capital. The cost of debt is the expected internal rate of return of bondholders . Therefore, we can again use the expectation of payments as in (10), but using the physical instead of the risk-neutral measure, and discounting payments by the expected return on debt-the cost of debt -to obtain the current debt value: This equation (14) implicitly defines the cost of debt . It is worth noting that the cost of debt is considered for a time horizon either until bankruptcy or to infinity. Calculating the cost of debt over a finite time horizon could yield different results. Instead of considering time-varying cost of capital, we refer to cost of capital as an average value over an unlimited time horizon.
Unfortunately, Equation (14) cannot be solved analytically for . However, the integrals can be solved to simplify the equation to (see the appendix): This equation can easily be solved iteratively for .
The cost of equity is determined analogously. The expected flow to shareholders at time , where expectations of the random EBIT flow are taken under the physical measure. However, payments are only received until a potential bankruptcy at time * .
Accordingly, the cost of equity, , is implicitly given by As the flow to equity is stochastic, solving the integral for it is more complicated than is solving the integral for the cost of debt. In the appendix we derive an implicit equation, which still involves the numerical computation of an integral.

Calibration Based on Market Price of Risk
Having derived an expression for the cost of debt capital in the model framework, in this section we analyze the applicability of the model in terms of input requirements, calibration, and outcome sensitivity. We have two general types of applications in mind: internal applications, for capital budgeting, performance measurement, etc., and external applications, for firm valuation.
According to (10), the model value of debt depends on the following input parameters: • current level of EBIT flow, 0 , • EBIT growth rate, , • notional debt, , • interest rate on corporate debt, , • bankruptcy costs, , • corporate tax rate, , • risk-free interest rate, , • market price of risk, , • correlation between asset returns and market returns, , and • EBIT and asset volatility, .
We can divide the parameters into two groups: Some are readily observable, while the others have to be estimated. Belonging to the first group are the company-specific parameters 0 , , , and , and the risk-free rate . The current level of EBIT flow can be taken from the latest financial report. 13 The same is true for the notional debt, while the interest rate is the nominal interest rate the company pays for its long-term debt. The corporate tax rate can also be deduced from the financial report, by relating paid taxes to earnings before taxes. The risk-free rate is the yield of long-term government bonds.
Within the second group, the EBIT growth rate has to be estimated. 14 While this is a standard requirement in firm valuation issues (e.g., Damodaran, 2002), an estimation suffers from considerable potential errors. Also an estimate for the bankruptcy costs is not easily carried out.
Approaches could be based on historical recovery rates of comparable companies. Furthermore, we have the market price of risk, based on the market risk premium, as a global parameter.
While accepted as one of the most important figures in finance, there is an extensive controversy about its value. 15 Additionally, the correlation between asset returns and market returns has to be estimated. For listed companies, the stock price time series can be used. Alternatively, average sector correlations can be applied. Within our sensitivity analysis, we put special emphasis 13 In practical applications, an average value over a few preceding years could be more appropriate.
14 As in the Gordon growth model, the EBIT growth rate is constant over time. 15 Welch (2000) reports large differences in the assessments of the market risk premium based on a survey of on the model behavior with respect to the uncertain input parameters , , , and .
Finally, the EBIT and asset volatility is left as an input parameter. In contrast to Cooper and Davydenko (2007), we do not attempt to estimate this value directly. While for listed companies, estimates could be based on stock price volatility, an estimation for private companies would be difficult. Instead, we imply a value for the volatility based on the other input data. As an identifying assumption, we assume that the debt of the firm is fairly priced, in other words, that the paid corporate interest rate reflects current market conditions. 16 This means, the debt value equals its face value . As we know the debt value according to this assumption, we can use the model equation for the debt value (10) to obtain an implied volatility which yields this value. In the appendix we can derive an expression for the unknown asset volatility : .
This equation can be solved iteratively for . Figure 1 shows the cost of debt for a typical company dependent on the market price of risk . The parameters are: current EBIT flow 0 = 5, EBIT growth rate = 1%, notional debt = 30, bankruptcy costs = 50%, tax rate = 30%, risk-free rate = 3%, and correlation = 0.6. The corporate interest rate varies from 4% to 7%. The graphs show the relative growth of the risk premium with respect to the total yield spread when the market price of risk increases. When the market price of risk is low, the major part of the yield spread refers to the default premium, which compensates for lower expected returns. The actual expected return is only a little above the risk-free rate, although the nominal interest rate might be considerably larger. This means that bondholders receive only a small premium as an add-on to the risk-free rate for bearing substantial default risk. For high values of the market price of risk on the other hand, the major part of the yield spread refers to the risk premium. Although the actual default risk might be low, bondholders receive large compensation for bearing that risk. For instance, at a market price of risk of = 0.25, the expected return for bondholders (and thus, the cost of debt) with a interest rate of 4% is 3.98%. This means that for bearing the risk of an expected loss of only 0.02%, they receive a risk premium above the risk-free rate of 0.98%. [ where is the normal density function. As can easily be verified, this covariance increases concavely with .
respect to the input parameters. Table 1 refers to a typical investment-grade firm (debt-toequity ratio 0.4), while Table 2 refers to a highly leveraged firm (debt-to-equity ratio 2.0). In both cases, the EBIT growth rate is 1%, the corporate tax rate is 30%, bankruptcy costs are 50%, the risk-free interest rate is 3%, the market price of risk is 0.25, and the correlation is 0.6.
The investment grade firm pays an interest rate of 4%, and the highly leveraged firm pays an interest rate of 7%.
[INSERT the unobservable parameters EBIT growth rate and bankruptcy costs. Shifting these parameters by 0.5 or 10 percentage points, respectively, leads to a change in the cost of debt of as little as one base point. The same is true for the corporate tax rate, which has no effect on the cost of debt. Also a change in the EBIT flow by 20%, leading to a similar change in the debt-to-equity ratio, leaves the cost of debt nearly unchanged. The effect of the yield spread-either by a change in the interest or in the risk-free rate-is already documented in Figure 2. Naturally, the cost of debt, in particular the relative proportion of the risk premium, is most sensitive with respect to the market price of risk, as demonstrated in Figure 1.
For the highly leveraged firm, a smaller proportion of less than 50% of the overall yield spread refers to the risk premium. This observation is in line with Figure 2, which shows a concave increase of the cost of debt with the corporate interest rate. Also for the highly leveraged firm, the sensitivity of the model with respect to the unobservable parameters and is very small.
Most crucial is the market price of risk.
The results are fairly in line with the figures reported by Cooper and Davydenko (2007)

Calibration Based on Cost of Equity
The market price of risk is a key input to the model, as it is prone to estimation errors, and the model output is relatively sensitive with respect to this parameter. As an alternative to a direct input of the market price of risk, we consider in this section an implied calculation based on an available estimate of the second component of the cost of capital, the cost of equity. The reasoning is that most applications focus on an accurate estimate of the cost of equity and give little attention to the cost of debt. In the following we therefore analyze how an existing estimate of the cost of equity can be used to calibrate the model and consistently obtain an estimate for the cost of debt without explicitly estimating the market price of risk.
The idea is simply to calibrate the model by choosing a value for the market price of risk so that the cost of equity calculated according to (16) equals the externally given value. As the external estimate for the cost of equity also depends on the market price of risk, we cannot expect to eliminate the associated estimation error. But we can use the model to obtain a consistent value for the cost of debt when the external value for the cost of equity is considered reliable.
According to (5), it is not the market price of risk alone, but instead its product with the correlation which determines the expected asset return. We can therefore also refrain from estimating and iteratively calculate an implied value for the product , as follows: • Given starting values 0 , 1 • For each iteration 18 Berg (2010) reports very similar figures based on the Merton model, calibrated to real-world data.
-Calculate an implied asset volatility according to (17) -Calculate model values for the bankruptcy threshold , the asset value , and the equity value -Calculate a value for the cost of equity by iteratively solving (16) -Calculate a new estimate for the correlation-adjusted market price of risk using regula falsi: Table 3 summarizes the results of an analysis when the model is calibrated based on a given value for the cost of equity. Parallel to Section 3.1, we consider an investment-grade firm and a highly leveraged firm. The externally given cost of equity for the investment-grade firm is assumed to be 7%, which is well in line with the basic setting of Table 1. The results are similar to the basic setting, however, the model output is now more sensitive. With respect to the unobservable parameters, EBIT growth rate and bankruptcy costs, the sensitivity is now twice as large, although it is still quite small in absolute terms. A change in of 0.5 percentage points or a change in of 10 percentage points leads to a change in the cost of debt of 2 base points.
Similar observations hold for the highly leveraged firm. Here, the externally given cost of equity is assumed to be 10%. Also for this leveraged firm, the sensitivity has roughly doubled.

Instantaneous Returns
Cost of debt and cost of equity are the components of the overall cost of capital for the company.
In theory and practice, the "weighted average cost of capital" (WACC) as a value-weighted average, incorporating the tax shield of debt financing, is in widespread use. In this section, we calculate this figure within our model framework.
In doing so, we distinguish between the long-term average cost of capital, as considered so far, and the instantaneous cost of capital. As discussed, for most applications, such as capital budgeting, long-term project analysis, or firm valuation, a long-term time horizon is required.
The cost of debt and cost of equity derived and analyzed in the preceding sections fulfill this requirement. Equations (15) and (16) For the variance rates, 20 where is the value of an arbitrary claim on the asset value-in particular, debt or equity . The derivatives of and with respect to are given by (see the appendix): and According to the foundations of the model, the EBIT flow, plus the change in asset value, are distributed among the claimants bondholders, shareholders, and government, and the bankruptcy costs: In the following we ignore bankruptcy costs to keep things in line with standard approaches. 21 Taking expectations yields where and denote the instantaneous returns on debt and equity, respectively. As the shareholders and government share claims on the identical remainder − − , their instantaneous returns are identical. Inserting = + /(1 − ) and + = /(1 − ) (according to (13)) yields The WACC is defined as the expected return on firm value, = + , after taxes, with taxes applied to the total EBIT without the tax shield of debt financing. The expected cash flow after taxes is (1 − ). Hence, the instantaneous WACC, , is given by This gives the textbook formula for the instantaneous WACC: Miles and Ezzell (1980) have shown that the textbook formula for the WACC is valid for all time horizons if the unleveraged cost of capital, the cost of debt, the tax rate, and the leverage of the firm are all constant. As this is not the case in our model, we cannot expect the textbook formula to hold for the long-term cost of debt, , and the long-term cost of equity, .

Long-Term Returns
The actual weighted average cost of capital for the firm, , is defined as the expected long-term return on firm value after taxes-analogous to the instantaneous case: The value of this actual WACC differs from the textbook WACC, Figure 3 shows the actual WACC and the textbook WACC in dependence on the leverage for a typical model setup with initial EBIT flow 0 = 5, EBIT growth rate = 1%, EBIT volatility = 20%, corporate tax rate = 30%, market price of risk = 0.25, and correlation = 0.6.
The interest rate is chosen so that the value of debt equals its respective face value, which varies at the x-axis. In this setting, total asset value equals 100.

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The dotted lines show the textbook WACC, while the solid lines refer to the actual WACC.
The grey pair of lines ignore bankruptcy costs ( = 0%), and the dark pair of lines incorporate bankruptcy costs with = 50%.
Without bankruptcy costs, the tax shield of debt financing leads to a monotonically negative relationship between leverage and the cost of capital, both for the actual WACC and the textbook formula. The textbook WACC with bankruptcy costs (solid grey line) also decreases monotonically. The actual WACC in the presence of bankruptcy costs, however, reaches a minimum at a certain leverage and starts to increase again for high values of leverage. This shape is a representation of the well-known trade-off theory. For small levels of leverage, the tax shield of debt financing leads to a decrease in the cost of capital with increasing leverage. Increasing leverage, on the other hand, leads to increasing bankruptcy costs. At a certain level of leverage, the trade-off between the positive effect of the tax shield and the negative effect of bankruptcy costs is optimal. For yet higher levels of leverage, the negative effect of bankruptcy costs starts to dominate, and the cost of capital increases again.
Comparing the actual WACC with the textbook WACC, the textbook formula overestimates the actual WACC for small levels of leverage, and underestimates it for higher levels of leverage. As Miles and Ezzell (1980)  Regarding the weighted average cost of capital (WACC), the textbook formula is valid only for instantaneous time horizons. As the debt-to-equity ratio is not constant in the model, the actual cost of overall capital can substantially differ from the textbook WACC.

Cost of Debt
From Equation (14), it follows where ( * ) is the density of bankruptcy (or, hazard rate) under the physical measure, with EBIT growth rate . The integral can be derived analogously to the pricing of barrier options: The value equals the expected value of a payment of 1 when the asset value for the first time hits the lower boundary , discounted with the (yet unknown) expected debt return, . This value is given by (see Rich, 1994) with defined as in (9). Hence, the cost of debt is defined by the equation which can be solved iteratively.

Cost of Equity
For the cost of equity, Equation (16) is figured out: ) .
For the former integral, note that the payoff to shareholders for each point in time, , equals the payoff of ( − ) down-and-out barrier options on the asset value following geometric Brownian motion, struck at zero, with barrier equal to the default boundary . 22 For a single , the discounted payoff can be calculated analogously to the pricing of such a barrier option, when we replace the risk-neutral interest rate by the discount rate , and set the underlying payout rate equal to − . The value of a zero-strike down-and-out call is given by (see Rich, 1994) [ with and Hence ) .
The determination of the cost of equity based on this equation requires an iterative procedure, where in each iteration step the numerical calculation of the integral has to be carried out.
22 Note that EBIT flow and asset value follow the same geometric Brownian motion, as = ( − ) .

Implied Asset Volatility
Given the current debt value , we can use (10) to calculate an implied value for , Inserting = and the threshold according to its definition (8) yields an expression for with the yet unknown asset volatility : .
We can now use the definition (11) of to iteratively calculate an implied value for the asset volatility: This equation can be solved iteratively for .

Derivatives with Respect to Asset Value
To calculate the derivatives of equity and debt value with respect to asset value, we first need the derivative of : where for simplicity of notation, = ( , , ).
Regarding the equity value, we can write (13) explicitly to obtain It follows using the definition = 1+ (8).
Regarding the debt value, deriving equation (10) yields  Table 2. Sensitivity analysis of the cost of debt for a typical highly leveraged firm. The basic parameters are initial EBIT flow 0 = 5, EBIT growth rate = 1%, corporate interest rate = 7%, relative bankruptcy costs = 50%, corporate tax rate = 30%, risk-free interest rate = 3%, market price of risk = 0.25, and correlation = 0.6. The model is calibrated (by means of volatility) so that the debt value equals its face value, = = 40. The additional columns show the implied volatility , bankruptcy threshold , asset value , absolute bankruptcy costs , equity value , governmental claim , firm value = + , and finally the cost of debt and the relative proportion of the risk premium with respect to the yield spread. The sensitivity analysis changes one of the seven input parameters, while the remaining parameters are kept constant. 3.5% 5.18% 48% 6% 3.52% 52% 9% 4.66% 42% 8% 3.82% 82% 11% 5.02% 51% Table 3. Sensitivity analysis of the cost of debt for a typical investment-grade firm (left part) and a typical highly leveraged firm (right part), with the model calibrated to an externally given cost of equity. The basic parameters are initial EBIT flow 0 = 5, EBIT growth rate = 1%, corporate interest rate = 4% (left) and = 7% (right), relative bankruptcy costs = 50%, corporate tax rate = 30%, and risk-free interest rate = 3%. The model is calibrated by means of market price of risk to a cost of equity = 7% (left) or = 10% (right). Then the model is calibrated (by means of volatility) so that the debt value equals its face value = = 20 (left) or = = 40 (right). The output columns show the the cost of debt and the relative proportion of the risk premium with respect to the yield spread . The sensitivity analysis changes one of the seven input parameters, while the remaining parameters are kept constant.  EBIT growth rate = 1%, relative bankruptcy costs = 50%, corporate tax rate = 30%, risk-free interest rate = 3%, and correlation = 0.6. The model is calibrated (by means of volatility) so that the debt value equals its face value = = 30. The four lines refer to four different levels of the corporate interest rate, ranging from = 4% to = 7%. Figure 2. Cost of debt with respect to the corporate interest rate. The basic parameters are initial EBIT flow 0 = 5, EBIT growth rate = 1%, relative bankruptcy costs = 50%, corporate tax rate = 30%, risk-free interest rate = 3%, market price of risk = 0.25, and correlation = 0.6. The model is calibrated (by means of volatility) so that the debt value equals its face value = = 30. The light straight line, as a benchmark, is the identity = . Accordingly, the actual dark line divides the yield spread in the risk premium (below) and the default premium (above).  The model is calibrated (by means of nominal interest rate) so that the debt value equals its face value, which varies at the x-axis. Total asset value is 100. The dark black lines represent relative bankruptcy costs = 50%, while the light grey lines represent no bankruptcy costs ( = 0). The solid lines refer to the actual WACC; the dotted lines refer to the "textbook WACC".