Valuing equity in distressed firms as a call option on assets

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Valuing equity in distressed firms as a

call option on assets

Afstudeeronderzoek Financiering en Belegging

Vrije variant

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Abstract

In this thesis the suitability of the equity-call model as a valuation tool of the equity of distressed firms is examined. From a theoretical perspective the equity-call model seems to be a useful addition to the tools available to valuation practitioners. Most importantly it simultaneously prices the equity and debt of a firm, taking account of the interactive effect on each other. This is in contrast to traditional valuation methods which price debt and equity separately. This property of the equity-call model provides insight into the key value drivers of the equity of distressed firms. However, the equity-call model does suffer from some limitations. Most notably the equity-call model makes simplifying assumptions about the capital structure of the firm.

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Acknowledgment

I would like to express my gratitude to everyone who made the completion of this thesis possible. Foremost I want to thank Duff and Phelps for the opportunity to work on my thesis and the use of resources while working as an intern. Special gratitude goes to Henk Oosterhout and Jochem Quaak my supervisors at Duff and Phelps for their help, suggestions and motivation while conducting the research for and writing this thesis. Furthermore I would like to thank all my colleagues at Duff and Phelps for the great time I had while working as an intern.

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1. Introduction

Ever since the renowned article by Black and Scholes [1973] corporate securities have been viewed as options on the assets of the firm. One of the consequences of this idea is that the value of equity can be seen as a call option on the firm’s assets, henceforth referred to as the ‘equity-call model’. Alternatively the value of debt is equal to holding risk free debt and writing a put option on the firm’s assets. In contrast to traditional valuation methods the equity-call model simultaneously prices the debt and equity of a firm. The equity-call model takes into account the interactive effects of each of these components of the firm’s capital structure on the other. Traditional valuation techniques on the other hand price each component of the firm’s capital structure separately. As will be explored in chapter 3 this interactive effect is most noticeable for distressed firms.

The use of the option pricing framework to determine the value of corporate securities is also known as contingent claims analysis. Much empirical research has been conducted into contingent claims analysis. However, the main focus has been on the application of contingent claims analysis in the context of credit risk analysis and the valuation of corporate debt. As will be discussed in chapter 4, little empirical research has been conducted to test if the equity-call model results in a equity value which is a good proxy for the market price of equity.

When option-pricing models are applied to the valuation of corporate securities the resulting option can be categorized as a real option in contrast to a financial option. The pricing of real options is not without difficulties. According to Teach [2003] real options are mathematically challenging and may lack simplicity. Furthermore Teach explains that there are a number of assumptions underlying option-pricing models that are violated when pricing real options. Copeland and Tufano [2004] discuss a number of technical issues that cause practitioners to shy away from the use of real options. Firstly the inputs required in option-pricing models are seen to be very difficult to estimate for real options. Secondly practitioners are concerned that reducing complex real world issues into standard option-pricing models will distort results.

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“highly leveraged financially distressed firms”1. Therefore further empirical research into the equity-call model is merited. The main question this thesis will try to answer is:

How suitable is the equity-call model as a valuation tool for determining the equity value of distressed firms?

The main question will be answered both by means of a theoretical and an empirical examination. Chapters 2 to 4 will examine the suitability of the equity-call model from a theoretical perspective. The theoretical concept underlying the equity-call model is examined in chapter 2. Chapter 3 looks into the pricing of the equity-call model. In this chapter the Black-Scholes-Merton option-pricing model is discussed. This pricing model will be used to apply the equity-call model. In addition this chapter examines the insights provided by, and the limitations of, the equity-call model. Which prior empirical tests of the equity-call model have been conducted is discussed in chapter 4. In addition, literature is discussed that gives insight into how previous studies have computed the inputs needed in the equity-call model.

How well the equity-call model performs as a valuation tool for the equity of distressed firms in practice is examined in chapter 5 and 6. The equity-call model is applied to a sample of distressed firms. In this paper the implied asset volatility resulting from the equity call model will be used to test the model. From theory it is known that the implied asset volatility is a proxy for the price of an option. For a perfect working model it is expected that the implied asset volatility is equal to the observed asset volatility in the market. In chapter 5 the methodology used tot test this hypothesis is discussed. Furthermore the selection of a sample of firms and the computation of the required inputs is discussed. This will give insight into the difficulties faced when applying the equity-call model in practice. In chapter 6 the results of the empirical test of the equity-call model will be discussed. Chapter 7 concludes.

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2. Equity as a call option on the firm’s assets

Practitioners can use a variety of valuation models to estimate the equity value of firms. The most widely used methods are the enterprise discounted cash flow method, the economic profit method and the adjusted present value method2. These frameworks all have in common that the method used to estimate the value of equity consists of two steps.

First these methods estimate the enterprise value, or the value of assets, of the firm. The manner in which the value of the firm’s assets is estimated differs for each method. However, all these methods are dependent on cash flow forecasts. The methods are flexible and capable of estimating the value of assets for firms with different characteristics. For distressed firms, which are characterized as firms with substantial amounts of leverage, the implementation of the above methods is not without difficulty. The going concern assumption underlying these valuation methods may not hold as the possibility of bankruptcy is very real. Therefore the future cash flows are uncertain and difficult to estimate. Furthermore, when computing the true taxes payable on the future earnings, the opportunity firms have to carry past losses forward to offset future profits must be kept in mind. Despite these complications the fundamentals of valuation still apply. The valuation estimates, however, are noisy, reflecting the real uncertainty of the future of these firms.

The second step in the valuation process is the same for each method. It involves subtracting the market value of debt from the value of the firm’s assets.

E = A – D (1)

where:

E = Market value of equity.

A = Value of the firm’s assets.

D = Market value of debt.

As a proxy for the market value of debt the book value of debt is often used, as the market value of debt is unobservable. Koller et al. [2005] state that this method leads to an acceptable estimate of the value of equity. In general, for healthy firms, the market value of debt will not differ

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substantially from the book value of debt. Slight differences do occur because of the difference between the historically determined interest payments on debt and the actual market interest rate. For distressed firms, however, the book value of debt is a poor proxy for the market value of debt as the possibility of defaulting on the debt is large. The possibility of default can greatly reduce the market value of debt. Furthermore the book value of debt of distressed firms is frequently greater than the value of the assets; these firms have a negative net worth. For these firms the book value of debt cannot be used as a proxy of the market value of debt in equation 1 as this would entail a negative equity value. It is thus necessary to estimate the market value of debt when valuing the equity of distressed firms. This is not without difficulties as distressed debt is often thinly traded, if traded at all. Therefore the estimation of the equity value of distressed firms using the above method, although possible, leads to noisy results.

Another valuation method though is available. As discussed by Damodaran [2002a] the equity of distressed firms can be seen as a call option on the asset’s of the firm. A major benefit of equity-call option method is that it simultaneously prices the equity and debt of a firm. The option value of equity may be in excess of the traditional methods as these traditional methods do not reflect the option that shareholders have to liquidate the firm’s assets. This concept is explained below.

Figure 1 shows the payoff to the firm, equity- and debtholders at different levels of the value of the firm’s assets, at the expiration date of the debt. It is assumed that the firm has a simple capital structure consisting of equity and a single issue of a zero coupon bond. The face value of debt is equal to K. If the value of the firm’s assets is A2, the payoff to equity is equal to the residual asset

value of the firm, A2 - K. As long as the value of the firm’s assets is above K, each increase in

asset value accrues directly to the shareholders. The shareholders have an unlimited upside potential. In other words equity is a residual claim. The shareholders have the right to all cash flows left over after other financial claims have been paid. The payoff to shareholders is equal to:

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Figure 1: Payoff to the firm, equity and debt at the expiration date of the debt.

On the other hand if the asset value falls below K, for example A1, the debt will not be able to be

repaid. The firm is bankrupt and on liquidation all assets are used to pay the debtors. The shareholders receive nothing. The “limited liability” aspect of equity, however, protects shareholders from losing more than their initial stake. The payoff to shareholders in this situation is:

Payoff = 0 if A < K (3)

In other words shareholders have a limited downside risk and an unlimited upside potential. This payoff pattern is equal to the payoff pattern of buying a call option on a stock3. Thus the value of equity (E) of a leveraged firm can be viewed as the value of a call option on the assets of the firm (A), where exercising the option requires that the firm be liquidated and the face value of the debt (K) to be paid off. K can be seen as the strike price of this option4. In this setting the life of the option is equal to the time to maturity of the single issue zero coupon bond. As will be discussed

3

See Hull [2006] chapter 1. 4

It is assumed that the reader understands the basic principals underlying options. For further reading see Hull [2006] chapter 8.

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in chapter 3, the payoff to debt is equal to the payoff of holding risk free debt and writing a put option on the firm’s assets

An important implication of viewing equity as a call option on the firm’s assets is that equity will have value even when the value of the firm’s assets is less than the present value of the face value debt. Figure 2 shows the variation in the value of equity and debt, when priced using the equity-call model, at varying levels of the value of the firm’s assets. Figure 2 demonstrates that equity has value, even for those firms where the value of the firm’s assets is less than the present value of debt. In this setting equity can be seen as an out of the money call option. Out of the money call options have value, all be it limited, due to the time value of an option. As long as the ‘equity-call option’ has not expired there is a possibility that the value of the firm’s assets will increase above the face value of debt. For a distressed firm this implies that equity has value because of the time premium on the option. The time premium embedded in equity is given by the dashed red area in figure 2. Figure 2 also shows the value of debt at varying values of the firm’s assets. The value of debt is always less than the present value of the face value of debt as the upside potential for debt is limited to the face value of debt. As debt has a fixed upside potential, the value of debt has a time discount. As long as the equity option has not expired, debtholders face the risk that the value of the firm’s assets will decrease below the face value of debt. The blue dashed area in figure 2 shows the time discount embedded in the value of debt. Figure 2 is further discussed in chapter 3.

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Figure 2: Variation in the value of equity and debt with the value of the firm’s assets.

Variation in the value of equity and debt with the value of the firm's assets

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 90 100 110 120

Value of the firm's assets

V a lu e Equity Value Debt Value Firm Value Intrinsic Value Present value of K Healthy firms

Distressed firms Dashed red area: Time

premium embedded in equity

Dashed blue area: Time discount embedded in debt.

PV of K

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3. Pricing the equity-call option

In order to apply the equity-call model to the valuation of equity an option-pricing model is needed. In this paper the Black-Scholes-Merton option-pricing model (Black and Scholes [1973] and Merton [1973]), henceforth the BSM model, will be used. From now on when referring to the equity-call model it is assumed that the BSM model is applied. The first section of this chapter discusses the factors that influence the value of the equity-call option after which the BSM model is introduced. Next the insights provided by the equity-call model will be explored. Finally the limitations of the equity-call model will be discussed.

3.1 Determinants of the value of the equity-call option

As discussed above the value of equity can be seen as a call option on the firm’s assets. A call option is an asset that gives the holder the right to buy the underlying asset at or before a certain date for a predetermined price. This predetermined price is known as the strike price, the date as the maturity. There are a number of variables that influence the value of a call option. Table 1 shows these variables both in the context of a financial call option and in the context of the equity-call option.

Variables in the context of a financial-call option

Variables in the context of the equity-call option

Value of the underlying asset Value of the firm’s assets

Strike price Face value of debt

Time to maturity Time to maturity of the debt

Volatility of the value of the underlying asset Volatility of the value of the firm’s assets Risk free interest rate Risk free interest rate corresponding to the

maturity of the debt

Dividends Interest payments on debt (cash outflows)

Table 1: Variables influencing the value of an option.

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Variable Equity Value

Increase in the value of the firm's assets +

Increase in the face value of debt

-Increase in the time to maturity of the debt +

Increase in the volatility of the value of the firm's assets +

Increase in the risk free interest rate +

Increse in the interest payments on debt (cash outflows)

-Table 2: Effect of an increase in a variable, ceteris paribus, on the value of equity

The value of the firm’s assets and the face value of debt: Equity derives value from the firm’s assets. Therefore its value is influenced by changes in the value of the firm’s assets and the price at which these assets can be bought, namely the face value of debt. When the equity-call option is exercised the payoff to shareholders is the amount by which the value of the firm’s assets exceeds the face value of debt. Consequently an increase in the value of the firm’s assets leads to an increase in the value of equity, while an increase in the face value of debt leads to a decrease in the value of equity. The equity-call model captures the financial risk of a firm through the value of the firm’s assets and the amount and time to maturity of debt.

The time to maturity of the debt: An increase in the maturity of the debt has a positive effect on the value of equity. As equity has a limited downside potential an increase in the life of the option increases the time frame over which the option can realize its upside potential.

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volatility, nevertheless, is generally an increase in the value of equity. This is further discussed in section 3.3.4.

The risk free interest rate: The affect of the risk free interest rate on the value of equity is less clear cut. Ceteris paribus, an increase in the risk free interest rate decreases the present value of the face value of debt which has to be paid when the option is exercised. However, an increase in the risk free rate generally results in a decrease in the value of the firms assets, which in turn decreases the value of equity. The net effect of an increase in the risk free interest rate is generally a decrease in the value of equity.

Interest payments on debt (cash outflows): Interest payments on debt, or cash out flows, reduce the value of the firm’s assets. These cash outflows are lost to the equity holders of the firm and thus reduce the value of equity.

3.2 The Black-Scholes-Merton option-pricing model

The basis for option-pricing models lies with the articles by Black and Scholes [1973] and Merton [1973]. The BSM model is based on the following set of assumptions5:

1. The underlying asset follows geometric Brownian motion. 2. The volatility of the value of the underlying asset is constant. 3. The short selling of the underling assets is permitted.

4. There are no transaction costs or taxes.

5. There are no dividends during the life of the derivative. 6. There are no riskless arbitrage opportunities.

7. The underlying asset is traded continuously.

8. The risk free rate of interest is constant and the same for all maturities.

The derivation of the BSM model begins with specifying a price process for the value of the underlying asset. Any variable whose value changes unexpectedly over time is said to follow a stochastic process. The BSM model specifies a continuous-variable, continuous-time stochastic process for the value of the underlying asset. The continuous-variable assumption implies that the value of the underlying asset is assumed to change in a continuous manner. There are no price

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jumps in the underlying asset. The continuous-time assumption implies that the value of the underlying asset can change at any time. The specific pricing process specified by the model is geometric Brownian motion. Most importantly the assumption of geometric Brownian motion imposes restrictions on the behavior of the underlying asset. First this process entails that the return on an underlying asset in a small period of time is normally distributed. Second it requires that the returns on the underlying asset are independent in two separate periods. For the future distribution of the value of the underlying asset this assumption implies a lognormal distribution.

The next step in de derivation of the BSM model involves the concept of no-arbitrage. No-arbitrage entails that it is impossible to make a riskless profit in the market. To ensure no arbitrage the following three conditions must hold.

1. Identical assets must trade at the same price on all markets (“law of one price”) 2. Assets with the same cash flow must trade at the same price.

3. An asset for which the future price is known must trade at its future price, discounted at the risk free interest rate.

To derive the BSM model a riskless portfolio composed of the underlying asset and a derivative on the underlying asset is set up. Key to making this portfolio riskless is the fact that both the value of the underlying asset and the value of the derivative on the underlying asset are dependent on the price movements of the underlying asset. It is therefore possible to set up a portfolio wherein, for a short period of time, the change in price of the underlying asset is exactly offset by the change in price of the derivative. This ensures that the price of this portfolio is known with certainty at the end of the short time period. The no arbitrage assumption then dictates that the expected returns on the portfolio must equal the risk free interest rate. The concept of forming a riskless portfolio, containing the underlying asset and a derivative on the underlying asset, in combination with the specified price process of the underlying asset leads to the BSM differential equation.

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neutral world investors never require a premium to take risks. In such a world the risk free rate is the appropriate expected return as well as the appropriate discount rate. It is possible to use this concept of risk neutral valuation to solve the BSM differential equation. The obtained solutions are, however, still valid in all worlds, irrespective of the risk preference assumptions.

By means of the BSM differential equation the pricing formula for a call option can be obtained. In the context of the equity-call option the value of equity, E, is equal to:

) 2 d ( N Ke ) 1 d ( AN E −rt − = (4) where: t t ) 2 / r ( ) K / A ln( 1 d 2

σ

+ + = t 1 d 2 d = −

σ

and

A = Value of the firm’s assets.

K = Face value of debt.

t = Time to maturity of the debt.

σ

= Volatility in the value of the firm’s assets.

r = Risk free interest rate corresponding to the maturity of the debt.

ln (x) = The natural logarithm of x.

N(x) = The cumulative probability distribution function for a standardized normal distribution.

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A = Value of the firm’s assets – Present value of interest payment. (5)

The remaining value of the firm’s assets only includes a risky component, with

σ

the volatility of the process followed by the risky component.

In a similar approach the annual interest payments, where the interest payments on debt are specified as a percentage of the value of the firm’s assets, are taken as the dividend yield in the option-pricing model for dividend paying stock. Equation 4 becomes:

) 2 d ( N Ke ) 1 d ( N Ae E −qt −rt − = (6) where: t t ) 2 / q r ( ) K / A ln( 1 d 2

σ

+ − + = t 1 d 2 d = −

σ

and

q = yearly interest payments as a percentage of the value of the firm’s assets. Other inputs are as above.

An alternative method advocated by Damodaran [2002b] involves adding the interest payments over the life of the equity-option to input K, the face value of debt. Instead of subtracting the present value of the interest payments from the value of the firm’s assets, the strike price of the equity-call option is increased by the amount of interest payments expected over the life of the option. As the interest payments occur at various times over the life of the equity-call option, cash flows occurring at various times are being mixed up when using this method. Input K becomes

K = Face value of debt + expected interest payments on debt (7)

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How this paper incorporates the interest payments on debt in the equity-call model is further discussed in chapter 5.

3.3 Insights provided by the equity-call model

In this section the insights provided by the equity-call model when priced with the BSM pricing model are discussed.

3.3.1 The expressions N(d1) and N(d2)

The expressions N(d1) and N(d2) in the BSM model can be viewed as probabilities in the risk neutral world. In order to interpret these terms equation 2 is rewritten to:

(

AN(d1)e KN(d2)

)

e

E = −rt rt − (8)

The expression N(d2) gives the probability in a risk neutral world that the firm’s asset value will exceed the face value of debt at maturity. It gives the probability that the equity-call option will be exercised and that the firm does not default on its debt. When an equity-call option is exercised the face value of debt must be paid. The term KN(d2), from equation 3, is the face value of debt times the probability that the face value of debt will be paid at maturity. It is the expected value of paying the face value of debt at maturity. The probability that the firm will default on its debt is given by the risk neutral default probability. The risk neutral default probability is equal to:

Risk neutral default probability = 1 – N(d2) (9)

This is the probability that the option will not be exercised and that the option expires worthless. This property from the BSM model is key to the use of the equity-call model in the practice of credit risk analysis6.

The expression N(d1) gives the probability that Aert will be received at maturity. Aertis the value of the firm’s assets at maturity as the expected return in the risk neutral world on any asset is the risk free rate. The term AN(d1)ert is the value of assets at maturity times the probability of receiving the firm’s assets at maturity. It is the expected value of receiving the firm’s assets at

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maturity. As a whole, equation 8, gives the present value of the difference between the expected value of receiving the firm’s assets at expiration and the expected value of paying the face value of debt at expiration. This is equal to the present value of equity.

Next the effect of two scenarios on the expressions N(d1) and N(d2) will be discussed. In the first scenario a healthy firm is considered. For such a firm the value of the firms assets, A, are large in comparison to the face value of debt, K. From equation 4 it is clear that the terms N(d1) and

N(d2) will approach to 1 as the ratio A/K increases. The value of equity approaches:

rt

Ke A

E= − − (10)

The term Ke−rt

can be seen as the present value of the face value of debt. This can also be seen from figure 2. The red line in this figure shows the variation in the value of equity for different values of the firm’s assets. In effect it is the value of an equity-call option at different underlying asset values. Healthy firms are situated on the right part of this line. For these firms the value of equity converges with the black dashed line originating at Ke−rt, the present value of the face value of debt. The black dashed line represents the intrinsic value of the option. This is the value resulting from the amount by which the value of the assets of the firm exceeds the present value of the face value of debt. In the case of healthy firms equity derives little value from the ‘option’ characteristic associated with equity. The equity-call model provides no added value to the traditional valuation methods as discussed in chapter 2.

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above the face value of debt before the equity-call option expires. The equity-call model is able to capture this time premium.

Figure 2 shows the subset of firms which are considered to be distressed. These distressed firms have an A/K ratio ranging from marginally above one up to zero. The lower the A/K ratio the more distressed a firm is. In practice, however, most distressed firms have an A/K ratio close to one. Firms with very low A/K ratios do not exist as these firms are forced into bankruptcy prior to maturity of the option. This is at odds with the equity-call model which assumes that firms do not go bankrupt prior to the maturity of the option. According to the equity-call model distressed firms with an A/K ratio approaching zero will have equity values and continue to exist until the maturity of the option. This limitation of the equity-call model will be explored in depth in section 3.4.2.

3.3.2 Vega and theta

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Figure 3: The theta and vega of the equity-call option.

Variation of theta with the value of the firm's assets

Value of the firm's assets Theta

Theta

0 _K

Variation of vega with the value of the firm's assets

Value of the firm's assets Vega

Vega

K 0

Source: Hull [2006] page 311 and 317.

3.3.3 The value of debt

From equation 1 it is known that the value of equity is equal to the difference between the value of the assets and the market value of debt. Equation 4 gives the value of equity as an option. Therefore the value of debt can be given as:

) 2 d ( N Ke ) 1 d ( AN A D= − + −rt (11)

Combining equation 11 with the well known put-call parity7 leads to an option-pricing equation for the market value of debt.

) 1 d ( AN ) 2 d ( N Ke Ke D= −rt − −rt − + − (12) 7

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where the inputs are as specified by equation 4. The value of debt is equal to holding risk free debt and writing a put option on the firm’s assets. The blue line in figure 2 gives the value of debt at various levels of the value of the firm’s assets. In effect the value of both debt and equity are contingent upon the value of the firm’s assets. In contrast to other valuation methods the equity-call model simultaneously prices the debt and equity of a firm. The equity-equity-call model takes into account the interactive effects of each of these components of the firm’s capital structure. This is also clear from figure 2. The time premium embedded in the value of equity, the red dashed area, is equal to the time discount embedded in the value of debt, the blue dashed area. This is further explored in the next section. Traditional valuation techniques on the other hand price each component of the firm’s capital structure separately.

3.3.4 The stockholder-bondholder conflict

From the above discussion it is clear that, given a constant value of the firm’s assets, a trade off occurs between the value of debt and equity. An increase in the value of equity entails a decrease in the value of debt. Naturally both share- and debtholders are interested in maximizing their claim on the firm’s assets, although this comes at a cost to the other claimant. This is known as the stockholder-bondholder conflict. The stockholder-bondholder conflict is characterized, for shareholders, by the asset substitution problem and the underinvestment problem. The asset substation problem entails that shareholders prefer projects with higher risk even if these projects have a negative net present value. The underinvestment problem is the opposite of the asset substitution problem, wherein shareholders refuse to invest in low-risk projects even if they have a positive net present value. As discussed by Chesney and Gibson-Asner [1999] debtholders at the same time try to protect their claim on the assets through safety covenants, convertible debt issues, callable bond issues and by shortening the maturity of existing debt. The equity-call model can explain the behavior exhibited by share- and debtholders which is associated with the stockholder-bondholder conflict.

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value of expressions N(d1) and N(d2). As explained above the value of equity is an increasing function of all three of these variables. Conversely the value of debt is a decreasing function of these variables. It is therefore in the interest of shareholders to try to increase the value of these three variables and vice versa for debtholders.

Both share- and debtholders have little power to influence the risk free rate. Shareholders, however, can increase the firm’s asset volatility by selecting risky investment projects. These risky projects can include projects with a negative net present value. These projects decrease the value of the firm’s assets but increase the claim shareholders have on the firm’s assets. The net effect can be an increase in the value of equity. The incentive to select risky investment projects is high for shareholders in distressed firms as the vega of the equity-call option is high for these firms. Shifts in asset volatility lead to relatively large shifts in equity value. At the same time shareholders refuse to invest in low-risk projects, even if these have a positive net present value. These projects increase the value of the firm’s assets but decrease the claim shareholders have on the firm’s assets. The net effect can be a decrease in the value of equity.

Debtholders on the other hand can try to influence firm investment policy through safety covenants and by exercising their convertible debt. At the same time they have an incentive to bring the maturity of the debt forward. This incentive is greatest for debtholders in distressed firms as the theta of the equity-call option is high for these firms. The equity-call model provides a valuable insight into the incentives faced by both share- and debtholders. These incentives are strongest for share- and debtholders in distressed firms as theta and vega are high for these firms.

3.4 Limitations of the equity-call model

There are a number of limitations associated with the equity-call model. First, like all models the equity-call model makes abstractions from reality. The model assumes a firm with a simple capital structure. Second, the equity-call model does not take into account real world bankruptcy processes. Last, the BSM option pricing model was derived to price financial options. When applying this model to real options, such as equity, some of the underlying assumptions of the model are violated. These limitations are discussed below.

3.4.1 Simple capital structure

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equity. Preferred equity, capital leases and other types of capital are not allowed for. Furthermore the equity-call model assumes that there is only one issue of debt outstanding which is a single issue of zero coupon bonds with no special features. This assumption aligns the features of debt more closely to the characteristics of an option. The maturity of the equity-call option is equal to the maturity of the zero coupon bonds. If the debt is coupon debt or more than one issue of debt is outstanding, the maturity of the option is difficult to define. Such a firm has to fulfill interest and principal payments at multiple maturities.

In practice, however, firms seldom have such simple capital structures. One way of dealing with this limitation is by extending the standard equity-call model. Numerous studies have extended the standard equity-call model in order to allow for more claimholders, multiple debt issues, coupon payments and special debt features. As discussed in Mason and Merton [1985] one of the benefits of these type of extended equity-call models are that they can deal with complexities encountered when valuing corporate securities that have special features, such as convertible debt. Ingersoll [1977], for example, develops a more advanced equity-call model to value convertible and callable corporate liabilities. More recently Ericsson and Reneby [1998] have developed a framework for valuing corporate securities using a portfolio of a down-and-out call option, a down-and-out binary option and a unit down-and-in claim. However, this approach leads to very complicated models with numerous extra inputs. Furthermore this approach entails the creation of a unique model for each firm to take into account all the types of claimholders and their specific features.

Alternatively all forms of capital, excluding equity, can be viewed as debt. The multiple debt issues found in real world firms can be repositioned into a single debt issue with a single maturity. This involves making adjustments to the manner in which the inputs are estimated. In this approach the standard equity-call model can be used to value a multitude of firms that do not fit perfectly into the framework described above. Damodaran [2002a] advocates the use of this method. In this paper this approach will be used and will be subjected to empirical testing in chapter 5 and 6.

3.4.2 Bankruptcy processes

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the firm is inadequate to meet its current obligations such as interest payments. Therefore bankruptcy can occur prior to maturity. Furthermore safety covenants give debtholders the right to force firms into bankruptcy when certain financial ratios are surpassed. Most often these ratios have to do with leverage and liquidity. Referring to section 3.3.1, this is the reason that firms with low A/K ratios do not exist in practice.

Brockman and Turtle [2003] try to model this aspect of the bankruptcy process. They argue that equity should be viewed as a down-and-out-call option on the firm’s assets. This is an option that ceases to exist when the asset value falls below a certain barrier level. At a certain barrier level of the value of the firm’s assets debtholders have the right to force the firm into bankruptcy. They view safety covenants as the main source of this barrier. A down-and-out-call option is less valuable than a standard option. Accordingly, they argue that the standard equity-call model overvalues equity and undervalues debt. They apply this model to a large selection of firms for which they calculate the implied barrier. They test the size of the calculated implied barriers against the null hypothesis that the implied barriers are zero. For a barrier level of zero the down-and-out call option collapses to the standard equity-call model. They conclude that the implied barriers are statistically and economically significant as they differ significantly from zero. However, I believe that this test only demonstrates that the standard equity-call model does not perform flawlessly. The implied barrier level captures any errors in the standard equity-call model. If the standard equity-call model does not perform flawlessly it is not strange to find that these barrier levels are significantly different from zero. Furthermore Brockman and Turtle [2003] do not indicate how a valuation practitioner can estimate the barrier levels from firm data.

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A real world example of the bankruptcy process is Eurotunnel. Eurotunnel is a heavily indebted firm which is facing bankruptcy. At the present time the various stock and debtholders of Eurotunnel are negotiating a debt restructuring plan to avoid bankruptcy. It is in the interest of all claimants to avoid bankruptcy because of the large loss of going concern value if Eurotunnel goes bankrupt. At the heart of the negotiations are two questions; how much are the assets of Eurotunnel as a going concern worth and how should the value be distributed between the claimants. The basis of the present plan involves reducing the amount of debt. The reduction of the amount of debt increases the value of equity, while debtholders lose part of their claim. To compensate for this loss the debtholders will receive convertible debt. If exercised this convertible debt will dilute the existing stockholders stake. During the negotiations both stock and debtholders try to maximize the value of their claims. The final debt restructuring plan will determine the value of the various claims. These real world complexities of the bankruptcy process are not captured by the equity-call model.

3.4.3 Violations of theoretical assumptions

The BSM model is built on the premises that the underlying asset is traded continuously and that short selling is allowed. In the case of the equity-call option this assumption becomes doubtful. In general, the firm’s assets are not actively traded, and neither is short selling possible. The assumption of continuous trading is needed to ensure the no arbitrage principle on which the BSM model is built.

The BSM model further assumes that the underlying asset price follows a continuous process. This implies that there are no price jumps in the underlying asset. For the firm’s assets it is doubtful if the price process is continuous, as the firm’s assets are not continuously traded.

The assumption that the asset volatility is known and constant is also debatable. Although the asset volatility can be assumed to be constant for a short period of time, over a longer period of time it will change as the operations of the firm change. Furthermore the BSM model requires a forward-looking asset volatility. As will be discussed in chapter 5 the estimation of the asset volatility is a big challenge.

3.5 Summary

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addition to the valuation toolbox. Specifically when valuing distressed firms the equity-call model provides an extra dimension compared to traditional valuation methods. The equity-call model is able to capture the time premium on equity. In chapter 3 the BSM model was introduced as the pricing model. An analysis of BSM model reveals that it has the right characteristics to value equity of distressed firms. Furthermore it gives insight into key value drivers of equity in distressed firms. The equity-call model, unlike traditional valuation methods, takes account of the interactive effects of debt and equity. This gives insight into the stockholder-bondholder conflict and the incentives faced by stock- and debtholders in distressed firms. Theoretically the equity-call model seems to be a suitable valuation tool for the valuation of equity of distressed firms. However, the equity-call model does have serious limitations. It assumes a simplified capital structure, does not take into account real world bankruptcy processes and violates some of the theoretical assumptions underlying the BSM model. The next step in assessing the suitability of the equity-call model involves empirically testing whether the equity-call model delivers

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4. Prior empirical literature

A large body of literature exists concerning contingent claims analysis. Much of this literature focuses on applying the equity-call model to the valuation of debt and the resulting use in credit risk analysis. As there are many types of debt with various features, extended models have been developed as explained in section 3.4.1. These extended models have been subjected to empirical testing, see for example Ericsson and Reneby [2003]. They test their extended model by pricing real world bonds. The yield spreads resulting from the estimates of the bond price are compared to real world yield spreads. They evaluate how well their extended model is able to capture the real world effect of credit risk on the price of the bonds. The main application of these extended models is in the modelling of credit risk. However, this and similar studies do not test whether the equity-call model results in a equity value which is a good proxy for the market price of equity. There has been a limited amount of studies that specifically test this aspect of the equity-call model.

Diba et al. [1995] use the equity-call model in ‘reverse’ to generate estimates of the market value of assets from observed stock market data and debt data for a sample of failed banks. From this market value of assets they subtract the face value of debt to obtain an estimate of the net worth of the banks. They compare this estimate with the Federal Deposit Insurance Corporation (FDIC) estimates of the (negative) net worth of these banks. The FDIC’s estimates are viewed as a proxy for the market value of net worth. They find that the FDIC’s estimates of negative net worth exceed estimates from the equity-call model. The most likely explanation given for their finding is that the equity-call model fails to capture institutional features of the banking industry. Specifically the model does not take into account the bankruptcy processes.

Baek et al. [2004] combine a traditional discounted cash flow type of model of total firm value developed by Schwartz and Moon [2000] with the debt-option pricing model (see equation 6) in order to calculate the equity value of IT companies. They subsequently compare this “fundamental” equity value with the market value of equity. They apply this methodology to three IT companies and find that two of the three firms studied are undervalued in the market.

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to asses if, as claimed by Damodaran [2002a], it is a valuable tool for valuation practitioners. Such a test can be viewed in line with the paper by Kaplan and Ruback [1995] who test how well different enterprise valuation techniques perform when applied to real world firms.

Kaplan and Ruback test how well the discounted cash flow method, under various assumptions, and comparable company valuation methods perform in the estimation of the enterprise value of firms. They use a sample of highly leveraged transactions for which the market value of the firm as well as cash flow information is available. Their error measure is the natural log of the difference between the estimated asset value and the observed transaction value. Kaplan and Ruback find that of the various methods evaluated the discounted cash flow method leads to the best results. A median error of 8% or less is found for the various discounted cash flow methods. The results found in the Kaplan and Ruback paper are used in chapter 6 to put the results found in this paper into perspective.

4.1 Estimating asset volatility

In chapters 5 and 6 of this paper the equity-call model is empirically tested. As will be discussed the test of the equity-call model involves comparing an observed asset volatility with the implied asset volatility. To interpret the results the observed asset volatility must be seen as a reasonable estimate of the ‘true’ asset volatility. However, the estimation of the asset volatility is not without difficulties. In theory, as discussed in section 3.4.3, the asset volatility has to be a forward-looking volatility and must be constant over the maturity of the option. This optimal asset

volatility, however, is not directly observable. First, the assets of the firm are generally not traded. As an alternative the traded debt and equity of the firm, the liability side of the balance sheet, can be used. Second, only historical data is available. Therefore a historical asset volatility is often used as a proxy. In order to estimate an up to date asset volatility it can be argued that a short historical time period over which to measure the asset volatility should be used. However, a short time period also entails fewer observations from which the asset volatility is estimated. This trade off must be kept in mind when estimating a historical asset volatility.

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all debt is priced and much of the priced debt is not traded continuously. Below a number of alternative methods for obtaining a proxy for the asset volatility are discussed.

Galai and Masulis volatility

Galai and Masulis [1976] derive an equation linking the asset volatility and the volatility of the value of equity. This equation allows for a direct link, using the equity-call model, between the equity volatility and the asset volatility. This equation is derived under a set of assumptions wherein both the capital asset pricing model and the option pricing model can be derived. Diba et al. [1995] use this method in their test of the equity-call model on failed banks. Delianedis and Geske [1998] also apply this equation in their study of risk neutral default probabilities. The equation is given as:

A E E A d N

σ

= ( ₁) (13)

where N(d1),

σ

_A, A and E are as defined in equation 4.

σ

_E is the volatility of the value of equity, henceforth equity volatility. This method results in a firm specific asset volatility.

Book value volatility

Alternatively, a time series of a proxy for the market value of a firm’s assets can be estimated. This is the approach used by Brockman and Turtle [2003]. They define the market value of a firm’s assets as the book value of assets less the book value of shareholders’ equity, plus the market value of equity. This market value of assets is computed on a quarterly basis for a period of ten years. A variance of quarterly percentage changes in asset values is then computed from this data. Their annual measure of asset volatility is then equal to the square root of four times the quarterly variance measure. This method is relatively easy to apply as the book value of debt is reported in the quarterly reports by firms. However, the book value of debt is not necessarily a good proxy for the volatility of the market value of debt. The book value of debt is not influenced by market forces. The book value of debt changes only when the capital structure of the firm is changed.

Peer firms volatility

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5 Methodology and Inputs

The most natural way to test the equity-call model would entail comparing the model-based value of equity with the observed market value of equity of distressed firms. To compute the inputs for the equity-call model requires compiling a time series of the value of a firm’s assets and relevant debt information. However, there are difficulties in computing the market value of the firm’s assets. Valuation practitioners usually estimate the market value of assets by means of a discounted cash flow analysis. However, not enough public data is available for the firms in the sample to be able to make any reasonable discounted cash flow analysis. Furthermore, in distressed firms the future cash flows are highly uncertain as discussed in chapter 2. An alternative method is to estimate the market value of assets as the sum of the market value of equity plus the market value of debt. This method, however, is internally inconsistent as it starts with one set of market values for debt and equity and, using the equity-call model, ends up with a different set of values.

To circumvent this problem this paper will test the equity-call model by solving it for the model-based asset volatility, or the ‘implied asset volatility’. In this manner the market value of equity is only an input, and not an input and an output, of the model. The only manner in which this can be achieved is through iteration as equation 2 cannot be solved directly for the asset volatility. The solver function in MS Excel is used to solve the equation for the implied asset volatility. The test of the equity-call model, as a valuation tool of the equity of distressed firms, then involves estimating the implied asset volatility and comparing it to an observed asset volatility. To interpret the results as a test of the equity-call model the observed asset volatility must be seen as being a reasonable proxy for the ‘true’ asset volatility. First though a sample of distressed firms is needed on which to test the equity-call model.

5.1 Firm selection

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of the firm as defined by Capital IQ. A cutoff ratio of 66% was taken to ensure a substantial amount of debt. Only 25 European firms were found with this characteristic. Second, European firms had significantly lower amounts of traded debt, as far as reported by Bloomberg. Third, the European firms lacked a common reporting standard. This made it difficult to determine the required inputs in a consistent manner across firms. The US firms on the other hand all follow consistent accounting standards and are required to file standardized annual reports with the SEC (Securities and Exchange Commission) known as SEC 10-K fillings. This enabled a uniform approach across firms in which to estimate the inputs. Furthermore the SEC fillings provided much information with regard to debt maturity, interest rates and coupon payments.

The US firms were further selected based on firm size, which was defined as the total enterprise value, defined by Capital IQ as the total book value of debt plus the market capitalization, on December 31st 2004. The cut off point of $200 million was chosen to ensure the companies in the sample were large enough to have traded debt. The next selection was on the basis of the industry group. All primary industry groups were selected with the exception of the financial industry group. This was done as the capital structure of financial firms is substantially different from other industries. Finally the firms were selected on the basis of a leverage measure. This measure was defined as the total debt divided by the total debt plus the market capitalization as defined by Capital IQ. The cut off point for this leverage ratio was set to 66%. This criterion ensured the firms in the selection had a substantial amount of debt. This selection resulted in a sample of 87 firms. This selection was further reduced to the final sample of 33 firms. This final selection was based on a manual analysis of each firm. Firms that had entered bankruptcy proceedings, had too little traded debt, or for which there was limited information available were discarded.

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with asset values above $10,000 million, while the smallest firm has an asset value of $231 million. The amount and quality of data available for the large firms in the sample, for example on debt prices, was often greater. However, the larger firms, in general, had more complicated capital structures making the estimation of the inputs more difficult and less precise.

Leverage measurea Retained Earningsb Value of assetsc

Median 82% -229 737 Mean 83% -640 2052 Maximum 99% 130 19,266 Minimum 71% -9,196 231 Notes: c

As defined in inputs in USD million.

b

As defined by Capital IQ in USD million.

Table 3: Summary statistics for the firms in the sample

a

The leverage measure is defined as: total book value of debt devided by total book value of debt plus market capitilization expressed as a percentage. As defined by Captial IQ.

Appendix B gives the industry concentration of the firms in the sample. The sample consists of firms in 17 different industry groups. By far the largest industry concentration is in media (7 firms, 21%). 11 industry groups contain only one firm.

The firms in the sample come from a variety of industries and differ greatly in size. All firms, however, have substantial amounts of debt in comparison to their value of assets. This makes them suitable candidates to test whether the equity-call model produces results which are a good proxy for the market values of equity. Nonetheless, the firm sample is small. Therefore caution must be taken when interpreting the results. Generalizing the results is not possible on the basis of such a small sample.

5.2 Inputs

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observed asset volatility to compare with the implied asset volatility. Appendix A provides detailed information on the computed inputs for all the firms in the sample. Before exploring the manners in which these inputs are estimated it is important to set the valuation date on which these inputs are based. Furthermore the different sources of capital in a firm may have to be specified and classified into debt or equity.

5.2.1 Valuation date

The valuation date for each firm is the end of book year date for the year 2004. This date ranges from the 30th of June 2004 up to the 29th of January 2005. The end of book year date is chosen in order to have the most accurate information regarding the outstanding debt of the firm as reported in the shareholders annual report or the annual report filled at the SEC, the so called 10-K form. Furthermore the year 2004 was chosen in order to be able to estimate post valuation date asset volatilities. That is asset volatilities as occurring in 2005.

5.2.2 Capital structure

Real world firms have many different sources of capital. The different capital sources have to be classified into debt or equity in order to be able to apply the model. All common shares outstanding are treated as equity. All the firms in the sample have one type of common shares outstanding. All long and short term debt is treated as debt. Furthermore any preferred shares are also treated as debt as they are senior to equity.

5.2.3 Market value of equity

The market value of equity is computed as the numbers of common shares outstanding as reported in Bloomberg multiplied by the share price as reported by Bloomberg on the valuation date.

5.2.4 Value of the firm’s assets

As discussed estimating the value of the firm’s assets by means of a discounted cash flow analysis is not possible. The asset value is instead estimated as the sum of the market value of equity and the market value of debt. The manner is which the market value of equity is estimated has already been discussed. This section will deal with the estimation of the market value of debt.

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in estimating the market value of debt for each outstanding debt issue. First, pricing information from Bloomberg is used if available. As most debt is often thinly traded it is not always possible to get a quote on the valuation date. Bloomberg pricing is used if a quote is available within a time frame of a month of the valuation date. When using the Bloomberg pricing no account is taken of the accrued interest payments. In the US interest payments are paid semi-annually. Therefore on average a quarter of a year accrued interest payments is not taken into account in the pricing of the Bloomberg quoted debt. However, as the estimation of the market value of debt in general suffers from lack of data and is therefore not very accurate, this is not seen as being problematic. If no Bloomberg information is available the fair market value of the debt issue as reported in the annual report is used. If no further information is available from the annual report, the face value of debt is used as an approximation of the market value of debt. The market value of debt is defined as the sum of the approximated market value of each single debt issue. The value of the firm’s assets is defined as the sum of the market value of debt and the market value of equity. Table 4 reports what method was used on average to price the debt, as a percentage of the total principal of debt. On average 57% of the face value of debt is priced using Bloomberg information. 28% of the face value of debt is priced against its face value.

Table 4: Method utilized to value debt

Bloomberg Annual report Face value

Mean 57% 15% 28%

Notes:

Method used, on average, as a percentage of the total principal of debt

5.2.5 Face value of debt

The face value of debt, hereafter FVD, is the strike price of the equity-call option. The BSM model assumes a single issue zero coupon bond. However, in practice firms have multiple types of debt. The face value of debt is equal to the sum of the principal due on the various debt issues of the firm. This input is defined as K1. The principal amount due on the various debt issues was

available from the annual reports.

5.2.6 Interest payments on debt

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The interest payments can be taken account of by subtracting the present value of the interest payments from the value of the firm’s assets. However, in order to implement this method, the timing of each of the interest payment cash flows must be known. As is discussed in section 5.2.7 a lack of data makes it difficult to utilize this method. A comparable method involves estimating the interest payments as a percentage of the value of the firm’s assets. This interest payment yield can then be used in equation 5. However, as the various debt issues in a firm come due at various times over the life of the equity-call option, it is difficult to estimate a yearly interest payment yield.

In this paper the interest payments are take account of by adding all the expected interest payments to the face value of debt of the firm, as given by equation 7. The downside of this method is that it mixes up cash flows which occur at different points in time. The FVD input which includes the interest payments on debt is defined as K2.

Information from the annual reports is used to estimate the interest payments on debt.

Furthermore the annual reports yield information on the time to maturity and interest rates of the various debt issues. It must be noted that this information is not complete for each debt issue. The interest payments are computed by multiplying the time to maturity of the debt, the interest rate and the principal of the debt. Both estimates of the FVD are evaluated in this paper.

Table 5 reports summary statistics for the inputs K1 and K2 as well as the effect of including

interest payments in the estimation of the FVD. On average the inclusion of interest payments adds 54% to the FVD, a substantial amount. This is not surprising as most of the debt on the books of the firms in the sample is interest paying debt. As input K2 is bigger for all firms than

input K1, a higher implied volatility is expected when applying this input to the equity-call model.

As discussed in section 3.1 a higher FVD results in a lower equity value. This is compensated by a higher implied volatility when the model is solved.

Also given are the A/K ratios for both inputs. As discussed in section 3.2 the A/K ratio shows how distressed the firms are. Furthermore the ratios indicate how sensitive the equity-call option is to changes in asset volatility and time to maturity. For an A/K ratio close to one the vega and theta of the equity-call option are high. For input K1 the median ratio is above one. From figure 2 we

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A/K2 ratio above one. For firms with A/K ratios below one the value of equity is only derived

from the time premium of the equity-call option. The firm with the lowest A/K2 ratio is FiberMark

at 0.41. This firm is very distressed as the value of its assets is only 41% of the required debt and interest payments.

Table 5: Summary statistics of inputs K1 and K2

K1 K2 IP % a A/K1b A/K2b Median 604 969 54% 1.22 0.79 Mean 1788 2717 51% 1.18 0.79 Max 19791 29201 82% 1.42 1.07 Min 215 323 12% 0.67 0.41 Notes:

a _{Interest payments as percentage of input K} 1 b

Ratio of market value of assets to input K1 and K2

5.2.7 Time to maturity of the debt

The BSM model only allows for one input for the time to expiration. It is therefore necessary to convert the multiple debt issues and interest payments into an equivalent single issue of a zero coupon bond. Damodaran [2002a] discusses two manners in which this can be done.

The first is a principal weighted maturity of the debt. This time to maturity is estimated by taking a weighted average of the maturity of the various debt issues in a firm. The weighting is done on the basis of the principal of each debt issue. This method does not take into account the various intermediate interest payments.

∑

= i i i P P M T₁ (14) where:

T1= Time to maturity measure 1.

Mi= Maturity of debt issue i.

Pi= Principal of debt issue i.

The second method which does take into account the intermediate interest payments is a principal weighted average of the durations of the different debt issues. This second method is a more accurate measure of the time to maturity of debt, especially in combination with input K2, as it

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payments. Input K2 also takes into account the intermediate interest payments. The first step to

estimating this input is to compute the duration of each debt issue. The duration is defined as:

B e c t D n i i t * y i i

∑

− = (15) where:

D = Duration of debt issue.

ti= Time to maturity of cash flow i.

ci= Cash flow i of debt issue (principal or interest payment).

y = Yield of debt issue.

B = Market value of debt issue.

However, the annual reports do not provide sufficient information to estimate a duration measure for each debt issue. For example the market value of the debt issue is not available, the exact cash flow dates are not known, or the appropriate yield is not known. For the debt issues for which these data are unavailable the time to maturity is taken as a proxy for the duration. To obtain a single time to maturity the durations of each debt issue are weighted on basis of the total principal of debt.

∑

= i i i 2 P P D T (16) where:

T2 = time to maturity measure 2

Di = Duration of debt issue i

Pi = Principal of debt issue i

Both measures of maturity are evaluated in this paper. Table 6 reports the summary statistics for the maturity measures. On average the input T2 estimates a shorter maturity. This can be

explained as input T2 takes into account the earlier maturities of the intermediate interest

payments. A reduction of the time to maturity is compensated with a higher implied volatility, given the observed value of equity. It is expected that input T2 will lead to a better results when

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T1 T2

Median 5.8 4.8

Mean 5.7 4.8

Maximum 11.6 11.6

Minimum 1.4 1.2

Table 6: Summary statistics of inputs T1 and T2.

5.2.8 The risk free interest rate

The risk free interest rate is taken to be the USD US treasury strips as reported by Bloomberg on the valuation date. The time to maturity of the USD US treasury strip should optimally be equal to the time to maturity of the debt. Bloomberg provides information on USD US treasury strips with a limited amount of different maturities. By means of linear extrapolation the implied interest rate of a USD US treasury strip with similar time to maturity as the debt is calculated. As there are two maturity measures there are also two sets of risk free interest for each firm, denoted as R1and R2.

5.2.9 Observed Asset volatility

As discussed in chapter 5 there are various manners by which the asset volatility has been measured in previous papers. The three procedures discussed in chapter 5; the GM volatility; the book value volatility; and the peer firm volatility each estimate a proxy for the ‘true’ asset volatility. The most reasonable of these approaches should be used to compare with the implied asset volatility. However, each of these asset volatility measures is dependent on the method used to compute it and the available data. Therefore the manner by which these proxies are computed is first discussed.

Valuing equity in distressed firms as a call option on assets