JSTOR

A. The Black and Scholes/Merton Model

The value of equity in the Black and Scholes (1973)/Merton (1974) (BSM) model is computed using the standard call option formula, with the exercise price set equal to the nominal amount (N ) of the discount bond with maturity T: where φ [·]represents the standard normal distribution function with d₁ and d₂ given in the appendix. All revenue generated by the assets is reinvested; therefore, in (1). The value of the bond is, of course, equal to the value of the assets less the value of equity: Since there are no taxes nor bankruptcy costs in this model, the value of the assets equals the value of the firm. Note that represents the risk‐adjusted probability of default up to date T.

B. The Briys and de Varenne Model

The Briys and de Varenne (1997) (BV) model differs from the BSM model in two respects: Default can occur prior to the maturity of the single bond issue and interest rates are stochastic. More precisely, default occurs if the value of the firm's assets at any time falls below where and P(t, T) is the value at t of a unit of risk‐free bond maturing at T. The value of the unit bond was derived by Vasicek (1977) and is reported in the appendix. In case of default prior to maturity, bondholders receive and equity holders ; in case of default at maturity, they receive and , respectively. The value of the bond is where P_E, d₁ − d₆, l₀, and q₀ are given in the appendix. The bond expression consists of five terms. The first captures the value of an otherwise identical credit‐risk‐free bond. The second reflects the loss in value at the maturity of the bond, corresponding to the short put present in the BSM model. The third captures the recovery in case of a default prior to maturity, and the last two capture the fact that sharing of any surplus in financial distress is assumed to deviate from the absolute priority rule ( would correspond to the case of no such deviations).

Given the assumed absence of bankruptcy costs or taxes, equity is, as in the BSM model, simply the residual claim to the firm's assets: The model is a direct extension of Nielsen et al. (1993), which differs from the model of Longstaff and Schwartz (1995) only by assuming that the default threshold, rather than being a constant, is linked to the value of a risk‐free bond. In contrast to these two models, the BV model readily allows the derivation of an equity formula for a firm with discount debt.

C. The Leland and Toft Model

Leland and Toft (1996) allow for taxes and bankruptcy costs. The continuously paid coupons C are tax deductible at a rate τ and the realized costs of financial distress amount to a fraction k of the value of the assets in default. In this setting, the value of the firm is equal to the value of assets plus the tax shield less the costs of financial distress: We let L denote the default barrier, that is, the critical asset value at which the equity holders voluntarily declare bankruptcy. The formulae for L and x are given in the appendix.

Shareholders are residual claimants to the value of the firm and so where the value of debt is given by The formulae for the functions I(ω_t) and J(ω_t), as well as for i(ω_t, s) and j(ω_t, s) used later, are given in the appendix. Note that the value of equity, the firm, and debt are independent of time.

Bonds are issued with maturity T, principal and coupon . This particular choice of bond principal and coupon is required to derive the preceding tractable debt formula. The value of a bond with remaining maturity is

A. Maximum‐Likelihood Estimation

The problem at hand is the maximum likelihood estimation of the unobserved asset value process (1). This problem was first studied by Duan (1994) in the context of deposit insurance. The estimation is carried out using a time series of stock prices, . We use subscript i to index daily observations, in contrast to subscript t, which refers to a point in time in years.

We need the likelihood function of the observed price variable. Defining f(·) as the conditional density for gives us the following log‐likelihood function for the observed equity vector E^obs: whereθ is a vector of parameters to be estimated. The choice of individual parameters to include in θ depends on the model and the data set.

To derive the density function for equity, a change of variables is made. This allows us to work with the well‐known density function g for a normally distributed variable, the log of the asset value: The (1‐period) conditional moments of the asset value distribution, m_i and s_i, for each model, are given in the appendix.

The change of variables results in where is the subset of the parameter vector necessary to price equity. Note that E_iE_i(ω_i, t_i; ·) refers to the equity formula, whereas denotes an observed market value. The function transforming equity to asset value is defined as , the inverse of the equity value function. Hence, there is a one‐to‐one correspondence between the stock price and the asset value ω_i (given ). By inserting (10) into (8), we obtain the log‐likelihood of the vector E^obs for a given choice of θ as

Differentiating the equity formulae (2), (5), and (6) with respect to the (log‐) asset value yields the desired results. The parameter vector, , is estimated by maximizing eq. (11) with respect to θ. The market price of risk is estimated as a consequence of the chosen estimation method, even though it is not relevant for pricing. Finally, an estimate of the value of assets is simply obtained using the inverse equity function: .

Once we have obtained the pair , we can compute the corresponding bond prices and credit spreads, (B, Σ). Following Lo (1986), we can calculate the asymptotic distributions of these estimators. For any differentiable function of a variable, it holds that the maximum‐likelihood (ML) estimator of that function is the function evaluated at the ML estimator of the variable. In this case, it also holds that the estimators are asymptotically normally distributed: where is the asymptotic standard deviation of .

B. The Volatility Restriction Method

We now review the traditional method of estimating the model from stock prices (see, for example, Jones et al. 1984; Ronn and Verma 1986; Ogden 1987; Delianedis and Geske 1999; and Hull 2002). For reasons that will become clear, we refer to this as the volatility restriction (VR) method. Only two unknowns are estimated: the current asset value, ω_n, and its volatility, σ. The following steps are typically carried out:

This method has several theoretical problems, as pointed out by Duan (1994). The stock price volatility typically is estimated assuming that it is constant,9 even though it is a known function of ω and t. Furthermore, the first equation is redundant, since it is used to derive the equity price formula in the second equation.10 Another disadvantage of this approach is that it does not allow the straightforward calculation of the distributions of the estimators for ω and σ.

A. Experiment Design

To measure the performance of the two estimation approaches, we implement them on simulated data for various firm scenarios. Firms are defined along two dimensions, financial risk and business risk. Following Merton (1974), we measure financial risk with the quasi‐debt ratio, , the ratio of risk‐free debt to the asset value at the beginning of the sample period. Business risk is measured by the instantaneous volatility of the asset value, σ.

Four base‐case scenarios are created by setting financial and business risk to be either “high” or “low.” A firm is considered to have high financial risk if its quasi‐debt ratio is 1 and low if it is 0.75. Scenarios are constructed by fixing ω₁ and changing N, the nominal amount of debt. Business risk is deemed high if and low if . A complete list of the parameters used to construct the base scenarios in the three models can be found in table 1. The values are chosen so as to be representative of values used in previous studies and available empirical evidence.11

The four base‐case scenarios imply values for firms' leverage, equity volatility, and spreads. For example, a firm with high financial but low business risk in the Briys and de Varenne model displays a leverage of 82% and a current equity volatility of 59%, whereas a low financial but high business risk firm has a leverage of 62% and an equity volatility of 76%. The resulting spreads are 270 and 351 basis points, respectively. Overall, spreads range from 104 to over 600 basis points. This ensures that our study covers a wide array of bonds, ranging from investment grade to speculative grade. We chose relatively high levels of financial and business risk. We expect estimation to be more difficult in such conditions and therefore an evaluation of estimator performance to be more informative. Intuitively, the higher is the uncertainty, the more difficult it should be to indirectly “observe” asset value using equity values.

Fixing an initial asset value ω₁ and interest rate r_i, we simulate 2,000 paths for the asset value (and, if applicable, also the interest rate). In the case of a constant interest rate, we evolve the following dynamics where is a standard normal variable. When both asset value and interest rates are random, the following set of dynamics is used:12

where is another standard normal variable. The variables and have correlation ρ.

For each asset value path, we first compute the corresponding stock price path. Second, we use this equity path to estimate the current asset value and the parameter vector θ. Finally, we use the estimates to price the bond (using eqq. [3], [4], or [7]) and compute the standard error of the estimates (using eq. [12]). These steps are repeated for each sample path to assess the sampling distribution for a given model and scenario.

B. Output

In this section, we discuss the output produced and the tests performed. Since the ultimate use of an implemented model is to price bonds, the first question to address is whether the price and spread estimators are unbiased in small samples. The metric reported in the tables is the mean error of an estimator. Since the true value of the asset value, the spread and the price, is different for each path, we report relative errors for these variables. Second, to measure efficiency, we report the corresponding standard deviation and mean absolute error of the estimators.

A third issue is whether the asymptotic distributions are useful approximations in small samples. We measure the skewness (Sk.) and kurtosis (Ku.) of the sampled distributions and perform Jarque‐Bera (JB) tests for normality.13 Even if the normality of a particular estimator can be rejected, its estimated standard deviation might be useful for hypothesis testing and to compute confidence intervals in small samples. Therefore, we report the mean estimated standard deviation (mean estimated std.) and, as a measure of its efficiency, its standard deviation (std. of estimated std.).14

To further pursue this issue, we carry out size tests ; that is, we count how often the true value of an estimated value falls outside the confidence interval, calculated using the estimated standard deviation (i.e. the population size). If its size is close to the chosen nominal size (we use 1%, 5%, and 10%), one may conclude that the asymptotic distribution is useful for calculating confidence intervals and conducting t‐tests. An asterisk indicates that we can not reject the null hypothesis of the population size equaling the nominal size.15

The price estimates ultimately depend on the estimates of the volatility and value of the assets. To help understand the results, we therefore report output for as well.16

No asymptotic distribution has been suggested for the volatility restriction method. Therefore, only bias and efficiency are reported for this approach.

A. Result Overview

First, we note, in table 2, that the bias of the maximum‐likelihood approach is negligible for practical purposes. This result holds for all models and scenarios. In contrast, the volatility‐restriction approach exhibits an average spread error of 23% (panel E). However, the bias varies across models and scenarios, which will be discussed in detail later.

It is also evident from table 2 that the ML approach is much more efficient than the VR approach; the standard deviation and mean absolute error of the latter estimators are several times higher (e.g., the standard deviation of the ML spread error is 13%, whereas that of the VR estimator is 54%). Again, this is true for all models and scenarios. Figure 1 provides a visual summary of the relative efficiency of the two empirical approaches for the three different models. The ML approach clearly dominates, although not as strikingly for the BSM model.

(1 of 2 images)

(58 KB)

Fig. 1.— Kernel density plots for credit spread errors. The solid line represents the density of the relative spread errors, , using the ML approach and the dotted line using the VR approach. The three panels plot the density pairs for the BSM, BV, and LT models, respectively.

Open New Window

The explanation for the failure of the VR approach is intuitive. A highly volatile historical stock price series translates into a high estimated asset volatility and vice versa. This is a direct effect of solving the system of equations. However, high stock volatility is not necessarily the result of high asset volatility, it could be the result of a historically high leverage. Therefore, in a situation where asset value and hence stock prices have risen over the sample period, leverage and stock volatility have fallen. Historical stock volatility, computed as the average of realized volatilities, therefore is higher than the current level. This translates into an excessive asset volatility estimate and thus a low bond price estimate. In sum, the described volatility restriction effect implies that increasing stock prices result in underpriced bonds, whereas decreasing stock prices produce overpriced bonds.

The reasons the VR approach tends to underprice are two: first, because stock prices increase on average in our simulations, and second, because the positive impact of leverage on stock volatility is more pronounced at high leverages. This effect can also explain why estimation is less successful when the financial risk is high (table 2, panels A and B). The higher is the leverage, the more pronounced the effect on stock volatility. In a low‐leverage firm, the assumption of constant equity volatility is less severe.

The volatility restriction effect is analyzed in table 3 with the LT model in the low business‐ and financial‐risk scenario. The upper panel displays the result for all 2,000 asset value paths, and the two lower panels for the split sample: the middle panel contains the results for increasing and the lower panel for decreasing paths. The resulting average stock volatilities are 51%, 48%, and 68%, respectively. As can be seen, increasing paths lead to too‐high spread estimates and vice versa. Most paths are increasing and hence VR overpredicts spreads on average.

It is also evident from table 3 that the problem just described is not present when applying the maximum‐likelihood approach. As illustrated by figure 2, the ML estimators are able to disentangle the effect of leverage on stock volatility from the impact of business risk. The ML relative spread errors (crosses) are evenly scattered around zero, regardless of terminal asset value. In contrast, the VR errors (circles) are higher for firms that ended up more valuable at the end of the estimation period. These are the firms for which leverage has fallen and where the VR method tends to overestimate asset risk and thus also bond yield spreads.

(59 KB)

Fig. 2.— Credit spread estimation and historical leverage. The y axis represents the percentage spread error, , and the x axis, the actual asset value at the end of the estimation period in the low financial‐ and asset‐risks scenario for the LT model. The crosses and circles represent pairs of terminal asset values and relative spread errors for the ML and VR approaches, respectively.

Open New Window

We now turn to the small‐sample distributions of the maximum‐likelihood estimators. Comparing columns Standard Deviation and Mean Estimated Std. in any table, the standard deviation estimator is unbiased except for the spread error. The reason for this result is that the spread is sometimes close to zero and, therefore, very small errors produce huge relative errors; the standard deviation estimator of the basis point error is unbiased. The variability of the estimated standard deviation of asset risk (Std. of Estimated Std.) is about half the mean estimate (see table 2, panel E). Again, the absolute estimator is much more precise.

However, by the Jarque‐Bera test, we often reject the hypothesis of normally distributed estimators. Yet, the size tests are quite successful, in the sense that the population size is close to the nominal size. Therefore, it appears that one can still use the estimated standard deviations to construct reliable confidence intervals and t‐tests. In this sense, the estimators are “sufficiently normal.”

B. Sample Size

In this section, we examine how the distributions of the maximum‐likelihood estimators depend on the length of the equity price sample (n). We investigate four sample sizes: 90, 250, 500, and 750 days. Results are displayed in table 4 for the Black and Scholes/Merton model; results are similar for all models.

Using a 3‐month sample size, the ML estimators are quite inefficient. Increasing the sample size to 250 days leads to improved efficiency and normality. As noted previously, estimators are unbiased and efficiency is improved by roughly one third. Moreover, the Jarque‐Bera test statistic has decreased; we are less confident in rejecting normality for the Black and Scholes/Merton estimators. The size tests leave a mixed impression; most population sizes approach the nominal ones, but in some cases, the convergence does not appear to be monotonic (e.g., the 1% size seems to oscillate). We cannot detect any statistically significant deviation from the nominal sizes 5% and 10%.

Turning to longer samples still, the same pattern can be observed. Efficiency (and the efficiency of the efficiency estimate, the Std. of Estimated Std.) improves monotonically, whereas the effect of the measures of normality is less clear cut.18 However, overall, the small sample distributions of estimators become more normal as the sample size increases.

C. Further Results

As noted previously, the bias of the volatility‐restriction approach varies across models and scenarios. In fact, implementation of the Leland and Toft model accounts for most of the bias; the average bias of the asset volatility is −0.1% in the BSM model, 0.4% in the BV model, and 5.3% in the LT model. The reason is that the (instantaneous) stock volatility in this model becomes extreme as asset values approach the barrier. This accentuates the volatility restriction effect. In the low business‐risk and high financial‐risk scenario, for example, the distress barrier is 687, which, when assets are worth 1,000, leads to a stock volatility of 95%. When assets are down to 800, volatility is 246%; at 750, it is 439%; and at 700, stock volatility reaches a staggering 2,046%. Thus, asset value paths that, at some point during the sample period were near the barrier, lead to a severe overestimation of the current stock volatility and badly underpriced bonds. In a scenario with a very low barrier, very few or no paths would pass close to the barrier, few extreme stock volatilities would be observed, and the effect just described would be mitigated.

This problem for the VR approach with the LT model arises because equity value drops to zero on default, and this causes extreme volatilities. The BV model incorporates deviations from the absolute priority rule, which tends to reduce volatility at the brink of bankruptcy. Shareholders do not face an all‐or‐nothing situation, which, in turn, prevents stock volatility from attaining extreme values.19 As a result, the VR approach is more successful with the BV model.

Of the three theoretical models, the implementation of the Briys and de Varenne model is the most efficient for the maximum‐likelihood as well as the volatility‐restriction approach. The mean spread error standard deviations are 7.5% and 17.3% for the two approaches with this model, whereas they are 17.7% and 28.9% with the BSM model. Recall that the two models have the same capital structure, but in addition, the BV model allows for stochastic interest rates and the possibility of early default. The explanation for estimator performance in this model is again related to the default barrier, although not through its effect in stock volatility.20 In the BV model, the barrier L is an exogenous fraction (δ) of risk‐free debt. The recovery to bondholders in financial distress is therefore also exogenous: . This implies that one crucial component of the bond price, the payoff in financial distress, is fixed and independent of the estimated value and risk of assets. Naturally, this benefits both estimation approaches, as evidenced by the low standard deviations. A closely related feature of the BV model is that the performance of some estimators actually improves as the risk and hence the spread increases; the riskier is the scenario, the more important the (exogenously specified) distress component of the bond price.21

The errors of the maximum‐likelihood approach are, as previously noted, independent of historical leverage. Errors do, however, depend on the realized volatility of the asset value path: An unusually high asset volatility results in an unusually high stock volatility which, in turn, translates into an excessive asset volatility estimate. After controlling for realized asset volatility, no further variables describing the realized stock value path have any explanatory power on the errors.

The maximum‐likelihood approach works well also for very risky firms, with spreads exceeding 1,000 basis points. Results, not reported here, show that the approach stays unbiased for all models, although efficiency may decrease somewhat. The performance of the VR approach, on the other hand, tends to deteriorate drastically as the spread increases.