Black-Scholes A Claimants Tool for Valuing Recovery in a Plan of Reorganization

Black-Scholes A Claimants Tool for Valuing Recovery in a Plan of Reorganization

Journal Issue: 
Column Name: 
Journal Article: 

By fate or by choice, common stockholders and subordinated debt holders in danger of being pushed (or crammed) out of the capital structure of a company emerging from chapter 11 bankruptcy protection are being given the opportunity to leverage what nominal value remains of their original capital into a long-term investment in the reorganized enterprise. An example of an increasingly common instrument provided to common equity holders or unsecured creditors in corporate reorganizations is the stock warrant, the right granted by the company to purchase a share of the reorganized firm at a specific price during a given time period.

Warrants as Compensation

In a number of recent major corporate reorganization plans, such as Flagstar Companies Inc. and Edison Brothers, debtors have utilized warrants to compensate either unsecured creditors or previously existing equity holders for their claims. Although warrants issued to such claimants are typically out of the money when issued, they have potential short- and long-term value to both debtor and claimant. In addition to offering unsecured creditors and equity holders some level of recovery through participation in the upside of the reorganized company, the issuance of warrants by the company can often serve as a catalyst to an expeditious confirmation of a plan of reorganization. The issuance of warrants also may be used as a "sweetener" or "nuisance fee" paid to gain the support of various classes of security holders and, therefore, prevent certain hurdles from arising that could unnecessarily lengthen and complicate the reorganization process.

In the short term, offering warrants as a form of capital recovery can benefit companies in the middle of reorganization efforts by reducing the likelihood of long, drawn-out legal battles that not only result in the accrual of significant professional and other administrative expenses, but may also potentially jeopardize existing agreements with other classes of creditors or providers of debtor-in-possession or exit financing. In the long term, if a company's turnaround is successful and the stock price is indicative of such, the issuance of warrants can be a good investment for both the issuer, in the form of additional equity funding at a low transaction cost, and the warrant holders, in the form of increased recovery.

In certain cases, the claimants have a good negotiating position because of the company's burden to demonstrate to banks, secured creditors and/or the bankruptcy court, via the reorganization plan, that they will be receiving long-term value in excess of the current liquidation value of the company's assets. As a result, in addition to having an established book equity position coming out of bankruptcy, the company will likely have a turnaround plan that follows an established timeline, and is based on sound economic assumptions. Such issues can facilitate financial settlements between the company and creditors by providing a fundamental basis for a valuation model from which to negotiate a fair recovery for unsecured creditors.

Leverage Effect of Warrants

Although common equity in the reorganized debtor is a more tangible asset in relation to recovery of capital, warrants can be an attractive alternative to common stock, particularly considering that, in most circumstances, there will be minimal, if any, dividend yield on common stock for the first few years following bankruptcy. Furthermore, warrants can be advantageous to claimants since warrants have similar leverage characteristics to ordinary call options that cause percentage price changes in stock warrants to always be higher than the correlating percentage changes in stock price.1

Valuing Warrants

The most commonly used analytic tool for valuing or pricing warrants is the Black-Scholes options pricing model. The Black-Scholes model was first published in 1973 by Fischer Black, Ph.D. and Myron S. Scholes, Ph.D., a 1997 Nobel Laureate in Economics for his collaborative work with Dr. Black and Robert C. Merton, Ph.D. in developing a new method to determine the value of derivatives. The work of these individuals has not only set the standard for various types of economic valuation, but it has also fostered the creation of new types of complex financial instruments that have facilitated more efficient corporate and individual financial risk management.

Variables to Determine Value

The use of the Black-Scholes model to price options such as calls, puts and warrants requires the use of several variables, some of which are relatively identifiable, and others that are determined through analysis, and may be the subject of negotiations between parties. In order to price a warrant using Black-Scholes, the following five variables need to be established as inputs to the formula, which ignores dividends, transaction costs and taxes:

  • Current Price of the Underlying Asset—The current stock price or, in the case of a bankruptcy, the net asset value per share after reorganization. In the case of a bankruptcy, the debtor and its financial advisor will have provided a valuation of the reorganized debtor. The creditor's financial advisor will be able to provide confirmation of such value.

  • Risk-free Rate of Return—The constant risk-free rate of return is a relatively certain variable based on existing market conditions; however, it can vary somewhat based on the duration of the negotiated expiration term of the warrants. Using the current yield on a "STRIP," or U.S. zero-coupon bond, that has a maturity term similar to that of the underlying option, is an appropriate rate of return because of a warrant's pricing similarities to zero-coupon debt, which is issued at a significant discount and accretes to its par value.

  • Variance of the Asset Returns—The variance or price volatility of a firm's stock price that a company will experience during the life of the warrant can be a very difficult assumption to establish because it requires the negotiating parties to reach a consensus on an appropriate benchmark from which the company's "pro forma" performance will be gauged. As a starting point, historical data can be used to determine comparable average rates of return and standard deviations based on companies with similar industry segments, capital structures and/or reorganization experience. Potential warrant holders also should consider a number of forward-looking risk factors including industry competition, general economic and market conditions and the post-reorganization capital structure that could significantly affect the company's ability to be competitive and achieve, on a timely basis, the operational and financial initiatives incorporated in its plan of reorganization.

  • Exercise Price and Expiration Term—The two remaining factors underlying a Black-Scholes valuation are exercise price and expiration term of the warrants. These pricing terms establish the potential rate of return to the prospective warrant holder and, conversely, the cost to the company of issuing the warrants. Consequently, they are open for the greatest amount of debate in negotiating the terms of a warrant issuance. Much like the pricing of an equity or bond security, determining the ideal exercise price and expiration term of a warrant is dependent on the negotiating parties' valuation of the asset as well as their required rate of return.

Sensitivity of Variables to Pricing of Warrants

For purposes of negotiation, an understanding of the effects of the variables that factor into a Black-Scholes valuation of warrants can be gained by performing various types of sensitivity analyses on the individual variables. The sensitivity analysis below demonstrates how changes in each of the individual variables, holding the other variables constant, affect the overall valuation of the warrant. Four of the five variables are positively correlated to changes in warrant price; the exception is exercise price, which at higher levels translates into a lower warrant price.

To illustrate this sensitivity, the following tables present the changes in warrant price based on variations of each respective variable, holding the other variables constant. The Base Case in the tables demonstrates that six-year warrants on a share of stock with a current price of $40 per share, exercise price of $50 per share, 7 percent risk-free rate of return and a 16 percent return variance will derive a valuation of $14.60 per warrant. As depicted in the first table, a higher stock price of $45 per share would increase the value of the warrant to $17.82 as the spread between the current price and the exercise price is decreased. Conversely, as shown in the second table, a higher exercise price of $60 per share would decrease the value of the warrant to $12.68 based on an increased spread between the current price and the exercise price. An increase in the term of the warrant from six years to eight years will increase the value of the warrant to $17.48 as a result of an extended time horizon for the price of the underlying stock to reach or grow beyond its exercise price. In the third table, an increase in the risk-free rate of return will also result in a higher warrant value because it has the effect of applying a greater discount to the present value of the future exercise price. In the fourth table, the value of the warrant is increased to $15.23 based upon an increase in the risk-free rate from 7 to 8 percent. Finally, increases in the estimated variance of returns on the related stock generate higher warrant values as a result of the increased probability that the stock will experience larger upward price movements. The value attributed to increased variance is also enhanced due to the fact that warrant holders' downside is limited compared to the potential upside from stock price increases. The final table shows that increasing the variance of returns from 16 to 36 percent increases the valuation of the warrant to $19.42.

of Warrant
Correlation: Positive
Value of Warrant $40.00
Correlation: Negative
Variable: TERM (years)
of Warrant
Correlation: Positive
of Warrant
Correlation: Positive
of Warrant
Correlation: Positive
Sensitivities for each variable based on other variables remaining constant at Base Case.

The warrant issuer must ensure that the warrants will add value to its enterprise by maximizing the proceeds it will potentially receive, while minimizing the cost of the warrants, including dilution issues that will affect current and future valuation of the company's common stock. On the other hand, potential warrant holders, or, in the context of a bankruptcy, creditors and pre-bankruptcy equity holders, must attempt to maximize the recovery of their investment in the quickest but most reasonable manner possible. Regardless of their perspective, the negotiating parties must determine how to maximize the value they receive by establishing the optimal time horizon based on the future performance of the reorganized company.


The Black-Scholes model is decidedly the most authoritative tool in the valuation of warrants. Understanding its sensitivity to the assumptions that drive the model is important to understanding the negotiation process. Inherent in the estimation of such variable assumptions is the possibility of overvaluing or undervaluing the warrants. If structured well, warrants can serve as an additional source of financing for the new company, and meaningful compensation to claimants of the bankrupt debtor.


1 Reilly, Frank K., Investment Analysis and Portfolio Management, 4th Edition, pp.783-805. Orlando: The Dryden Press, 1994. Return to article

Journal Date: 
Sunday, November 1, 1998